Nanodispersions: Interactions, Stability, and Dynamics

Nanodispersions

Eli Ruckenstein · Marian Manciu

Nanodispersions Interactions, Stability, and Dynamics

123

Eli Ruckenstein Department of Chemical and Biological Engineering University at Buffalo The State University of New York 303 Furnas Hall Buffalo NY 14260-4200 USA [email protected]

ISBN 978-1-4419-1414-9 DOI 10.1007/978-1-4419-1415-6 Springer New York Dordrecht Heidelberg London

Marian Manciu Department of Physics University of Texas at EI Paso 500 W. University Avenue EI Paso TX 79968 Physical Science Bldg. 210 USA [email protected]

e-ISBN 978-1-4419-1415-6

Library of Congress Control Number: 2009939271 c Springer Science+Business Media, LLC 2010  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book contains a number of papers published by Ruckenstein and coworkers on the topic of nanodispersions. Aerosols are the focus of the first chapter which features a model for the sticking probability as the main contribution. One concludes that, when the particles are small enough, the dissociation rate can become sufficiently large for doublets to reach equilibrium with single particles. However, above a critical radius for the particles, the doublets become stable and their concentration increases with time, providing nuclei for aerosol growth. The second chapter examines the deposition of Brownian particles on surfaces when the interaction forces between particles and collector play a role. When the range of interactions between the two (which can be called the interaction force boundary layer) is small compared to the thickness of the diffusion boundary layer of the particles, the interactions can be replaced by a boundary condition. This has the form of a first order chemical reaction, and an expression is derived for the reaction rate constant. Although cells are larger than the usual Brownian particles, the deposition of cancer cells or platelets on surfaces is treated similarly but on the basis of a Fokker-Plank equation. Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. Stern has noted that the traditional Poisson-Boltzmann approach leads to an ionic density in the vicinity of the interface which exceeds the available volume. This anomaly can be corrected by taking into account the hydration of ions. This issue is examined in Chapter 4. This chapter also examines the Helfrich force [Helfrich, W. Z. Naturforsch. 1978, 33a, 305], a repulsion generated by thermally undulating interfaces. Together with the hydration force, this force is responsible for the stability of neutral lipid bilayers. In contrast to the Helfrich theory, which is valid only at large separations between bilayers, the present theory also provides the exponential behaviour that is observed experimentally at small separations. An equation is derived for the force generated between two charged plates immersed in an electrolyte solution, which contains small charged particles, by coupling double layer and depletion forces. It is also shown via Monte Carlo simulations that in concentrated colloidal systems, pairwise repulsive interactions between particles can generate a collective attractive interaction. Chapter 5 is concerned with specific ion effects due to ion hydration forces and is based on the Structure Making / Structure Breaking (SM/SB) model. The structure making ions prefer the bulk water and are repelled by the surface, while the structure breaking ions disturb the order of bulk water and are expelled toward the interface. The model was used to calculate the force between two parallel plates. There are numerous experiments that point out that the v

vi

cations (except H+ ) are repelled by the air-water interface, whereas the anions (except F− ) are accumulated there. It is shown that the simple SM/SB model can account for these observations quantitatively. Chapter 6 emphasizes that, as the double layer force, the hydration force is a consequence of electrostatic interactions. In the latter case, the interactions are between neighboring dipoles, and a theory in which the hydration and double layer forces are treated in a unified manner is proposed. The dipoles of the surface align the dipoles of the neighboring water molecules; the dipoles of the partially aligned water molecules, in turn, partially align the next layer of molecules, and so on. Thus a decaying polarization field is generated. The overlap of the polarization fields of two plates brought close enough generates repulsion. Whereas in the traditional continuum theory a uniform dipolar field on a surface does not generate an electric field outside, the discrete nature of the water molecules is considered to be responsible for the generation of a local field, which is calculated by assuming that water is organized in ice-like layers. In addition, in our “polarization model” the surface charge is increasingly neutralized with increasing electrolyte concentration and replaced with dipoles. As a consequence, the repulsive force due to charges is diminished whereas that due to dipoles is increased as the ionic strength increases. At low ionic strengths the repulsion is decreased as the ionic strength increases (particularly because of the screening of the electric field). At high ionic strengths the increasing number of surface dipoles generated by the increasing ionic strength causes an increase in repulsion. Chapter 7 examines a number of consequences of the polarization model presented above. One of the most significant is the existence of a minimum in the repulsive interactions between colloidal particles as a function of electrolyte concentration, an old experimental result, which remained unexplained for 75 years. Voet reported that stable solutions of various metals (Pt, Pd) and salts (sulfides, halides) in highly concentrated solutions of acids (sulfuric, phosphoric) coagulated upon the dilution with water, hence by decreasing the electrolyte concentration [Voet, A. Thesis, Amsterdam, 1935 See also: Kruyt, H. R. Colloid Science; Elsevier: Amsterdam, 1952]. This effect can explain (i) the light scattering observations regarding the apoferritin molecules at various NaCH3COO concentrations which reveal high repulsion at high ionic strengths, as well as (ii) the restabilization of protein-covered latexes at high ionic strengths. The polarization model is combined with thermal undulations to explain the behaviour of common and Newton black films as well as the swelling of neutral lipid bilayers induced by simple salts. This theory, combined with ion-hydration forces, is extended to rough silica surfaces, which disorganize the structure of the interfacial water. The last chapter, Chapter 8, is devoted to the interactions between grafted polyelectrolyte brushes, which exhibit peculiar repulsive and attractive interactions. Several approaches are suggested. First, a general theory is developed, based on a simple cubic lattice model and matrix formalism, which is used to calculate the steric interaction between two grafted polyelectrolyte brushes. Another approach based on an approximate partition function is suggested to treat the steric repulsion between grafted polymer brushes, leading to simple equations for both neutral and charged grafted brushes. A general formalism for double layer interactions between polyelectrolyte brushes is proposed and it is shown that the net interactions can become attractive, due to the bridging generated by the polyelectrolyte chains between plates. Consequently, the repulsive steric interactions compete with attractive bridging forces. A simple model to calculate the available area for competitive adsorption of molecules is also proposed.

Preface

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

1 Coagulation, dissociation and growth of aerosols . . . . . . . . . . . . . . . . . .

1

2 Dynamics of deposition of Brownian particles or cells on surfaces . . . . . . . .

67

3 Stability of dispersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

4 Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects . . . . . . . . . . . . . . . .

325

5 Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

379

6 Polarization Model: a unified framework for hydration and double layer interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

459

7 Polarization Model and ion specificity: applications . . . . . . . . . . . . . . . .

511

8 Polymer brushes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

607

vii

Introduction to CHAPTER 1 Coagulation, dissociation and growth of aerosols

1.1

1.2

1.3

1.4

G. Narsimhan, E. Ruckenstein: “The Brownian Coagulation of Aerosols over the Entire Range of Knudsen Numbers: Connection between the Sticking Probability and the Interaction Forces” JOURNAL OF COLLOID AND INTERFACE SCIENCE 104: 2 (1985) 344–369. G. Narsimhan, E. Ruckenstein: “Monte Carlo Simulation of Brownian Coagulation over the Entire Range of Particle Sizes from Near Molecular to Colloidal: Connection between Collision Efficiency and Interparticle Forces,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 107: 1 (1985) 174–193. G. Narsimhan, E. Ruckenstein: “Dissociation Kinetics of Doublets of Aerosol Particles,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 116: 1 (1987) 278–287. G. Narsimhan, E. Ruckenstein: “A Possible Nucleation Type of Mechanism for the Growth of Small Aerosol Particles,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 116: 1 (1987) 288–295.

Traditional approaches to Brownian coagulation account for the interaction between particles through a phenomenological sticking probability, which is usually assumed to be unity. A model is proposed for the Brownian coagulation coefficient of electrically neutral aerosol particles, which takes into account the interparticle van der Waals attraction and Born repulsion [1.1]. A “sphere of influence”, which has a thickness equal to the correlation length of the relative Brownian motion of two particles is suggested; within this region, the relative motion of the particles is considered free molecular, whereas outside this region it is described by a Fokker-Plank equation. For very small particles, the sticking probability becomes vanishingly small, while for sufficiently large particles it is virtually unity. In contrast to earlier models, the expression derived for the coagulation coefficient of large particles displays the proper continuum and free molecular limits and agrees well with the Fuchs empirical formula [1.1]. Upper and lower bounds of the ratio between the coagulation coefficient and the Smoluchowski coagulation

coefficient are expressed as functions of the Knudsen number for various Hamaker constants [1.1]. The model was tested using Monte-Carlo simulations (for selected values of the Hamaker constant) over the entire range of the Knudsen number [1.2]. For large particles, the Brownian coagulation coefficient agrees well with the Fuchs interpolation formula and with the analytical expression obtained in [1.1]. For sufficiently small particles [1.2], the coagulation coefficient agrees very well with the lower bound obtained analytically [1.1]. For intermediate particle sizes, the range of interparticle attractive forces becomes comparable to the particle size. This enhances the coagulation coefficient, which acquires values even greater than for the free molecular limit. The coagulation of aerosols is typically regarded as a result of the short-range van der Waals attraction between particles; when two particles come sufficiently close to each other, they can coagulate. This is true if the particles are large enough and hence the potential well due to van der Waals and Born interactions is sufficiently deep; it is not accurate if the particles are too small and the potential well so shallow that the particles are able to escape from it. In the latter case, the coagulated aerosol doublets can acquire enough energy from the collision with other molecules to dissociate. A novel theoretical treatment, based on the assumption that the time scale of oscillations within the well is much shorter than the time scale of Brownian motion, allows one to average the Fokker-Plank equation over the positions of the doublet and leads to a one-dimensional Fokker-Plank equation in terms of the energy of the relative motion of the constituents of the doublet [1.3]. The average lifetime of a doublet can thus be calculated. It is shown that in air at 298 K, the average dissociation time increases dramatically, from 10 - 7 to 10 - 1 seconds, as the radius changes from 15 to 50 Ǻ. The doublets formed by particles with a radius larger than 50 Ǻ are found to be extremely stable [1.3]. Very small aerosol particles can be generated via physical or chemical nucleation, and their subsequent growth is due either to condensation on the surface of the particles or to the Brownian coagulation of the particles themselves. If

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_1, © Springer Science+Business Media, LLC 2010

2

the particles are sufficiently large, the latter process is irreversible (i.e., the sticking probability is unity). However, as noted above, for very small particles there is a nonnegligible chance for the doublet to dissociate, because the interaction between the two particles leads to a shallow potential well. The rate of formation of doublets is calculated as suggested in [1.1] and the rate of dissociation is calculated as indicated in [1.3]. The difference between the

Nanodispersions

rates of coagulation and dissociation increases as the particle size increases, becoming zero at a critical particle radius. For particles that are sufficiently small compared to the critical radius, the doublets become unstable and can reach a dynamic equilibrium with the single particles. For particles that are large compared to the critical size, the doublets are stable and their concentration increases with time. They provide nuclei for aerosol growth [1.4].

Coagulation, dissociation and growth of aerosols

3

4

Nanodispersions

Coagulation, dissociation and growth of aerosols

5

6

Nanodispersions

Coagulation, dissociation and growth of aerosols

7

8

Nanodispersions

Coagulation, dissociation and growth of aerosols

9

10

Nanodispersions

Coagulation, dissociation and growth of aerosols

11

12

Nanodispersions

Coagulation, dissociation and growth of aerosols

13

14

Nanodispersions

Coagulation, dissociation and growth of aerosols

15

16

Nanodispersions

Coagulation, dissociation and growth of aerosols

17

18

Nanodispersions

Coagulation, dissociation and growth of aerosols

19

20

Nanodispersions

Coagulation, dissociation and growth of aerosols

21

22

Nanodispersions

Coagulation, dissociation and growth of aerosols

23

24

Nanodispersions

Coagulation, dissociation and growth of aerosols

25

26

Nanodispersions

Coagulation, dissociation and growth of aerosols

27

28

Nanodispersions

Coagulation, dissociation and growth of aerosols

29

30

Nanodispersions

Coagulation, dissociation and growth of aerosols

31

32

Nanodispersions

Coagulation, dissociation and growth of aerosols

33

34

Nanodispersions

Coagulation, dissociation and growth of aerosols

35

36

Nanodispersions

Coagulation, dissociation and growth of aerosols

37

38

Nanodispersions

Coagulation, dissociation and growth of aerosols

39

40

Nanodispersions

Coagulation, dissociation and growth of aerosols

41

42

Nanodispersions

Coagulation, dissociation and growth of aerosols

43

44

Nanodispersions

Coagulation, dissociation and growth of aerosols

45

46

Nanodispersions

Coagulation, dissociation and growth of aerosols

47

48

Nanodispersions

Coagulation, dissociation and growth of aerosols

49

50

Nanodispersions

Coagulation, dissociation and growth of aerosols

51

52

Nanodispersions

Coagulation, dissociation and growth of aerosols

53

54

Nanodispersions

Coagulation, dissociation and growth of aerosols

55

56

Nanodispersions

Coagulation, dissociation and growth of aerosols

57

58

Nanodispersions

Coagulation, dissociation and growth of aerosols

59

60

Nanodispersions

Coagulation, dissociation and growth of aerosols

61

62

Nanodispersions

Coagulation, dissociation and growth of aerosols

63

64

Nanodispersions

Coagulation, dissociation and growth of aerosols

65

66

Nanodispersions

Introduction to CHAPTER 2 Dynamics of deposition of Brownian particles or cells on surfaces

2.1

E. Ruckenstein, D. Prieve: “Rate of deposition of Brownian particles under the action of London and double-layer forces,” JOURNAL CHEM. SOC. FARADAY TRANSACTIONS II, 69 (1973) 1522–1536. 2.2 E. Ruckenstein, D.C Prieve: “Adsorption and desorption of particles and their chromatographic separation,” AIChE JOURNAL 22 (1976) 276–283 2.3 E. Ruckenstein, D.C. Prieve: “On reversible adsorption of hydrosols and repeptization,” AIChE JOURNAL 22 (1976) 1145–1147. 2.4 D.C. Prieve, E. Ruckenstein: “Effect of London forces upon the rate of deposition of Brownian particles,” AIChE JOURNAL 20(1974) 1178–1186. 2.5 D. C. Prieve, E. Ruckenstein: “Role of surface chemistry in particle deposition,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 60 (1977) 337–348. 2.6 D.C. Prieve, E. Ruckenstein: “The double-layer interaction between dissimilar ionizable surfaces and its effect on the rate of deposition,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 63 (1978) 317–329. 2.7 E. Ruckenstein: “Reversible rate of adsorption or coagulation of Brownian particles- effects of the shape of the interaction potential,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 66 (1978) 531–543. 2.8 E. Ruckenstein, D.C Prieve: “Dynamics of cell deposition on surfaces,” JOURNAL OF THEORETICAL BIOLOGY 51 (1975) 429–438. 2.9 E. Ruckenstein, A. Marmur, W.N.Gill: “Coverage dependent rate of cell deposition,” JOURNAL OF THEORETICAL BIOLOGY 58(1976)439–454. 2.10 A. Marmur, W. N. Gill, E. Ruckenstein: “Kinetics of cell deposition under the action of an external field,” BULLETIN OF MATHEMATICAL BIOLOGY 38 (1976) 713–721. 2.11 E. Ruckenstein: “Thermodynamic insights on macroemulsion stability,” ADVANCES IN COLLOD AND INTERFACE SCIENCE 79 (1999) 59–76. 2.12 E. Ruckenstein: “On the stability of concentrated, non-aqueous dispersions,” COLLOIDS AND SURFACES 69 (1993) 271–275.

The rate of deposition of Brownian particles is predicted by taking into account the effects of diffusion and convection of single particles and interaction forces between particles and collector [2.1] -[2.6]. It is demonstrated that the interaction forces can be incorporated into a boundary condition that has the form of a first order chemical reaction which takes place on the collector [2.1], and an expression is derived for the rate constant. The rate of deposition is obtained by solving the convective diffusion equation subject to that boundary condition. The procedure developed for deposition is extended to the case when both deposition and desorption occur. In the latter case, the interaction potential contains the Born repulsion, in addition to the London and double-layer interactions [2.2]-[2.7]. Paper [2.7] differs from [2.2] because it considers the deposition at both primary and secondary minima. Papers [2.8], [2.9] and [2.10] treat the deposition of cancer cells or platelets on surfaces. For a constant amount of nonionic surfactant, the interfacial tension at the planar oil-water interface, for the same amounts of oil and water, passes through a minimum when plotted against the hydrophilic-lipophilic balance (HLB). The emulsion stability passes through maxima in the W/O and O/W ranges and through a minimum between the two at the phase inversion point. The minima in the two cases coincide. These observations are explained on the basis of thermodynamics. The stability of macroemulsions can be correlated with the surface excess of surfactant, which also passes through two maxima and a minimum between them [2.11]. In paper [2.12] it is shown that the van der Waals interactions between two particles can be decreased if they are covered with layers that have a Hamaker constant which is near to that of the suspending liquid. It is also suggested that, at sufficiently high concentrations, the collective behavior of the colloidal particles can generate repulsion when the pairwise interactions are attractive. These two effects are suggested to be responsible for the kinetic stability of the system, and a methodology for achieving kinetic stability is provided.

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_2, © Springer Science+Business Media, LLC 2010

Dynamics of deposition of Brownian particles or cells on surfaces

69

70

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

71

72

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

73

74

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

75

76

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

77

78

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

79

80

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

81

82

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

83

84

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

85

86

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

87

88

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

89

90

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

91

92

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

93

94

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

95

96

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

97

98

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

99

100

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

101

102

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

103

104

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

105

106

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

107

108

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

109

110

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

111

112

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

113

114

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

115

116

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

117

118

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

119

120

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

121

122

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

123

124

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

125

126

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

127

128

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

129

130

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

131

132

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

133

134

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

135

136

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

137

138

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

139

140

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

141

142

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

143

144

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

145

146

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

147

148

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

149

150

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

151

152

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

153

154

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

155

156

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

157

158

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

159

160

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

161

162

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

163

164

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

165

166

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

167

168

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

169

170

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

171

172

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

173

174

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

175

176

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

177

178

Nanodispersions

Advances in Colloid and Interface Science 79 Ž1999. 59]76

Thermodynamic insights on macroemulsion stability Eli RuckensteinU Chemical Engineering Department, State Uni¨ ersity of New York at Buffalo, Clifford C. Furnas Hall, SUNY-Buffalo, Box 604200, Buffalo, NY 14260-4200, USA

Abstract Experiments reported in the literature for nonionic surfactants have revealed that: Ž1. the interfacial tension at the planar surface between oil and water, at a constant amount of surfactant and equal volumes of water and oil, passes through a minimum when plotted against the HLB Žthe hydrophilic]lipophilic balance.; Ž2. under the same conditions, the emulsion stability against HLB passes through two maxima, one in the WrO range and the other in the OrW range, and a minimum between the two at the phase inversion point; Ž3. the minima under Ž1. and Ž2. coincide; Ž4. the phase inversion temperatures of emulsions, formed of oil and water as distinct phases, or microemulsions and their excess phase, and the corresponding microemulsions are the same. These observations are explained on the basis of thermodynamics. It is also shown that the stability of emulsions can be correlated with the surface excess of surfactant, which also passes through two maxima and a minimum between the two when plotted against HLB or temperature. In addition, a two ratio approach is used to determine when the Bancroft rule is obeyed or violated in microemulsions. Q 1999 Elsevier Science B.V. All rights reserved. Keywords: Macroemulsion stability; Hydrophilic]hydrophobic balance ŽHLB.; Interfacial tension vs. HLB; Surface excess vs. HLB; Surface excess vs. temperature; Phase inversion temperature; Bancroft rule

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 U

Tel.: q1 716 645 2911; fax: q1 716 645 3822; e-mail: [email protected]

0001-8686r99r$ - see front matter Q 1999 Elsever Science B.V. All rights reserved. P I I: S 0 0 0 1 - 8 6 8 6 Ž 9 8 . 0 0 0 7 9 - 7

179

Dynamics of deposition of Brownian particles or cells on surfaces 60

E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

2. Effect of HLB Ž h. on the interfacial tension and the stability of macroemulsions 2.1. The interfacial free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Surface excess against HLB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Phase inversion temperature of macro and microemulsions . . . . . . . . . . . . . . 3.1. The interfacial tension against temperature . . . . . . . . . . . . . . . . . . . . . 3.2. Macroemulsions and microemulsions . . . . . . . . . . . . . . . . . . . . . . . . . 4. The Bancroft rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. 61 . 61 . 65 . 67 . 67 . 69 . 73 . 75 . 75

1. Introduction While kinetics are mostly responsible for the stability of macroemulsions, thermodynamics plays a role because adsorption of the surfactant or polymer on the interface of the globules affects the stability in a major way. In what follows it will be shown that thermodynamics can explain some of the overall features observed experimentally regarding the stability of emulsions. Two attempts have been made to relate macroemulsions to thermodynamics. One group of authors tried to explain the surfactant performance through the interfacial tension w1]4x. They observed that a lowering of the interfacial tension sometimes increases the stability, but that a too-low interfacial tension has the opposite effect. Interesting experiments have been carried out by Boyd et al. w5x, who prepared emulsions containing equal volumes of oil and water and a fixed weight amount of mixtures of nonionic dispersants. They observed that increasing the hydrophilic]lipophilic balance ŽHLB. of the dispersant, the initial rate of coalescence displayed two minima and a maximum at the phase inversion point. The first minimum was for a WrO emulsion and the second for an OrW emulsion. Berger et al. w4x, using a series of polypropylene glycol ethoxylates as dispersants Ž2 wt.% with respect to water., mineral oil as the oil phase and water containing 2 wt.% NaCl as the water phase, repeated the kind of experiments reported by Boyd et al. w5x. Equal volumes of the two phases have been mixed and the time in which complete separation took place was used as a measure of stability. They noted that for the same weight amount of dispersant, the emulsion stability passed, with increasing HLB, through a maximum in the WrO range, followed by a minimum at the phase inversion point and again a maximum in the OrW range. The interfacial tension between the two phases before emulsification exhibited a minimum for the HLB corresponding to the phase inversion point. It is clear that the stability depends upon the HLB and is not directly related to the interfacial tension. A second group of researchers has noted a parallelism between emulsions and microemulsions formed with surfactants possessing a polyŽethylene oxide. head group w6,7x. The emulsions were of the oil in water ŽOrW. kind at low tempera-

180

Nanodispersions E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

61

tures and of the water in oil ŽWrO. kind at relatively high temperatures. Similarly, an OrW microemulsion was in equilibrium with excess oil at low temperatures; a WrO microemulsion coexisted with excess water at high temperatures and a middle phase microemulsion coexisted with both excess phases in an intermediate range of temperatures. Shinoda and Saito w6,7x noted that the phase inversion temperature provided by the water]oil]polyŽethylene oxide. surfactant phase diagram coincides with the phase inversion temperature of the emulsion formed by mixing a microemulsion with its excess phase, and that the emulsion was of the same kind as the microemulsion. While microemulsions are thermodynamically stable, and the stability of emulsions has a kinetic origin, in both cases the adsorption of the dispersant upon the interface of the globules is responsible for stability. For this reason it appears natural to attempt to explain the above equality between the two inversion temperatures on the basis of surfactant adsorption. In addition, both the micro and macro-emulsions obey in many cases the Bancroft rule w8,9x, which indicates that the phase in which a larger amount of dispersant is present becomes the continuous phase; there are, however, some violations of this rule which will be discussed later in the paper. The scope of the present review is to emphasize that thermodynamics can explain the above experimental observations. The next section ŽSection 2., which is based on ref. w10x, will be concerned with the effects of HLB Ždenoted in what follows h. on the interfacial tension and on the stability of macroemulsions, the goal being to explain the observations of Boyd et al. w5x and of Berger et al. w4x. Section 3, which is based on ref. w11x, will examine the effect of temperature on the interfacial tension at the oil]water interface by assuming that no microemulsion or emulsion is formed, as well as its effect on the stability of emulsions. Shinoda and Saito’s observations regarding the equality of the two inversion temperatures will be thus explained. Finally, the Bancroft rule w8,9x, and some of the violations of this rule, will be examined in the spirit of ref. w12x. 2. Effect of HLB (h) on the interfacial tension and the stability of macroemulsions [10] 2.1. The interfacial free energy At constant temperature, the interfacial tension g of a water]oil system containing a single surfactant solute can be calculated at thermodynamic equilibrium starting with the Gibbs adsorption equation dg s yG d m

Ž1 .

where G is the surface excess of the surfactant and m is its chemical potential. In order to characterize the tendency of dispersants to generate OrW or WrO emulsions, Griffin w13,14x suggested an empirical scale which he called hydrophilic]lipophilic balance ŽHLB. which will be denoted as h. Surfactants with

181

Dynamics of deposition of Brownian particles or cells on surfaces E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

62

low values of h are more compatible with the oil phase Žare more hydrophobic. and tend to stabilize the water in oil ŽWrO. emulsions, while those with high values of h are more compatible with water Žare more hydrophilic. and tend to stabilize the ŽOrW. emulsions. The surfactants with intermediary values of h are ineffective as stabilizers. Davies w15]17x proposed a group contribution method to evaluate h from the molecular structure of the surfactants. While the HLB scale does not account for the effect of temperature and the nature of the oil, it is useful at room temperature and as a qualitative concept and will be used in this section. Consider a planar interface between water and oil, and a surfactant with a polyŽethylene oxide. head group distributed at equilibrium between the two phases. It is assumed that no macro or micro-emulsification nor solubilization takes place. For sufficiently small values of h, the surfactant is partitioned mostly in the oil phase, while for sufficiently large values of h, it is partitioned mostly in the water phase. The surfactant is mostly nonaggregated at small values of h, because it is present in oil where only small aggregates are formed w18,19x, but becomes aggregated above the CMC for large values of h, when the surfactant is mostly partitioned in water. The case in which the concentration in the water phase is below the CMC will be considered first. For dilute systems, one can write 0 m s m0W q kT lnC W s mO q kT lnCO ,

Ž2 .

where C is the concentration of the surfactant, the superscript 0 refers to the standard state, the subscripts W and O to water and oil, respectively, k is the Boltzmann constant and T is the absolute temperature. Consequently, the concentrations in the oil and water phases are related via the equation CO s C W e Dm 0 r kT ,

Ž 3.

where 0 Dm 0 s m0W y mO .

Ž4 .

A mass balance for equal volumes of water and oil leads to CO q C W s 2C,

Ž 5.

where C is the surfactant concentration per unit volume of the system, which was kept constant in the experiments w4,5x which we attempt to interpret. Eliminating CO between Eq. Ž3. and Eq. Ž5., yields CW s

2C 1 q e Dm 0 r kT

.

Ž6 .

The change of g with h is given by the expression dg dh

s yG

dm dh

,

Ž7 .

182

Nanodispersions E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

63

in which, because dm

s

dh

d m0W dh

q

kT dC W CW d h

and dC W dh dm dh

CW

sy

kT 1 q eyD m

0

r kT

d m 0W

1

s

0 . d Ž m 0W y mO

1

1 q e Dm 0 r kT d h

q

dh 1

, 0 d mO

1 q ey Dm 0 r kT d h

.

Ž 8.

The standard chemical potential in water decreases with increasing hydrophilicity because of increasing favorable interactions between surfactant and water, and that in oil increases. Consequently, d m0W dh

- 0 and

0 d mO

dh

) 0.

Ž9.

For low values of h, Dm 0 is positive and large, because m 0W has its largest and its smallest value, and Eq. Ž8. becomes

0 mO

dm dh

,

0 d mO

dh

) 0.

Žsmall h .

Ž 10a .

For large values of h, Dm 0 is negative and large in absolute value and one can similarly conclude that dm dh

f

d m0W dh

- 0.

Ž large h .

Ž 10b .

The surface excess G is a positive quantity even when the interactions between surfactant and water become as favorable as those between surfactant and oil. In the latter case Dm 0 s 0 and the concentrations in the two phases become equal. G does not vanish in that case, because the interactions between water and head group and between oil and hydrocarbon chain are stronger than those between water and hydrocarbon chain and oil and head group, respectively. It is, however, small because the interactions at the interface are not much stronger than those in the bulk. Consequently, dg dh dg dh

- 0, )0

Ž small h . Ž large h .

Ž 11a . Ž 11b .

183

Dynamics of deposition of Brownian particles or cells on surfaces E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

64

and it is clear that g passes through a minimum. The position of the minimum is given by d mrd h s 0, hence by the equation d m 0W

1

1 q e Dm 0 r kT d h

q

0 d mO

1

1 q ey Dm 0 r kT d h

s 0.

Ž 12.

0 Because the derivatives d m0W rd h and d mO rd h have opposite signs, there is a value h c of h for which g is minimum. If near the minimum, the curves m 0W and 0 mO are symmetrical, then Dm 0 s 0 at the minimum and the concentrations of surfactant in oil and water become equal. Even when the curves are not completely symmetrical, dgrd h f 0 for Dm 0 s 0, because G is small for h s h c and the factor which multiplies G in Eq. Ž7. is also small. A qualitative plot of g against h is presented in Fig. 1. If the concentration in water surpasses the CMC, aggregates are formed and the above equations are no longer valid. A conclusion can be, however, reached by combining Eq. Ž1. and Eq. Ž2. to obtain an expression wEq. Ž13.x in terms of the oil phase where no important aggregation takes place:

dg dh

s yG

ž

0 d mO

dh

q

kT dCO CO d h

/

.

Ž 13.

0 With increasing hydrophilicity, hence increasing h, mO increases and CO decreases. Consequently, 0 d mO

dh

) 0 and

dCO dh

-0

Ž 14.

0 and g can pass through an extremum. At low values of h, mO is expected to increase more rapidly than at high values; in contrast, the concentration CO is

Fig. 1. Qualitative representation of the interfacial tension at a planar oil]water interface against HLB Ž h..

184

Nanodispersions E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

65

expected to vary more rapidly at high values of h, because of the aggregation of the surfactant as micelles in the aqueous phase. As a result, dm

f

0 d mO

) 0, Žsmall h . dh kT dCO f - 0 Ž large h . dh CO d h

dh dm

Ž 15a . Ž 15b .

and g passes through a minimum. A similar conclusion can be reached by assuming that the concentration of the nonaggregated surfactant is equal to the CMC s C W1 , hence that

m s m0W q kT lnC W 1 .

Ž 16a .

Combining Eq. Ž1. and Eq. Ž16a. yields dg dh

s yG

ž

d m 0W dh

q

kT dC W 1 CW1

dh

/

.

Ž 16b .

With increasing hydrophilicity, m 0W decreases and C W1 increases, hence d m0W dh

- 0 and

dC W 1 dh

) 0.

Ž 17.

At low values of h, m 0W is expected to decrease less rapidly than at high values; in contrast, C W1 is expected to increase more rapidly at low values of h. Consequently, dm dh dm dh

f f

kT dC W 1 CW1 d m0W dh

dh

,

Žsmall h .

Žlarge h .

Ž18a .

Ž 18b .

and g passes through a minimum. It is clear from the above considerations that there is no parallelism between the stability of emulsions, which passes through two maxima and a minimum between them, and the interfacial tension g which passes through a minimum. However, a parallelism is expected to occur between the amount of surfactant adsorbed per unit area and the stability of emulsion, because when the former is larger, the latter is higher due to greater steric repulsion. 2.2. Surface excess against HLB The discussion which follows implies that the mixing process by which the emulsions are generated is the same, because in order to compare the stabilities,

Dynamics of deposition of Brownian particles or cells on surfaces

185

E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

66

the surface areas generated by mixing should be nearly the same. G is expected to have low values for low and large values of h, as well as for that moderate value of h at which the concentrations of the surfactant in the two phases become equal Ž Dm 0 s 0.. At low values of h, the surfactant is mostly distributed in oil and the strong interactions between oil and surfactant make the adsorbability of the latter low. At large values of h, the surfactant is mostly distributed in water and the strong interactions between water and surfactant lead again to a low adsorbability. Finally, at a moderate value h 0 of h, the concentrations of surfactant in oil and water become equal Ž Dm 0 s 0. and the surface excess low because the adsorbability is moderate. Between the low value of h and h 0 , it is convenient to write

G s G Ž CO ,h . ,

Ž 19.

where CO s CO Ž h.. Consequently, dG dh

s

­G

ž / ­CO

dCO h

dh

q

­G

ž / ­h

Ž 20. CO

An increase in the concentration CO at constant hydrophilicity increases the surface excess, because more surfactant is adsorbed on the interface; similarly an increase in hydrophilicity at constant CO increases the amount adsorbed, because of increased favorable interactions with water. However, CO decreases with h because an increasing amount of surfactant is distributed in the water phase. Consequently, ­G

ž / ­CO

) 0, h

­G

ž / ­h

) 0 and CO

dCO dh

- 0.

Ž 21.

On the basis of these inequalities, one can conclude that G passes through an extremum. The extremum is a maximum because: Ži. G is small for small values of h, due to the low adsorbability caused by the strong interactions between surfactant and oil; Žii. it increases with h, due to the increasing favorable interactions between the head group of the surfactant and water; Žiii. passes through a maximum when the decrease of CO with increasing h compensates for the ratio of increases of G with h and CO ; it becomes again small for h s h 0 . Between h s h 0 and large values of h, it is convenient to write

G s G Ž C W ,h . and considerations similar to the preceding ones lead to the conclusion that G passes through a maximum. Above the CMC, C W should be replaced by C W1. The region between low values of h and h 0 corresponds to WrO emulsions and that between h 0 and large values to OrW emulsion. The highest stabilities occur at the maxima of G . The phase transition occurs at h s h 0 , and the emulsions are very unstable at low and high values of h, as well as around h s h 0 . Because the

Nanodispersions

186 E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

67

concentrations for h s h 0 are equal, h 0 s h c , where h c is the value of HLB at which the interfacial free energy exhibits a minimum. Fig. 2 provides a qualitative representation of G against h. It should also be noted that the presence of micelles intensifies the coalescence of the globules of oil of an OrW emulsion because of the depletion effect w10x.

3. Phase inversion temperature of macro and microemulsions [11] 3.1. The interfacial tension against temperature This time the temperature is the field variable and the Gibbs adsorption equation has the form dg s yG d m y sdT ,

Ž 22.

where s is the surface excess entropy per unit area. At sufficiently low temperatures, the hydrophilic surfactants have strong hydrogen bonding interactions with the water molecules and are mainly distributed in the water phase. ŽThere are cases in which the surfactant is distributed in the oil phase over the entire range of temperature. Such cases will be discussed later in the paper.. The increase in temperature weakens the hydrogen bonding, thus increasing the hydrophobicity of the surfactant molecules. In other words, the surfactant is increasingly distributed in the oil phase as the temperature increases and the following inequalities can be written: 0 d m0W rdT ) 0, d mO rdT - 0

Fig. 2. Qualitative representation of the surface excess G against HLB Ž h..

Ž 23.

187

Dynamics of deposition of Brownian particles or cells on surfaces E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

68

and dC W rdT - 0, dCO rdT ) 0.

Ž 24.

One can define a temperature T0 at which the concentrations in the two phases become equal. This temperature is provided by the equation: 0 Ž m0W Ž T0 . s mO T0 . .

Ž25 .

Considering equal volumes of oil and water and concentrations in water below the critical micelle concentration ŽCMC., Eqs. Ž2., Ž5. and Ž22. lead to: dg dT

½

s yG klnC W q q

1 q e Dm 0 r kT dT

Dm 0

1 1qe

d m 0W

1

yD m 0 r kT

T

5

y s.

q

1

0 d mO

1 q ey Dm 0 r kT dT Ž26 .

At low temperatures, m0W is small, because of the strong interactions between 0 the surfactant and water molecules, and mO large. Consequently, Dm 0 is negative and large in absolute value, C W f 2C and

CO f 0

and Eq. Ž26. leads to dg dT

d m 0W

s yG kln2C q

ž

dT

/

ys

Ž 27a .

0 At high temperatures, m0W is large and mO small, hence Dm 0 is positive and large,

C W s 0 and

CO f 2C

and Eq. Ž26. becomes dg dT

s yG kln2C q

ž

0 d mO

dT

/

y s.

Ž 27b .

Because the surfactants considered here are very sensitive to temperature, 0 d m0W rdT and
G

d m0W dT

4 s and G

0 d mO

dT

4s

188

Nanodispersions E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

69

and the entropy s can be neglected in Eqs. Ž27a.,Ž27b.. As a result, dgrdT - 0, Ž low temperatures . dgrdT ) 0, Ž high temperatures .

Ž 28.

and g passes through a minimum. The minimum is expected to be located near the temperature T0 , because at this temperature G is small Žsee the next subsection. and the factor multiplying G in Eq. Ž26. is also small. The latter factor is small, 0 because Dm 0 s 0 and d m 0W rdT and d mO rdT have opposite signs. If the concentration in the water phase becomes greater than the CMC, one can make the approximate assumption that the concentration of the nonaggregated surfactant remains constant and equal to the critical micelle concentration, C W1 . One thus obtains dg dT

s yG

ž

d m 0W dT

q klnC W 1 q

kT dC W 1 CW1

dT

/

y s.

Ž 29.

The hydrogen bonding between the head group of the surfactant and water is very sensitive to temperature; for this reason d m 0W rdT is expected to be larger at low temperatures. Because the hydrophobicity of the surfactant increases with temperature, dC W 1rdT - 0. In reality, the critical micelle concentration passes through a minimum with increasing temperature w20x. The minimum is, however, located around 558C for nonionic surfactants. As a result, the interfacial tension is expected to pass through a minimum. 3.2. Macroemulsions and microemulsions When the minimum against temperature of the interfacial tension between the two phases separated by a planar surface has a positive value ŽFig. 3., their mixing will lead to an emulsion whose stability depends upon the magnitude of the surface excess. When the surfactant is sufficiently hydrophilic, it will be mostly distributed in water at low temperatures, and the surface excess will be small because of the high compatibility of the surfactant with the water phase. At high temperatures, the surfactant will be mostly distributed in the oil phase and the surface excess will be small because of the high compatibility between the surfactant and oil. At the moderate temperature T0 at which the concentrations in the two phases become equal, the surface excess is again small. For the region between the low temperature and T0 it is convenient to write that G s G Ž C W ,T . where C W is the total concentration below the CMC and the concentration of the nonaggregated surfactant Žassumed equal to the CMC. above the CMC. Consequently, dG dT

s

­G

ž / ­C W

dC W T

dT

q

­G

ž / ­T

Ž 30. CW

Dynamics of deposition of Brownian particles or cells on surfaces E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

70

Fig. 3. Qualitative representation of the interfacial tension g against temperature, Ža. emulsions; Žb. microemulsions.

At constant temperature, G increases with C W , while at constant C W , it increases with T, because of progressively stronger interactions between surfactant and oil. Hence ­G

ž / ­C W

) 0 and T

­G

ž / ­T

) 0. CW

Because ŽdC W rdT . - 0, G passes through an extremum. The surface excess G is small at low temperatures, increases with temperature, because of increasingly weaker interactions between the surfactant and water, passes through a maximum,

189

190

Nanodispersions E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

71

Fig. 4. Qualitative representation of the surface excess G against temperature.

when the decrease of C W with temperature compensates for the ratio of increases of G with T and C W , and finally becomes small when T reaches T0 . Similarly, between T0 and high temperatures it is convenient to write G s G Ž CO ,T .. Considerations similar to the previous ones indicate that G passes through a maximum between the two temperatures. Fig. 4 presents a qualitative picture regarding the dependence of G on T. When the minimum of g with respect to T is negative ŽFig. 3., the interfacial tension is negative in the range of temperatures in which the curve g s g ŽT . intersects the abscissa. A negative g makes the water]oil interface unstable to thermal and mechanical perturbations and a spontaneous emulsification with the formation of globules of oil in water and water in oil takes place. At low temperatures, because of coalescence, only a fraction of the globules of oil survive

191

Dynamics of deposition of Brownian particles or cells on surfaces E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

72

after they attain the equilibrium radius; an OrW microemulsion in equilibrium with excess oil free of water is thus generated. At high temperatures, a WrO microemulsion in equilibrium with excess water free of oil is formed, and, at intermediate temperatures, a middle phase microemulsion in equilibrium with both excess phases is produced. The large interfacial area created increases the amount of surfactant adsorbed and decreases its concentrations in the bulk, thus increasing the interfacial tension at the surface of the globules and between the microemulsion and excess phaseŽs. to small positive values. At low temperatures, a microemulsion of the OrW type coexists with excess oil, while at high temperatures, a WrO microemulsion coexists with excess water, for reasons which will be discussed in Section 4. A phase inversion occurs at a temperature corresponding to the minimum interfacial tension, which is located in the middle of the middle phase microemulsion range. The interfacial tension g at the planar interface has a minimum near the temperature T0 . Indeed, at the latter temperature G is small, Dm 0 s 0 and 0 because d m 0W rdT and d mO rdT have opposite signs and s is also small Žbecause G is small., dgrdT f 0. The temperature T0 is provided by Eq. Ž25. and is independent of the concentration of surfactant. In other words, the two minima of Fig. 4 which correspond to the phase inversion temperatures of a macroemulsion Žthe curve with a positive minimum. and microemulsion Žthe curve with a negative minimum. are the same. When emulsions are generated from a microemulsion and its excess phase, the emulsion is of the same kind as the microemulsion, the phase inversion temperature is obviously located in the middle of the middle phase microemulsion range and the above conclusion remains valid. The above results explain the observation of Shinoda and Saito w6,7x that the phase inversion temperature ŽPIT. of emulsions can be provided by the ternary equilibrium phase diagram. The sufficiently strong hydrophobic surfactants are preferentially distributed in the oil phase over the entire range of temperatures. One such surfactant, C 12 E 5 Ža nonionic surfactant of the polyoxyethylene glycol ether type., was investigated by Binks w21x, and its behavior will be explained in Section 4. As in the previous cases, the inequalities Ž23. and Ž24. remain valid; however, Dm 0 does not change sign but is always positive. For concentrations below the CMC, the inversion temperature Tin can be obtained on the basis of Eq. Ž26., from the minimum of the interfacial tension. Neglecting the entropy s and klnC W compared to the other terms, one obtains d m0W

1 1 q e Dm 0 ŽT in .r kT in q

ž / dT

q T in

0 d mO

1 1 q ey Dm 0 ŽT in .r kT in

1

Dm 0 Ž Tin .

1 q eyD m 0 ŽT in .r kT in

Tin

ž / dT

TsT in

Ž 31a .

s0

0 If the derivatives d m 0W rdT and
192

Nanodispersions E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

73

inversion temperatures are expected to be the same for macro and micro-emulsions. It is likely that, because of the interactions involved, the adsorbability is such that G against temperature can be again represented qualitatively as in Fig. 4. For concentrations greater than the CMC, Eq. Ž29. leads, after neglecting the entropy s, to the following expression for the inversion temperature: d m0W

ž / dT

q k lnC W 1Ž Tin . q TsT in

kTin C W 1 Ž Tin .

ž

dC W 1 dT

/

s0

Ž 31b .

Ts T in

It should be noted that an inversion temperature may not exist for a too-hydrophobic, or a too-hydrophilic surfactant.

4. The Bancroft rule In most cases, in both micro- and macroemulsions, the phase in which the surfactant is most soluble constitutes the continuous phase. This rule was formulated by Bancroft w8,9x in 1913 for macroemulsions in the form ‘a hydrophile colloid will tend to make water the dispersing phase while a hydrophobe colloid will tend to make water the dispersed phase’. Winsor w22x extended this rule to microemulsions. There are, however, some violations of the rule. Smith and Lim w23x investigated the morphologies and inversions of emulsions formed by pairs of conjugate liquids in the system C 4 H 9 OC 2 H 4 OH-n-C 10 H 22 -aqueous NaCl solution as a function of volume fraction of oil or water. They noted that the emulsions obeyed the Ostwald w24,25x stereometric hard sphere model and violated the Bancroft rule, because they inverted at a volume fraction of the dispersed phase of about 0.6. Ross and Kornbrekke w26x, mixing two immiscible conjugate solutions of benzene]ethanol]water, observed that, for any volume ratio of the two liquids, only a probability of obtaining one or the other type of emulsion can be determined. In the above cases, relatively weak surface active compounds were employed and the mixing process, hence hydrodynamics, was responsible for the outcome. In contrast, Binks w21x investigated a system containing a nonionic surfactant of the polyoxyethylene glycol ether type, namely C 12 E 5 , which is a stronger surface active agent. Regarding the phase behavior, he noted an OrW microemulsion coexisting with excess oil at low temperatures, a WrO microemulsion in equilibrium with excess water at high temperatures, and a middle phase microemulsion in equilibrium with both excess phases at intermediate temperatures. This is as expected. However, the Bancroft rule was violated at low temperatures, because the continuous phase was water and the surfactant was preferentially distributed in the oil phase. Regarding the macroemulsions, Binks noted that at low surfactant concentrations the Bancroft rule was violated since the higher concentration of surfactant was in the oil phase, which was the dispersed and not the continuous phase. Obviously, the Bancroft rule is valid for microemulsions and macroemulsions for different reasons. Thermodynamics is responsible for it in the former and kinetics

193

Dynamics of deposition of Brownian particles or cells on surfaces 74

E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

in the latter case. For macroemulsions three regimes were identified w12x. At low surfactant concentrations, or for weak surface active agents, the type of emulsion is controlled by the mixing process Žhydrodynamics.; at sufficiently high concentrations it is controlled by the mass transfer Žonly in this case the Bancroft rule is valid., and at even higher concentrations by hole nucleation in black films. Some of the violations of the rule noted by Binks for macroemulsions were at low surfactant concentrations Žthe mixing controlled regime for which the rule is not valid.. The macroemulsion formed by mixing a microemulsion with its excess phase was of the same kind as the microemulsion because, for the surfactants and concentrations employed, the mass transfer is responsible for the outcome, and the Bancroft rule should be obeyed. Indeed, it is obeyed because the microemulsion contains much more surfactant than the excess phase and the rule indicates that it should become the continuous phase. Regarding the phase behavior, it is clear that the state with the lowest free energy will be achieved. Detailed calculations are however complicated and for the time being a simple, intuitive approach will be employed. Winsor w22x explained qualitatively the phase behavior in terms of the R1 ratio, which compares the surfactant ]oil interactions to those between surfactant and water. Two ratios are, however, needed to describe the behavior of a system w12x. One of them is the ratio R1 R1 s A sorAsw ,

Ž 32.

where A so is a measure of the strength of the surfactant ]oil interactions and A sw of the surfactant ]water interactions in the bulk phases. Obviously, the surfactant will be preferentially distributed in the water phase when R1 - 1 and in the oil phase when R1 ) 1. The second ratio R 2 indicates the preferred curvature and can be obtained on the basis of an approximate expression for the free energy of formation of a microemulsion. The packaging of the surfactant molecules at the surface of the globules is expected to provide a major contribution to the free energy of the system. Because the globules are small Ž; 10 nm., the thickness of the surfactant layer cannot be neglected compared to the radius of the globules and the interfacial free energy of a globule can be evaluated as Fint s A hw g hw q A co gco ,

Ž33 .

where A hw and A co are the surface areas of the globule between the head groups and water, and between hydrocarbon chains and oil, respectively, and g hw and gco are the corresponding interfacial tensions, which, because the globules are small, are curvature dependent. At low temperatures, the interactions between the head groups of the surfactant and water are strong because of hydrogen bonding, and hence g hw - gco . In this case, Fint is smaller when the head groups are located on the external larger area and an OrW microemulsion in equilibrium with excess oil will be generated Žthe curvature of the globules is positive.. At temperatures sufficiently high for the hydrogen bonding to be sufficiently weakened, g hw ) gco

194

Nanodispersions E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76

75

and a WrO microemulsion in equilibrium with excess water will be formed Žthe curvature of the globule is negative.. At moderate temperatures, g hw f gco , and a middle phase microemulsion Žzero curvature. will be generated. The ratio R 2 between the interactions between the hydrocarbon chains of the surfactant molecules of the interfacial layer and oil and those between the head groups in the same layer and water determine the kind of microemulsion and is expected to vary in the same direction as g hw rgco . The ratio R1 determines the distribution of the surfactant between water and oil, while the ratio R 2 determines the type of microemulsion. When both ratios are smaller or larger than unity, Bancroft’s rule is obeyed, while when one of them is larger and the other one smaller than unity, it is violated. In Bink’s experiments, water was the continuous phase even though the surfactant was preferentially distributed in oil, because g hw - gco Ž R 2 - 1., while A so ) A sw Ž R1 ) 1. w12x.

5. Conclusion In conclusion, thermodynamics provides some information about the overall behavior of macroemulsions, suggests to correlate the stability of macroemulsions to the surface excess, relates the phase inversion temperature or the phase inversion HLB to the minimum of the interfacial tension at a planar interface between oil and water and explains why the phase inversion temperature is the same for macroemulsions and the corresponding microemulsions.

References w1x M. Bourrel, A. Graciaa, R.S. Schechter, W.H. Wade, J. Colloid Interface Sci. 72 Ž1979. 161. w2x H.L. Rosano, D. Jon, J. Oil Am. Chem. Soc. 59 Ž1982. 360. w3x A. Graciaa, Y. Barakat, R.S. Schechter, W.H. Wade, S.J. Yiv, J. Colloid Interface Sci. 89 Ž1982. 217. w4x P.D. Berger, C.A. Hsu, J.P. Arendell, SPE ŽSociety Petroleum Engineers. 16285, San Antonio, TX, 1987. w5x J. Boyd, C. Parkinson, P. Sherman, J. Colloid Interface Sci. 41 Ž1972. 359. w6x K. Shinoda, H. Saito, J. Colloid Interface Sci. 26 Ž1968. 70. w7x K. Shinoda, H. Saito, J. Colloid Interface Sci. 30 Ž1969. 258. w8x W.D. Bancroft, J. Phys. Chem. 17 Ž1913. 501. w9x W.D. Bancroft, J. Phys. Chem. 19 Ž1915. 275. w10x E. Ruckenstein, Langmuir 4 Ž1988. 1318. w11x E. Ruckenstein, Langmuir 13 Ž1997. 2494. w12x E. Ruckenstein, Langmuir 12 Ž1996. 6351. w13x W.C. Griffin, J. Soc. Cosmet. Chem. 1 Ž1949. 311. w14x W.C. Griffin, J. Soc. Cosmet. Chem. 5 Ž1954. 249. w15x J.T. Davis, Proceedings 2nd International Congress on Surface Activity, London, 1957, p. 417. w16x J.T. Davis, E.K. Rideal, Interfacial Phenomena, Academic Press, NY, 1963, p. 129. w17x J.T. Davis, in: J.F. Danielli, K.G.A. Parkhurst, A.C. Riddford ŽEds.., Progress in Surface Science, vol. 2, Academic Press, NY, 1964, p. 129. w18x A.S. Kertes, H. Gutmann, in: E. Matijevic ŽEd.., Surface and Colloids Science, Interscience, NY, 1975, p. 8.

Dynamics of deposition of Brownian particles or cells on surfaces 76

w19x w20x w21x w22x w23x w24x w25x w26x

E. Ruckenstein r Ad¨ . Colloids Interface Sci. 79 (1999) 59]76 E. Ruckenstein, R. Nagarajan, J. Phys. Chem. 84 Ž1980. 1349. P. Becher, in: M.J. Schick ŽEd.., Nonionic Surfactants, M. Dekker, NY, 1967, p. 478. B.P. Binks, Langmuir 9 Ž1993. 25. P.A. Winsor, Chem. Rev. 68 Ž1968. 1. D.H. Smith, K.H. Lim, J. Phys. Chem. 94 Ž1990. 3746. W. Ostwald, Kolloid Z. 6 Ž1910. 103. W. Ostwald, Kolloid Z. 7 Ž1910. 64. S. Ross, R.E. Kornbrekke, J. Colloid Interface Sci. 81 Ž1981. 58.

195

196

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

197

198

Nanodispersions

Dynamics of deposition of Brownian particles or cells on surfaces

199

200

Nanodispersions

Introduction to CHAPTER 3 Stability of dispersions

3.1

E. Ruckenstein, R. Nagarajan: “Critical micelle concentration. A transition point for micellar size distribution,” JOURNAL OF PHYSICAL CHEMISTRY 79 (1975) 2622–2626. E. Ruckenstein, R. Nagarajan: “Critical micelle concentration and the transition point for micellar size distribution,” JOURNAL OF PHYSICAL CHEMISTRY 85 (1981) 3010. I.V. Rao and E. Ruckenstein: “Phase behaviour of mixtures of sterically stabilized colloidal dispersions and free polymer,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 108 (1985) 389–402. E. Ruckenstein and I.V. Rao: “Effect of solvent on the stability of mixtures of sterically stabilized dispersions and free polymers,” COLLOIDS AND SURFACES 17 (1986) 185–205. E. Ruckenstein: “Thermodynamics of microemulsions: relation between their structure and phase behaviour,” FLUID PHASE EQUILIBRIA 20 (1985) 189–206. E. Ruckenstein: “Thermodynamics of microemulsion II,” Handbook of Microemulsion Science and Technology, edited by P. Kumar and K. L. Mittal, Marcel Dekker, New York, 1999, 45–58. R. Nagarajan, E. Ruckenstein: “Molecular theory of microemulsions,” LANGMUIR 16: 16 (2000) 6400– 6415. E. Ruckenstein: “Phase transitions in insoluble monolayers of surfactants,” Encyclopedia of Surface and Colloid Science (2007) Taylor & Francis. E. Ruckenstein, M. Manciu: “On the Stability of Lyotropic Lamellar Liquid Crystals and the Thicknesses of Their Lamellae,” LANGMUIR 17 (2001) 5464–5475.

The effect of addition of a free polymer to a sterically stabilized non-aqueous colloidal dispersion is investigated using a second-order perturbation approach, which allows one to determine the conditions for phase separation. It is 3.2 shown that the separation into a disordered dilute dispersion and an ordered concentrated dispersion occurs at a limiting value of the free polymer concentration, which decreases with increasing molecular weight [3.3]. The effect of the 3.3 solvent on the stability of sterically stabilized dispersions containing free polymers is also investigated, and it is shown that the limiting concentration of the free polymer (at which separation occurs) decreases with increasing mo3.4 lecular weight of the polymer; with increasing particle size; with increasing temperature; with thinner steric layers; and with better solvents for the free polymers, in agreement with experiment [3.4]. 3.5 In contrast to macroemulsions, which are kinetically stable, microemulsions are dispersions of oil in water or water in oil, stabilized by a surfactant, which are thermodynami3.6 cally stable. They often contain a short chain alcohol (referred to as cosurfactant) and a salt. The microemulsions can be of the droplet type, either with spherical oil droplets dispersed in a continuous water phase ( O/W microemul3.7 sions) or with spherical water droplets dispersed in a continuous oil phase (W/O microemulsions). The droplet type microemulsion can be either a single phase or part of a two 3.8 phase system wherein the microemulsion coexists with an excess of dispersed phase (an upper phase of excess oil in the case of O/W and a lower phase of excess water in the 3.9 case of W/O microemulsions).There are also non-droplet type microemulsions characterized by randomly distributed oil and water micro-domains and exhibiting bicontinuity in both oil and water. Such a system is known as a middle The critical concentration, C crit , is defined as the concentration phase microemulsion, that occurs as part of a three phase at which there is a change in the size distribution of the sur- system with the middle phase coexisting with an upper oil factant aggregates from a monotonically decreasing curve phase and a lower water phase. A novel thermodynamic (below C crit), to one that possesses a maximum and a minimum treatment that explains the thermodynamic stability and above the critical concentration. The transition in shape oc- phase behaviour of microemulsions ( [3.5] and [3.6] ) is curs at a size distribution that displays an horizontal inflexion suggested. Thermodynamic equations are derived regardpoint. The critical micelle concentration (CMC) has a value ing the single phase microemulsions, the transition to a miclose to the critical value C crit [3.1] and [3.2]. croemulsion in equilibrium with an excess dispersed phase

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_3, © Springer Science+Business Media, LLC 2010

202

and the transition from a two phase system to one in which a middle phase microemulsion coexists with both excess phases. A predictive molecular thermodynamics approach is developed for microemulsions, to determine their structural and compositional characteristics [3.7]. The theory is built upon a molecular level model for the free energy change. For illustrative purposes, numerical calculations are performed for the system water, cyclohexane, sodium dodecyl sulfate as surfactant, pentanol as cosurfactant and NaCl as electrolyte. The droplet radius, the thickness of the surfactant layer at the interface, the number of molecules of various species in the droplets, and the distribution of the components between droplets and the continuous phase are calculated. The theory also predicts the transition from a mi-

Nanodispersions

croemulsion to one in equilibrium with an excess phase as well as the transition from a two-phase droplet-type microemulsion to a three-phase system involving a bicontinuous microemulsion. In addition, the persistence length of the bicontinuous microemulsion, the surface tension between phases, and the curvature elasticity are also calculated [3.7]. A thermodynamic treatment, similar to that used for microemulsions, as well as an approximate statistical mechanical one, are developed to explain the phase transition in monolayers of insoluble surfactants [3.8]. A similar thermodynamic approach is applied to multilamellar liquid crystals, and it is shown that, for a given set of interactions and bending moduli, only narrow ranges of the thicknesses of the water and oil layers are allowed [3.9].

Stability of dispersions

203

204

Nanodispersions

Stability of dispersions

205

206

Nanodispersions

Stability of dispersions

207

208

Nanodispersions

Stability of dispersions

209

210

Nanodispersions

Stability of dispersions

211

212

Nanodispersions

Stability of dispersions

213

214

Nanodispersions

Stability of dispersions

215

216

Nanodispersions

Stability of dispersions

217

218

Nanodispersions

Stability of dispersions

219

220

Nanodispersions

Stability of dispersions

221

222

Nanodispersions

Stability of dispersions

223

224

Nanodispersions

Stability of dispersions

225

226

Nanodispersions

Stability of dispersions

227

228

Nanodispersions

Stability of dispersions

229

230

Nanodispersions

Stability of dispersions

231

232

Nanodispersions

Stability of dispersions

233

234

Nanodispersions

Stability of dispersions

235

236

Nanodispersions

Stability of dispersions

237

238

Nanodispersions

Stability of dispersions

239

240

Nanodispersions

Stability of dispersions

241

242

Nanodispersions

Stability of dispersions

243

244

Nanodispersions

Stability of dispersions

245

246

Nanodispersions

Stability of dispersions

247

248

Nanodispersions

Stability of dispersions

249

250

Nanodispersions

Stability of dispersions

251

252

Nanodispersions

Stability of dispersions

253

254

Nanodispersions

Stability of dispersions

255

256

Nanodispersions

Stability of dispersions

257

258

Nanodispersions

Stability of dispersions

259

260

Nanodispersions

Stability of dispersions

261

262

Nanodispersions

Stability of dispersions

263

264

Nanodispersions

Stability of dispersions

265

266

Nanodispersions

Stability of dispersions

267

268

Nanodispersions

Stability of dispersions

269

270

Nanodispersions

Stability of dispersions

271

272

Nanodispersions

Stability of dispersions

273

274

Nanodispersions

Stability of dispersions

275

276

Nanodispersions

Stability of dispersions

277

278

Nanodispersions

Stability of dispersions

279

280

Nanodispersions

6400

Langmuir 2000, 16, 6400-6415

Molecular Theory of Microemulsions R. Nagarajan* Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802

E. Ruckenstein Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York 14260 Received December 2, 1999. In Final Form: May 25, 2000 In this paper, a molecular thermodynamic approach is developed to predict the structural and compositional characteristics of microemulsions. The theory can be applied not only to oil-in-water and water-in-oil droplet-type microemulsions but also to bicontinuous microemulsions. This treatment constitutes an extension of our earlier approaches to micelles, mixed micelles, and solubilization but also takes into account the self-association of alcohol in the oil phase and the excluded-volume interactions among the droplets. Illustrative results are presented for an anionic surfactant (SDS)-pentanol-cyclohexane-waterNaCl system. Microstructural features including the droplet radius, the thickness of the surfactant layer at the interface, the number of molecules of various species in a droplet, the size and composition dispersions of the droplets, and the distribution of the surfactant, oil, alcohol, and water molecules in the various microdomains are calculated. Further, the model allows the identification of the transition from a twophase droplet-type microemulsion system to a three-phase microemulsion system involving a bicontinuous microemulsion. The persistence length of the bicontinuous microemulsion is also predicted by the model. Finally, the model permits the calculation of the interfacial tension between a microemulsion and the coexisting phase.

1. Introduction Microemulsions are thermodynamically stable dispersions of oil and water stabilized by a surfactant and, in many cases, also a cosurfactant.1-4 The microemulsions can be of the droplet type, either with spherical oil droplets dispersed in a continuous medium of water (oil-in-water microemulsions, O/W) or with spherical water droplets dispersed in a continuous medium of oil (water-in-oil microemulsions, W/O). The droplet-type microemulsions can be either a single-phase system or part of a two-phase system wherein the microemulsion phase coexists with an excess dispersed phase (an upper phase of excess oil in the case of O/W and a lower phase of excess water in the case of W/O microemulsions). There are also nondroplet-type microemulsions, referred to as middle-phase microemulsions. In this case, the microemulsion phase is part of a three-phase system with the microemulsion phase in the middle coexisting with an upper phase of excess oil and a lower phase of excess water. One possible structure of this middle-phase microemulsion, characterized by randomly distributed oil and water microdomains and bicontinuity in both oil and water domains, is known as the bicontinuous microemulsion. Numerous experimental studies have shown1,2,4 that one can achieve a transition * To whom correspondence should be addressed ([email protected]). (1) Rosano, H. L., Clausse, M., Eds. Microemulsion Systems; Marcel Dekker: New York, 1987. (2) Friberg, S. E.; Bothorel, P. Microemulsions: Structure and Dynamics; CRC Press: Boca Raton, FL, 1987. (3) Ruckenstein, E.; Chi, J. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1690. Ruckenstein, E. Fluid Phase Equilib. 1985, 20, 189; J. Colloid Interface Sci. 1998, 204, 142. (4) Verhoeckx, G. J.; De Bruyn, P. L.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1987, 119, 409. Overbeek, J. Th. G.; Verhoeckx, G. J.; De Bruyn, P. L.; Lekkerkerker, H. N. W. J. Colloid Interface Sci. 1987, 119, 422; De Bruyn, P. L.; Overbeek J. Th. G.; Verhoeckx, G. J. J. Colloid Interface Sci. 1989, 127, 244.

Figure 1. Schematic representation of the transition from a two-phase system with an oil-in-water droplet microemulsion to a three phase system with a bicontinuous microemulsion and to a two-phase system with a water-in-oil droplet microemulsion.

from a two-phase system involving an O/W droplet-type microemulsion to a two-phase system involving a W/O droplet-type microemulsion by passing through a threephase system containing a bicontinuous microemulsion (Figure 1). Such phase transitions can be realized in many different ways, e.g., by increasing the concentration of added electrolytes when the surfactant employed is ionic, by increasing the temperature when the surfactant is nonionic, and by increasing the concentration of alcohol (typically butanol to hexanol) when the surfactant is ionic, nonionic, or zwitterionic. Theoretical studies of microemulsions have focused on various features of the complex systems mentioned

10.1021/la991578t CCC: $19.00 © 2000 American Chemical Society Published on Web 07/14/2000

281

Stability of dispersions Molecular Theory of Microemulsions

above.3-10 However, currently there exists no model which can predict the detailed phase behavior and the microstructural characteristics of the microemulsion systems starting from the composition and molecular structures of the constituent molecules. Such models were formulated in our earlier work for micelles, vesicles, and mixed micelles in aqueous media and for the solubilization of oil inside such aggregates.11-15 These models are extended here to microemulsions. On this basis, a predictive approach for the various characteristics of microemulsions, such as the droplet radius, the number of molecules of various species in a droplet, and the size and composition dispersions of the droplets, is developed. The model also allows the prediction of the interfacial tension between the microemulsion and the coexisting phase. Further, the model allows one to examine the transition from a twophase system involving a droplet-type microemulsion to a three-phase system involving a bicontinuous microemulsion. The persistence length in bicontinuous microemulsions is also predicted. In developing a molecular treatment for microemulsions, we take into account the chain packing, which depends on the curvature of the aggregates, the self-association of alcohol in the oil, the penetration of the interfacial layer region of the microemulsion by the oil molecules, their effects on chain packing and interfacial energies, and the hard-core interactions among the droplets. Since the present treatment of microemulsions is an extension of our approach to micellization and solubilization, many of the above aspects of the model that were presented in detail in earlier papers are only briefly summarized here. Two additional features included in this paper are selfassociation of alcohol in oil domains and hard-sphere repulsive interactions among droplets. To illustrate the present theory, calculations were carried out for a system consisting of an anionic surfactant, oil, water, alcohol, and electrolyte. Obviously, this model can be readily applied to simpler microemulsions free of alcohol, wherein the phase transitions are caused by varying the electrolyte concentration or the temperature.

Langmuir, Vol. 16, No. 16, 2000

Figure 2. Geometrical variables RW and RO characterizing the size of O/W and W/O droplets in microemulsions.

(with k ) S for surfactant, A for alcohol, O for oil, and W for water), among which gkI molecules are present in the interfacial layer and the remaining ones are present in the core containing mainly the dispersed phase. For the reason mentioned above, there are no water molecules in the interfacial layer (i.e., gWI ) 0). The radii RO and RW provide the boundaries of the interfacial layer. The volume of the aggregate Vg can be related to the radii RO and RW via

Vg )

(5) Scriven, L. In Micellization, Solubilization and Microemulsions; Mittal, K. L., Ed.; Plenum Press: New York, 1977; p 277. (6) Talmon, Y.; Prager, S. J. Chem. Phys. 1978, 69, 2984. (7) de Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982, 86, 2284. (8) Ruckenstein, E. In Progress in Microemulsions; Martellucci, S.; Chester, A. N., Eds.; Plenum Press: New York, 1989; pp 1 and 41. (9) Peck, D. G.; Schechter, R. S.; Johnston, K. P. J. Phys. Chem. 1991, 95, 9541. (10) Borkovac, M. J. Chem. Phys. 1989, 91, 6268; Adv. Colloid Interface Sci. 1992, 37, 195. (11) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1979, 71, 580; Langmuir 1991, 7, 2934. (12) Nagarajan, R. Langmuir 1985, 1, 331; in Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 54. (13) Rao, I. V., Ruckenstein, E. J. Colloid Interface Sci. 1986, 113, 375. (14) Nagarajan, R.; Ruckenstein, E. Sep. Sci. Technol. 1981, 16, 1429; in Surfactants in Solution. Theoretical and Applied Aspects; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; p 923. Nagarajan, R.; Chaiko, M. A.; Ruckenstein, E. J. Phys. Chem. 1984, 88, 2916. (15) Ruckenstein, E. In Progress in Microemulsions; Martellucci, S., Chester, A. N., Eds.; Plenum Press: New York, 1989; p 31; Colloids Surf. 1992, 65, 95.

4πRO3 3

(2.1)

for the water-in-oil droplets and

2. Geometric Characteristics of Microemulsions Both O/W and W/O droplet structures have a core region surrounded by an interfacial film, depicted schematically in Figure 2. The interfacial film excludes the headgroups of the surfactant and the alcohol. This region is analogous to the micelle core and, as confirmed by many experiments, is free of water molecules and is thus hydrophobic in character. Each droplet contains gk molecules of kind k

6401

Vg )

4πRW3 + gSIvSP + gAIvAP 3

(2.2)

for the oil-in-water droplets, where vSP and vAP are the volumes of the polar headgroups of the surfactant and alcohol molecules, respectively. The volume VI of the interfacial layer per surfactant molecule is given by

VI )

gSIυST + gAIυAT + gOIvO 4π |RO3 - RW3| ) (2.3) 3gSI gSI

where υST, υAT, and υO are the volumes of surfactant tail, alcohol tail, and oil, respectively. The surface area of the droplet in contact with water, per surfactant molecule, is given by

a)

4πRW2 gSI

(2.4)

Combining eqs 2.3 and 2.4 yields

(

RO 3VI 3 ) RW a|RO - RW| 4

)

1/2

-

1 2

(2.5)

For the sake of simplicity, the thickness of the interfacial layer |RO - RW| is taken to be the composition average of the surfactant tail length lS and the alcohol tail length lA:

282 6402

Nanodispersions Langmuir, Vol. 16, No. 16, 2000

|RO - RW| )

Nagarajan and Ruckenstein

lS + (gAI/gSI)lA

(2.6)

1 + (gAI/gSI)

The determination of this thickness from free energy minimum considerations require a more detailed treatment of the interfacial film than justified by the available experimental information. Consequently, a particular O/W or W/O droplet can be completely characterized by three geometrical variables. A set of convenient variables is (i) the surface area of the droplet in contact with water per surfactant molecule, a, (ii) the ratio of alcohol-to-surfactant molecules in the interfacial layer, gAI/gSI, and (iii) the ratio of oil-tosurfactant molecules in the interfacial layer, gOI/gSI. For a flat interfacial layer, denoting the layer thickness by |RO - RW|F and the area per surfactant molecule of the flat interface by aF, one can write the following on the basis of molecular packing considerations:

aF|RO - RW|F ) VI

(2.7)

Therefore, for a flat interface, only two geometrical variables are needed for its complete specification, instead of the three needed for a spherical droplet. 3. Size Distribution of Droplet-Type Microemulsions For a W/O microemulsion, the size and composition distribution of the aggregates can be written as for micelles, mixed micelles, and solubilization. However, whereas the earlier expressions were formulated for dilute solutions where interaggregate interactions are negligible, the size and composition distribution for microemulsions will take into account the hard-sphere repulsions among the droplets. For a single-phase microemulsion, the mole fraction XgO of water-in-oil droplets is given by the following expression (see Appendix A for the derivation):

XgOγgO ) ΩgSIfWOgW

(

Ω ) (XSWγSW)(XAWγAW)gAI/gSI(XOOγOO)gOI/gSI exp ∆µ°gI )

gAI gOI 1 µ°gI - µ°SW µ°AW µ° gSI gSI gSI OO

)

∆µ°gI kT

(3.1)

In the preceding equations, the subscript g refers to a droplet containing a total of g molecules (g ) gS + gA + gO + gW), XgO is the mole fraction of aggregates of size g in the continuous oil (O) phase, XSW and XAW are the mole fractions of singly dispersed surfactant and alcohol molecules, respectively, in the dispersed water phase, µ°SW and µ°AW are their standard chemical potentials, and γSW and γAW are their activity coefficients. XOO denotes the mole fraction of oil in the continuous oil phase, µ°OO is its standard chemical potential, and γOO is its activity coefficient. It is important to emphasize that the alcohols used as cosurfactants have high solubility in the oil phase because they are present there as both single molecules and aggregates. Therefore, in calculating XOO, one must consider the self-association of alcohol in the oil phase. The activity coefficients arise because of the hard-sphere interactions among the droplets. Therefore, in the dispersed phase, which is free of droplets, the activity coefficients are equal to unity, whereas, in the continuous phase, which contains the droplets, the activity coefficients differ from unity.

The fractional saturation of water in the oil phase is denoted fWO. For a single-phase microemulsion, fWO is less than unity. As the amount of water in the phase increases, fWO progressively increases and reaches the constant value of unity when an excess aqueous phase coexisting with the microemulsion phase appears, creating a two-phase system. The standard free energy difference ∆µ°gI in eq 3.1 is due to the transfer of one surfactant molecule and (gAI/gSI) alcohol molecules from water and (gOI/gSI) oil molecules from pure oil to the interfacial layer of the microemulsion droplet. Expressions for ∆µ°gI are provided in section 7 of the paper. When eq 3.1 is used to calculate the size and composition distributions of droplets in a single-phase O/W microemulsion, XgO and γgO should be replaced by XgW and γgW W O and the factor f gWO should be replaced by f gOW , where fOW denotes the fractional saturation of oil in the water phase. As before, fOW is less than unity for a single-phase microemulsion and reaches the constant value of unity when a two-phase system comes into existence with an excess oil phase coexisting with the microemulsion phase. To predict the characteristics of a microemulsion on the basis of the size distribution relation (3.1), only two control variables, XSW and XAW, must be specified. This is equivalent to fixing the total amounts of surfactant and alcohol in the microemulsion. Detailed calculations of the size and composition distributions involved in eq 3.1 are not required. Instead, as shown in the following section and Appendix C, the size and composition dispersions can be obtained from calculations involving only the maximum term of the size distribution. By calculating how the size and composition corresponding to the most populous aggregates (i.e., the maximum in Xg) vary when the surfactant concentration is slightly perturbed, one can obtain the variance in the size and composition of aggregates with respect to their average values. In eq 3.1, the activity coefficients appear as a result of the hard-sphere repulsions among the droplets. Since the calculations focus on the most populous aggregates, the hard-sphere repulsions will be expressed in terms of a single droplet size corresponding to the most populous aggregates. One can derive expressions for the activity coefficients γkO of a component k in the continuous phase O starting from an equation for the osmotic pressure of a hard-sphere fluid,3,4 such as that based on the Carnahan-Starling equation of state (see Appendix B for the derivation):

vk 1 + ΦgO + ΦgO2 - ΦgO3 ΦgO (3.2) VgO (1 - ΦgO)3

ln γkO ) -

where VgO is the volume of a droplet and ΦgO is the volume fraction of the droplets in the microemulsion. Combining eq 3.2 with the Gibbs-Duhem equation, we obtain the following expression for the activity coefficient γgO of the droplet (Appendix B):

(

ln γgO ) ln

ΦgO/VgO

∑ ΦkO/vk k*g

XgO

)

+

7ΦgO - 3ΦgO2 - ΦgO3 (1 - ΦgO)2 (3.3)

where ΦkO is the volume fraction of component k in the continuous phase O. Equation 3.1 is an implicit relation because the activity coefficients γgO and γgW depend on the volume fractions

283

Stability of dispersions Molecular Theory of Microemulsions

Langmuir, Vol. 16, No. 16, 2000

of the droplets and hence on the mole fractions XgO and XgW, respectively. The volume fractions of the droplets can be related to the mole fractions of various components and their respective volumes through the relations

ΦgW )

XgWVgW XgWVgW + XSWvS + XWWvW + XAWvA

ΦgO )

XgOVgO

XgOVgO +

∑jXjAOvA + XOOvO

(3.4)

In the first expression, XWW can be replaced by (1 - XgW - XAW - XSW) because the sum of all mole fractions should be unity. In the second expression, the total mole fraction ∑jXjAO of alcohol in the oil phase accounts for the alcohol present as both monomers and aggregates in the oil phase and is calculated on the basis of the continuous-association model16 as described in Appendix D. For given values of the control variables XSW and XAW, the maximum in XgO (or XgW) was found, using the IMSL subroutine ZXMWD, by solving the implicit eq 3.1 in combination with eq 3.4. As mentioned in section 2, the area per surfactant molecule a, the alcohol-to-surfactant ratio gAI/gSI, and the oil-to-surfactant ratio gOI/gSI in the interfacial layer were selected as the three independent variables with respect to which the maximization was carried out. The total volume fraction ΦS of surfactant present in the microemulsion is given by

gSIvS ΦS ) ΦgO (for W/O) VgO

gSIvS ΦS ) ΦgW (for O/W) VgW (3.5)

because the amount of singly dispersed surfactant molecules is relatively small and can be neglected. The total volume fraction of alcohol in the microemulsion, ΦA, can be obtained from a species balance, since the concentrations of alcohol in the water domain, the oil domain, and the interfacial layer are known. 4. Size and Composition Dispersions of Droplets The size and composition dispersions of droplets can be estimated by using the maximum-term method, without performing detailed distribution calculations. This is accomplished by adapting the approach used for micellar systems,17,18 as described in Appendix C. The aggregate corresponding to the maximum of XgO or XgW is considered to provide the number-average size and composition of the equilibrium droplets. For each component k ()S, A, O, or W) present in the microemulsion, the mean-square deviation σn2(k) from the number-average number of molecules gn(k) can be shown to be

σn(k) gn(k)

)

(

∆ lngn(k) ∆ ln(gn(k)XgM)

)

1/2

(4.1)

where XgM ()XgO or XgW) is the mole fraction of the droplet corresponding to the maximum in the distribution, gn(k) are the numbers of molecules of kind k in that droplet, and ∆ denotes the difference between two values. Con(16) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1969; Chapter 7. Acree, W. E., Jr., W. E. Thermodynamic Properties of Non-Electrolyte Solutions; Academic Press: New York, 1984; Chapter 8. (17) Nagarajan, R. Langmuir 1994, 10, 2028. (18) Nagarajan, R. In Structure-Performance Relationships in Surfactants; Esumi, K., Ueno, M., Eds.; Marcel Dekker: New York, 1997; pp 1-81.

6403

sequently, by calculating the maximum XgM and the corresponding numbers of molecules gn(k) for two slightly different conditions (for example, through marginal variations in XSW), one can estimate σn(k) for each of the four components k and thus provide information about the size and composition dispersions of the droplet. The root-mean-square deviation in the droplet radius σR is obtained from purely geometric considerations using the calculated values of σn(k). Equating the volume of the droplet to the volumes contributed by all the component molecules yields

4πR3 ) gn(k) vk 3 4π(R + σR)3 4πR3 ) σn(k) vk 3 3

∑

∑

(4.2)

By rearranging eq 4.2, we obtain the following expression for the dispersion in the droplet radius:

(

σR ) 1+ R

)

∑σn(k) vk 1/3 - 1 ∑gn(k) vk

(4.3)

5. Calculation of Interfacial Tension The interfacial tension γF refers to that at the flat interface (denoted by subscript F) that separates the microemulsion from the coexisting excess dispersed phase. The flat interface can be treated as a planar infinite aggregate. The chemical potentials of the molecules in this interfacial layer should be equal to the chemical potentials of the molecules in the bulk phases and hence

( )

( )

( )

gAI µAW gOI µOO 1 (µ°gI)F γFaF µSW ) + + gSI F kT kT kT gSI F kT gSI F kT (5.1)

In the preceding equation, the first term on the left-hand side is the reference state chemical potential per surfactant molecule of an infinite-size aggregate and the second term involving γF accounts for the interfacial tension to which this infinite aggregate is subjected. The terms on the righthand side represent the chemical potentials of the surfactant, alcohol, and oil in the bulk phase in equilibrium with the molecules at the interface. Using eq 3.1, the above equation can be rewritten as

[

( )

γFaF (∆µ°gI)F gAI ) - ln γSWXSW + ln gAWXAW + kT kT gSI F gOI ln γOOXOO ) -ln ΩF (5.2) gSI F

( )

]

where (∆µ°gI)F is the standard free energy difference associated with the transfer of one surfactant molecule and (gAI/gSI)F alcohol molecules from their infinitely dilute states in water and (gOI/gSI)F oil molecules from a pure oil phase to the flat interfacial layer. For given chemical potentials of various components in the system, the composition of the flat interfacial film (namely, aF, (gAI/ gSI)F, and (gOI/gSI)F) is obviously different from the composition of the interfacial film of droplets in the microemulsion phase (namely, a, gAI/gSI, and gOI/gSI). 6. Bicontinuous Microemulsions No well-defined structural characterization of a bicontinuous microemulsion is currently available. However, it has been suggested that oil and water domains are

284 6404

Nanodispersions Langmuir, Vol. 16, No. 16, 2000

Nagarajan and Ruckenstein

randomly distributed within the bicontinuous microemulsion separated by locally planar interfaces.3-8 A characteristic domain size or persistence length ξK has been defined7 such that, for length scales smaller than ξK, the interface is essentially flat. Using this concept, one can establish conditions for the formation of a three-phase system containing a bicontinuous microemulsion phase coexisting with excess oil and water phases. One can also calculate, on the basis of the present molecular theory, the persistence length and the interfacial tensions between microemulsion and excess phases. Considering a planar configuration, the size distribution of microemulsion domains in a bicontinuous microemulsion coexisting with two excess phases can be written, in a form similar to eq 3.1, as

Xg )

1 gSI Ω K F

(

)

∆µ°gI ΩF ) XSWXAW(gAI/gSI)FXOO(gOI/gSI)F exp kT

( )

( )

F

( )

gAI gOI 1 (∆µ°gI)F ) (µ°gI)F - µ°SW µ°AW µ° gSI F gSI F gSI F OO (6.1) In these equations, the standard free energy terms are approximated as for a flat interface since the domains are considered locally flat and the activity coefficients have been taken as unity since there are no droplets in the water and oil domains. The factor K accounts for the fact that some molecules must remain at the corners and edges of the surfaces separating the oil and the water domains and not on the flat regions. Thus, K is a measure of the free energy cost of having molecules in such locations compared to those in the flat regions. While, for droplet microemulsions, an optimal value for gSI could be obtained by maximizing Xg in eq 3.1, there is no maximum for Xg in eq 6.1, since ΩF is independent of gSI. The equilibrium characteristics of the microemulsion are obtained by maximizing ΩF for given values of the control variables XSW and XAW. One may note the analogy between the above equation and the equation Xg ) (1/K)Yg, where Y ) X1 exp(-∆µ°cyl/kT), governing the formation of rodlike micelles.11,18 The equilibrium structural characteristics of rodlike micelles are obtained from the maximum of Y or, equivalently, minimum of ∆µ°cyl, which is the standard state free energy difference between a molecule in the cylindrical part of the micelle and a singly dispersed molecule. K is the factor that accounts for the free energy cost of having molecules at the two globular ends of the cylinder rather than in the cylindrical part of the micelle and is thus analogous to the K of eq 6.1. Since Y is independent of g, the size distribution Xg does not have a maximum and rodlike micelles of all aggregation numbers are present in appreciable concentrations. Y is the analogue of ΩF in eq 6.1. The maximum of ΩF is determined with respect to the two independent variables, the area per surfactant molecule of the interface, aF, and the alcohol-to-surfactant ratio in the interfacial layer, (gAI/gSI)F. In contrast to that in the droplet-type microemulsion, the oil-to-surfactant ratio in the interfacial layer, (gOI/gSI)F, is not an independent variable but is determined by the packing constraint (eq 2.7) for flat layers. The maximum value of ΩF will be close to but smaller than unity, since otherwise the domain size distribution would diverge. Since the domain size gSI can assume all

possible values, the average domain size of the bicontinuous microemulsion is given by

(gSI)avg )

∑gSIXg ) 1 ∑Xg 1 - ΩF

(6.2)

where the last equality is obtained by performing the analytic summation of the two factors in the above expression on the basis of eq 6.1. The equilibrium area per surfactant molecule being aF, the total surface area of the average domain is aF(gSI)avg, which can be used to obtain the following expression for the persistence length ξK:

6ξK2 ) (gSI)avgaF )

aF 1 - ΩF

(6.3)

The numerical constant 6 in the preceding equation implies the cubic domain assumed by de Gennes and Taupin,7 and the surface area aF(gSI)avg is that of a cube of characteristic size ξK. For other domain geometries, such as the polyhedra considered by Talmon and Prager,6 this numerical coefficient will be different. The relative proportions of the oil and water domains are related to the persistence length ξK and the volume fraction ΦS of the surfactant present through the species balance relation

ξK )

6Φ(1 - Φ)vS ΦSaF

(6.4)

In this equation, Φ stands for the volume fraction of the oil domain and includes the hydrophobic tails of the surfactant and alcohol in the interfacial layer, while (1 Φ) stands for the volume fraction of the water domain and includes the headgroups of the surfactant and alcohol in the interfacial layer. For given values of the control variables XSW and XAW, the maximum of ΩF (eq 6.1) and the equilibrium characteristics of the flat layer can be calculated by employing the IMSL optimization subroutine ZXMWD. The concentration XSW does not affect the equilibrium characteristics aF, (gAI/gSI)F, and (gOI/gSI)F of the interfacial layer but does influence the magnitude of ΩF and the total amount of surfactant in the bicontinuous microemulsion. The persistence length can be calculated from eq 6.3 using the optimized value of ΩF. For a specified total volume fraction ΦS of surfactant in the microemulsion, the proportions of the oil and the water domains in the microemulsion can then be determined from eq 6.4. Finally, the volume fraction of the total alcohol ΦA in the microemulsion can be calculated from a species balance, since the concentrations of alcohol in the water and the oil domains and in the interfacial layer are known. To determine whether a droplet-type microemulsion coexists in equilibrium with an excess dispersed phase or a bicontinuous microemulsion coexists with both excess phases, one has to carry out the maximizations of Xg (eq 3.1) and ΩF (eq 6.1) for given values of XAW and XSW. For low values of XAW as well as for sufficiently large values of XAW, a maximum in Xg occurs, indicating the formation of a two-phase droplet-type microemulsion system. However, for intermediate values of XAW, no maximum of Xg exists and the radius of the equilibrium droplet diverges to infinity. In this region, a three-phase system involving a bicontinuous microemulsion is favored. The interfacial tension at the flat interface is calculated using eq 5.2, but with the activity coefficients equated to unity. Hence

Stability of dispersions

285

Molecular Theory of Microemulsions

[

Langmuir, Vol. 16, No. 16, 2000

( ) ( ) ]

γFaF (∆µ°gI)F gAI ) - ln XSW + ln XAW + kT kT gSI F gOI ln XOO ) -ln ΩF (6.5) gSI F Since for physical reasons ΩF is close to unity, the logarithmic term in the above equation can be expanded in series, and after taking into account eq 6.3, one obtains

γF )

kT kT kT (-ln ΩF) ) (1 - ΩF) ) aF aF 6ξK2

(6.6)

This expression was suggested previously by de Gennes and Taupin7 on the basis of scaling arguments. Experimentally, one can measure three different interfacial tensions in the three-phase system, namely, at the microemulsion-excess water phase boundary (γMW), at the microemulsion-excess oil phase boundary (γMO), and between the excess oil and excess water phases (γOW). On the basis of intuitive arguments, it has been suggested8 that

γF ) γOW

γMW ) ΦγOW

γMO ) (1 - Φ)γOW (6.7)

7. Free Energy of Formation of the Interfacial Layer The equations developed in previous sections can be used to calculate the structural features of microemulsions, provided explicit expressions for the standard free energies of transfer of surfactant and alcohol molecules from their infinitely dilute states in water and of oil molecules from the pure oil phase to the interfacial layer of the microemulsion droplets are available. Such expressions are given below for spherical layers of O/W droplets and W/O droplets and also for flat layers. The difference in the standard state free energy consists of a number of contributions:

∆µ°gI ) (∆µ°gI)tr + (∆µ°gI)def + (∆µ°gI)int + (∆µ°gI)steric + (∆µ°gI)dipole + (∆µ°gI)ionic + (∆µ°gI)mix (7.1) which account respectively for the transfer of surfactant and alcohol tails from water to the interfacial film, the deformation of the tails to satisfy packing constraints, the free energy of formation of the oil-water interface, the steric, dipolar, and ionic interactions among the headgroups, and the free energy of mixing of surfactant, alcohol, and oil in the film region. These contributions were previously examined in our treatments for micelles, mixed micelles, and solubilization in micelles.11,18 Consequently, only the modifications required for the treatment of microemulsions are emphasized below. 7.1. Transfer Free Energy of Surfactant and Alcohol Tails. For the interfacial layer that contains gSI surfactant and gAI alcohol molecules, the transfer free energy per surfactant molecule is obtained by adding the contributions of the surfactant and alcohol molecules:

( )

(∆µ°gI)tr (∆µ°gI)tr,S gAI (∆µ°gI)tr,A ) + kT kT gSI kT

(7.2)

The expressions for the transfer free energy of methyl and methylene groups obtained previously11 are used to calculate the transfer free energies of the surfactant and alcohol tails. The expression for a methylene group of an

6405

aliphatic tail at temperature T (in kelvins) has the form

(∆µ°gI)tr/kT ) 5.85 ln T + 896/T - 36.15 - 0.0056T (7.3) while that for a methyl group in the aliphatic chain has the form

(∆µ°g)tr/kT ) 3.38 ln T + 4064/T - 44.13 + 0.02595T (7.4) For the short-chain alcohols considered here, the methylene group adjacent to the hydroxyl group is considered to make no contribution to the transfer free energy because it is located close to the hydrophilic domain of the aggregate. High concentrations of electrolyte affect the transfer free energy, and the extra contribution due to the presence of NaCl at 25 °C is given by19

(∆µ°gI)ex tr /kT ) (-0.322 - 0.064nC)Cadd

(7.5)

where Cadd is the molarity of NaCl in the aqueous domain of the microemulsion and nC is the number of methyl and methylene groups of the surfactant or alcohol tails. Since no direct information on this extra contribution at other temperatures is available, the temperature dependence of the methyl and methylene group contributions in the presence of added electrolyte in the above equation will be taken as equal to that in salt-free systems. 7.2. Deformation Free Energy of Surfactant and Alcohol Tails. As in the treatment of mixed micelles,12,18 this free energy contribution is calculated by adding the contributions from both surfactant and alcohol tails. Since the thickness of the interfacial layer |RO - RW| is larger than the extended length lA of the alcohol tail, a part of the interfacial film is not accessible to the alcohol tails. Such a packing problem for molecules of differing tail lengths was treated in an earlier paper,12 giving

(

(∆µ°gI)def 10π2 |RW - RO|2 gAI lA2 ) P + kT 80 gSI N L2 N L2 S

A

)

(7.6)

where NS ()lS/L) and NA ()lA/L) are the number of segments of the surfactant and alcohol tails, L ()4.6 Å) is the effective segment length (defined such that L2 is the cross-sectional area of the tail), and P is the packing factor defined as

P)

VI a|RW - RO|

(7.7)

For flat interfaces, P ) 1, because of the packing constraint provided by eq 2.7. 7.3. Interfacial Free Energy of the Layer-Water Contact. This free energy contribution is calculated using the expression

(

(∆µ°gI)int σagg gAI ) a - aoS a kT kT gSI oA

)

(7.8)

where σagg is the macroscopic interfacial tension between the hydrophobic domain and water, a is the surface area per surfactant molecule of the interfacial layer, and aoS and aoA are the areas per molecule shielded from contact (19) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1973; 2nd ed., 1980.

Nanodispersions

286 6406

Langmuir, Vol. 16, No. 16, 2000

Nagarajan and Ruckenstein

with water by the headgroups of surfactant and alcohol molecules, respectively. For surfactants, aoS is equated to the smaller between apS and L2, as described in refs 11 and 18, where apS is the cross-sectional area of the polar headgroup. For alcohols, the cross-sectional area of the hydroxyl headgroup apA is about 8 Å2, which is smaller than L2; hence, apA ) aoA ) 8 Å2. The interfacial tension σagg depends on the composition of the hydrophobic domain of the layer and thus on the surfactant tail-water, alcohol tail-water, and oil-water interfacial tensions. Since the surfactant and alcohol tails considered here are aliphatic chains, their interfacial tensions against water are practically the same and are taken to be σSW, for which expressions are provided in refs 11 and 18. If the oil in the interfacial layer is a saturated hydrocarbon, the oil-water interfacial tension σOW is practically the same as the tail-water interfacial tension. However, if the oil is an aromatic hydrocarbon, its interfacial tension against water is appreciably different from that between the tails and water. Depending upon the oil-water interfacial tension, a preferential location of the oil molecules in contact with water will occur, and this affects the overall interfacial tension σagg. For a volume fraction ηO of the oil in the interfacial layer, the interfacial tension σagg can be estimated using the approach of Prigogine,20,21 as in our earlier paper dealing with solubilization.11 In that approach, the interfacial tension of a solution was obtained by first calculating the composition of the surface monolayer, thus accounting for the preferential location of any one component at this surface monolayer. It is not possible to identify a bulk region and surface monolayer region in the present case, since the interfacial layer itself consists only of a monolayer. Nevertheless, the formal application of the Prigogine approach provides a way to at least approximately take into account the specificity of the oil. Thus, the interfacial layer is considered to be a solution with a uniform oil volume fraction of ηO. Corresponding to this composition, the volume fraction of the oil in the surface monolayer ηSO is determined using the equation

[

(ηSO/ηO)vST/vO

]

σSW - σOW 2/3 ln ) vS + S kT (1 - ηO)/(1 - ηO)

[

χOS

] [

]

3 1 ((1 - ηO) - ηO) - χOS ((1 - ηSO) - ηSO) (7.9) 4 2

where the parameter χOS denotes the interaction parameter for the oil and the surfactant and alcohol tails. After eq 7.9 is solved for ηSO, the interfacial tension σagg is calculated using the equation

( ) ( [

)

σagg - σSW 2/3 1 - ηSO vST S vS ) ln + 1(η - ηO) + kT 1 - ηO vO O

]

1 3 χOS (ηSO)2 - (ηO)2 (7.10) 2 4

The parameter for the interaction between the oil and the tails χOS is expressed in terms of the solubility parameters H H δH S , δA , and δO of the surfactant tail, the alcohol tail and the oil, respectively, using the expression (20) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Wiley: New York, 1966. (21) Siow, K. S.; Patterson, D. J. Phys. Chem. 1973, 77, 356.

χOS )

H 2 vO(δH O - δmix) kT

H H H δH mix ) ηSδS + ηAδA + ηOδO (7.11)

where ηS, ηA, and ηO are the volume fractions of the surfactant tail, alcohol tail, and oil, respectively, in the interfacial layer. The volume fractions are readily calculated from the respective molecular volumes and the composition of the interfacial layer. The solubility parameters22,23 are estimated to be 16.76 MPa1/2 for the surfactant tail, 9.1 MPa1/2 for the pentanol tail, and 16.8 MPa1/2 for cyclohexane (selected as the oil for illustrative purposes). 7.4. Headgroup Steric Interactions. This contribution is expressed as that for mixed surfactants. Because, in the present case, the free energy is expressed per surfactant molecule (not surfactant + alcohol molecule), one can write

(

) (

)

(∆µ°gI)steric gAI apS gAI apA )- 1+ ln 1 kT gSI a gSI a

(7.13)

where, as mentioned before, apS is the cross-sectional area of the polar headgroup of the surfactant and apA is the cross-sectional area of the hydroxyl headgroup. 7.5. Headgroup Dipole Interactions. For zwitterionic surfactants, the dipole-dipole interactions are evaluated using the expression

(∆µ°gI)dipole 2πe2d 1 ) kT aδkT H

(7.14)

where d is the dipole length, δ is the distance from the oil-water contact surface at which the dipole is located, aδ is the surface area per surfactant molecule evaluated at the distance δ, and H is a curvature correction given by one of the following expressions

H ) 1 + (d + δ)/RW, aδ ) a(1 + δ/RW)2 for O/W droplets H ) 1 - (d + δ)/RW, aδ ) a(1 - δ/RW)2 for W/O droplets H ) 1, aδ ) a

for a flat film

(7.15)

The dielectric constant  is a function of temperature24 and is evaluated using the relation

 ) 87.74 exp[-0.0046(T - 273)]

(7.16)

7.6. Headgroup Ionic Interactions. Ionic interactions arise at the interfacial layer if the surfactant has a charged headgroup and are calculated using the expression

( ) ∆µ°gI kT

{ [ ( ( )) ]

S S 2 1/2 + 1+ ionic 2 2 2Cg 2 S 2 1/2 1 1 S 2 1/2 1+ -1 ln + 1 + S 2 κS 2 2 2 (7.17) ) 2 ln

[( ( ) )

]

[

( ( ) ) ]}

where (22) Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand Reinhold: New York, 1970. (23) Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC Press: Boca Raton, FL, 1983. (24) CRC Handbook of Chemistry and Physics, 60th ed.; CRC Press: Boca Raton, FL, 1980.

287

Stability of dispersions Molecular Theory of Microemulsions

S)

4πe2 κaδkT

κ)

(

8πnoe2 kT

)

Langmuir, Vol. 16, No. 16, 2000

6407

1/2

no )

C1ion + Cadd NAv (7.18) 103

The area per molecule aδ which appears in the preceding equation is evaluated at a distance δ from the surface in contact with water and is curvature dependent. Expressions for aδ are given by eq 7.15. The distance δ is estimated as the distance from the surface in contact with water to the surface where the center of the counterion is located. κ is the reciprocal Debye length, no is the number of counterions in solution per cubic centimeter, C1ion is the molar concentration of the singly dispersed ionic surfactant molecules in water, Cadd is the molar concentration of the salt added to the surfactant solution, and NAv is Avogadro’s number. The last term in the right-hand side of eq 7.17 provides a curvature correction to the ionic interaction energy. For normal droplets, Cg ) 2/(RW + δ); for reverse droplets, Cg ) -2/(RW - δ); and for flat layers, Cg ) 0. 7.7. Free Energy of Mixing Inside the Layer Region. This free energy contribution is estimated using the Flory-Huggins expression25

[

]

(∆µ°gI)mix gAI gOI ) ln ηS + ln ηA + ln ηO + kT gSI gSI

[

vS

H 2 H 2 (δH gAI (δH S - δmix) A - δmix) + vA + kT gSI kT

]

H 2 gOI (δH O - δmix) vO (7.19) gSI kT

8. Predictions for Droplet-Type Microemulsions Calculations were carried out for a system consisting of the anionic surfactant sodium dodecyl sulfate, 1-pentanol (cosurfactant), cyclohexane, and water containing 0.3 M NaCl. As mentioned at the very beginning, the choice of this system was dictated by the possibility of identifying various types of phase behaviors for the same chemical components by merely changing the amount of added alcohol. In all calculations, we assumed the coexistence of an excess dispersed phase. This means that the droplet microemulsion phase is part of a two-phase system and that the amount of dispersed phase present in the droplet is the maximum achievable. The total surfactant volume fraction ΦS in the microemulsion is kept constant at 0.01 and the concentration XSW of the singly dispersed surfactant is negligibly small compared to the total amount of surfactant in the microemulsion. The alcohol-to-surfactant ratio in the microemulsion and, correspondingly, the alcohol-to-surfactant ratio in the interfacial layer of the droplet increase with increasing XAW. The radius RW of the water-oil interface (see Figure 2 for the definition of RW) predicted by the model is plotted as a function of the alcohol-tosurfactant volume ratio in the microemulsion in Figure 3. The calculations show that, for XAW less than 0.00172 and, correspondingly, for ΦA/ΦS less than 2.0 and gAI/gSI less than 1.012, an O/W droplet-type microemulsion coexists with an excess oil phase. When XAW is larger than 0.00184 (and, correspondingly, when ΦA/ΦS is larger than 3.15 and gAI/gSI is larger than 1.13), a W/O droplet-type microemulsion coexists with an excess water phase. For (25) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.

Figure 3. Predicted radii of microemulsion droplets as a function of the volume ratios of alcohol to surfactant in microemulsions. The predictions are for a system consisting of the surfactant SDS, 1-pentanol, 0.3 M NaCl, water, and cyclohexane. The total volume fraction of the surfactant in the microemulsions is 0.01. Filled circles denote O/W droplets, and open circles correspond to W/O droplets.

intermediate values of XAW (corresponding to ΦA/ΦS in the range 2.0-3.15), no maximum in the aggregate size distribution occurs because the equilibrium aggregate radius diverges. For these conditions, a three-phase system involving a bicontinuous microemulsion comes into existence. The radius of the droplet increases significantly as the two-phase system (corresponding to either O/W or W/O droplet microemulsions) nears the transition to a three-phase system. The number of surfactant molecules in the interfacial layer of a droplet is plotted in Figure 4 for the systems described in Figure 3. Detailed information regarding the composition of the microemulsions is provided by Figure 5. The volume fractions of oil, water, and alcohol plotted in the figures represent the total amounts in the microemulsion, and the volume fraction of surfactant ΦS is 0.01 in all cases. As the alcohol-to-surfactant ratio increases, the volume fraction of oil increases while that of water decreases, as has been generally observed in numerous experiments.1,3 The total volume fractions of the surfactant (ΦS) and alcohol (ΦA) determine the volume fraction of oil solubilized or water solubilized in the microemulsion. Figure 6 displays the predicted interfacial layer thickness and the area per surfactant molecule at the oil-water interface of the droplets as functions of the alcohol-tosurfactant ratio in the microemulsion. The calculations indicate that the area per surfactant molecule of the droplet is not a constant but increases with an increasing amount of alcohol. The extent of penetration of the oil into the interfacial layer is illustrated in Figure 7. The oilto-surfactant ratio in the interfacial layer exhibits a monotonic increase as the alcohol-to-surfactant ratio in the microemulsion increases. The volume fraction of droplets in the microemulsion phase predicted by the model is plotted in Figure 8, which

288 6408

Nanodispersions Langmuir, Vol. 16, No. 16, 2000

Figure 4. Numbers of surfactant molecules in the interfacial layers of droplets as a function of the volume ratios of alcohol to surfactant in microemulsions. All the system characteristics are identical to those described for Figure 3. Filled circles denote O/W droplets, and open circles correspond to W/O droplets.

Figure 5. Volume fractions of oil (squares), water (triangles), and alcohol (circles) in microemulsions containing O/W and W/O droplets as a function of the volume ratios of alcohol to surfactant in the microemulsions. Filled symbols refer to O/W droplet microemulsions, and open symbols, to W/O droplet microemulsions. All the system characteristics are identical to those described for Figure 3. In all cases, the volume fraction of the surfactant in the microemulsions is 0.01.

shows that the volume fraction increases as one moves from the two-phase toward the three-phase system. The interfacial tension at the flat surface between the micro-

Nagarajan and Ruckenstein

Figure 6. Thicknesses of the interfacial layers of droplets (circles) and areas per surfactant molecule of the droplets at an oil-water interface (squares) as functions of the volume ratios of alcohol to surfactant in microemulsions for systems containing O/W (filled symbols) and W/O (open symbols) droplets. All the system characteristics are identical to those described for Figure 3.

Figure 7. Alcohol-to-surfactant mole ratios and oil-to-surfactant mole ratios in the interfacial layers of O/W (filled symbols) and W/O (open symbols) droplets as functions of the volume ratios of alcohol to surfactant in microemulsions. All the system characteristics are identical to those described for Figure 3.

emulsion phase and the coexisting excess phase calculated with eq 5.2 is plotted in Figure 9, which reveals that the interfacial tension decreases as one moves from the twophase to the three-phase system.

289

Stability of dispersions Molecular Theory of Microemulsions

Langmuir, Vol. 16, No. 16, 2000

6409

Table 1. Predicted Size and Composition Dispersions of Microemulsion Droplets variable

O/W microemulsion

W/O microemulsion

〈RW〉 , Å 〈gSI〉 〈gAI/gSI〉 〈gOI/gSI〉 ΦS ΦA ΦO ΦW σn(S)/gn(S) σn(A)/gn(A) σn(O)/gn(O) σn(W)/gn(W) σR/〈RW〉

233.9 11 940 0.9228 0.9965 0.01 0.0158 0.0963 0.8779 0.219 0.241 0.274

282 16 460 1.152 1.226 0.01 0.0329 0.8206 0.1365 0.266 0.278 0.260 0.326 0.096

0.0822

Figure 8. Volume fractions of droplets as a function of the volume ratios of alcohol to surfactant in microemulsions. All the system characteristics are identical to those described for Figure 3. Filled symbols are for O/W microemulsions, and open symbols refer to W/O microemulsions.

Figure 10. Volume fraction of oil in a bicontinuous microemulsion as a function of the volume ratio of alcohol to surfactant in the microemulsion. The values of XAW and gAI/gSI are respectively 0.001 72 and 1.036 (filled squares) and 0.001 84 and 1.074 (open squares). In all cases, the volume fraction of surfactant is 0.01 and the volume fraction of water is found by subtraction from unity. The system consists of SDS, 1-pentanol, cyclohexane, water, and 0.3 M NaCl. Also shown are the corresponding results for O/W droplet microemulsions (filled circles) and for W/O droplet microemulsions (open circles).

Figure 9. Oil-water interfacial tensions at the flat surfaces between microemulsion phases and excess dispersed phases as a function of the volume ratios of alcohol to surfactant in microemulsions. All the system characteristics are identical to those described for Figure 3. Filled symbols are for O/W microemulsions, and open symbols refer to W/O microemulsions.

Estimates of the size and composition dispersions of droplets in O/W and W/O droplet-type microemulsions, obtained using the maximum-term method described in Appendix C, are summarized in Table 1. The calculations for the O/W droplet-type microemulsions were carried out

for XAW ) 0.001 62, while the calculations for the W/O droplet-type microemulsion were for XAW ) 0.00185. In all cases, the volume fraction of the surfactant in the microemulsion ΦS was kept constant at 0.01. The average radii of the droplets, the average numbers of surfactant, alcohol, and oil molecules in the interfacial layers of the droplets, the overall compositions of the microemulsions, and their dispersions are listed in Table 1. The values of σn/gn for all components are generally small, implying relatively narrow composition dispersions of the droplets. Consequently, the standard deviations σR of the droplet radii are also small. 9. Predictions for Bicontinuous Microemulsions As mentioned previously, for alcohol concentrations XAW in the water domain in the range 0.001 72-0.00 184 (corresponding to a ΦA/ΦS in the microemulsion between

290 6410

Nanodispersions Langmuir, Vol. 16, No. 16, 2000

Figure 11. Dependence of the volume fractions of oil (open symbols) and alcohol (filled symbols) in a bicontinuous microemulsion as a function of the mole fraction of alcohol in the water phase. The different symbols refer to different volume fractions of oil and water domains in the microemulsion: Φ ) 0.25 (circles), 0.45 (squares), 0.55 (triangles), and 0.75 (diamonds). The system consists of SDS, 1-pentanol, cyclohexane, water, and 0.3M NaCl.

2.0 and 3.25), three-phase systems involving a bicontinuous microemulsion are preferred thermodynamically. The overall composition limits within which a bicontinuous microemulsion can be generated are presented in Figure 10. The two lines correspond to XAW ) 0.001 72 and 0.001 84. For overall phase compositions between these two lines and for ΦS ) 0.01, a three-phase system is favored thermodynamically. As XAW increases from 0.00172 to 0.00184, the area per surfactant molecule aF increases from 59.06 to 59.88 Å2, while the alcohol-to-surfactant ratio in the interfacial layer (gAI/gSI)F increases from 1.036 to 1.074. All the bicontinuous microemulsions marked by points on a line are characterized by the same values of XAW, aF, and (gAI/gSI)F. One can observe from Figure 10 that, for a given ΦA/ΦS, there is a unique value of ΦO for an O/W or W/O droplet microemulsion, but a range of ΦO values lying between the two lines is possible for a bicontinuous microemulsion. The volume fraction of oil ΦO in the bicontinuous microemulsion, for a given value of ΦA/ΦS, is determined by the concentration XAW of the alcohol in the coexisting water phase. The volume fractions of oil ΦO (unfilled symbols) and alcohol ΦA (filled symbols) in the bicontinuous microemulsion are plotted in Figure 11 as a function of XAW, with the volume fraction of surfactant ΦS kept constant at 0.01. Because the sum of all volume fractions should be unity, the volume fraction of water ΦW can be determined. Each symbol in Figure 11 corresponds to a given volume fraction of oil in the bicontinuous microemulsion: Φ ) 0.25 (circle), 0.45 (square), 0.55 (triangle), and 0.75 (diamond). For a given value of ΦO, as XAW increases, ΦA increases, implying a corresponding decrease in ΦW (because all volume fractions sum to unity). This predicted increase in the oil-to-water ratio in a bicon-

Nagarajan and Ruckenstein

Figure 12. Dependence of the droplet radii in two-phase systems containing O/W (filled circles) or W/O microemulsions (open circles) and of persistence length/2 of a bicontinuous microemulsion (filled squares) on the volume fraction of oil in the microemulsion. In all cases, the volume fraction of the surfactant is 0.01. The system consists of SDS, 1-pentanol, cyclohexane, water, and 0.3 M NaCl.

tinuous microemulsion system with increasing alcohol content constitutes a well-known experimental result. The persistence length of the bicontinuous microemulsion predicted by the model is plotted in Figure 12 as a function of the volume fraction of oil in the microemulsion for XAW ) 0.001 72. The calculated persistence lengths for XAW ) 0.001 84 practically coincide with the former, because the areas per surfactant molecule in the interfacial layer aF are close to one another for the two values of XAW. The persistence length has a maximum when the oil and water domains have equal volume fractions. The radii of the O/W droplets increase with an increasing amount of alcohol in the low alcohol content range, while the radii of the W/O droplets decrease with an increasing amount of alcohol in the high alcohol content range. The interfacial tension calculated using eq 6.6 is plotted in Figure 13 as a function of the volume fraction of oil in the bicontinuous microemulsion. The calculated interfacial tensions for the two values of XAW ) 0.001 72 and 0.001 84 practically coincide (as expected, since the persistence lengths also practically coincide). The interfacial tension has a minimum when the oil and water domains have equal volume fractions. 10. Conclusions In this paper, a predictive molecular thermodynamic approach is developed to calculate the structural and compositional characteristics of microemulsions. The theory applies not only to oil-in-water and water-in-oil droplet-type microemulsions but also to bicontinuous microemulsions. The treatment is an extension of our earlier theories for micelles, mixed micelles, and solubilization but also takes into account the self-association of alcohol in oil and the volume-excluded interactions among

291

Stability of dispersions Molecular Theory of Microemulsions

Langmuir, Vol. 16, No. 16, 2000

G ) NOOµOO + NSOµSO +

6411

∑NjAOµjAO + NWOµWO + ∑NgOµgO (A.1)

where µ refers to the chemical potentials of given kinds of species. The last subscript on N and µ stands for the nature of the continuous phase, with O denoting the oil. The subscripts O, W, S, A, and jA refer to molecules of oil, water, surfactant, alcohol, and alcohol aggregates containing j alcohol molecules, respectively. The summations over j extend from 1 to ∞ in all the equations in this paper. A summation over g in the last term of eq A.1 accounts for the size and the composition distribution of droplets. (c) Equilibrium Condition. The chemical potentials of the singly dispersed alcohol molecules and alcohol aggregates are related via the expression

µjAO ) jµ1AO

(A.2)

which, combined with eq A.1, yields

G ) NOOµOO + NSOµSO + NAOµ1AO + NWOµWO +

∑NgOµgO

(A.3)

where Figure 13. Interfacial tension at the oil-water interface in a bicontinuous microemulsion system (filled squares) as a function of the volume fraction of oil in the microemulsion. In all cases, the volume fraction of surfactant is 0.01. The system consists of SDS, 1-pentanol, cyclohexane, water, and 0.3 M NaCl. Also shown are the interfacial tensions at the flat surface between the O/W droplet microemulsion phase and the excess oil phase (filled circles) and between the W/O droplet microemulsion phase and excess water phase (open circles) in two-phase systems.

the droplets. Illustrative numerical results are provided for an anionic surfactant (SDS)-pentanol-cyclohexanewater-NaCl system. Microstructural features including the droplet radius, the thickness of the interfacial layer, the numbers of molecules of various species in a droplet, the size and composition dispersions of the droplet, and the distributions of the surfactant, oil, alcohol, and water molecules in the various microdomains have been calculated. Further, the model allows the identification of the transition from a two-phase to a three-phase system. The characteristic persistence length of the bicontinuous microemulsion is also predicted. Finally, the model permits the calculation of the interfacial tension between the microemulsion and the coexisting phase. Appendix A: Equilibrium Size Distribution of Droplets A.1. A Water-in-Oil Droplet Microemulsion. (a) Description of the Phase. The microemulsion system is composed of NgO droplets of various sizes g (each consisting of surfactant, alcohol, oil and water molecules) and outside the droplets, NOO oil molecules, NSO surfactant molecules, NAO alcohol molecules, and NWO water molecules. The subscript g for a droplet denotes the total number of molecules of different kinds present in it (i.e., g ) gS + gA + gO + gW). The NAO alcohol molecules outside the droplets are present both as singly dispersed molecules and as aggregates. The number of alcohol aggregates containing j alcohol molecules is denoted by NjAO. (b) Free Energy of the Phase. The total Gibbs energy G of the microemulsion can be written as

NAO )

∑jNjAO

(A.4)

For a given composition of the system

NkO +

∑gkNgO ) constant

k ) O, S, A, W (A.5)

The condition δG ) 0, subjected to the mass constraints of eq A.5, leads to

µgO ) gOµOO + gSµSO + gAµ1AO + gWµWO

(A.6)

The chemical potentials of the components in the oil phase other than the droplets are expressed in the form

µkO ) µ°kO + kT ln(γkOXkO)

k ) S, 1A, W, O (A.7)

The standard chemical potential µ°OO corresponds to the pure oil phase, the standard chemical potential µ°1AO refers to the standard state of the monomeric alcohol in pure alcohol, and the standard chemical potentials µ°SO and µ°WO of the surfactant and water, respectively, correspond to their infinitely dilute solution states in the oil phase. For the droplets, the chemical potential is written as

µgO ) µ°gO + kT ln(γgOXgO)

(A.8)

where µ°gO is the standard chemical potential of the aggregate defined as that of an isolated aggregate in the continuous oil phase and γgO is the activity coefficient arising from the hard-sphere interactions. The standard chemical potential µ°gO of the aggregate can be split into a part due to the interfacial layer (denoted with the subscript I) and another part due to the dispersed water phase of the droplet (denoted with the subscript W). Thus, one can write

µ°gO ) µ°gI + gOWµOW + gSWµSW + gAWµAW + gWWµWW (A.9) where µ°gI is the standard chemical potential of the interfacial layer of the droplet, while µkW are the chemical potentials of the components k present in the dispersed

292 6412

Nanodispersions Langmuir, Vol. 16, No. 16, 2000

Nagarajan and Ruckenstein

water phase. The number of molecules gOW, gSW, gAW, and gWW, which refer to the oil, surfactant, alcohol, and water molecules in the dispersed water phase, are related to gOI, gSI, gAI, and gWI, the number of molecules in the interfacial layer of the droplet, via

gkI ) gk - gkW

k ) S, A, O, W

(A.10)

(d) Relation between Chemical Potentials. The chemical potentials µkO of all the components k in the oil phase are equal to the chemical potentials µkW of the same components in the dispersed water phase. For oil, surfactant, and alcohol, one can write

µOO ) µ°OO + kT ln(γOOXOO) ) µOW ) µ°OW + kT ln(γOWXOW) (A.11) µSO ) µ°SO + kT ln(γSOXSO) ) µSW ) µ°SW + kT ln(γSWXSW) (A.12) µ1AO ) µ°1AO + kT ln(γ1AOX1AO) ) µAW ) µ°AW + kT ln(γAWXAW) (A.13) where XSW, XAW, and XOW are the concentrations of the singly dispersed surfactant, alcohol, and water, respectively, in the dispersed water phase. µ°SW, µ°AW, and µ°OW are their standard chemical potentials for infinitely dilute solutions in water, while γSW, γAW, and γOW are the activity coefficients in the dispersed water phase. For the fourth component, water, the concentration of singly dispersed water XWO will be smaller than the value at saturation Xsat WO if the microemulsion is a single-phase system and becomes equal to Xsat WO when an excess water phase is present. Therefore

µWO ) µ°WO + kT ln(γWO Xsat WO fWO) ) µWW ) µ°WW + kT ln(γWWXWW) (A.14) where

fWO ) XWO/Xsat WO

XgWγgW ) ΩgSIfOWgO

(A.16)

In eq A.16

Ω ) XSWγSW(XAWγAW)gAI/gSI(XOOγOO)gOI/gSI exp(-∆µ°gI/kT) (A.17) and

gAI gOI 1 ∆µ°gI ) µ° - µ°SW µ° µ° (A.18) gSI gI gSI AW gSI OO The standard free energy difference ∆µ°gI is due to the transfer of one surfactant molecule and gAI/gSI alcohol molecules from water and gOI/gSI oil molecules from pure oil to the interfacial layer of the droplet. Equations A.16A.18 imply that the number of water molecules in the interfacial layer (gWI) is zero.

(A.19)

where

Ω ) XSWγSW(XAWγAW)gAI/gSI(XOOγOO)gOI/gSI exp(-∆µ°gI/kT) (A.20) and

∆µ°gI )

gAI gOI 1 µ° - µ°SW µ° µ° (A.21) gSI gI gSI AW gSI OO

As in the previous case, fOW is less than unity for a singlephase microemulsion but becomes equal to unity when an excess oil phase coexists. The activity coefficients γSW and γSA of the singly dispersed surfactant and alcohol molecules and the activity coefficient γgW of the aggregates differ from unity, since the water phase contains the droplets which contribute to solution nonideality through the hardsphere interactions. For the same reason, the activity coefficient γOO of the oil in the dispersed oil phase is unity since no droplets are present in the oil phase. Appendix B: Calculation of Activity Coefficients for Multicomponent Mixtures Containing Hard Spheres Let us consider a solution composed of hard-sphere droplets of a single size g present in a multicomponent solvent. The expression for the osmotic pressure due to the hard spheres allows us to calculate the chemical potentials of the components in the mixed solvent. Subsequently, the Gibbs-Duhem equation is used to calculate the chemical potentials of the hard-sphere droplets.26 The mole fraction Xi of the components and their volume fraction Φi are related via the expressions

Xi )

(A.15)

(e) Size and Composition Distribution. One can now introduce into the equilibrium relation (A.6) the relations between the various chemical potentials, eqs A.7-A.15, to obtain the size and the composition distribution function of the droplets. For a single-phase microemulsion

XgOγgO ) ΩgSI fWOgW

A.2. An Oil-in-Water Droplet Microemulsion. Similarly, for the oil-in-water droplet-type microemulsion, the size and composition distribution of droplets is given by

Φi/vi

∑Φi/vi

Φi )

Xivi

∑Xivi

(B.1)

where vi are the volumes of the various components including the hard-sphere aggregates. For all components other than i ) g (which refers to the hard spheres), considering their partial molar volumes equal to the molar volumes of the pure components, one can write

( ) ∂µi ∂p

T,Nj

) vi

µi(Φg ) 0) - µi(Φg) ) Πvi (B.2)

where µi is the chemical potential of component i in the solution, while Π is the osmotic pressure due to the hardsphere droplets of volume fraction Φg. The CarnahanStarling equation of state leads to

Π ) kT

Φg 1 + Φg + Φg2 - Φg3 Vg (1 - Φ )3

(B.3)

g

where Vg is the volume of a single hard-sphere droplet. Therefore, the chemical potential of component i is given by (26) Reed, T. M.; Gubbins, K. E. Applied Statistical Mechanics; Butterworth-Heinemann: Boston, MA, 1973. Overbeek, J. Th. G. Faraday Discuss. Chem. Soc. 1978, 65, 7.

293

Stability of dispersions Molecular Theory of Microemulsions

Langmuir, Vol. 16, No. 16, 2000

vi 1 + Φg + Φg2 - Φg3 µi ) µ°i + kT ln Xi - kT Vg (1 - Φ )3

(B.4)

g

The first term on the right-hand side µ°i is the standard chemical potential of component i, the second term is the ideal mixture contribution, and the third term is the nonideality caused by the hard-sphere interactions. The chemical potential of the hard-sphere droplet (component g) is obtained using the Gibbs-Duhem equation

∑Xi dµi ) 0

dµg ) -

Xi dµi ∑ i*g 1-

(B.5)

Xi ∑ i*g

[( )

µg ) µ°g + kT ln Xg +

Φg/Vg

Xg

∂ ln gn(k) ∂ ln

∑gkXg

)

∑gkXg - ∂ ln ∑Xg ) 1 - gn(k) gw(k) ∂ ln ∑gkXg ∂ ln ∑gkXg (C.4) ∂ ln

The root-mean-square deviation of gk from its numberaverage value gn(k) is given by

σn2(k) )

∑ i*g

Φi/vi

]

7 - 3Φg - Φg2

+ Φg

(1 - Φg)2

σn(k) gn(k) (B.6)

In obtaining the expression for the activity coefficient part of the chemical potential, we have considered droplets of a single size represented by the most populous size (corresponding to the maximum in the size distribution). A more formal equation allowing for droplets of various sizes can be written according to the Mansoori-Carnahan-Starling equation of state for mixtures of hard spheres.26 The results based on such an expression are not expected to be essentially different from those obtained on the basis of a single droplet size. Appendix C: Approximate Method for Obtaining Size and Composition Polydispersity of Droplets Using the Maximum-Term Method The size and composition distribution of droplets will be first rewritten in a general form applicable to both oil-in-water and water-in-oil microemulsions (by deleting the second subscripts):

Xg ) XSgSXAgAXOgOXWgW Z(gS, gA, gO, gW)

(C.1)

where Z represents all the other factors that appear in the expression for the size distribution. The number- and weight-average numbers of the various constituents in the droplet are denoted by gn(k) and gw(k), respectively and are defined as

gn(k) )

Further, from the definition of the number average (eq C.2), one can write the following derivative, which combined with eq C.3, leads to

∑[gk - gn(k)]2Xg ) g (k) g (k) - [g (k)]2 n w n ∑Xg (C.5)

which, combined with eqs C.4 and C.5, leads to

and has the form

kT ln

6413

∑gkXg ∑Xg

gw(k) )

∑gk2Xg ∑gkXg

(C.2)

where k stands for the four components present in a droplet (S, A, O, and W). In the above equation, as elsewhere in this paper, we employ the single-sum notation to represent the summation over all sizes and compositions of the droplets. One can write the following relations involving the derivatives by combining eqs C.1 and C.2:

∑

∑gkXg ) g (k) n ∑Xg ∂ ln ∑gkXg ∑gk2Xg ) ) gw(k) ∂ ln Xk ∑gkXg ∂ ln Xg ) ∂ ln Xk

(C.3)

)

[(

1-

∂ ln gn(k) ∂ ln

) ] [

∑gkXg

-1

1/2

-1

≈

∂ ln gn(k) ∂ ln

]

1/2

∑gkXg

(C.6)

Replacing the derivatives in eq C.6 by difference terms and restricting ourselves to the maximum term, we obtain eq 4.1 of section 4. Appendix D: Model for Alcohol Self-Association in Oil The partitioning of alcohol into the oil, water, and interfacial layer domains of a microemulsion controls whether a two-phase or a three-phase microemulsion system is formed, as well as the microscopic characteristics of the microemulsion phases. For the typical alcohols used, the amount of alcohol present in the oil domain can be large and comparable to the amount present in the interfacial layer. This is in contrast to the behavior of the surfactant, most of which remains at the interfacial layer and only a negligibly small amount of which are partitioned into the oil and the water domains. Therefore, the accurate accounting of the partitioning of alcohol into the oil domain is a necessary part of any quantitative theory of microemulsions. Such a theory must account for the facts that the alcohol is present in the oil phase as both monomers and aggregates and that the self-association of alcohol in the oil is responsible for its appreciable presence in the oil domain. The thermodynamic treatments of the self-association of alcohols in hydrocarbon solvents16 involve various levels of complexity and detail, depending on which of the following physical features are accounted for: (a) The alcohol molecules exist as self-associated aggregates. (b) The association can be represented by linear polymers, cyclic polymers, or aggregates of one or more distinct sizes. (c) The alcohol molecules can associate with the oil molecules to form oil-alcohol complexes. (d) The size differences between the various aggregates are appreciable and contribute to the solution properties. (e) In addition to the chemical association phenomenon, physical interactions between the various aggregates and the oil molecules constituting the solvent medium are also important. To keep the molecular theory of microemulsions at the simplest level possible, we treat the alcohol-oil systems by placing the emphasis on the most important source of nonideality, namely, the self-association of alcohol.16 D.1. Formulation of the Self-Association Model. Consider a mixture of NOO oil molecules and NAO alcohol molecules constituting a binary system from the stoichio-

294 6414

Nanodispersions Langmuir, Vol. 16, No. 16, 2000

Nagarajan and Ruckenstein

metric point of view. The alcohol molecules are present as NjAO aggregates of size j (j ) 1, for the monomer), making the mixture a multicomponent system from the viewpoint of the true chemical species present. The true mole fractions of the aggregates of size j and of the oil are

XjAO )

NjAO NOO +

∑NjAO

XOO )

NOO NOO +

∑NjAO

(D.1)

while the stoichiometric mole fractions of alcohol and oil are

XA )

∑

NAO jNjAO ) NAO + NOO NOO + jNjAO

∑

XO ) 1 - XA

j ) 1, 2, 3, ...

(D.3)

where the association equilibrium constants are considered to be the same for all association steps. If we denote the stepwise association equilibrium constant by KX, the mole fraction of alcohol aggregates of size j is

XjAO ) KXj-1X1AOj

(D.4)

Because KXX1AO must be less than unity to ensure that all mole fractions remain finite

X1AO

∑XjAO ) 1 - KXX1AO

X1AO

∑jXjAO ) (1 - K X X

1AO)

2

(D.5) XOO ) 1 -

∑j

XjAO )

1 - KXX1AO - X1AO 1 - KXX1AO

(D.6)

The average aggregation number of alcohol aggregates is therefore given by

∑jXjAO ) 1 javg ) 1 K XX1AO ∑XjAO

(D.7)

and the ratio of the total number of alcohol molecules to those in monomeric form is

∑

NAO jXjAO 1 ) ) N1AO X1AO (1 - KXX1AO)2

1 + 2KXXA - (1 + 4KXXA - 4KXXA2)1/2 2KXXA(1 + KX)

(D.9)

The alcohol-partitioning calculations allow us to calculate XOO, provided the self-association equilibrium constant KX is known. D.2. Determination of KX from Thermodynamic Data for Alcohol-Hydrocarbon Mixtures. Using the available excess Gibbs energy and excess enthalpy data for alcohol-hydrocarbon mixtures,16 the value of KX can be estimated as described below. The Gibbs energy of the

∑NjAOµjAO ) NOOµOO + NAOµ1AO

(D.10)

where µjAO ) jµ1AO. The standard chemical potential of oil µ°OO is taken to correspond to the pure oil, while the standard chemical potential of the monomeric alcohol µ°1AO is taken to be that of the monomers in pure alcohol. One can find µ°1AO by equating the chemical potential µ/1AO of the pure alcohol to the chemical potential of the monomers defined in terms of their standard state and their concentration X/1AO in pure alcohol:

µ/1AO ) µ°1AO + kT ln X/1AO

(D.11)

The monomeric mole fraction X/1AO in the pure alcohol is obtained from eq D.9 by setting the stoichiometric alcohol mole fraction XA equal to unity:

X/1AO )

1 1 + KX

(D.12)

The excess Gibbs energy is thus provided by the expression

GE ) G - [NOO µ°OO + NAO µ/1AO + kT(NOO ln XO + NAO ln XA)] (D.13) and the molar excess Gibbs energy by

[ ( )

( )]

X1AO XOO GE ) kT XA ln / + XO ln NOO + NAO XO X1AO

(D.14)

The excess enthalpy can be related to the number of alcohol-alcohol hydrogen bonds formed in the alcoholhydrocarbon solution compared to those formed in pure alcohol. Considering the enthalpy of formation of a hydrogen bond hA to be independent of the degree of aggregation, the excess enthalpy is given by

HE ) hA[

∑(j - 1)NjAO - ∑(j - 1)N/jAO]

(D.15)

where N/jAO denotes the number of aggregates with a degree of association j in pure alcohol. Because ∑jN/jAO ) NAO and ∑jNjAO ) NAO, eq D.7 can be written as

NAO

(D.8)

Finally, the stoichiometric mole fraction XA can be used to calculate the true mole fraction X1AO of the monomeric species using the expression

X1AO )

G ) NOOµOO +

(D.2)

The self-association of alcohol in oil is represented by the association equilibria

A1 + Aj a Aj+1

solution is given by

∑NjAO

)

∑jNjAO ) ∑jXjAO ) 1 1 K N X XX1AO ∑ jAO ∑ jAO

(D.16)

Therefore

∑NjAO ) NAO(1 - KXX1AO) ∑N/jAO ) NAO(1 - KXX*1AO)

(D.17)

and, consequently, the molar excess enthalpy is given by the relation

HE ) hAXAKX(X1AO - X/1AO) NOO + NAO

(D.18)

The thermodynamic models available in the literature16 employed a constant value of -6000 cal/mol for the hydrogen-bonding energies hA of various alcohols. Therefore

Stability of dispersions

295

Molecular Theory of Microemulsions

d ln KX d(1/T)

)-

Langmuir, Vol. 16, No. 16, 2000

hA R

(D.19)

where R is the gas constant. By fitting the available excess Gibbs energy data to eq D.14, the excess enthalpy data to eq D.18, the average aggregation number of the alcohol to eq D.7, and the ratio between the total and the monomeric alcohol to eq D.8 and incorporating the temperature dependence provided by eq D.19, we have developed the following expression for an alcohol with ncA carbon atoms:

KX ) 92ncA

-0.47

1 1 exp 3000 T 323

[

(

)]

potential of the alcohol in the oil phase to the chemical potential of the alcohol in the water phase, one obtains

XAW XSAW

)

X1AO X/1AO

) (1 + KX)X1AO

The preceding equation is valid for aliphatic alcohols ranging from propanol to decanol. For example, at 25 °C, the association equilibrium constants are 104.6, 94.2, 86.4, and 80.4, respectively, for butanol, pentanol, hexanol, and heptanol. D.3. Use of the Alcohol Self-Association Model. In performing the computations for microemulsions, one first estimates KX using eq D.20 for a given chain length of alcohol and a given temperature. By equating the chemical

(D.21)

where XSAW is the saturation concentration of alcohol in water when the water phase coexists with a pure alcohol phase. At 25 °C, the solubility of aliphatic alcohols in water is given19 by the expression

ln XSAW ) 1.40 - 1.38ncA (D.20)

6415

(D.22)

To estimate XSAW at any other temperature, the temperature dependence of alcohol solubility is assumed comparable to that of aliphatic hydrocarbons and the relations provided for the methyl and methylene groups by eqs 7.3 and 7.4, respectively, are used to obtain this temperature dependence. For a given value of XAW, the mole fraction X1AO of the monomeric alcohol in the oil phase is calculated by combining eqs D.21 and D.22. Finally, the true mole fraction of oil XOO is calculated using eq D.6. LA991578T

296

Nanodispersions

Stability of dispersions

297

298

Nanodispersions

Stability of dispersions

299

300

Nanodispersions

Stability of dispersions

301

302

Nanodispersions

Stability of dispersions

303

304

Nanodispersions

Stability of dispersions

305

306

Nanodispersions

Stability of dispersions

307

308

Nanodispersions

Stability of dispersions

309

310

Nanodispersions

Stability of dispersions

311

312

Nanodispersions

313

Stability of dispersions 5464

Langmuir 2001, 17, 5464-5475

On the Stability of Lyotropic Lamellar Liquid Crystals and the Thicknesses of Their Lamellae Eli Ruckenstein* and Marian Manciu Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received March 13, 2001. In Final Form: June 19, 2001 The stability of lamellar lyotropic liquid crystals was investigated using a thermodynamic formalism. The interfacial tension γ was expressed in terms of the thicknesses δ1 and δ2 of the two kinds of lamellae. For given thicknesses, the system might be stable if γ g 0. In adition, the derivatives ∂f/∂δi of the excess free energy f, due to van der Waals, hydration, steric, electrostatic, and undulation interactions, must be negative for the lamellar structure to be stable. The thermodynamic conditions employed show that the domain of existence of the lamellar phase is restricted to narrow ranges of layer thicknesses and parameters of interaction. The effect of the bending modulus and of the ionic strength on the stability was examined. The thicknesses of the two kinds of lamellae were calculated using equilibrium and mass balance equations. It was shown that the nonlinearity of these equations could be responsible for the observed deviations from the “ideal dilution law”. A comparison between the calculated and experimental results was made.

Introduction Lyotropic liquid crystals are composed of a surfactant, a cosurfactant (typically a medium chain length alcohol), water, and oil. It is well-known that, in relatively narrow ranges of composition and temperature, a lamellar structure that exhibits a long-range translational order along one direction (LR) can be formed.1 It was also shown that the lamellar phase could preserve its orientational order upon its dilution with water.1 This was initially supposed to be a consequence of the repulsion between the charged interfaces formed by ionic surfactants. However, it was subsequently found that the lamellar phase can be hyperswollen by adding large amounts of an oil-cosurfactant mixture, a system for which the electrostatic interactions do not play a role;2 a hyperswelling in water was also observed for nonionic surfactants.3 It was first suggested by Helfrich4 that the confinement of the thermal undulations of uncharged phospholipid bilayers by the neighboring bilayers, in a lipid/water lamellar liquid crystal, leads to a long-range repulsive force, which can be responsible for the extensive swelling of the lipid bilayers in water.5 However, for most bilayers, their high bending modulus KC provides a small contribution of the entropic confinement (which is inversely proportional to KC) to the free energy; hence only a limited swelling can occur.6 It was shown by Safinya et al.7 that the addition of pentanol to an uncharged lipid (dimyristoylphosphotidylcholine)/water multilayer decreases the rigidity of the bilayers and consequently the equilibrium separation distance increases from d ≈ 40 to d g 200 Å. It is then tempting to attribute the high swelling (distances over 5000 Å)2 of uncharged lyotropic liquid * To whom correspondence may be addressed: e-mail, feaeliru@ acsu.buffalo.edu; phone, (716) 645 2911/2214; fax, (716) 645 3822 (1) Ekwall, P. In Advances in Liquid Crystals; Brown, G. H., Ed.; Academic: New York, 1975. (2) Larche, F. C.; Appell, J.; Porte, G.; Bassereau, P.; Marignan, J. Phys. Rev. Lett. 1986, 56, 1700. (3) Strey, R.; Schomacker, R.; Roux, D.; Nallet, F.; Olsson, V. J. Chem. Soc., Faraday Trans. 1990, 86, 2253. (4) Helfrich, W. Z. Naturforschung 1978, 33a, 305. (5) Harbich, W.; Helfrich, W. Chem. Phys. Lipids 1984, 36, 39. (6) Petrache, H. I.; Gouliaev, N.; Tristram-Nagle, S.; Zhang, R.; Sutter, R. M.; Nagle, J. F. Phys. Rev. E 1998, 57, 7014. (7) Safinya, C. R.; Sirota, E. B.; Roux, D.; Smith, G. S. Phys. Rev. Lett. 1989, 62, 1134.

crystals to a low value of the interface bending modulus, which provides a large entropic confinement repulsion. However, the matter is more complex in this case than in that of lipid bilayers. The lipid bilayers are generally formed before their swelling in water, and as a first approximation one can consider that the total area and the thickness of the bilayers remain constant during the formation of the lamellar liquid crystal for any water dilution. In this case, even for strong repulsions between the layers, the system remains stable. Indeed, Palmer and Schmitt8 observed that charged lipid bilayers swelled indefinitely in distilled water but collapsed steadily upon exposure to solutions of progressively higher ionic strength. In contrast, the lyotropic liquid crystals are selfassembled after the mixture of the constituents is allowed to equilibrate for a long time (typically, days), and there are additional contributions to the free energy due to the formation of the lamellae. In this case, a strong repulsion between layers will increase the free energy of the lamellar phase and as a result a nonlamellar phase may be formed. It was observed experimentally that the addition of salt, which screens the electrostatic interactions, and hence reduces repulsion, to a dilute ionic surfactant/cosurfactant/ water system, can stabilize the lamellar phase.9 Dilute lamellar phases typically occur in relatively narrow ranges of temperature and composition, and the addition of a cosurfactant or the increase of temperature (which by decreasing the bending modulus enhances the long-range undulating repulsion between layers, thus increasing the free energy) can lead to a phase transition. The domains of existence of the lamellar liquid crystals can be in principle identified by calculating the free energies of the various possible phases. Because of the difficulty in performing such a calculation, a first purpose of this paper is to employ the thermodynamic formalism developed by Ruckenstein10 to extract some information about the domain of stability of the lamellar phase. A second goal is to determine the widths of the water and oil lamellae and to explain the deviation from the so-called “ideal dilution law”. In a crude model, one can (8) Palmer, K. J.; Schmitt, F. O. J. Cell. Comput. Physiol. 1941, 17, 385. (9) Guerin, C.; Bellocq, A. M. J. Phys. Chem. 1988, 92, 2550. (10) Ruckenstein, E. J. Chem. Phys. 1998, 109, 6995.

10.1021/la010385e CCC: $20.00 © 2001 American Chemical Society Published on Web 08/09/2001

314

Nanodispersions

Stability of Lyotropic Lamellar Liquid Crystals

Langmuir, Vol. 17, No. 18, 2001

assume that the entire surfactant and cosurfactant is adsorbed on the interface and hence the total area of the interface, S, for a system of volume V is proportional to the volume fraction of surfactant and cosurfactant, φ. Because for planar interfaces V ) Sd/2, the repeat distance d (the sum of the thicknesses of the hydrophobic and hydrophilic layers) becomes inversely proportional to φ. Consequently, the product between d and φ should be independent of φ. This behavior is sometimes called the “ideal dilution law”. However, it was well established that deviations from the ideal dilution law occur frequently,1 the area per headgroup of surfactant being variable during swelling, and dependent on the surfactant/cosurfactant ratio. These deviations were accounted by Strey et al.,3 who carried out experiments with the binary water-C12E5 system, by noting that the amplitude of the thermal undulations increased with the repeat distance d. If one considers the total area of the interface a constant S which depends only on the number of surfactant molecules, the projection S0 of the total area on a plane perpendicular to d will decrease with increasing d. Hence, the apparent area per surfactant molecule, which is defined as the ratio between S0 and the total number of surfactant molecules, decreases with increasing d, while the ideal dilution law implies that the apparent area per surfactant molecule is a constant. The excess area, defined as ∆S ) S - S0, was related to the bending modulus of the interface,3 and the experimental results for the deviations from the ideal dilution law were used to determine KC.3,11 However, it should be noted that there are binary systems for which the deviations from the ideal dilution law occur in the opposite directions. For instance, in the binary systems of fatty acid alkali soaps/water, the apparent area per headgroup increases with water dilution, because of the incorporation of water in the interface.1 In the second part of the paper, it will be shown that the partition of the alcohol between the phases can explain the deviations from the ideal dilution law for a water/ oil/surfactant/cosurfactant lyotropic lamellar liquid crystal. I. Thermodynamic Stability of Lyotropic Liquid Crystals I.1. Thermodynamic Equations. We briefly describe the thermodynamic formalism of Ruckenstein.10 Let us consider a liquid crystal of volume V per unit area, in equilibrium, and denote by δ1 the average thickness of the water layer and by δ2 the average thickness of the oil layer. The Helmholtz free energy of the system can be written as the sum of a free energy F0 of a hypothetical system in which the lamellae are treated as bulk, planar, phases and a free energy F1, which accounts for the smallness of the lamellae, the interactions between them, and their thermal undulations. One can write

(

)

∑

V + µidNi δ1 + δ2 Vδ1 Vδ2 p1d - p2d (1a) δ1 + δ2 δ1 + δ2

dF0(δ1,δ2,V,Ni) ) 2γd

(

) (

)

where γ is the interfacial free energy between the two kinds of lamellae, µi is the chemical potential of species i (at pressure p1 in the layers of thickness δ1 and pressure p2 in the layers of thickness δ2), and Ni is the number of (11) Roux, D.; Nallet, F.; Freyssingeas, E.; Porte, G.; Bassereau, P.; Skouri, M.; Marignan, J. Europhys. Lett. 1992, 17, 575.

5465

molecules of species i in the volume V, and

(

dF1(δ1,δ2,V) ) d f(δ1,δ2)

)

V δ1 + δ2

(1b)

where f is the free energy per unit area of one pair of lamellae 1 and 2, due to their thinness, interaction between them, and their thermal undulation (V/(δ1 + δ2) is the number of pairs of lamellae). It should be emphasized that the pressures p1 and p2 are not equal to the external pressure because they are defined on the basis of the free energy F0. However, at thermodynamic equilibrium one can write

dF0 + dF1 ≡ dF(Ni,V) )

∑ µi* dNi - p dV

(2)

where p is the external pressure and µi* is the chemical potential of species i at the pressure p. By equating the coefficients of dV, dNi, dδ1, and dδ2 from eqs 1 and 2, after some algebra the following equations are obtained10

p1 ) p +

∂f ∂δ1

(3a)

p2 ) p +

∂f ∂δ2

(3b)

2γ ) -f + δ1

∂f ∂f + δ2 ∂δ1 ∂δ2

µi*(p) ) µi(pj)

(3c) (3d)

The free energy per unit area for a pair of successive lamellae, which includes the effect of thermal undulations, is computed as follows. Let us denote by U(x1,x2) the interaction potential between planar water and oil lamellae of thicknesses x1 and x2, which include the van der Waals, hydration, steric, and electrostatic interactions

U(x1,x2) ) fvdw + fhydration + fsteric + fe

(4)

The first quantitative method to account for the undulation repulsion between membranes, due their mutual confinement, was developed by Helfrich,4 which showed that the entropic contribution to the free energy, due to the confinement of the thermal fluctuations by the neighboring lamellae, for membranes interacting via a hard-wall potential, is given by ∆F ∝ (kT)2/KCσ2, where k is the Boltzmann constant, T is the absolute temperature, KC is the bending modulus, and σ is the root-mean-square fluctuation of the position of the membrane. There have been a number of theoretical attempts to extend the Helfrich theory to bilayers interacting via arbitrary potentials.12-15 Here we will employ a variational procedure developed recently by the authors.15 The free energy is composed of the energy of interaction between lamellae, averaged over the distance distributions, and an entropic term, due to the confinement of one interface by its neighbors. Denoting with P1(x1) and P2(x2) the distribution of distances through the water and oil lamellae, respectively, and considering that the free energy due to the entropic confinement of the membranes is inversely (12) Sornette, D.; Ostrowsky, N. J. Chem. Phys. 1986, 84, 4062. (13) Evans, E. A.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7132. (14) Podgornik, R.; Parsegian, V. A. Langmuir 1992, 8, 557. (15) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 2455.

Stability of dispersions 5466

315

Langmuir, Vol. 17, No. 18, 2001

Ruckenstein and Manciu

tively. Then, the minimization of f(δ1,δ2,σ1,σ2) of eq 5 with respect to σ1 and σ2 provides the free energy f for any values of δ1 and δ2. I.2. The Interaction Potential U(x1,x2). Let us consider the contributions to the interaction potential U(x1,x2), for fixed x1 and x2 thicknesses of the planar water and oil layers, respectively (see Figure 1). In the hypothesis of pairwise additivity, the van der Waals excess interaction for a pair of lamellae in a periodic multilayer is given by (see Appendix A)

fvdW ) -

Aeff

{

∞

1

+

12π (x2 - th)2

∑ n)1

1

Figure 1. A sketch of a lyotropic lamellar liquid crystal, composed of water, oil, a surfactant (open circles), and cosurfactant (full circles). The distribution of the surfactant and cosurfactant between the water and oil phases and interface is discussed in section II.

(

1

f(δ1,δ2,σ1,σ2) )

∫ dx1 dx2 P1(x1)P2(x2)U(x1,x2) + (kT)2 64KCσ1

2

+

(kT)2 64KCσ22

(5)

where x1 and x2 are the fluctuating thicknesses of the water and oil layers, respectively. In a first approximation one might consider that σ1 and σ2 are independent of each other. The energy is increased by distributions with large values of σ, because in this case a larger part of the interface is in a region of stronger repulsive interactions; on the other hand the entropic term decreases when the σ values are large. Consequently, there are optimum values of σ1 and σ2 for which the free energy is minimum. The following asymmetric truncated Gaussian function will be employed for the distance distributions (i ) 1, 2)15

≡ (n(x1 + x2) + x2 - th)2 (n(x1 + x2))2 ∞ Aeff 1 1 + + 2 2 12π (x + t ) n)1 (n(x + x ) - x - t ) 1 h 1 2 1 h

{

∑

(

Pi(xi) )

(

(xi - δi′) 1 exp Ca 2σi2

2

0 < x i < δi′ Pi(xi) )

(

(xi - δi′)2 1 exp Ca 2(Rσi)2

(n(x1 + x2) + x1 + th)

(n(x1 + x2))2

)}

(6)

where R is an asymmetry coefficient of the distribution; in the calculations, the value R ) 1.4, suggested by Monte Carlo simulations,16 will be employed. The constant Ca and the position δi′ of the maximum of the distribution can be calculated as a function of the average distance δi, through normalization and distance averaging, respec-

(7)

where Aeff ) A11 + A22 - 2A12 is an effective Hamaker constant. In the calculation of the van der Waals attraction, the headgroups of thickness th of the surfactant will be considered as belonging to the water layer, since their dielectric properties are closer to water than to hydrocarbon. The van der Waals energy in eq 7 is symmetric in the distances (x1 + th) and (x2 - th) and can be interpreted as an interaction between lamellae (the terms under the sum) plus a contribution, the first term, due to the thinness of the lamellae; this term will be called effective self-energy (the true van der Waals self-energy is given by eq A.1 in Appendix A). When one of the thicknesses x1 or x2 f ∞, the corresponding sum vanishes; however, the effective self-energy of the finite layer provides a negative contribution to the free energy. It will be shown later that this contribution is essential for the stability of the lamellar phase. The second contribution to the energy comes from the hydration repulsion between the interfaces of the water layers, which is given by

( )

xi > δi′

(16) Gouliaev, N.; Nagle, J. F. Phys. Rev. Lett. 1998, 81, 2610.

2

fhydration ) AH exp -

) )

2

-

Pi(xi) ) 0 xi < 0

)}

2

-

1 proportional to the root-mean-square fluctuations of the distribution of the distances, (σ1)2 and (σ2)2, through water and oil, respectively, one can write, using Derjaguin approximation, that

+

(n(x1 + x2) - x2 + th)2

x1 λH

(8)

The interaction parameters AH and λH were determined for various lipid bilayers;17 λH is typically between 1 and 3 Å, while AH was found within a rather broad range, around 10-1 J/m2. The third contribution to the free energy comes from the steric repulsion in the oil layers between the hydrocarbon tails of the surfactant adsorbed on the interface. Extending the simple formula suggested by de Gennes18 for the steric force between long-chain polymer-covered surfaces, to relatively short chains, one can write

fsteric ) 2L(kT) 4 2L C 5 x2 - th s3

[ ((

)

5/4

) ((

-1 +

)

4 x 2 - th 7 2L

7/4

)]

-1

(9)

(17) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (18) de Gennes, P. G. Adv. Colloids Interface Sci. 1987, 27, 189.

Nanodispersions

316 Stability of Lyotropic Lamellar Liquid Crystals

Langmuir, Vol. 17, No. 18, 2001

for (x2 - th) < 2L and

fsteric ) 0 for (x2 - th) g 2L, where x2 - th represents the actual thickness of the hydrocarbon lamella, L is the maximum length of the surfactant tail, and s is the separation distance between the headgroups of the surfactant along the interface. The proportionality coefficient C can be determined from a comparison with experimental data and was shown to be of the order of unity;19 here it will be taken as unity. The electrostatic contribution to the energy, in the presence of an electrolyte, depends on the behavior of the surface charge density, σe, when two surfaces approach each other. Assuming that the surface charge density remains constant, an excellent approximation for the electrostatic contribution to the energy is given by20

{ (

2nkT fe(x1) ) 2y0 ln κ

B + y0 coth 1 + y0

( ) κx1 2

)

-

ln(y02 + cosh(κx1) + B sinh(κx1)) + κx1

}

where y0 ) tanh(eψ0/kT) and B ) (1 + y02 cosech2(κx1/ 2))1/2. In the above expressions, n is the concentration of the 1:1 electrolyte, κ ) (2e2n/0kT)1/2 is the reciprocal Debye length, e is the protonic charge, 0 is the permittivity constant,  is the dielectric constant, and ψ0 is the surface potential when the plates are infinitely far apart, which is related to the surface charge density via the expression σe ) (4ne/κ) sinh(eψ0/2kT). The electrostatic contribution to the energy when the only ions present in the solution are those of the counterions of the surfactant molecules, hence in the absence of an added electrolyte, is calculated by integrating the electrostatic pressure from infinity to the distance x1. Denoting the surface charge σe ) ade/A, where ad is the degree of dissociation and A is the area per surfactant molecule adsorbed on the interface, the electrostatic energy per unit area is given by (see Appendix B)

fe(x1) ) -

20(kT)2 e

2

∫∞

x1

K2(x) dx

(11a)

where

adxe2 Kx Kx tan ) 2 2 4A0kT

( )

of δ1 and δ2. While γ can be very small (∼10-5 N/m), it cannot be negative, because then the interface will become unstable and a different phase (microemulsion, cubic, hexagonal, etc.) will be generated. Consequently, a necessary condition for the stability of the lamellar phase is

2γ ) -f + δ1

∂f ∂f + δ2 g0 ∂δ1 ∂δ2

(11b)

The case ad ) 0 corresponds to a noncharged interface (fe ) 0). I.3. Necessary Conditions for Stability. In a system with a fixed number of layers, such as the phospholipid bilayers, the equilibrium position (corresponding to the minimum of the free energy, F, of the whole system) is obtained when the free energy per unit area for the pair water/bilayer, f, is a minimum. This is no longer true when the number of pairs of layers is variable. In this case, at thermodynamic equilibrium one should use eq 3c. From this equation, if the interactions between lamellae are known, one can calculate the surface tension γ as a function (19) Ruckenstein, E.; Li, B. Q. J. Chem. Phys. 1997, 107, 932. (20) Gregory, J. J. Chem. Soc., Faraday Trans. 1973, 69, 1723.

(12a)

When the lamellar liquid crystal is in equilibrium with an excess phase i (either water or oil) and one assumes that the concentrations of the surfactant and cosurfactant are the same in the excess phase and in the lamellae of the same kind, the equality of the chemical potentials leads (because of eq 3d) to p ) pi, which when used in eqs3a or 3b provides the condition for equilibrium with an excess phase i, ∂f(δ1,δ2)/∂δi ) 0. After the lamellar phase reaches the equilibrium with an excess phase, δi can no longer increase, since this will always increase the total free energy of the system. Consequently, the lamellar phase might exist only in the regions where

∂f(δ1,δ2)/∂δi e 0 (10)

5467

(12b)

i ) 1, 2 The domains in which ∂f(δ1,δ2)/∂δi > 0 correspond to unstable lamellar phases; in such cases, an excess phase will separate until thicknesses of the lamellae, which satisfy the conditions ∂f(δ1,δ2)/∂δi ) 0, will be reached. We will present now detailed calculations for a model lyotropic lamellar liquid crystal, using for the interaction parameters typical values from the literature: the thickness of the headgroups th ) 8 Å;6 the effective Hamaker constant Aeff ) 1.0 × 10-20 J.6,15 A recent comparison with experiment of the force per unit area and of the rootmean-square fluctuation of the separation distance between two neutral phospholipid bilayers as a function of the average separation distance provided the following values for the parameters involved in the hydration force: the decay length λH = 1.5 Å and the preexponential factor AH = 0.035 J/m2.15 There are no experimental results for the lamellar lyotropic liquid crystals; however one can expect a comparable value for the decay length but a smaller one for the preexponential factor, because the former is due to the structure of water and the latter to the headgroup of the surfactant and the presence of cosurfactant. Here we will employ λH ) 1.5 Å and AH ) 0.02 J/m2. A value of L ) 8 Å for the length of the surfactant tails and a value of s ) 7 Å for the distance between the headgroups along the interface will be used. First, the surface charge will be set equal to zero, which corresponds to nonionic surfactants. The elastic bending modulus KC for lipid bilayers was found to be of the order of 10 × 10-20 J.6 For the lyotropic lamellar liquid crystals, the additional presence of a cosurfactant at the interface should lead to smaller values. Various experimental measurements21 provided values ranging between 0.08 and 5 × 10-20 J. Figure 2 presents (-f) (the negative of the excess free energy per pair of layers), as a function of the average thicknesses of the layers, δ1 and δ2, for KC ) 5 × 10-20 J. From conditions (12a) and (12b), one can see that the lamellar phase cannot exist for positive values of f. To simplify the figure, we set f ) 0 in Figure 2 when f was positive. In the case considered, the free energy has a minimum for δ1 ) 22 Å and δ2 ) 39 Å. Since both (21) Quist, P. O. Langmuir 1995, 11, 2201.

Stability of dispersions 5468

Langmuir, Vol. 17, No. 18, 2001

Figure 2. The negative of the excess free energy (-f) vs the thicknesses of water (δ1) and oil (δ2) layers. The interaction parameters are given in the text.

derivatives are vanishing for these values of δ1 and δ2, this point represents a triple point, where the lamellar phase is in thermodynamic equilibrium with both water and oil excess phases. Since eq 3c assigns a unique value to the interfacial tension (2γ ) -f), the triple point can be reached only for certain concentrations of the constituents. There are however points in the vicinity of this minimum, where both derivatives of f are nonpositive and hence the lamellar phase might be stable for lower values of the interfacial tension than for the value at the triple point. In Figure 3, the derivatives of the free energy with respect to δ1 (Figure 3a) and δ2 (Figure 3b) are plotted as functions of the distances δ1 and δ2 for the same parameters as in Figure 2. The lamellar phase might exist only in the regions where the derivatives are negative, which are the flat regions in parts a and b of Figure 3 (to simplify the figures, we set the values of the derivatives equal to zero in the regions where they are negative). One can see from the two-dimensional projections (Figure 3c) that condition (12b) is very restrictive, since most of the δ1, δ2 plane is inaccessible to the lamellar phase. As already noted, when ∂f/∂δi vanishes, the system is in equilibrium with an excess phase of kind i, and consequently the thickness of layer i cannot increase any longer. Figure 3c reveals that the above vanishing does not occur if at least one of the thicknesses is sufficiently small. This can be easily understood in terms of the interaction between lamellae of the same kind. While the repulsive interaction depends only on the separation between lamellae, the van der Waals attraction increases with their thickness (see eq 9). Therefore, thick lamellae of one kind are bound, because of increased van der Waals attraction, and the system cannot be swollen any longer in the other kind of liquid, while a system with thin lamellae of one kind might be unbound and hence be swollen in the other kind of liquid. In a lipid/water system, the thickness of the bilayers is constant, and the unbinding transition can occur, in principle, by varying the Hamaker constant or temperature.22 For lyotropic lamellar liquid crystals, hyperswelling in a liquid of one kind might also occur if the lamellae of the other kind are thin enough and hence unbound. However, thin lamellae might lead to a positive contribution to the free energy (since the repulsive forces overcome at short distances the van der Waals attraction), and hence the lamellar phase can become unstable. (22) Lipowsky, R.; Leibler, S. Phys. Rev. Lett. 1986, 56, 2541.

317 Ruckenstein and Manciu

To complete the analysis, we present in Figure 4 the domains in which both conditions (12a) and (12b) are fulfilled for various values of KC, namely, (10, 5, 3, and 2) × 10-20 J. The figure indicates that outside the marked region the lamellar phase is certainly unstable; however, inside the marked region, the lamellar phase might compete for stability with another unknown phase, which might have a lower free energy. Figure 4 indicates that the domains that might be accessible to the lamellar phase in the δ1,δ2 plane are rather narrow. For high values of KC, hence low undulation repulsion, it is possible to generate lamellar liquid crystals when both kinds of lamellae are thin. This occurs because, if the undulation repulsion is negligible, there is always an absolute minimum of f at small thicknesses, since both the hydration repulsion through water and steric repulsion through oil have shorter ranges than the van der Waals attraction. Conditions (12a) and (12b) limit the existence of a lamellar phase only to a narrow region in the vicinity of this minimum, and further addition of water or oil will lead to the separation of an excess phase. However, there are other regions which are in principle accessible to the lamellar phase; they correspond to systems with thin lamellae of one kind and thick lamellae of the other kind, the effective self-van der Waals energy of the thin lamellae ensuring the thermodynamic stability of the system. When KC decreases, the undulation repulsion increases, and this has two consequences. On one hand, to maintain stability, the thickness of the thin lamellae should increase in order to diminish the effect of repulsion. This in turn increases their attraction and, hence, stimulates their binding. On the other hand, the repulsion between the thin lamellae increases, and this favors swelling. The results presented in Figure 4 show that, in a relatively broad range for KC, it is possible to obtain a lamellar phase, if the lamellae of one kind are thin enough to be unbound. However, the contribution to the free energy due to their thinness (the self-energy) should remain sufficiently negative, to counteract the positive free energy due to the repulsion between them. When KC further decreases, the undulation repulsion might become so large that no thin layers of one kind can provide enough effective self-energy for the formation of a stable lamellar liquid crystal. Indeed, for KC ) 1 × 10-20 J, the undulation repulsion between layers becomes so strong that the lamellar phase was found unstable in all the domains of the δ1,δ2 plane investigated (δ1 e 200 Å, δ2 e 200 Å). Let us now consider the same system in the presence of an 1:1 electrolyte, with KC ) 10 × 10-20 J and the surface charge density, at large separation distance δ1, σe ) 1e/ 200A2 ) 0.08C/m2. A higher bending modulus was selected to take into account the effect of the charge on the rigidity of the interfaces. In Figure 5 we present the region of stability of the lamellar phase, for various electrolyte concentrations. For n ) 0.5 M, it is possible to obtain a lamellar phase with thin oil lamellae (δ2 ≈ 30 Å) and thick water layers (δ1 > 40 Å), and also with thin water layers (δ1 ≈ 40 Å) and thick oil layers (δ2 > 170 Å). In the latter case, the oil layers were thick enough for their attraction to overcome the electrostatic repulsion. When the electrolyte concentration was decreased, the electrostatic repulsion increased, and no lamellar phase containing thin water lamellae could be formed. However, a lamellar phase with thick water lamellae could be generated, when the thickness of the oil layers was sufficiently small to provide, via their self-energy, the required negative contribution to the free energy. Our calculations indicated that the thickness of the (thin) oil lamellae was almost

318 Stability of Lyotropic Lamellar Liquid Crystals

Nanodispersions Langmuir, Vol. 17, No. 18, 2001

5469

Figure 3. (a) The derivative of f with respect to δ1. The lamellar phase cannot exist in the regions where the derivative is positive. (b) The derivative of f with respect to δ2. The lamellar phase cannot exist in the regions where the derivative is positive. (c) The contour plot of the derivatives of f with respect to δ1 and δ2, respectively. At line 1, ∂f/∂δ1 ) 0 and at line 2, ∂f/∂δ2 ) 0 (inside each of the marked domains delimited by lines 1 and 2, an excess phase separates). The intersections of the line 1 and 2 are triple points, where the lamellar phase is in equilibrium with both excess phases.

319

Stability of dispersions 5470

Langmuir, Vol. 17, No. 18, 2001

Figure 4. Domains in which the lamellar phase fulfills the conditions (12a) and (12b) and might be stable, for (1) KC ) 10 × 10-20 J, (2) KC ) 5.0 × 10-20 J, (3) KC ) 3.0 × 10-20 J, and (4) KC ) 2.0 × 10-20 J. For KC < 1.0 × 10-20 J, no domain of stability was found in the region investigated in the δ1,δ2 plane (δ1 e 200 Å, δ2 e 200 Å).

Figure 5. Possible domains of stability of the lamellar phase for charged lyotropic liquid crystals, in the presence of electrolyte: (1) n ) 0.5 M; (2) n ) 0.2 M; (3) n ) 0.1 M; (4) n ) 0.05 M. No domain of stability for the lamellar phase was found in the region investigated for n < 0.01 M.

independent of the electrolyte concentration n (δ2 ≈ 30 Å), while the minimum thickness of the water lamellae was strongly dependent on n (δ1 > 40 Å for n ) 0.5 M, δ1 > 70 Å for n ) 0.2 M, δ1 > 90 Å for n ) 0.1 M and δ1 > 120 Å for n ) 0.05 M). A further decrease of the electrolyte concentration increased the free energy and provided negative values for the interfacial tension calculated with eq 3c and, hence, made the system unstable. For n ) 0.01 M, no region of stability was found in the δ1,δ2 plane investigated (200 Å × 200 Å). Most challenging are the systems that have ionic surfactants and are free of electrolyte. In this case it is possible to obtain a swollen phase, by the addition of a mixture of cosurfactant and oil.2,23,24 The thickness of the water lamellae, evaluated from the composition of the liquid crystal and the repeat distance, is about 17 Å.2 If (23) Larche, F. C.; El Qebbaj, S.; Marignan, J. J. Phys. Chem. 1986, 90, 707. (24) Safinya, C. R.; Roux, D.; Smith, G. S.; Sinha, S. K.; Dimon, P.; Clark, N. A.; Bellocq, A. M. Phys. Rev. Lett. 1986, 57, 2718.

Ruckenstein and Manciu

one assumes that the surface charge is σe ) 1e/200A2, then the osmotic pressure due to counterions exceeds by 1 order of magnitude the van der Waals attraction, and the system cannot be stable. However, at the distances involved, the dielectric constant of water should be much smaller than that of the bulk water, since the water molecules are polarized by the headgroups of the surfactant adsorbed on the interface. In addition, while the dissociation increases the entropy and thus decreases the free energy, the repulsion between the surfaces generates a positive free energy, which can dominate. For this reason, the degree of dissociation which optimizes the free energy is expected to depend on the distance between the interfaces. When the latter distance is small, the dissociation can be low, because the dissociation will generate too strong a repulsion. A theory for the interaction between layers at low separations will be presented in a separate publication. Here a comparison between the results of the present calculations and experiment will be used to evaluate the degree of dissociation. Stable lamellar liquid crystals can be obtained for KC ) 10 × 10-20 J and a surface charge σe ) 1e/50000A2. In this case, ad ≈ 10-3 and the domain in which the lamellar phase might be stable corresponds to a thickness between 17 and 19 Å of the water lamellae; the system, however, can swell indefinitely in oil, the self-energy of the water lamellae providing the necessary negative contribution to the free energy. In summary, information about the interactions between layers allowed one to identify the domains where the lamellar phase is unstable. A positive derivative of f with respect to δ1 or δ2 implies that the region is inaccessible to a lamellar phase. In this case, a water or oil phase will separate until the allowed values for δ1 and δ2 will be reached. In addition, a negative surface tension indicates that another phase (cubic,hexagonal, microemulsion, etc.) is stable. It should be, however, emphasized that when γ > 0, a phase other than the lamellar one may be the thermodynamically stable one. It is generally believed that an extreme swelling is a consequence of a strong repulsion between layers and hence that a low bending modulus favors swelling. This is not, however, always accurate, because a high repulsion can lead to a large positive contribution to the free energy that makes the lamellar phase unstable. II. The Distribution of the Components in the Lamellar Liquid Crystal II.1. Uncharged Lyotropic Lamellar Liquid Crystals. A common procedure to evaluate the thicknesses of the water and oil layers was to assume that all the molecules of surfactant and cosurfactant are adsorbed on the interface, where they occupy constant areas. While very simple, this procedure is not always accurate, because the cosurfactant is also present in the water and particularly in the oil phase.23 Larche et al. explained the deviation from the ideal dilution law by taking into account the distribution of the cosurfactant (pentanol) between the interface and the oil (decane) phase.23 A more complete procedure to compute the average thicknesses of the water and oil layers will be presented below. It is based on eq 3c, mass balances of components, and phase equilibrium equations. The calculations indicated that the interfacial tension of lamellar liquid crystals is very low, of the order of 10-5 N/m. It will be shown that γ ) 0 is always an excellent approximation of eq 3c.

320

Nanodispersions

Stability of Lyotropic Lamellar Liquid Crystals

Langmuir, Vol. 17, No. 18, 2001

For an uncharged lyotropic lamellar liquid crystal, the equilibrium concentrations are given by the expressions25

XiS ) XiW

f iW fiS

eγτia/kT e-(µi

0,S-µ 0,W)/kT i

(13)

where a is the area occupied at the interface by a molecule of water, τia is the area occupied at the interface by a molecule of species i, XiS is the surface fraction occupied by species i (XiS ) τiNiS/N, where NiS is the number of molecules of species i at the interface and N the total number of molecules of water which completely cover the interface, in the absence of surfactant and cosurfactant), XiW is the mole fraction of species i in the aqueous phase, µiW and µiS are the standard chemical potentials in the water phase and at the interface, respectively, and fiW and fiS are the activity coefficients in the water phase and at the interface, respectively. At room temperature (T ) 300 K), and for γ ≈ 10-5 N/m and a ) 10 Å2, one obtains γa/kT ≈ 2 × 10-4 and the exponential involving γ can be approximated by unity. We will assume that the activity coefficients are unity in water, and the activity coefficients at the interface are computed using the expression

( ) ( )

∂

ln fiS )

∆GE kT ∂NiS

(14) T,P,NSj*i

we will examine the swelling of the quaternary mixture C12E6/pentanol/water/decane. The changes in the standard chemical potentials of the components between the water phase and interface, ∆µi ) µiO,W - µiO,S, and the values of the pair interaction parameters Hij can be determined in principle from the fit of eq 13 to experimental data regarding the oil/water interfacial tension at low concentrations of surfactant and cosurfactant. The earlier calculations of Ruckenstein and Rao25 provided the values ∆µS ) 6500 cal/mol for C12E6 and ∆µW ) -718 cal/mol for water. For pentanol we selected the value ∆µA ) 3000 cal/mol, which is compatible with the experimental data for the lowering of the oil/water interfacial tension by alcohol.27 The surface interaction parameters HijS for different pair of molecules were taken as 0.5,25 the value KX ) 94.2 was selected for the association constant of pentanol in oil,28 and (XAW)SAT was taken as 4.1 × 10-3.28 We denote by NS, NA, NW, and NO the total number of molecules of surfactant, cosurfactant, water, and oil, and by VS, VA, VW, and VO their volumes per molecule. The volumes and the area occupied on the interface were estimated from the density of alkanes and the earlier calculations of Nagarajan and Ruckenstein.29 The values employed were VW ) 30 Å3, VO ) 320 Å3, VS ) 650 Å3, VA ) 180 Å3, a ) 10 Å2, τS ) 3.4 and τA ) 1.0. By denoting NSS, NAS, and NWS the number of molecules of surfactant, cosurfactant, and water, respectively, adsorbed on the interface, and with NAO the number of cosurfactant molecules in the oil phase, the system of equations to be solved is

where for the excess free energy ∆GE the following expression for ternary mixtures was employed26

∆GE ) H12S N1S X2S + H13S N1S X3S + H23S N2S X3S kT (15) where HijS is the interaction parameter for the pair i-j at the interface. We assume that no surfactant is present in the oil phase, and the main problem to be solved is to determine the relation between the concentrations of alcohol in the water and oil phases. A theory for the partition of the alcohol between oil and water, at chemical equilibrium, which takes into account the self-association of alcohol in the oil phase, is detailed in Appendix C. The equation that relates the mole fraction of the alcohol in the oil phase to the mole fraction of the alcohol in the water phase has the form

XSS )

(NS - NSS) (NW + NS + NA - NWS - NSS - NAS - NAO) fS W fSS

XAS )

XAO

)

NOO

+

NAO

)

(

1 + KX 1 -

XA

W

)

2

fA W fAS XWS )

where the subscripts stand for the type of molecule (A for alcohol and O for oil), the superscripts stand for the phase (O for oil and W for water), KX is the self-association equilibrium constant, and (XAW)SAT is the saturation concentration of alcohol in water (see Appendix C). Equations 13-16 with γ ) 0 and mass balances can be used to determine the distribution of molecules between the interface and the water and oil phases. In what follows, (25) Ruckenstein, E.; Rao, I. V. J. Colloid Interface Sci. 1987, 117, 104. (26) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The properties of gases and liquids; McGraw-Hill: New York, 1987.

e-(µA

O,S-µ O,W)/kT A

NAO NO + NAO

e-(µW

O,S-µ O,W)/kT W

×

(17b)

(NW + NS + NA - NWS - NSS - NAS - NAO) fWS

×

(17a)

(NW - NWS)

(16)

(XAW)SAT

O,S-µ O,W)/kT S

(NW + NS + NA - NWS - NSS - NAS - NAO)

fW W

(XAW)SAT

e-(µS

(NA - NAS - NAO)

XAW NAO

5471

×

(17c)

)

(NA - NAS - NAO) 1 (XAW)SAT (NW + NS + NA - NWS - NSS - NAS - NAO)

(

1 + KX 1 -

1 (XAW)SAT

(NA - NAS - NAO) S

(NW + NS + NA - NW -

NSS

-

NAS

-

NAO)

)

2

(17d) (27) Aveyard, R.; Briscoe, B. J. J. Chem. Soc., Faraday Trans. 1972, 68, 478. (28) Nagarajan, R.; Ruckenstein, E. Langmuir 2000, 16, 6400. (29) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1979, 71, 580.

321

Stability of dispersions 5472

Langmuir, Vol. 17, No. 18, 2001

Ruckenstein and Manciu

where the surface fractions are given by

XSS )

XAS )

XWS )

τSNSS

(17e)

τSNSS + τANAS + NWS τANAS

(17f)

τSNSS + τANAS + NWS NWS

(17g)

τSNSS + τANAS + NWS

The activity coefficients on the interface are calculated using the expressions

(

fSS ) exp HSAS(XAS - XAS XSS) + HSWS(XWS XWSXSS) - HAWS(XAS XSS)

(

)

τS (17h) τW

τA + HAWS(XWS τS τA XWS XAS) - HSWS(XWSXSS) (17i) τS

fAS ) exp HSAS(XSS - XSSXAS)

(

)

1 + HAWS(XAS τS 1 1 XAS XWS) - HSAS(XSSXAS) (17j) τA τS

fWS ) exp HSWS(XSS - XSS XWS)

)

The following mass balance relations complete the system of equations

NS ) NSS + NSW

(17k)

NA ) NAS + NAW + NAO

(17l)

NW ) NWS + NWW

(17m)

where it was assumed that no oil is present at the interface or in the water phase and also that no water or surfactant is present in the oil phase. The repeat distance d and the average thicknesses of the layers, δ1(water) and δ2(oil) can be obtained from the relation between the total interface area and total volume

d)

2(NSVS + NAVA + NWVW + NOVO)

δ1 ) d

δ2 ) d

a(τSNSS + τANAS + NWS) NSWVS + NAWVA + NWWVW NSVS + NAVA + NWVW + NOVO

(18a)

(18b)

NOOVO + NAOVA + NSSVS + NASVA + NWSVW NSVS + NAVA + NWVW + NOVO (18c)

where it was considered that the interface (including the polar headgroups) belongs to the oil layer. In Figure 6 the repeat distance (Figure 6a) and the thickness of the oil lamellae (Figure 6b) are plotted as functions of φ (the ratio between the volume of surfactant + cosurfactant and the total volume), for fixed volume

Figure 6. The repeat distance d (a), the thickness of the oil layer δ2 (b), and the product dφ (c) vs φ (volume ratio of surfactant and cosurfactant) for various volume ratios r between surfactant and cosurfactant, for the quaternary system C12E6/pentanol/ water/decane described in text: (1) r ) 1.0; (2) r ) 2.0; (3) r ) 3.0; (4) r ) 4.0.

ratios, r, between surfactant and cosurfactant. The initial volumes of water and oil were taken each as 10% of the total volume of the system; further, the system was diluted by addition of water and the new equilibrium values were calculated, using eqs 17a-m and 18a-c. One can see that the thickness of the oil layers does not remain constant upon dilution. This is a consequence of the nonlinearity of the system of eqs 17, which takes into account the partition of the alcohol between oil, water, and interface and the dependence of the surface activity coefficients on the surface concentrations of all the components. Figure 6c, in which the product between the repeat distance d and φ as a function of φ is plotted, clearly shows that there are deviations from the ideal dilution law; during the water dilution, alcohol and surfactant molecules leave the interface, and this shrinks its area. II.2. Charged Lyotropic Lamellar Liquid Crystals. In the case of a charged system, two changes in the system of eqs 17 must be made.30,31 First γ should be replaced in eq 17a, for the surfactant, by γ + ∫ψ)0ψ0 σ dψ. Second, the mole fraction of the surfactant in water, XSW, should be replaced in the same equation by the mole fraction of the surfactant in water, in the vicinity of the interface (XSW)*. Assuming that the surfactant in the aqueous phase is totally dissociated, the concentration of surfactant near the interface (XSW)* can be related to the average surfactant concentration in the water phase, XSW, by using the equilibrium and mass balance relations

kT ln(XSW)* + eψ0 ) kT ln(XS(x)) + eψ(x)

(19a)

with (30) Ruckenstein, E.; Krishnan, R. J. Colloid Interface Sci. 1980, 76, 201. (31) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987.

322

Nanodispersions

Stability of Lyotropic Lamellar Liquid Crystals

Langmuir, Vol. 17, No. 18, 2001

∫-δ /2 XS(x) dx ) XSW

1 δ1

δ1/2

5473

(19b)

1

where XS(x) is the local concentration of the surfactant in the water phase. One thus obtains

W

(XS )* ) (XS

(

)

eψ0 kT ) eψ(x) δ1/2 dx exp -δ1/2 kT δ1 exp -

W

∫

(

)

(20)

In the absence of electrolyte, both integrals ∫ψ)0ψ0 σ dψ and ∫-δ1/2δ1/2 dx exp(-eψ(x)/kT) can be carried out analytically (see Appendix B), and eq 17a is replaced by

(aB′ kT )

Figure 7. The repeat distance, as a function of dodecane weight fraction, for a quaternary system SDS/pentanol/water/dodecane: curve 1, computed using eqs 17 and 18; curve 2, computed by assuming that the entire surfactant and cosurfactant are adsorbed on the interface. The circles represent the experimental result of Safinya et al.24

XSS ) A′ exp

(NS - NSS) (NW + NS + NA - NWS - NSS - NAS - NAO) fSW fS S

e-(µS

O,S-µ O,W)/kT S

(17a′)

where

A′ )

( ) ( ( ) ( ) ) Kδ1 2 Kδ1 Kδ1 sin + 2 2

Kδ1 cos2

Kδ1 cos 2

(21a)

and

B′ )

(

( ) ( ( ))

80(kT)2 Kδ1 Kδ1 tan 2 2 2 eδ 1

Kδ1 1 ln 1 + tan2 2 2

III. Conclusions

-

)

K2δ12 (21b) 8

with K the solution of

( )

Kδ1 Kδ1 σeδ1 e tan ) 2 2 40kT

calculated using a very simple model, in which the entire surfactant and cosurfactant was assumed to be adsorbed on the interface, and the total area of the interface was calculated as NSτSa + NAτAa. It is of interest to note that this simple evaluation provides a smaller value for the total area (larger repeat distance) than the more involved calculation in which the partition of surfactant and alcohol between phases was taken into account. The apparent paradox (less adsorbed surfactant/alcohol molecules and larger area in the latter case) is explained by the fact that eqs 17 predict that about 25% of the sites of the interface are occupied by water molecules.

(21c)

The calculated results in the absence of electrolyte will be now compared with the experimental results obtained regarding a lamellar lyotropic liquid crystal SDS (sodium dodecyl sulfate)/pentanol/water/dodecane swollen in a mixture of dodecane and pentanol.24 The weight fraction water/surfactant was 1.552; from the dilution line in the phase diagram, we calculated that the initial concentration of pentanol in the oil-free system was 29 wt % and the concentration of pentanol in the dodecane-based diluant was 8 wt %. The experimental values for the repeat distance were obtained from the X-ray diffraction spectrum (Figure 2 in ref 24) for various dodecane concentrations. In the calculations, we used ∆µS ) 6960 cal/mol for SDS25 and, as in section II.1, ∆µW ) -718 cal/mol for water and ∆µA ) 3000 cal/mol for pentanol. The area occupied on the surface was considered τSa ) 34 Å2 for SDS and τAa ) 10 Å2 for pentanol.25 The values VS ) 480 Å3 and VO ) 380 Å3 were employed.29 A very low surface charge, σe ) 1e/ 50000A2, as suggested by the stability analysis of section I, was employed. In Figure 7 the calculated repeat distance (curve 1) is compared with the experimental result of Safinya et al.26 Curve 2 represents the repeat distance

In the first part of the paper, a thermodynamic formalism developed earlier10 was used to obtain information about the domains of stability of the lamellar phase. It was shown that, for a set of interaction parameters between layers and bending modulus of the interface, only certain thicknesses are allowed for the water and oil layers. In the second part, using the equilibrium relations for the various components present in the water and oil phases and on the interface and coupling them with mass balances, one could calculate the thicknesses of the water and oil layers as functions of the component concentrations. It was shown that the deviation from the ideal dilution law can be accounted for by the partition of the components between phases. A comparison between the calculated and experimental results was made. Appendix A: Van Der Waals Free Energy for a Multilayer, Formed by Alternating Water and Oil Planar Lamellae A complete expression for the van der Waals interaction energy can be obtained, in the hypothesis of pairwise additivity, by adding the self-energy of the planar layers

fI,1 ) -

A11 1 12π X 2 1

fI,2 ) -

A22 1 12π X 2 2

(A.1)

323

Stability of dispersions 5474

Langmuir, Vol. 17, No. 18, 2001

Ruckenstein and Manciu

(where A11 and A22 are Hamaker constants and X1 and X2 are the thicknesses of the layers) to the interaction energy between all the pair of layers, computed as for layers separated by vacuum. The van der Waals interaction between two planar layers of thickness X1 and X2, separated by a distance d, is given by31

(

A12 1 1 1 f12 ) + 12π d2 (d + X1 + X2)2 (d + X1)2 1 (A.2) (d + X2)2

)

The total van der Waals energy (per pair of successive water and oil layers and unit area) is computed by adding all the interactions between each of the layers of the pair and all the layers in the system to their interaction and their self-energies. When two layers are in contact, it was considered that a small (but finite) distance e, which is related to the closest approach between molecules, separates the layers. By rearranging the terms and taking the limit e f 0, the excess van der Waals energy, dependent on the thicknesses X1 and X2, acquires the form

fvdW ) -

Aeff

{

∞

1

+

12π (X2)2

∑

n)1

(

≡-

Aeff

+

-

(n(X1 + X2) + X2)2

{

1

12π (X )2 1

∞

+

∑

(

n)1

2 (n(X1 + X2))2

1

+

(n(X1 + X2) - X1)2

1

)}

2

-

(n(X1 + X2) + X1)2

)}

(n(X1 + X2))2 (A.3)

where Aeff ) A11 + A22 - 2A12 is an effective Hamaker constant.

(

2

eF0 ∂ ψ(x) eψ(x) )exp 2 0 kT ∂x

)

(B.1)

where F0 is the concentration of counterions at the middle distance between the planes, x is the position coordinate, with x ) 0 at the middle distance,  is the dielectric constant of water, and 0 is the permittivity constant. The solution of this equation, for the boundary conditions

ψ(0) ) 0,

∂ψ(x) ∂x

|

x)0

)0

is32 (32) Cowley, A. C.; Fuller, N. L.; Rand, R. P.; Parsegian, V. A. Biochemistry 1978, 1, 7, 3163.

( )

(B.3)

δ1 being the distance between the planes and σe the surface charge density. From (B.1) and (B.2), the following relation between F0 and K is obtained

F0 )

20kT e2

K2

(B.4)

The osmotic pressure between the charged plates is given by32

p ) F0kT )

20(kT)2 e2

K2

(B.5)

The integral ∫ψ)0ψ0 σe dψ can be obtained by combining (B.2) and (B.3) ψ0

σe dψ )

(

∫

K

K)0

σe

dψ dK ) dK

( )

(

( ))

80(kT)2 Kδ1 Kδ1 Kδ1 1 tan - ln 1 + tan2 2 2 2 2 2 eδ 1

-

K2δ12 8

)

(B.6) where K is the solution of eq B.3 for the surface charge density, σe, and ψ0 is the surface potential (at x ) δ1/2). It will be assumed that the surfactant molecules in the aqueous phase are completely dissociated and that their presence in the water phase does not affect the potential given by eq B.2, since their concentration is very low. The condition of equilibrium and the mass balance of the surfactant in the water phase lead to the following expression for the concentration (XSW)* of surfactant ions near the interface, as a function of the average concentration of the surfactant in the water phase

Appendix B. Electrostatic Interaction in a Electrolyte-free Lamellar Liquid Crystal In this Appendix, equations will be derived for the double layer interaction between two charged, planar surfaces in an electrolyte-free system. We assume that the potential ψ(x) obeys the Poisson-Boltzmann equation

(B.2)

Kδ1 Kδ1 σeδ1e tan ) 2 2 40kT

∫

(n(X1 + X2) - X2)2

2kT ln(cos(Kx)) e

where K is given by

ψ)0

1

1

ψ(x) )

W

(XS )* ) XS

( ) ( ( ) ( ) ) Kδ1 2 Kδ1 Kδ1 sin + 2 2

Kδ1 cos2

W

Kδ1 cos 2

(B.7)

Appendix C: Distribution of Alcohol between Water and Oil Phases In what follows, we provide the derivation of an equation, which relates the concentration of alcohol in water and the oil phases, on the basis of a model developed earlier.28 Let us assume that NOO molecules of oil and NAO molecules of alcohol are present in the oil phase. Denoting by NjAO the number of aggregates containing j molecules of alcohol, the mole fraction of the aggregates of size j is given by

XjAO

)

NjAO NOO +

∑j

(C.1) NjAO

The association equilibrium for alcohol aggregates is written

324

Nanodispersions

Stability of Lyotropic Lamellar Liquid Crystals

A1 + Aj T Aj+1

Langmuir, Vol. 17, No. 18, 2001

(C.2)

where the association equilibrium constant is considered to be independent of j. Denoting by Kx the stepwise association equilibrium constant, the mole fraction of alcohol aggregates of size j is given by

XjAO )

(KxX1AO)j

(C.3)

Kx

X1AO )

1 + 2KxXAO - x1 + 4KxXAO - 4KX(XAO)2 2KxXAO(1 + Kx)

∑j ∑j

jXjAO )

X1AO

X1AO

XOO ) NOO

+

∑j NjA 1-

NOO

∑j XjAO )

(1 -

+

∑j

XAW

-

X1AO)

(1 - KxX1AO)

(XAW)SAT

∑j

X1AO (1 - KxX1AO - X1AO)

(C.6)

1 - KxX1AO 1 - KxX1AO - X1AO

)

N1AO

∑j jXjAO X1AO

)

1 (1 - KxX1AO)2

XAO )

)

) (1 + Kx)X1AO

(C.13)

(C.14)

Kx ) 92(nC)-0.47 e3000((1/T)-(1/323))

XAW XAO )

NAO NOO + NA

X1AO 1 - 2KxX1AO + Kx(Kx + 1)(X1AO)2 (C.9)

The latter equation can be inverted as

(C.15)

Because Kx and (XAW)SAT are known, a relation between the molar fraction of alcohol in water and oil, XAW and XAO, respectively, can be established. Using eqs C.13 and C.9, one obtains

(C.8)

Using eqs C.7 and C.8, the mole fraction of alcohol can be calculated as a function of the mole fraction of monomers

NAO NOO + NAO

(X1AO)*

ln(XAW)SAT ) 1.40 - 1.38nC

(C.7)

The ratio between the total number of alcohol molecules and the number of alcohol monomers is given by

NAO

X1AO

where nC is the number of carbon atoms. The stepwise association equilibrium constant Kx for aliphatic alcohols ranging from propanol to decanol can be computed using the empirical expression28

∑j NjAO) )

NOO X1AO

)

where (XAW)SAT is the saturation concentration of alcohol in water when the water phase coexists with a pure alcohol phase. At 25 °C, the solubility of alcohol in water is given by the expression33

and further

N1AO ) X1AO(NOO +

(C.12)

(C.5)

from where one obtains

NjAO ) NOO

1 1 + Kx

)

NjAO

KxX1AO

(C.11)

By equating the chemical potential of alcohol in the oil phase to the chemical potential of alcohol in the water phase, one obtains

∑j NjAO

)1O

(X1AO)* )

(C.4)

KxX1AO)2

The true mole fraction of oil is then

NOO

µA ) µ1AO + kT ln(X1AO)*

The monomeric mole fraction in pure alcohol is obtained from eq C.10 in the limit XAO f 1

(1 - KxX1AO)

(1 -

(C.10)

The standard chemical potential of the monomeric alcohol in oil is considered equal to that of the monomers of alcohol in alcohol and is related to the chemical potential of the pure alcohol, µA, and the concentration of monomers in pure alcohol via

Since the summation over all mole fractions should remain finite, KxX1AO < 1 and

XjAO )

5475

) O

(XAW)SAT

(

1 + Kx 1 -

XAW (XAW)SAT

)

2

(C.16)

LA010385E (33) Tanford, C. The hydrophobic effect; John Wiley & Sons: New York, 1980.

Introduction to CHAPTER 4 Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

E. Ruckenstein, D. Schiby: “Effect of the Excluded Volume of the Hydrated Ions on Double-Layer Forces,” LANGMUIR 1 (1985) 612–615. M. Manciu, E. Ruckenstein: “Lattice Site Exclusion Effect on the Double Layer Interaction,” LANGMUIR 18 (2002) 5178–5185. M. Manciu, E. Ruckenstein: “Free Energy and Thermal Fluctuations of Neutral Lipid Bilayers,” LANGMUIR 17 (2001) 2455–2463. M. Manciu, E. Ruckenstein: “The Interaction between Two Fluctuating Phospholipid Bilayers,” LANGMUIR 18 (2002) 4179–4182. M. Manciu, E. Ruckenstein: “Specific Ion Effects in Common Black Films: The Role of the Thermal Undulation of Surfaces,” LANGMUIR 20 (2004) 1775–1780. H. Huang, E. Ruckenstein: “Interaction Force between Two Charged Plates Immersed in a Solution of Charged Particles. Coupling between Double Layer and Depletion Forces,” LANGMUIR 20 (2004) 5412–5417. H. Huang, E. Ruckenstein: “Thermodynamically stable dispersions induced by depletion interactions” JOURNAL OF COLLOID AND INTERFACE SCIENCE 290 (2005) 336–342. J. Feng and E. Ruckenstein: “Attractive interactions in dispersions of identical charged colloidal particles: a Monte-Carlo simulation,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 272 (2004) 430–437.

A long time ago, Stern noted that the traditional assumption that the ions interact only with a “mean” electrical field (the Poisson - Boltzmann approach) leads to an ionic density in the vicinity of the interface that exceeds the available volume. A simple way to avoid this difficulty is to consider that the ions are hydrated, and therefore there are fewer positions available to them in the vicinity of charged surfaces [4.1]. When compared to the traditional Poisson-Boltzmann result, this correction leads to an increase in the repulsive force at

short separations [4.1]. This approach is extended to account for the ion competition for hydrating water molecules [4.2]. A position in the water lattice is considered available to an ion only when it is “free” and has a certain number of “free” neighboring water molecules (i.e., water molecules which do not participate in the hydration of other ions). Whereas at high surface potentials, this “excluded” volume effect increases the repulsion as compared to the traditional double layer force, at low surface potentials it can either increase or decrease the repulsion (the latter case occurs when the hydrated coion is larger than the hydrated counterion). The strong dependence of this effect on the hydration number of ions provides a possible explanation for ion-specific effects, which will be examined in the next chapter. Other interactions that cannot be explained in the framework of the Poisson-Boltzmann approach include the repulsion between thermally undulating interfaces (the Helfrich force, which is partially responsible for the stability of neutral lipid bilayers [Helfrich, W. Z. Naturforsch. 1978, 33a, 305]). The Helfrich theory predicts that this repulsion is inversely proportional to the third power of the separation distance, while experimental evidence indicates an exponential dependence at small separations. The free energy of entropic confinement can be calculated exactly for harmonic interactions. Assuming that the same expression for the free energy remains valid when the interactions are slightly non-linear, the minimization of the total free energy leads, for the thermal undulation force, to an exponential behavior at short separations (as observed experimentally) and a power law dependence at large separations (as also predicted by Helfrich) [4.3]. Another approach considers that the undulating membrane is composed of independent pieces of a suitable area, which is selected in such a manner that, for an harmonic potential, the entropic confinement pressure of the separate pieces to be equal to the undulating repulsion. A harmonic approximation of any non-harmonic interaction potential allows one to calculate the area of the independent pieces and the corresponding repulsion force [4.4].

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_4, © Springer Science+Business Media, LLC 2010

326

It is well known that the traditional double layer theory is not accurate for small separations and high ionic strengths. However, experiment revealed that, for common black films, it is inaccurate even for moderate ionic strengths and large separation distances. Furthermore, strong specific ionic effects are exhibited by black films at large separation distances, again in contrast with the traditional theory. It is shown that by taking into account the thermal undulations of the black films and the traditional forces (double layer, hydration and van der Waals) and considering that different ions provide different bending moduli to the membrane, one can explain the dependence of the thickness of the films on the external pressure for various uni-univalent electrolytes [4.5]. An expression is derived for the force generated between two charged plates immersed in an electrolyte solution that contains small charged particles by coupling the double layer and depletion forces. The effects of particle charge, particle charge sign, particle size and particle volume fraction are examined [4.6]. When the depletion interactions are taken into account, together with double layer, van der Waals interactions and entropic effects, an increase in the volume frac-

Nanodispersions

tion of the small particles leads to a more negative free energy of the system, which stabilizes a dispersion of large particles. This effect can explain the formation of gels observed experimentally in concentrated emulsions [4.7]. Iridescence was observed in a lattice-structured dispersion for distances between particles of the same order of magnitude as the wavelength of light, distances at which the van der Waals interactions are negligible. The collective interactions between identically charged monodisperse particles have been examined via Monte-Carlo simulations, using the Debye-Hückel pair potential. When the number of charges per particle and the particle volume fraction become sufficiently large, the pair long-range electrostatic repulsion generates an effective attractive interaction, because of many-body effects [4.8]. Small charges per particle and small particle volume fractions lead to disordered structures, while larger ones lead to ordered structures. Depending on these parameters, disordered liquid-like structures, ordered structures dispersed in disordered ones, disordered structures dispersed into ordered ones, and ordered crystal-like structures are observed [4.8].

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

327

328

Nanodispersions

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

329

330

Nanodispersions

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

5178

331

Langmuir 2002, 18, 5178-5185

Lattice Site Exclusion Effect on the Double Layer Interaction Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received February 21, 2002. In Final Form: April 22, 2002 A lattice model for an electrolyte solution is proposed, which assumes that the hydrated ion occupies τi (i ) 1, 2) sites on a water lattice. A lattice site is available to an ion “i” only if it is “free” (it is occupied by a water molecule, which does not hydrate an ion) and has also at least (τi - 1) first-neighbors free. The model accounts for the correlations between the probabilities of occupancy of adjacent sites and is used to calculate the “excluded volume” (lattice site exclusion) effect on the double layer interactions. It is shown that at high surface potentials the thickness of the double layer generated near a charged surface is increased, when compared to that predicted by the Poisson-Boltzmann treatment. However, at low surface potentials, the diffuse double layer can be slightly compressed, if the hydrated co-ions are larger than the hydrated counterions. The finite sizes of the ions can lead to either an increase or even a small decrease of the double layer repulsion. The effect can be strongly dependent on the hydration numbers of the two species of ions.

I. Introduction The distribution of ions near a charged surface immersed in an electrolyte solution was first calculated on the basis of the Poisson-Boltzmann equation, assuming that the ions are point charges subjected to a mean electrostatic field. One of the first corrections of the theory was a result of the observation that, at high charges, the PoissonBoltzmann equation predicts a too high density of counterions in the vicinity of the surface, whose volume exceeds the available volume. For this reason, Stern1 suggested to consider in the vicinity of the surface a layer (the Stern layer) that contains bound counterions and is not available to the thermally moving counterions, and Bikerman2,3 proposed a modified Boltzmann expression which accounted for the finite volume of ions. Ruckenstein and Schiby derived4 an expression for the electrochemical potential, which accounted for the hydration of ions and their finite volume. The modified PoissonBoltzmann equation thus obtained was used to calculate the force between charged surfaces immersed in an electrolyte. It was shown that at low separation distances and high surface charges, the modified equation predicts an additional repulsion in excess to the traditional double layer theory of Derjaguin-Landau-Verwey-Overbeek. The volume exclusion effect has received recently renewed attention.5-9 Paunov et al.5 argued that the excluded volume should be taken as eight times the real volume of the ions (this corresponds to the first-order correction in the virial expansion for a hard-sphere fluid) * Corresponding author: e-mail address, feaeliru@ acsu.buffalo.edu; phone, (716) 645 2911/2214; fax, (716) 645 3822. (1) Stern, O. Z. Electrochem. 1924, 30, 508. (2) Bikerman, J. J. Philos. Mag. 1942, 33, 384. (3) Rowlinson, J. S. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992. (4) Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612. (5) Paunov, V. N.; Dimova, R. I.; Kralchevsky, P. A.; Broze, G.; Mehreteab, A. J. Colloid Interface Sci. 1996, 182, 239. (6) Kralj-Iglic, V.; Iglic, A. J. Phys. II 1996, 6, 477. (7) Borukhov, I.; Andelman, D.; Orland, H. Phys. Rev. Lett. 1997, 79, 435. (8) Marcelja S. Nature 1997, 385, 689. (9) Lue, L.; Zoeller N.; Blankschtein, D. Langmuir 1999, 15, 3726.

and calculated on this basis the repulsive force. While this approximation is reasonable at very low concentrations, it breaks down when the volume of the spheres becomes about 10% of the total volume (since it predicts a maximum compaction of 12.5%). For hydrated ions with a large hydration radius, such as Na, this corresponds to a maximum electrolyte concentration of about 1 M. In principle, the model can be improved by including in the “excluded volume”, in addition to the term proportional to the ion density, higher-order corrections proportional to powers of the ion density, which are related to the virial expansion for hard-sphere fluids. However, only the first eight virial coefficients have been calculated,10 and the power series converges slowly at high densities. To avoid this, one can account for the volume exclusion effect for hard-sphere fluids starting from an equation of state. The latter can be obtained with reasonable accuracy from the interpolation of the virial expansion, either using Pade´ approximants,11 or other ad hoc analytical continuations of the series, a procedure first employed by Carnahan and Starling.12 The method was employed by Lue et al.9 to calculate the counterions distribution starting from the corresponding modified Poisson-Boltzmann equation. The advantage of this method consists of its easy application to ions of different sizes, since good approximations of the equation of state (and hence of the excess chemical potential with respect to an ideal solution) are known for mixtures of hard spheres of different sizes.13 The main question is whether the hydrated ions behave as hard spheres; while this seems plausible for ions much larger than the water molecules, it is probably not entirely applicable to small ions, whose hydration shells continuously change. Marcelja calculated recently the double layer interaction8 using the anisotropic hypernetted chain method and potentials of mean force between pairs of ions in water, provided by Monte Carlo simulations. This (10) van Rensburg, E. J. J. J. Phys. A: Math. Gen. 1993, 26, 4805. (11) Ree, F. H.; Hoover, W. G. J. Chem. Phys. 1967, 46, 4181. (12) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (13) Boublik, T. J. Chem. Phys. 1970, 53, 471. Masoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523.

10.1021/la020194r CCC: $22.00 © 2002 American Chemical Society Published on Web 06/18/2002

332

Nanodispersions

Double Layer Interactions

Langmuir, Vol. 18, No. 13, 2002 5179

method implicitly accounted for the excluded volume and also for other correlations between ions; however, it required accurate information about the interactions between pairs of ions in water. It was shown14 that the method is very sensitive to the details of the potential of the mean force between pairs of ions and that different models can lead to even qualitatively different results. Another approach that accounted for the volume exclusion effect was based on lattice models.4 Recently, KraljIglic and Iglic6 and, independently, Borukhov et al.7 derived an expression for the counterions distributions, starting from a modified Poisson-Boltzmann equation, for a lattice whose cell size was that of a hydrated ion (both kind of ions were considered of the same size). The lattice cell can be either occupied or not by a hydrated ion. This type of lattice allows for a maximum compaction of unity. For this reason, in this case the excess chemical potential (with respect to an ideal solution) is, in general, important either for very large ions at sufficiently large concentrations or for small ions at high ionic strengths and high surface potentials. It is not yet clear how this model6,7 can be extended to electrolytes with hydrated ions of different sizes. A modality to overcome these difficulties was proposed earlier by Ruckenstein and Schiby,4 who considered that the ions of different kinds (i ) 1, 2) occupy τi sites on a water lattice. Since the size of the cell of the water lattice is independent of the size of the ions, the model can be easily extended to any number of ions of different kinds. To compare the prediction of various models, let us first consider the case in which the ions of different kinds have the same volume (τ1 ) τ2 ) τ). For n hard spherical particles of radius R and volume τv (v being the volume of a cell of the lattice) and for large values of τ (the continuum limit), the volume V′ available to the particles can be calculated as follows.15 The center of a new particle cannot be located within a distance 2R from an existing particle; hence each existing particle can be thought of as carrying a “forbidden region” of radius 2R (volume 8τv), where a new particle cannot be placed. However, by subtracting 8nτv from the total volume V of the system, one obtains an underestimate of the available volume, since the “forbidden regions” of the existing particles overlap (and hence the overlapping regions are subtracted twice). Assuming that the particles are uniformly distributed in the available space, the overlapping volume per particle is15 34τv nτv/V′ = 34 (τv) × (τη) (where η ) nv/V ) n/N is the ratio between the total volume occupied by the ions and the total volume of the system, τη is the ratio between the total volume of the hydrated ions and the total volume of the systems, and N is the total number of sites (n ions + (τ - 1)n water molecules hydrating the ions + free water molecules). Since the second correction should be counted only once per each pair of particles, the available volume is15 2

V′ ) V(1 - 8τη + 17(τη) - ...)

(1)

The first three terms of the series are overestimating the available volume, since they do not account for the overlapping of the “forbidden regions” for triplets, and a new correction (proportional to the third power of density) should be included, and so on. The calculation of the remaining terms of the (slowly) converging series is however increasingly difficult. The first two terms of the (14) Otto, F.; Patey, G. N. J. Chem. Phys. 2000, 113, 2851. (15) Ruckenstein, E.; Chi, J. C. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1690.

series (1) constitute the result proposed recently for the volume available to a hard sphere fluid by Paunov et al.5 On the other hand, if the hydrated ions are totally deformable (hence, they can occupy a site if the site is free and other (τ - 1) sites are free, regardless of their positions), the available volume is given by15

V′ ) V(1 - τη)

(2) 6

which is the result of Kralj-Iglic and Iglic or Borukhov et al.7 However, the hydrated ions are probably not totally rigid (since the hydration shell changes its shape and its water molecules, which are bound to the ion in a transient manner only. To account for this, Ruckenstein and Schiby4 considered that the ion of volume τv can occupy a site of the water lattice only if the site and its (τ - 1) neighbors are free (i.e., free of ions and hydration molecules). The probability for a randomly chosen site to be free is (1 τη). Considering that the probabilities of occupancy of neighboring sites are independent, the available volume is4

V′ ) V(1 - τη)τ

(3a)

The result can be extended to ions of different sizes τi

Vi′ ) (1 - τ1η1 - τ2η2)τi (i ) 1, 2)

(3b)

where ηi ) ni/N, ni being the number of ions of kind “i”. In this model, the “available volume” is different for each kind of ion. For τ ) 1, eq 3a becomes identical to eq 2 and for not too large values of τ (τ < 8) predicts an available volume between those of eqs 1 and 2 and, hence, can be interpreted as corresponding to a partially deformable hydrated ion. However, for large values of τ, the probability of occupation of one site becomes strongly affected by the occupation of their adjacent sites. The purpose of this paper is to calculate the electrochemical potential and the double layer repulsion using a lattice model, applicable to hydrated ions of different sizes, that accounts for the correlation between the probabilities of occupancy of adjacent sites. As the other lattice models,4-7 this model accounts only for the steric, excluded volume effects due to ionic hydration. In fact, short-ranged electrostatic interactions between the ions and the dipoles of the water molecules, as well as the van der Waals interactions between the ions and the water molecules, are responsible for the formation of the hydrated ions. The long-ranged interactions between charges are taken into account through an electrostatic (mean field) potential. The correlation between ions is expected to be negligible for sufficiently low ionic concentrations. II. The Lattice Model In the model of Ruckenstein and Schiby, the ion can replace a water molecule on the lattice of water only if the site and (τ - 1) of its first neighbor sites are “free” (i.e., occupied by water molecules which do not hydrate other ions); the probability of occupation of any of the τ sites was taken to be the same.4 Here we will account to some extent for the correlation between the probabilities of occupancy of adjacent sites, assuming a coordination number w in the water lattice. The number of sites available to an ion is (in the large N limit) the total number of sites, N, times the probability that a randomly chosen site can be occupied by an ion. This implies that the site and at least (τ - 1) of its first

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

5180

Langmuir, Vol. 18, No. 13, 2002

Manciu and Ruckenstein

Figure 1. A schematic planar projection of the direct bonding of a water molecule in an ice lattice. A central site (0) has four first-neighbors (1), which are adjacent to the central site and to three other molecules (2). Each of the 12 second neighbors (2) of the central site (0) is adjacent to only one first-neighbor (1) of the central site. Some of the third-order neighbors (3) of the central site (molecules separated through three successive bondings by the central site) are, however, adjacent to two second neighbors. Only one of the third neighbors is drawn.

neighbors are “free” water molecules. It will be assumed that the probability of occupation of any first neighbor (of a central site) is independent of each other. This is obviously not valid for a normal liquid, which has a large number of nearest neighbors. Indeed, at close compaction, each of the 12 first neighbors of a selected site has as first neighbors (is adjacent to) other 3 first neighbors of the same selected site, and hence their occupation probabilities are not independent, because the ions that may occupy the first neighbors compete for hydration molecules. However, for sparse lattices, it is a reasonable approximation. Indeed, for ice (with a coordination number of 4), all the first and second neighbors of the central site are distinct (i.e., none of the first neighbors is adjacent to other first neighbors of the same central site and each of the second neighbors of the central site is adjacent to only one of the first neighbors of the same site; see Figure 1). Each of the third neighbors of the central site is, however, adjacent to two second neighbors of the central site; this leads to a small correlation between the occupancy of the second neighbors, which will be, however, neglected. A molecule of liquid water has between 4.4 and 4.9 first neighbors,16 which is much closer to the coordination in ice (4) than to that in a normal liquid (∼12). We will therefore assume that for a water lattice the probability of occupation of one of the w sites first-neighboring a central site is independent of the occupation of the other (w - 1) sites. In what follows an expression will be established for the probability for a random site to be available to a hydrated ion, by assuming first that a single kind of ion is present. The chance for a site, chosen randomly (which will be called central site), to be occupied by an ion is η, and the probability for the site to be “free” is

pf,0 ) 1 - τη

333

(4a)

Let us now obtain an expression for the probability for an ion to occupy one site adjacent to a free central one. (16) Eisenberg, D.; Kauzmann, W. The structure and properties of water; Oxford University Press: New York, 1969.

An ion surrounded by w water molecules can be hydrated by (τ - 1) molecules in Cwτ-1 distinct configurations, where Cwτ-1 ) w!/[(w - τ + 1)!(τ - 1)!] is the number of combinations in which (τ - 1) objects (water molecules hydrating the ion) can be selected out of w objects (available water molecules). However, because one of the adjacent molecules (the central site) is “free”, there are only Cw-1τ-1 configurations possible for the hydrated ion to sit on the lattice (the number of combinations in which (τ - 1) hydrating water molecules can be chosen out of the remaining (w - 1) available water molecules). It will be assumed that all the possible configurations of the ion hydrated by the (τ - 1) water molecules are equally probable and that the number of hydrated configurations of an ion is not affected by its concentration. The last approximation is valid only for small ion concentrations and is reasonable because the density of ions is in general much smaller than unity; only the density of hydrated ions (τη) can become comparable to unity. The probability for an ion to occupy a site adjacent to a free central one (poi,1) is therefore

poi,1 ) η

Cw-1τ-1 Cwτ-1

)η

w-τ+1 w

(4b)

and consequently, the probability for the site to be not occupied by an ion (pfi,1) is given by

pfi,1 ) 1 - η

w-τ+1 w

(4c)

We will now calculate the probability for a water molecule, first-neighboring a free central site, not to participate in the hydration of an ion. Since one of its neighbors (the central site) is a free water molecule, it can hydrate only an ion occupying one of the remaining (w 1) adjacent sites (which are second-neighbors of the central site). Even if an ion occupies one of these sites, it can be hydrated by (τ - 1) out of the other (w - 1) of its firstneighbors, which are third-order-neighbors of the central site (the third-order-neighbors are connected to the central site through three successive bonds, regardless of their mutual angles). Since there are Cw-1τ-1 configurations in which the ion can be hydrated by the water molecules located in the third-neighboring sites of the central site, out of a total of Cwτ-1 configurations of possible hydration of an ion, the probability for an ion to occupy a site second neighbor to a central site, and not to be hydrated by the molecule occupying the first-neighboring site, is

poi,2 ) η

Cw-1τ-1 Cwτ-1

)η

w-τ+1 w

(4d)

Therefore, a water molecule first-neighboring a free central site does not hydrate an adjacent ion, located on a second-neighboring site of the central site, in the following two circumstances: (1) there are no ions occupying any of the (w - 1) sites, and (2) there is one or more ions occupying some or all of the (w - 1) positions, ions which are hydrated by water molecules located on the third-order-neighboring sites of the central site. Since the probabilities of occupation of the first neighbors of each site are assumed independent of each other, the probability for none of the (w - 1) sites to be occupied by ions is (1 - η)(w-1) and the probability for an ion to occupy one site and the other (w - 2) sites to be free of ions is η(1 - η)(w-2). Because there are (w - 1) modes in which an ion can occupy one of the (w - 1) sites, the probability for an

334

Nanodispersions

Double Layer Interactions

Langmuir, Vol. 18, No. 13, 2002 5181

ion to occupy one of the (w - 1) sites and to be not hydrated by the water molecules first neighboring the central site is (w - 1)η(1 - η)w-2[(w - τ + 1)/w]. Adding the probabilities for ions to occupy two, three, ..., (w - 1) sites, none of them being hydrated by the water molecules adjacent to the central free site, one obtains the following expression for the probability for a water molecule, first neighboring the central free site, not to hydrate any ion

pfh,1 ) (1 - η)w-1 + (w - 1)η(1 - η)w-2 ... + η w-1

w-1

(

w-τ+1 w

)

w-1

Cw-1kηk(1 - η)w-1-k ∑ k)0

(

(

w-τ+1 w

)

where the volumes available to each species, Vi′ ) PiV, are different (τ1 * τ2). III. The Excess Chemical Potential The chemical potential for a fluid of n identical particles interacting with a hard-core potential (which is zero at separations larger than the distance of closest approach and infinity for smaller separations) can be calculated using the equation

+ µ ) -kT

)

w-τ+1 w

)

(4e) Z ) 〈z〉n/n!

(4f)

Hence, the probability for all w first neighbors of the central site to be free is (pf,1)w and the probability for (w - 1) to be free and one not free (i.e., either occupied by an ion or hydrating another ion) is w(pf,1)w-1(1 - pf,1) (there are w places that can be occupied by a nonfree water molecule). The last term of the sum is the probability for (τ - 1) water molecules to be free and (w - τ + 1) to be occupied, which is Cwτ-1pf,1τ-1(1 - pf,1)w-τ+1. Therefore, the probability for a central site (chosen randomly) and at least (τ - 1) of its first neighbors to be “free” is given by

Cwτ-1pf,1τ-1(1 - pf,1)w-τ+1) (4g) When two ionic species, of hydrated sizes τ1 and τ2 are present, the probabilities for a randomly chosen central site to be free is given by

(5a)

the probability for a first-neighboring site not to be occupied by an ion by

pfi,1 ) 1 - η1

w - τ1 + 1 w - τ2 + 1 - η2 w w

(5b)

and the probability for a water molecule from a firstneighboring site of a central free one not to participate in the hydration of an ion by

pfh,1 ) (1 - η1 - η2)w-1 + (w - 1)(1 - η1 - η2)w-2 × w - τ1 + 1 w - τ2 + 1 η1 + η2 + ... (5c) w w

[(

)]

) (

With pf,1 ) pfi,1pfh,1, the probability for a random central site to be available to an ion is

Pi ) pf,0(pf,1w + wpf,1w-1(1 - pf,1) + ... + Cwτi-1pf,1τi-1(1 (i ) 1, 2)

- pf,1)

w-τi+1

) (5d)

(6b)

In eq 6b, 〈z〉 represents the individual partition function, averaged over all possible configurations

〈z〉 )

1 z′ h3

p ∫∫∫ exp(- 2mkT ) d3p × 2

〈∫∫ ∫ ( ) 〉 exp -

V

U 3 z′V′ d x ) 3 (6c) kT Λ

where z′ accounts for the internal degree of freedom of the particle, h is the Planck constant, and Λ is the thermal de Broglie wavelength. The individual partition function is proportional to the available volume V′, which is the volume available to a particle, out of the total volume V, when the other (n - 1) particles occupy nonoverlapping positions. Using eqs 6a-c, one obtains for the chemical potential the expression

P ) pf,0(pf,1w + wpf,1w-1(1 - pf,1) + ... +

pf,0 ) 1 - τ1η1 - τ2η2

(6a)

V,T

where k is the Boltzmann constant, T the absolute temperature, and Z the partition function

k

Consequently, the probability for one neighbor of the central site neither to be occupied by an ion nor to participate in the hydration of another ion is given by

pf,1 ) pfi,1pfh,1

(∂ ln∂n Z)

µ ) - kT ln

( )

()

z′ n n + kT ln ) µ0 + kT ln ) V′ V′ Λ3 µid + µexcees (6d)

where

( )

µid ) - kT ln

z′ n + kT ln V Λ3

(6e)

is the chemical potential in an ideal mixture and

µexcess ) kT ln(V/V′)

(6f)

is the excess chemical potential, due to the hard-core interactions. Since for a lattice the available volume is V′ ) PV, where P is the probability for a randomly chosen site to be occupied by an ion, eq 6f becomes

µexcess ) -kT ln P

(6g)

The excess chemical potential for a dilute hard-sphere fluid in the Paunov et al. approximation (eqs 1 and 6f) is given by5

µexcessP ) -kT ln(1 - 8τη)

(7a)

for a lattice model in which each ion occupies one site (Iglic-Iglic model) (eqs 2 and 6f) by6,7

µexcessII ) -kT ln(1 - τη)

(7b)

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

5182

Langmuir, Vol. 18, No. 13, 2002

335

Manciu and Ruckenstein

for the Ruckenstein and Schiby lattice model (eqs 3 and 6f) by4

µexcessRS ) -τkT ln(1 - τη)

(7c)

and for the present lattice model (eqs 4e and 6f) by

µexcessMR ) -kT ln[pf,0(pf,1w + wpf,1w-1(1 - pf,1) + ... + Cwτ-1pf,1τ-1(1 - pf,1)w-τ+1)] (7d) The excess chemical potential of a hard-sphere fluid can be calculated on the basis of an equation of state using the expressions17

Aexcess ) nkT µexcess )

(

∫0

τη

(

)

pV -1 nkT d(τη) τη

)

∂Aexcess 1 Aexcess + kTτη n ∂τη

(8)

where p is the pressure, Aexcess represents the excess Helmholtz free energy, and τη is the fraction of the volume occupied by the hydrated ions. Using the CarnahanStarling equation of state12 2

1 + τη + (τη) - (τη) pV ) nkT (1 - τη)3

3

(9)

the excess chemical potential becomes

µexcessCS ) kT

τη(8 - 9τη + 3(τη)2) (1 - τη)3

(7e)

In Figure 2 the excess chemical potentials provided by the models listed above (eqs 7a-e) are compared. We used w ) 5, which is the closest integer to the estimate of the number of first neighbors in water (4.4-4.9),16 and (a) τ ) 6, (b) τ ) 5, and (c) τ ) 4. A more realistic model (but more tedious to calculate) could have been obtained by the statistical averaging of systems with different values of w and τ. When the ion is hydrated by all the adjacent molecules (w ) τ - 1), there is a strong lattice-site exclusion effect at high ionic concentrations, since the central site can be available to an ion only when both its first and second neighbors are free of ions (a total of 1 + w + w(w - 1) ) w2 + 1 sites). On the other hand, when the hydration number (τ - 1) is low, almost any “free” site can be occupied by an ion, since in this case there is a high chance to find at least (τ - 1) free water molecules around a selected site. In the next section it will be shown that there is a strong dependence of the site-exclusion effect on the hydration number and, hence, that specific ion effects can be important in the double layer interactions. IV. Excluded-Site Effect on the Double Layer Repulsion IV.1. General Equations. At thermodynamic equilibrium, the electrochemical potential, (eq 6d) is con(17) Khoshbarchi, M. K.; Vera, J. H. Fluid Phase Equilib. 1998, 142, 131.

Figure 2. The excess chemical potential relative to that of an ideal solution, for the models discussed in the text (eqs 7a-e) and (a) τ ) 6, (b) τ ) 5, and (c) τ ) 4.

stant in the system

µi0 + kT ln(ni(x)/V) - kT ln Pi(n1(x),n2(x),ψ(x)) + qiψ(x) ) constant (10) (i ) 1, 2) where µi0 is the standard chemical potential of species i, qi is the charge of an ion of species i (qi ) (-1)i+1e, with e the protonic charge), ψ is the mean potential, and x

Nanodispersions

336

Double Layer Interactions

Langmuir, Vol. 18, No. 13, 2002 5183

denotes the distance from one of the two charged, planar surfaces separated by a distance l. It will be assumed that the surface charge and potential are negative; hence the subscript 1 denotes the counterions and the subscript 2 is for co-ions. Consequently, the concentration of each species of ions for a 1:1 electrolyte is given by

ni(x) ) n

Pi(x) Pi(x)∞)

(

exp -

qiψ(x) kT

)

(11a)

(i ) 1, 2) where n1 ) n2 ) n for x ) ∞. A modified Poisson-Boltzmann equation is obtained by replacing in the Poisson equation for parallel plates

∂2ψ(x) 2

)

∂x

e(n1(x) - n2(x)) 0V

(11b)

n1 and n2 by their expressions (11a), 0 being the dielectric constant of water. The above equation has to satisfy the boundary conditions ∂ψ(x)/∂x|x )l/2 ≡ 0 at the middle distance between the plates, and either ψS ) ψ(x)0) ) const (for constant surface potential) or ∂ψ(x)/∂x|x)0 ) const (for constant surface charge density), where ψS represents the surface potential. Caution should be taken when calculating the doublelayer force between two parallel plates. It is clear that the force is not proportional to the excess concentration of ions at the middle distance (with respect to the concentration of ions at infinity), since this Langmuir equation involved the assumption of ions of negligible sizes. We will use instead the procedure introduced by Verwey and Overbeek,18 which is based on general thermodynamic principles, and does not imply the Boltzmann distribution of ions.19 The force, per unit area, between two parallel plates separated by a distance l is given by

pψ(l) ) 2

∂ ∂l

∫0ψ

S

σ(ψ) dψ

(12a)

at constant surface potential ψS and

pσ(l) ) -2

∂ ∂l

∫0σ

S

ψ(σ) dσ

(12b)

at constant surface charge density σS, where the surface charge density is related to the surface potential through the expression

∂ψ(x) σ | )∂x x)0 0

(13)

The relation between the surface charge density and surface potential, as a function of the distance l, is obtained using the additional boundary condition required by symmetry ∂ψ(x)/∂x|x)l/2 ) 0. Consequently, eqs 12 can be integrated numerically to provide the value of the double layer force. IV.2. Distribution of Ions near a Charged Surface. The distribution of counterions in the vicinity of a single charged surface, at a high surface potential (ψs ) 0.1, Figure 3a) is represented as a function of the distance x (18) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (19) For a detailed discussion, see: Ruckenstein, E. Adv. Colloid Interface Sci. 1998, 75, 169.

Figure 3. (a) Ion distributions in the vicinity of a single charged surface at high potentials (ψS ) 0.1 V), for large (τ1 ) 6, τ2 ) 2) and small (τ1 ) 2, τ2 ) 6) counterions: n/V ) 1.0 M and T ) 300 K. (b) Ion and charge distributions (relative to those predicted by the Boltzmann distribution) as functions of the mean potential. Large counterions (τ1 ) 6, τ2 ) 2): n/V ) 1.0 M and T ) 300 K. (c) Ion and charge distributions (relative to those predicted by the Boltzmann distribution) as functions of the mean potential. Small counterions (τ1 ) 2, τ2 ) 6): n/V ) 1.0 M and T ) 300 K.

from the surface for w ) 5, τ1 ) 6, and τ2 ) 2 (large counterions) and for τ1 ) 2, τ2 ) 6 (small counterions). In both cases, the densities of counterions and co-ions in the vicinity of the surface are smaller than those predicted by the Poisson-Boltzmann equation, since the agglomeration

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

5184

Langmuir, Vol. 18, No. 13, 2002

Manciu and Ruckenstein

of the finite-size counterions near the surface reduces the available volume. Let us examine now the effect of the excluded volume at low surface potentials. In the linear approximation of the Poisson-Boltzmann expression, the increase in the number of counterions in the vicinity of the interface equals the decrease in the number of co-ions. If the co-ions have a larger size, one expects the available volume near the surface to be larger than that in the bulk. As a result, a concentration of ions in excess to that predicted by the Poisson-Boltzmann equation is expected to occur in the vicinity of the surface, when the volume exclusion is taken into account. Of course, when the potential of the surface becomes sufficiently large, the linear approximation fails, and the (exponential) increase in the counterions density in the vicinity of the surface predicted by the Poisson-Boltzmann equation largely exceeds the depletion of co-ions and the available volume is expected to become smaller than that in the bulk. To illustrate this, let us use the simple approximation suggested by Bikerman2

(

) (

n1(ψ)v1 + n2(ψ)v2 1qiψ V ni(ψ) ) n exp v1 + v2 kT 1-n V

(

)

)

(

)

(xλ - 2λl ) l cosh( ) 2λ

cosh ψ(x) ) ψS

(16)

and eq 13 yields

σS )

0ψS l tanh λ 2λ

( )

(17)

The double layer forces obtained from eqs 12, 16, and 17 are

pψ(l) )

0ψS2 2λ2 cosh2

l 2λ

( )

(18a)

at constant surface potential and

1 ) n(v1 - v2) eψ 1+ V kT 1 1 1 ψ ) 2 ψ (15) 2 n(v v ) λ λ′ 1 2 eψ 1+ V kT

(

IV.3. Modification of the Double Layer Repulsion Due to the Finite Volume of Ions. In the linear approximation of the Poisson-Boltzmann equation, the potential between two surfaces, separated by the distance l is given by

(14)

(where vi is the volume of the hydrated ions), which combined with Poisson equation becomes, in the linear approximation

∂2ψ 2ne2 ) ψ 2 0kT ∂x

337

)

When v1 > v2, the effective Debye-Hu¨ckel length λ′ (which now depends on ψ(x)) is larger than that obtained for the Poisson-Boltzmann equation. Consequently, the diffuse double layer is larger in the vicinity of a charged surface, as predicted earlier.4-7,9 However, when v2 > v1 (small counterions), λ′ < λ and the diffuse double layer is compressed. The effect is proportional to the ionic strength and is, in general, small for typical electrolyte concentrations, since |n(v1 - v2)/V| , 1 and |eψ/kT| < 1. Let us now calculate the dependence of the ions and charge densities on the mean potential in the present lattice model. For large counterions (Figure 3b) the ratios between counterion and co-ion densities and those calculated with the Poisson-Boltzmann equation are monotonic decreasing functions of the mean potential ψ. For small counterions, however, at low potentials (Figure 3c), the two ratios slightly exceed unity, because the available volume (as compared to V′ at ψ ) 0) increases. The ratio between the local charge density (which is proportional to η1 - η2) and that calculated from the PoissonBoltzmann equation also exceeds slightly unity in a range of potentials. The subunit ratios at very low potentials occur because the available volumes are different for the two types of ions and depend differently on the potential ψ; one of them grows initially slowly and decreases later slowly, while the other grows initially quicker but decreases later also quicker.

pσ(l) )

σS2 20sinh2

(2λl )

(18b)

at constant surface charge. At constant surface charge, an increase in the Debye length implies an increase in the repulsion at any distance. However, this is not true for constant surface potential, since the function (l/2λ)2 cosh-2(l/2λ) (for fixed l) has a maximum at l/2λ = 1.2. Consequently, an increase in the effective Debye length corresponds to an increase in repulsion only at large separations (l > 2.4λ) but to a decrease in repulsion at smaller separations. At sufficiently high surface charges or potentials, the thickness of the double layer (the effective Debye length) increases when the finite volumes of the ions are taken into account. The replacement of λ by a larger value in eq 18b leads to a higher repulsion at any separation distance. This suggests that the accounting of the ion sizes will lead to a higher repulsion at a high constant surface charge. Figure 4a (n/V ) 0.01 M, σS ) constant ) 0.32 C/m2) confirms this expectation. The replacement of λ with a larger value in eq 18a suggests that, at high constant surface potentials, the double layer interaction will be decreased at short separation distances by the finite sizes of the ions. This expectation is confirmed by Figure 4b (n/V ) 0.01 M, ψS ) constant ) 0.1 V). The modification of the double layer repulsion due to the sizes of the ions is, however, much smaller when the counterions are smaller and is strongly dependent on the hydration number, when the latter has a value near to w. At low surface charges, the effective Debye length can either increase or decrease, depending on the relative sizes of the two species of ions. Consequently, the double layer repulsion might become smaller than that predicted by the Poisson-Boltzmann equation not only at constant surface potential but also at constant surface charge. In Figure 5, the double layer repulsion (relative to that predicted by Poisson-Boltzmann equation) is presented for (a) a small constant surface potential ψS ) 0.02 V and (b) a small constant surface charge density σS ) 0.032 C/m2. Even at the relatively high ionic strength selected (n/V ) 1.0 M), the decrease of the repulsion is small.

338

Nanodispersions

Double Layer Interactions

Figure 4. (a) Double layer force per unit area for constant surface charge density (σS ) 0.32 C/m2), n/V ) 0.01 M and T ) 300 K. (b) Double layer force per unit area for constant surface potential (ψS ) 0.1 V), n/V ) 0.01 M and T ) 300 K.

V. Conclusions The lattice model of Ruckenstein and Schiby,4 which accounted for the sizes of the ions, was extended to account for the correlations between the probabilities of occupancy of neighboring sites. In the present model, the excess chemical potential depends strongly on the number of water molecules that hydrate an ion; hence slightly different hydration numbers can lead (at high potentials) to large differences in the double layer repulsion. It was also shown that, at high surface potentials, the thickness of the double layer near a charged surface is increased

Langmuir, Vol. 18, No. 13, 2002 5185

Figure 5. Ratio between the double layer force with site exclusion (modified Poisson-Boltzmann) and the double layer force provided by the Poisson-Boltzmann equation: (a) constant (small) surface potential (ψS ) 0.02 V); (b) constant (small) surface charge density (σS ) 0.032 C/m2), n/V ) 1.0 M and T ) 300 K.

(compared to that predicted by the Poisson-Boltzmann equation). However, at low surface potentials the diffuse layer might be slightly compressed, when the co-ions are larger than the counterions. The increase in the effective Debye length (thickness of the double layer) leads to an increase in the double layer repulsion at constant surface charge at any distance but to a slight decrease of the double layer repulsion at constant surface potential and small separation distances and an increase at large separations. LA020194R

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Langmuir 2001, 17, 2455-2463

339

2455

Free Energy and Thermal Fluctuations of Neutral Lipid Bilayers Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received November 27, 2000. In Final Form: January 31, 2001 A new method is proposed to calculate the free energy of lamellar liquid crystals lipid bilayers/water. The root-mean-square fluctuation of the distance between two neighboring bilayers is calculated for a number of distributions by minimizing the total free energy. Analytical solutions for the free energy are derived for a Gaussian distribution of distances, which are compared with numerical solutions obtained for more realistic distributions, which account for the correlation between the fluctuations of neighboring bilayers. Calculations are performed for typical values of the interaction parameters, and the comparison with experiment provides a more than qualitative agreement. The calculations also show that at infinite separation distance there is a minimum of zero free energy and that a local minimum can occur at a finite distance, which can be stable or unstable. The two minima are separated by a potential barrier. An unbinding transition occurs when the free energies of the two minima are equal to zero.

I. Introduction It is well-known that the lipid molecules placed in water form multilamellar vesicles, which can be regarded locally as smectic liquid crystals, with stacks of bilayers aligned at constant separation, along a director axis. An important problem is to calculate the free energy of the system, from which one could derive the equilibrium separation distance and the corresponding binding free energy. It was recently noted1,2 that the experimental data for both the interaction force per unit area (pressure) and the root-mean-square fluctuation of the distance between two neighboring bilayers as functions of the average separation between them are not satisfactorily explained by the existing theories.3-5 For neutral bilayers, there are no long-range doublelayer forces which, coupled with the van der Waals attraction, could explain the stability of the lamellar structure. At small separations, the required repulsion is provided by the hydration force, which was investigated both experimentally6-8 and theoretically.9,10 However, it was experimentally observed that the lipid bilayers could be swollen in water up to very large interlayer distances,11 where the short-range exponential hydration repulsion becomes negligible. Helfrich was the first to suggest3 that the entropic confinement (due to neighboring layers) of thermally * To whom correspondence may be addressed. E-mail address: [email protected]. Phone: (716) 645 2911/2214. Fax: (716) 645 3822. (1) Petrache, H. I.; Gouliaev, N.; Tristram-Nagle, S.; Zhang, R.; Sutter, R. M.; Nagle, J. F. Phys. Rev. E 1998, 57, 7014. (2) Gouliaev, N.; Nagle, J. F. Phys. Rev. E 1998, 58, 881. (3) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (4) Evans, E. A.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7132. (5) Podgornik, R.; Parsegian, V. A. Langmuir 1992, 8, 557. (6) Le Neveau, D. M.; Rand, R. P.; Parsegian, V. A.; Gingell, D. Biophys. J. 1977, 18, 209. (7) Parsegian, V. A.; Fuller N.; Rand, R. P. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 2750. (8) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (9) Marcelyia, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (10) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (11) Harbich, W.; Helfrich, W. Chem. Phys. Lipids 1984, 36, 39.

excited out-of-plane undulations of the interfaces of the lipid bilayers (lamellae) leads to a long-range repulsion force, inverse proportional to the third power of the distance, which could be responsible for the hyperswelling of the bilayers. His original method was to assume a linear interaction between lamellae, dependent on an unknown elastic constant B. Because of the linearity, the oscillation modes are not coupled, and this allowed him to compute the free energy as a sum over the free energies of the individual modes. He showed that the unknown elastic constant could be obtained in a self-consistent manner, either from the bulk properties of the multilayer system or from the root-mean-square fluctuation σb of the bilayer positions.3 Since at large distances the van der Waals forces are inversely proportional to the fifth power of the distance, Helfrich showed that the entropic confinement repulsion would always dominate at large enough separations3 and, consequently, that the separation between the bilayers can extend at infinity. Experiments appeared to confirm this conclusion.11 Some other experiments, however, indicated that the separation between membranes remained finite even in the presence of excess water.1,12 A possible explanation could be that the Helfrich theory, which involves in the calculation of the entropic confinement only a hard-wall (steric) repulsion, does not account for other forces between membranes. The role of undulation on the equilibrium of lipid bilayers was also examined by Lipowsky and Leibler,13 who used a nonlinear functional renormalization group approach, and by Sornette,14 who employed a linear functional renormalization approach. It was theoretically predicted that a critical unbinding transition (corresponding to a transition from a finite to an infinite swelling) can occur by varying either the temperature or the Hamaker constant. However, the renormalization group procedures do not offer quantitative information about the systems, when they are not in the close vicinity of this critical point. (12) Lis, L. J.; McAlister, M.; Fuller N.; Rand, R. P.; Parsegian, V. A. Biophys. J. 1982, 37, 657. (13) Lipowsky, R.; Leibler, S. Phys. Rev. Lett. 1986, 56, 2541. (14) Sornette, D. J. Phys. C: Solid State Phys. 1987, 20, 4695.

10.1021/la0016266 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/21/2001

340

2456

Nanodispersions

Langmuir, Vol. 17, No. 8, 2001

The effect of the nonlinear interaction forces between the lipid bilayers on the undulation was investigated by several groups.4,15 An important advance was made in the perturbative theory of “soft confinement” of Podgornik and Parsegian.5 In their work, the nonlinear interactions between bilayers (due to DLVO forces plus hydration repulsion) was considered as a small perturbation to a hard wall (steric) confinement. Using a variational method due to Feynman,16 they obtained an equation for the unknown interaction constant B of an hypothetical harmonic potential, as a function of the separation distance a, by minimizing the free energy with respect to both σb and B. The quantity σb represents the root-mean-square fluctuation of the distribution of the positions of membranes, F(u), u being the displacement of a point of the membrane surface from the average position, located at half-distance between its two neighboring bilayers, assumed rigid. A Gaussian distribution of membrane positions was employed for averaging the interaction energy in the partition function, and then σb2 was calculated by assuming that F(u) should verify a diffusiontype equation, with the boundary conditions F(u) ) 0 at u ) (a. The latter distribution was employed because the long tails of the Gaussian will lead to a divergent energy for the hard (rigid wall) confinement, which is their unperturbed state. The purpose of this article is to present a variational analysis which no longer considers that the fluctuating bilayer is confined by two rigid walls. A Gaussian, a truncated Gaussian, or an asymmetrical Gaussian distribution is assumed for the fluctuating distances between two neighboring bilayers. In the case of a linear interaction between neighboring lipid bilayers, Helfrich has demonstrated that the repulsive free energy due to confinement is inversely proportional to σb2. While this result is strictly valid for a harmonic interaction potential (linear force), we assume that it can be extended to any interaction. We will examine later under what conditions this approximation is accurate. The free energy of a bilayer in a given potential is composed of the energy of interaction with the other membranes and an entropic term, due to the bilayer confinement, which is inverse proportional to σb2. The energy is increased by a distribution which has a large value of σb, because then a large part of the bilayer is in a region of stronger repulsive interactions; on the other hand the entropic term decreases with increasing σb. Consequently, there is an optimum σb for which the free energy is minimum. Instead of computing via optimization the hypothetical elastic constant B corresponding to the interaction potential of the problem, as in the work of Podgornik and Parsegian,5 we express B in terms of σb via the equipartition principle and calculate σb which minimizes the free energy. In addition, whereas in the Podgornik and Parsegian work, the calculation involved a small perturbation of a rigid wall confinement, in the present approach the hard wall confinement is not used as a base state. However, while the confining bilayers are not considered rigid, the exclusion of permeation between two neighboring membranes provides indirectly such a confinement. Most importantly, the present approach can employ an asymmetrical distance distribution for the fluctuating distances between two neighboring bilayers,17 which cannot be used (15) Sornette, D.; Ostrovski, N. J. Chem. Phys. 1986, 84, 4062. (16) Feynman, R. Statistical Mechanics; Benjamin: Reading, MA, 1972.

Manciu and Ruckenstein

when one assumes that a bilayer is confined by two rigid walls. The former distribution is expected, because the fluctuations of neighboring membranes are increasingly correlated, as the distance between them becomes smaller. II. Free Energy of Entropic Confinement for a Linear Interaction Following Helfrich,3,18 one can compute the free energy of confinement, starting from the Hamiltonian of the confined bilayer

H)

( (

∫dx dy 12 Kc

∂2u(x,y) ∂x

+

2

∂2u(x,y) ∂y

2

)

2

+

)

1 Bu2(x,y) 2 (1)

where Kc is the bending modulus of a bilayer, B is an unknown constant of the assumed linear interaction force between bilayers, and u(x,y) denotes the displacement along z from the average position z ) 0 of a point of a bilayer of coordinates x and y. Denoting by u˜ (qx,qy) the Fourier transform of u(x,y), the partition function acquires the form

Z)

( (

1

∫∏du˜ exp -β ∑ 2 u˜ 2(Kcbq4 + B) q ,q x

y

∏ q ,q x

y

(

))

)

2πkT

Kcq4 + B

)

1/2

(2)

where q2 ) qx2 + qy2. The difference in the free energies of the free bilayer (which formally corresponds to B f 0) and the confined one, per unit area, can be computed by replacing the summation over qx,qy with an integral

∆f ) kT

1 (2π)2

∫0

∞

dq2πq ln

(

)

Kcq4 + B Kcq4

1/2

)

( )

kT B 8 Kc

1/2

(3)

The free energy due to the entropic confinement alone is obtained from eq 3 by subtracting the interaction energy per unit area

fC )

( )

kT B 8 KC

1/2

-

1 B〈u2〉 2

(4)

The unknown coupling constant B can be related to the mean-square fluctuation of the bilayer position σb2 ) 〈u(x,y)2〉 by using the equipartition principle

u˜ 2

(12 K q c

4

+

1 1 B ) kT 2 2

)

(5)

which leads to the following relation between σb2 and B:

σb2 ) 〈u2〉 )

∑ u˜ 2 ) q ,q x

y

kT (2π)

∞ ∫ 0 2

2πq dq Kcq4 + B

)

kT 8(KcB)1/2 (6)

Introducing eq 6 in eq 4, yields:5,18 (17) Gouliaev N.; Nagle, J. F. Phys. Rev. Lett. 1998, 81, 2610. (18) Sornette, D.; Ostrovsky, N. Micelles, Membranes, Microemulsions and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; SpringerVerlag: 1994.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Free Energy of Neutral Lipid Bilayers

fc )

Langmuir, Vol. 17, No. 8, 2001

(kT)2

341

2457

(7)

128Kcσb2

a result which is essentially due to Helfrich.3 The Helfrich formula for the free energy (eq 7) is strictly valid only for a linear interaction. However, eq 7 constitutes a good approximation, even for nonlinear interactions, as long as the interaction energy between layers is much smaller than the elastic energy of the modes corresponding to bending, hence if B , Kcq4. In this case, the coupling between modes is minimal and eq 5 is approximately valid. Using the expression of B from eq 6, the above condition reduces to an inequality for the wavelength λxy in the xy plane

( )

λxy 8Kc , 2π σ kT

(

1/2

λxy )

)

2π q

(8)

which indicates that eq 7 can be used for bilayers that possess a high bending modulus. For lipid bilayers, a typical value of the bending modulus is Kc ) 1.0 × 10-19 J,1 and the right-hand quantity becomes ≈102. III. Thermal Fluctuations for Soft Confinement As already stated, the free energy due to the entropic confinement remains of the form of eq 7 even for nonlinear interactions, if the inequality (8) is satisfied. To compute the total free energy, one should calculate the pair distribution function of the distances between neighboring fluctuating bilayers. If the distribution of bilayer positions F(u) is Gaussian with the root-mean-square σb and the undulations of neighboring bilayers are totally uncorrelated, then the distribution of distances z between two neighboring bilayers (see Figure 1) is the convolution of two Gaussians, which is again a Gaussian

P(z) )

∫-∞ ∫-∞ du1 du2 2πσ1 2 e-u ∞

∞

1

2/2σ 2 b

e-u2 /2σb × 2

2

b

δ(z - (a - (u1 - u2))) )

2 2 1 e-(z-a) /2σ (9) 1/2 (2π) σ

where σ ) 21/2σb. Let us denote by V the interaction potential for planar bilayers. Assuming that the fluctuations of adjacent bilayers are totally uncorrelated, and using locally the planar approximation, the free energy per unit area can be written in terms of the distance distribution function P(z)

f(a,σ) )

(kT)2

∫ dz P(z)V(z) + 64K σ2

(10)

Figure 1. (a) Gaussian distributions of displacements from equilibrium positions, F(u1) and F(u2), for two neighboring bilayers, situated at the average distance a apart. (b) Distributions of distances between bilayers. (1) a Gaussian and a truncated Gaussian (the same distribution as Gaussian, except P(z) ) 0 for z < 0) and (2) an asymmetric distribution (R ) 1.4).

First, we will assume that the fluctuations of neighboring bilayers are independent of each other, as long as the distance between them is positive, but forbid those fluctuations which involve their interpenetration. In this case, the distance distribution function is a truncated Gaussian (see Figure 1b). This is equivalent to superimposing an additional hard-wall potential to the interaction potential, and we will show that, in the asymptotic limit, the results are in qualitative agreement with those of Podgornik and Parsegian.5 Second, we will consider that the degree of correlation between the fluctuating bilayers depends on their separation distance, with total correlation as z f 0 and no correlation as z f ∞. This provides an asymmetric distribution (Figure 1b), which will be described using an asymmetry coefficient R in a truncated Gaussian distribution

c

where a ) 〈z〉 is the average separation between neighboring bilayers and σ2 ≡ 〈(z - 〈z〉)2〉 is the root-mean-square fluctuation of the distribution P(z). The minimum of the free energy with respect to σ for any interaction potential V(z) and all separation distances

∂f(a,σ)/∂σ ) 0

(11)

provides an expression for σ. However, the assumption that the fluctuations of neighboring bilayers are totally uncorrelated leads to the unphysical result that they can permeate each other (P(z) > 0 for z < 0). One can approximately account for the correlation of the motions of the neighboring bilayers as follows.

P(z) ) P(z) )

1 -(z-a′)2/2σ′2 e , C

1 -(z-a′)2/2(Rσ′)2 e , C

0 < z e a′ a′ < z < ∞

(12)

where a′ is the position of the maximum and the constant C is determined through normalization. The average separation distance a ) 〈z〉 and the variance of the distribution (the mean-square fluctuation) σ2 ) 〈z2〉 - 〈z〉2 can be computed as functions of the parameters a′, R, and σ ′. Recent Monte Carlo simulations of bilayer fluctuations17 revealed that the distance distribution function is asymmetric. While this result is expected, since it implies that the correlation between fluctuating bilayers decreases

342

2458

Nanodispersions

Langmuir, Vol. 17, No. 8, 2001

Manciu and Ruckenstein

with their separation, there are no quantitative analytic theories to predict the degree of asymmetry from the bilayer interaction parameters. However, the simulations show that the asymmetry is stronger only when a few bilayers are present. For a large number of bilayers, the distribution function tends to a limiting distribution which can be approximated by eq 12, with R ≈ 1.4. Let us now present explicit calculations for each distribution. III.1. Gaussian Distribution. The Gaussian distribution allows one to obtain analytical expressions for σ and for the free energy, when only the hydration force is present. It will be shown later that, for small distances, these expressions provide also good approximations for the truncated and asymmetric Gaussian distributions and that the van der Waals attraction does not modify substantially the expression of σ. The hydration interaction energy between two neighboring planar bilayers,6-10 separated by a distance z, is given by

VH(z) ) AHe-z/λ

(13)

The free energy per unit area f(a,σ) acquires the form

f(a,σ) )

∫-∞ (2π)dz1/2σ e-(a-z) /2σ ∞

2

2

AHe-z/λ +

(kT)2 64Kcσ2

AH e-a/λ eσ /2λ + 2

2

)

(kT)2 64Kcσ2

(14)

Figure 2. Equation 15 (circles): (a) asymptotic approximation for small values of a (eq 16), continuous line; (b) asymptotic approximation for large values of a (eq 20), continuous line.

(15)

Let us now calculate the free energy for small values of a. Equations 14 and 16 lead to

and the minimum with respect to σ leads to

()

4 ln

(

)

(kT)2 σ σ2 a + 2 - ln ) 2 λ λ 2λ 32AHKcλ

The solution of the above equation is plotted in Figure 2, for T ) 300 K and the following typical values of the parameters:1,8,19 Kc ) 1 × 10-19 J, AH ) 0.035 J/m2, and λ ) 1.5 Å. One can see from Figure 2a that σ remains finite even when a ) 0 (i.e., the neighboring bilayers touch each other). In previous theories, which include the hard wall confinement,3,5 the free energy diverges for a f 0; in the present case, it remains finite because the repulsion is provided by the hydration force, which is finite at any distance. One can see from Figure 2a that for separation distances less than 10 Å, the quadratic term can be neglected compared to the logarithms, and eq 15 becomes

σ2 )

λkT ea/2λ 1/2 4(2AHKc)

(16)

A recent improvement of the Caille theory of X-ray scattering by smectic liquid crystals20,21 opens the possibility to obtain σ ) σ(a) from the diffraction spectrum. Extensive experimental measurements on lipid bilayers1 indicated that, at low separations, there is an exponential dependence as predicted by eq 16. However, to obtain reliable interaction parameters from the fit of the experimental data, one should identify the validity domain of eq 16. As shown in Figure 2a, eq 16 is valid only for small values of a. (19) McIntosh, T. J.; Simon, S. A. Biochemistry 1993, 32, 8374. (20) Caille, A. C. R. Seances Acad. Sci., Ser. B 1972, 174, 891. (21) Zhang, R.; Sutter, M.; Nagle, J. F. Phys. Rev. E 1994, 50, 5047.

f(a) ) AH e-a/λ eσ /2λ + 2

2

( )

AH kT 1/2 16(2 ) Kcλ2

1/2

e-a/2λ (17)

an equation which contains two repulsive contributions to the free energy. The first is due to hydration, which is enhanced by an exponential factor eσ2/2λ2 due to undulation; this enhancement was also suggested by Sornette and Ostrovsky.15 The other term is due to the entropic confinement. Both terms have exponential decays, but with decay lengths of λ and 2λ, respectively. The exponential decay in the second term is due to the exponential dependence of σ on the separation a. A similar functional form for the free energy, with two exponential decays, was proposed by Evans and Parsegian4,22

fEP(a) ) AH e-a/λ +

( )

πkT AH 16 Kcλ2

1/2

e-a/2λ

(18)

The differences consist in the absence of the undulation enhancement in the first term of eq 18 and the numerical constant in the second term. The variational treatment of hydration confinement by Podgornik and Parsegian5 provided an expression similar to eq 18 in the limit of small a; however the coefficient multiplying the exponential of the second term was dependent on a. (22) Evans, E.; Needham, D. J. Phys. Chem. 1987, 91, 4219.

343

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Free Energy of Neutral Lipid Bilayers

Langmuir, Vol. 17, No. 8, 2001

2459

For large values of a, the quadratic term dominates the left-hand term of eq 15; neglecting the other terms, one can write

σ ) (2aλ)1/2

(19)

A better approximation is obtained by evaluating the neglected terms using eq 19. This leads to

(

σ ) 2aλ + 2λ2 ln

(

(kT)2 128AHKca

))

1/2

(20)

2

Figure 2b shows that the asymptotic value of σ provided by eq 20 is in very good agreement with eq 15 for a > 30 Å. Using eqs 14 and 20, the free energy acquires the form

f(a) )

(kT)2 128Kca

2

(kT)2

(

+

2

(kT)2

)

(21)

64Kc 2aλ + 2λ ln 128AHKca2

For large values of a, the logarithm in the second term becomes negligible, and the free energy becomes proportional to a-1

f(a) ≈

(kT)2 128Kcaλ

(22)

The derived equations involve the assumption that the bilayers can permeate each other. It will be shown later that, when the permeation is forbidden, σ becomes proportional to a for large values of a, as predicted by the existing hard wall confinement theories.3,5 However, for small values of a, eq 15 constitutes a very good approximation. III.2. Truncated Gaussian Distribution. As already noted, to avoid permeation, the integration range should start from zero and, when only the hydration forces are taken into account, eq 10 becomes

f(a,σ) )

e-(a-z) /2σ′ ∫0∞ dz C 2

AHe-a/λeσ′ /2λ 2

2

AHe-z/λ +

2

∫0

∞

(kT)2 64Kcσ2

)

(kT)2 dz -(a′-z)2/2σ′2 e + (23) C 64Kcσ2

with a′ ) a - (σ ′ 2/λ), where σ ′ is the root-mean-square fluctuation of the Gaussian distribution, which is different from σ, the root-mean-square fluctuation of the truncated Gaussian distribution However, when σ ′ is at least a few times smaller than a

∫-∞0 dz e-(a-z) /2σ′ 2

2

,

∫0∞ dz e-(a-z) /2σ′ 2

2

and one can write that σ = σ ′ and C = σ(2π)1/2. When, in addition, σ ′ is a few times smaller than a′, the integral of the first term of eq 23 is close to unity, and the results are well approximated by eq 15. However, σ ′ is not always smaller than a′, and then numerical methods are necessary to obtain the values of σ which minimize the free energy in eq 23. This happens either when a f 0 or, at large separations, when σ2/λ becomes comparable or even larger than a. In these cases, the results differ from those obtained in the previous section.

Figure 3. Root-mean-square fluctuation σ vs average separation distance a, for a Gaussian and a truncated Gaussian distribution. (a) Kc ) 1 × 10-19 J and λ ) 1.5 Å: truncated Gaussian, (1) AH ) 0.01 J/m2, (2) AH ) 0.035 J/m2, (3) AH ) 0.1 J/m2; Gaussian, (4) AH ) 0.01 J/m2, (5) AH ) 0.035 J/m2, (6) AH ) 0.1 J/m2. (b) Kc ) 1 × 10-19 J and AH ) 0.035 J/m2: truncated Gaussian, (1) λ ) 0.5 Å, (2) λ ) 1.5 Å, (3) λ ) 3 Å; Gaussian (4) λ ) 0.5 Å, (5) λ ) 1.5 Å, (6) λ ) 3 Å. In the inserts, the same functions are presented for small values of a at a larger scale.

The results of the numerical computations are compared with the analytical ones provided by eq 15 in Figure 3, for various values of the hydration interaction constant AH (Figure 3a) and of the decay length λ. In all the cases investigated, a linear asymptotic behavior was obtained for σ as a function of a, as suggested by Helfrich. As already noted, this occurs because the truncation is equivalent to a hard wall repulsion, which limits the fluctuations of the bilayers. However, the proportionality coefficient µ ) σ2/a2 differs from those obtained from the theories for hard wall confinement3,5 and is dependent on the parameter AH of the hydration force. It is of interest to note that the asymptotic behavior of σ(a) is almost independent of λ for the truncated Gaussian distribution, but almost independent of AH for the Gaussian distribution (see eq 20). III.3. Asymmetric Distribution. When in eq 12 σ ′ is at least a few times smaller than a′, one can neglect the integral from -∞ to 0, and the following expressions are obtained for C, a′, and σ2 ) 〈z2〉 - 〈z〉2:

C)

(π2)

1/2

a′ ) a - σ ′

(

σ ′(R + 1)

(π2)

1/2

(R - 1)

σ2 ) σ ′ 2 R 2 - R + 1 -

2 (R - 1)2 π

(24)

)

344

2460

Nanodispersions

Langmuir, Vol. 17, No. 8, 2001

Manciu and Ruckenstein

tion. The bilayers are composed of hydrophilic headgroups, which are hydrated by water, and of hydrophobic hydrocarbon chains. A complete theory should take into account that their dielectric properties are different, by using different Hamaker constants.22 The problem can be simplified by considering that, for the purpose of van der Waals interaction, the polar headgroups, being hydrophilic, are part of the water lamellae. Denoting by th the total thickness of the headgroups in the lamellae and by tc the thickness of the layer of hydrocarbon chains, the van der Waals attraction between two bilayers separated by a distance z, measured in water, becomes (we include only the interactions between the nearest neighbors)

VvdW(z) ) Figure 4. Root-mean-square fluctuation σ vs the separation distance a, for (1) eq 15 and (2-4) σ calculated for truncated Gaussian distribution with various asymmetry coefficients: (2) R ) 1; (3) R ) 1.4; (4) R ) 2.0; (5) Helfrich proportionality relation, σ2 ) µa2 (µ ) 0.183); (6) approximate solution for small distances, eq 16.

The following approximate expression can be written for the free energy:

f(a,σ ′) )

∫0

∞

dz P(z)AHe-z/λ + (kT)2

≈ 2 64Kc R2 - R + 1 - (R - 1)2 σ ′ 2 π 2 2 2AHe-a/λeσ′ /2λ ∞ 2 2 dz e-(a′′-z) /2σ′ + 1/2 0 (R + 1) (2π) σ ′ (kT)2 (25) 2 64Kc R2 - R + 1 - (R - 1)2 σ ′ 2 π

(

)

∫

(

)

where

a′′ ) a - σ′

(π2)

1/2

(R - 1) -

σ′ 2 λ

It should be emphasized that eqs 24 and 25 are only approximate. In the calculation, we used eqs 10 and 12 for the numerical minimization of f(a,σ ′), with respect to σ ′, for various values of the asymmetry coefficient R. The results are presented in Figure 4. For a smaller than about 20 Å, eq 15, and for a smaller than about 10 Å, eq 16, constitute accurate approximations. The figure indicates that again σ is proportional to a in the asymptotic limit a f ∞, with a proportionality coefficient dependent on the degree of asymmetry R. The value µ ) 0.183 was derived for the hard wall confinement.5 Recent Monte Carlo simulations of soft confinement of bilayers (interacting via van der Waals and hydration forces) provided lower values for the proportionality coefficient µ ) 0.06-0.12.17 The results in Figure 3 show that, for a truncated Gaussian distributions, involving only hydration interactions, even lower values are obtained. However, using asymmetric Gaussian distributions, as in Figure 4, the values of µ, which are in the range 0.06-0.08, becomes compatible with those obtained from Monte Carlo simulations. IV. The Inclusion of the van der Waals Interaction Let us now complete the calculation by adding the van der Waals interaction potential to the hydration interac-

-

(

)

H 1 1 2 + (26) 12π (z + th)2 (z + th + 2tc)2 (z + th + tc)2

where H is the Hamaker constant. Typical values for the Hamaker constant were estimated to range from 10-21 to 10-20 J.1,19,22,23 The values of the bending modulus are either not known for some lipids or uncertain by a factor of 4, around 10-19 J, for other lipids.1 The values of the parameters of the hydration force were estimated from the fit of experimental data1,19 with a formula derived by Evans and Parsegian4 (reproduced here as eq 18). The values lie in the rather large range of λ ) 1.3-2.1 Å and AH ) 6.0 × 10-3 to 2.5 × 10-2 J/m2. In addition to depending upon the expression employed in the fitting of the data, large differences in AH, by orders of magnitude, can be obtained whether the distance z in the hydration force is measured from the boundary between the hydrocarbon chains and headgroups or between the headgroups and water.1 In this section we present the results of the calculations performed for σ and the free energy for T ) 300 K and the following values of the parameters: H ) 1.0 × 10-20 J, tc ) 37.8 Å, th ) 7.6 Å, Kc ) 1.0 × 10-19 J, λ ) 1.5 Å, and AH ) 0.035 J/m2. The root-mean-square fluctuation of the distance distributions, σ, will be computed for the following cases: (1) using the analytical formula (15), which involves only the hydration force (H ) 0); (2) for a truncated Gaussian distribution, with only the hydration force present (H ) 0); (3) for a truncated Gaussian distribution, with hydration force and van der Waals interactions with a Hamaker constant H ) 1.0 × 10-20 J; (4) for an asymmetric Gaussian distribution with R ) 1.4 and H ) 0; (5) for an asymmetric Gaussian distribution with R ) 1.4 and H ) 1.0 × 10-20 J; (6) for an asymmetric Gaussian distribution with R ) 2.0 and H ) 0; (7) for an asymmetric Gaussian distributions with R ) 2.0 and H ) 1.0 × 10-20 J; (8) using Helfrich’s proportionality relation, with µ ) 0.183. The results of these computations are presented in Figure 5, which shows that (i) σ is only slightly affected by the van der Waals interactions, but depends on the distribution considered, and (ii) for a < 20 Å, the results for all the distributions employed are well approximated by the simple expression, eq 15, which can be used to extract some of the interaction parameters from the experimental values of σ(a) for small values of a. The free energy calculations are presented in Figure 6 as a function of the average distance a, for the following cases: (23) Petrache, H. I.; Tristram-Nagle, S.; Nagle, J. F. Chem. Phys. Lipids 1998, 95, 83.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Free Energy of Neutral Lipid Bilayers

Figure 5. Root-mean-square fluctuation σ vs separation distance a, for Kc ) 1 × 10-19 J, th ) 7.6 Å, tc ) 37.8 Å, AH ) 0.035 J/m2, λ ) 1.5 Å for: (1) eq 15 (H ) 0); (2) truncated Gaussian distribution, H ) 0; (3) truncated Gaussian distribution, H ) 1.0 × 10-20 J; (4) asymmetric Gaussian distribution, R ) 1.4, H ) 0; (5) asymmetric Gaussian distribution, R ) 1.4, H ) 1.0 × 10-20 J; (6) asymmetric Gaussian distribution, R ) 2.0, H ) 0; (7) asymmetric Gaussian distribution, R ) 2.0, H ) 1.0 × 10-20 J (8) Helfrich proportionality relation (µ ) 0.183). In the inserts, the same functions are presented for small values of a at a larger scale.

Figure 6. Free energy per unit area of bilayer, for H ) 1.0 × 10-20 J, Kc ) 1 × 10-19 J, th ) 7.6 Å, tc ) 37.8 Å, AH ) 0.035 J/m2, λ ) 1.5 Å: (1) Kc f ∞ (energy of planar bilayers); (2) calculated by adding the entropic term (eq 7 with σ2 ) µa2) to the energy of planar bilayers; (3) eq 18; (4) truncated Gaussian distribution; (5, 6) asymmetric Gaussian distribution with (5) R ) 1.4 and (6) 2.0.

(1) Kc f ∞. In this case, the entropic term vanishishes and the free energy is provided by the energy of planar (nonundulating) bilayers. (2) The free energy is obtained by adding to the energy obtained under (1) the entropic term, using for σ the relation σ2 ) µa2 suggested by Helfrich,3 with the value µ ) 0.183.5 This procedure neglects both the fluctuation enhancement of the energy and the more involved dependence of σ on a, obtained through the minimization of the free energy with respect to σ. (3) The free energy was computed by adding to the Evans-Parsegian formula (eq 18) the van der Waals attraction between nonundulating bilayers. This kind of calculation, which was often used to fit the experimental data, offers results in qualitative agreement with our approach only for small values of a. However, even for small a, the fluctuation enhancement of the energy (as in eq 17) is neglected. It should be noted that the free energy

Langmuir, Vol. 17, No. 8, 2001

345

2461

computed in this manner would always have a stable minimum at finite distance, since the exponential has a shorter range than the van der Waals attraction. (4) The free energy is computed via minimization, by using the truncated Gaussian distribution in eq 10. (5, 6) The free energy is computed by using the asymmetric Gaussian distribution, with R ) 1.4 and 2.0, respectively, in eq 10. The addition of the hard wall entropic confinement free energy to the interaction energy, as under (2), only raises somewhat the minimum, but almost does not modify its position. In contrast, in all the other cases, which take into account the other interactions in the calculation of entropy, considerable shifts of the equilibrium distance and of the value of the free energy at the minimum occur. This observation indicates that a common procedure, to add to the conventional interactions, calculated as for planar layers, the free energy due to the hard wall entropic confinement is inaccurate. The asymptotic value of the free energy is zero for a f ∞. Since the entropic free energy has a slower decay, for all distance distributions employed (4-6), than the van der Waals energy, the free energy will be positive at sufficiently large distances. This does not mean, however, that an unbinding of the bilayers will occur, since a negative free energy minimum (hence an absolute minimum) can exist at a finite distance. Reciprocally, the existence of a local minimum for a given set of parameters does not necessarily imply that the bilayers are bound, since the local minimum can be positive and hence unstable. However, this unstable minimum can be separated from the absolute minimum (which is zero and located at infinity) by a potential barrier and, if the latter is sufficiently high, the metastable state can survive for a long time. Similarly, if the minimum is negative but small in absolute value, a perturbation can cause the bilayers to extend to the unstable state at infinity. These results can explain the contradictory experimental observations11,12 for the same phospholipid bilayers. The first authors11 observed that the bilayers extend at very large separations, while the latter12 authors observed that they swell up to small distances. In Figure 7 we present the free energy for an asymmetric Gaussian distribution (R ) 1.4) as a function of distance for various values of the Hamaker constant (with all the other parameters unchanged). For H > 3.825 × 10-21 J, a stable minimum is obtained at a finite distance. For H < 3.825 × 10-21 J, the stable minimum is at infinite distance; however, for 3.825 × 10-21 J > H > 3.45 × 10-21 J, a local (unstable) minimum is still obtained at finite distance. For H ) 3.825 × 10-21 J, a critical unbinding transition occurs, since the minima at finite and infinite distances become equal. However, these two minima are separated by a potential barrier, with a maximum height of 1.68 × 10-7 J/m2, located at a separation distance of 90 Å. The results remained qualitatively the same for any combination of the interaction parameters. V. Comparison with Experimental Data A common procedure to determine the values of the interaction parameters between bilayers is to fit the experimental data with a functional form predicted by theory. It was shown1,19 that the equilibrium separation distance between bilayers as a function of the osmotic pressure applied can be well fitted by expression (18). A recent improvement of Caille theory20,21 of X-ray scattering by smectic liquid crystals allowed the simultaneous measurement of the average separation distance and the

Nanodispersions

346

2462

Langmuir, Vol. 17, No. 8, 2001

Figure 7. Free energy per unit area, computed for an asymmetric Gaussian distribution, for Kc ) 1 × 10-19 J/m2, th ) 7.6 Å, tc ) 37.8 Å, AH ) 0.035 J/m2, λ ) 1.5 Å, R ) 1.4, and various values of H: (a) from bottom to top, H ) 10.0, 7.0, 5.0, 4.0, and 3.0 × 10-21 J; (b) from bottom to top, H ) 4.0, 3.825, 3.6, 3.45, and 3.0 × 10-21 J. For H > 3.825 × 10-21 J, the minimum at finite distance is stable. For H < 3.825 × 10-21 J, the minimum at infinity is the stable one. For H < 3.45 × 10-21, the minimum at infinity is the only one.

root-mean-square fluctuation σ as a function of the applied osmotic pressure, and the authors1,2 noted disagreement between the existing theories and experiment. Figure 8 reproduces the experimental data of ref 1 for EPC (egg phosphatidylcholine) (circles) and DMPC (1,2dimyristoyl-sn-glycero-3-phosphatidylcholine) (squares). Figure 8a presents p ) p(a) and Figure 8b presents σ ) σ(a). Petrache et al.1 have shown that agreement can be obtained by fitting eq 18 to the pressure data, for various values of H ((1.65-7.13) × 10-21 J) and Kc ((0.5-2.0) × 10-19 J). In all the cases, the fitting provided λ ) 2.0 ( 0.3 Å. The experimental data for σ ) σ(a) are well described by the exponential function σ2 ∝ ea/λfl for distances between 5 and 20 Å, with λfl ≈ 6 Å.1 Combining the Helfrich expression (eq 7) with that of Evans and Parsegian (eq 18), one obtains σ2 ∝ ea/2λ, which is in disagreement with experiment because λfl * 2λ. Now σ and the pressure (computed as the derivative of the free energy per unit area) will be calculated, using the procedure outlined in this article. For the Hamaker constant and the bending modulus, typical values from literature, namely, 1.0 × 10-20 and 1.0 × 10-19 J, respectively, will be used. For th and tc we employed the values obtained from X-ray data,1 th ) 7.6 Å and tc ) 37.8 Å for EPC and th ) 7.6 Å and tc ) 36.4 Å for DMPC, respectively. Because the hydrocarbon thicknesses tc of EPC and DMPC produced almost no difference (see eq 27), in the following only the results for the EPC are presented (tc ) 37.8 Å). For the degree of asymmetry R the value of 1.4, which is in agreement with the Monte Carlo simulations,17 was selected.

Manciu and Ruckenstein

Figure 8. Experimental data from ref 1 for EPC (circles) and DMPC (squares) bilayers compared with predictions obtained using some typical values for the interaction parameters: (1) Kc ) 1.0 × 10-19 J, H ) 1.0 × 10-20, th ) 37.8 Å, tc ) 7.6 Å, R ) 1.4, AH ) 0.035 J/m2, λ ) 1.5 Å; (2) Kc ) 0.5 × 10-19 J, AH ) 0.01 J/m2, and the other parameters as in (1).

It is clear that one cannot use the values of the parameters obtained through the fitting of a different expression to experimental data, to compare the present calculations to the same experimental data. One can proceed in two different ways: either to determine all the parameters from fitting the present equations to the experimental data or to use the experimental data for one of the physical quantities (p or σ) to determine some parameters and to verify if the second (σ or p) is in agreement. Because the number of parameters is large, it is obvious that one can obtain “agreement” with experiment by fitting the present equations. It is, however, more meaningful to employ the second procedure. The values of λ ) 1.5 Å and AH ) 0.035 J/m2 provided agreement with the p ) p(a) data (curve 1 of Figure 8a). Because for these values, the σ experiments were underestimated, a lower value for Kc was chosen. Selecting Kc ) 0.5 × 10-19 J required a value of AH of 0.01 J/m2 to preserve the position of the minimum of the free energy. Under these conditions (all the other parameters remaining unchanged), curve 2 was obtained. While in better agreement with σ(a), the agreement with p(a) was somewhat deteriorated. There is no doubt that better agreement can be further achieved by fitting the parameters; however, even using some typical values from literature for the parameters, a more than qualitative agreement with experimental data was obtained. VI. Conclusions A new variational analysis of the interactions of fluctuating bilayers is proposed, in which the entropic

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Free Energy of Neutral Lipid Bilayers

confinement energy is considered inverse proportional to the root-mean-square fluctuation of the distribution function of the distance between two successive bilayers. This approximation can be used, even for nonlinear interactions, when the bending elastic energy of the bilayer provides the dominating contribution to the mode energy. For a given functional form of the distribution, σ was obtained by minimizing the free energy. Analytical results were derived for a Gaussian distribution and compared with numerical results obtained for truncated and asymmetric Gaussian distributions. It is shown that, in the asymptotic limit, the present approach recovers the results of other existing theories, but important differences occur at relatively short distances. As reported in the literature,1,2 there are no theories or simulations, which are in agreement with the experimental data for both the pressure and the root-mean-

Langmuir, Vol. 17, No. 8, 2001

347

2463

square fluctuations of the distance between two neighboring bilayers, as a function of the average distance, for the same set of interaction parameters. Comparison with recent experimental data shows that, with the present approach, a more than qualitative agreement can be obtained. The present approach recovers the well-known result that a transition from a bounded to an unbounded state can be achieved by varying the Hamaker constant. However, the stable and the unstable states (which can be either at infinity or at finite distance) are separated, in the vicinity of the transition point, by a potential barrier. This might explain the contradictory experimental observations on infinite or finite swelling of some lipid bilayers.11,12 LA0016266

348

Nanodispersions

© Copyright 2002 American Chemical Society

MAY 28, 2002 VOLUME 18, NUMBER 11

Letters The Interaction between Two Fluctuating Phospholipid Bilayers Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received February 12, 2002. In Final Form: April 5, 2002 The dependence of the interaction force between two undulating phospholipid bilayers and of the rootmean-square fluctuation of their separation distances on the average separation can be determined once the distribution of the intermembrane separation is known as a function of the applied pressure. However, most of the present theories for interacting membranes start by assuming that the distance distribution is symmetric, a hypothesis invalidated by Monte Carlo simulations. Here we present an approach to calculate the distribution of the intermembrane separation for any arbitrary interaction potential and applied pressure. The procedure is applied to a realistic interaction potential between neutral lipid bilayers in water, involving the hydration repulsion and van der Waals attraction. A comparison with existing experiments is provided.

The role of thermal undulations on the interaction between elastic phospholipid bilayers has received considerable attention because of its relevance to biological processes. Helfrich suggested that,1 for membranes interacting via a rigid wall potential, the attenuation of the thermal undulation of a membrane by the neighboring membranes generates an entropic repulsion force (fluctuation pressure), pfl ) R(kT)2/KCd3, where k is the Boltzmann constant, T the absolute temperature, KC the bending modulus of the elastic membrane, and d the average distance between membranes. Helfrich also provided the first estimates of R, ranging from R ) 0.841 to R ) 0.0242;2 more accurate values were obtained later, either assuming that the distribution of the membrane positions satisfies a diffusion equation (R ) 0.0854)3 or via a diagrammatic expansion (R ) 0.0797),4 in good * Corresponding author: e-mail address, feaeliru@ acsu.buffalo.edu; phone, (716) 645 2911/2214; fax, (716) 645 3822. (1) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (2) Helfrich, W.; Servuss, R.-M. Nuovo Cimento 1984, 3D, 137. (3) Podgornik, R.; Parsegian, V. A. Langmuir 1992, 8, 557. (4) Bachmann, M.; Kleinert, H.; Pelster, A. Phys. Lett. A 1999, 261, 127.

agreement with Monte Carlo simulations (R ) 0.079).5 The role of thermal fluctuations for membranes interacting via arbitrary potentials, which constitutes a problem of general interest, is however still unsolved. Earlier treatments 6,7 coupled the fluctuations and the interaction potential and revealed that the fluctuation pressure has a different functional dependence on the intermembrane separation than that predicted by Helfrich for rigid-wall interactions. The calculations were refined later by using variational methods.3,8 The first of them employed a symmetric functional form for the distribution of the membrane positions as the solution of a diffusion equation in an infinite well.3 However, recent Monte Carlo simulations of stacks of lipid bilayers interacting via realistic potentials indicated that the distribution of the intermembrane distances is asymmetric;9 the root-meansquare fluctuations obtained from experiment were also shown to be in disagreement with this theory.10 (5) Janke W.; Kleinert, H. Phys. Lett. A 1986, 117, 353. (6) Sornette D.; Ostrowsky, N. J. Chem. Phys. 1986, 84, 4062. (7) Evans E. A.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7132. (8) Manciu M.; Ruckenstein, E. Langmuir 2001, 17, 2455. (9) Gouliaev N.; Nagle, J. F. Phys. Rev. Lett. 1998, 81, 2610.

10.1021/la0201568 CCC: $22.00 © 2002 American Chemical Society Published on Web 05/03/2002

349

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

4180

Langmuir, Vol. 18, No. 11, 2002

Letters

The variational method proposed earlier by the authors relied on Monte Carlo simulations to select an intermembrane distance distribution function.8 The purpose of this paper is to present a new approach, in which the interaction between two membranes, in the presence of thermal fluctuations, is calculated directly by employing a suitable approximate partition function. Thus, the asymmetry of the distance distributions results in a natural manner from the calculation. First, let us summarize the results obtained for a fluctuating membrane placed in a harmonic potential.1,11 Assuming that the total area of the membrane is constant, the Hamiltonian is given by

H )

∫ dx dy

( (

∂2u(x,y) ∂2u(x,y) 1 KC + 2 ∂x2 ∂y2

)

2

+

)

(ballistic) pressure exerted by an 1D gas of planar pieces in a rigid-wall confinement, is of the form pfl ) R(kT)2/ KCd3, for a suitable choice of the area of the pieces (different from eq 6). However, Monte Carlo simulations for membranes interacting via realistic potentials12 indicated that in these cases the ballistic term, due to the collisions between a membrane and the rigid walls, provides only a negligible contribution. In what follows, the ballistic contribution to the fluctuation pressure will be neglected. When two fluctuating bilayers interact, the intermembrane distance distribution is still given by eq 5, since the interaction depends only on the distance z between the bilayers. For an arbitrary interaction (per unit area) U(z) and a constant applied pressure p, the Boltzmann distribution of distances between the small independent surfaces of area S0 can be calculated using the enthalpy (per unit area) H(z) ) U(z) + pz instead of the energy

1 Bu2(x,y) (1) 2 where KC is the curvature elastic modulus of a bilayer, B is the spring constant (per unit area of the bilayer) of the harmonic potential, and u(x,y) denotes the displacement along the normal to the x-y plane from the average position u0 of a point of the membrane of coordinates x and y. Denoting by u˜ (qx,qy) the Fourier transform of u(x,y), the average energy of each mode for a membrane of unit area was obtained from the equipartition principle as1,11

〈u˜ 2〉

(12 K q

4

C

+

1 1 B ) kT 2 2

)

(2)

F(z) )

(

S0(U(z) + pz) 1 exp N kT

B0 )

〈u˜2〉 ) ∑ q ,q x

y

kT (2π)

∫ 2 0

∞

2πq dq KCq4 + B

kT

)

8(KCB)1/2

(

(u - u0) 1 exp 1/2 (2π) σ 2σ2

)

( ) ( SB 2πkT

1/2

exp -

(4)

SB(u - u0)2 2kT

)

(5)

if the area S of the independent, nonundulating small membranes is given by

S)

( )

KC kT )8 2 B Bσ

dz2

|

(8a)

z)zm

dH(z)/dz ) 0

(8b)

C ) H(zm)

(8c)

In this case

2

This distribution is formally identical with the normal (Boltzmann) distribution of small independent, planar membranes of area S in the potential (1/2)B(u - u0)2

F(u) )

d2H(z)

where zm ) z0′ is the solution of

(3)

and the distribution of the positions of the membrane is Gaussian

F(u) )

(7)

where the constant N can be obtained by normalization. The last task is to calculate the area of the independent pieces S0, which is provided by eq 6 only for a harmonic interaction. To accomplish this, we will seek a harmonic potential H0(z) ) (1/2)B0(z - z0′)2 + C (with B0, z0′, and C independent of z), which best approximates H(z) and use S0 ) 8(KC/B0)1/2. A simple procedure consists of using the parabolic approximation of the enthalpy in the vicinity of its minimum at zm

Hence the mean square fluctuation is given by

σ2 ) 〈(u - u0)2〉 )

)

1/2

(6)

where eq 3 was used to derive the last equality. The treatment of fluctuating membranes as small planar pieces moving independently was employed by Helfrich and Servuss,2 who showed that the kinetic (10) Petrache, H. I.; Gouliaev, N.; Tristram-Nagle, S.; Zhang, R.; Suter, R. M.; Nagle, J. F. Phys. Rev. E 1998, 57, 7014. (11) Sornette D.; Ostrowsky, N. In Micelles, Membranes, Microemulsions and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D. Eds.; Springer-Verlag: Berlin, 1994.

A more rigorous procedure to calculate the harmonic approximate of the enthalpy is to seek the minimum of the expression

W(B0,z0′,C) )

1 N

(

∫ exp -

)

S0H0(z) (H(z) - H0(z))2 dz kT (9)

with respect to B0, z0′, and C. It will be shown later that for not too small pressures almost identical results are obtained by both procedures for usual interaction potentials. For illustration purposes, we will apply the method described above to a potential typical for the interaction between two neutral, planar, lipid bilayers in water

( )

U(z) ) A exp -

(

AH z 1 1 + λ 12π (z + th)2 (z + th + 2tc)2 2 (10) (z + th + tc)2

)

where z represents the thickness of the water layer, λ is the decay length of the hydration force, A is a hydration (12) Gouliaev N.; Nagle, J. F. Phys. Rev. E 1998, 58, 881.

Nanodispersions

350

Letters

Langmuir, Vol. 18, No. 11, 2002 4181

parameter, AH is the Hamaker constant, and the thicknesses th and tc are defined below. An accurate theory for the van der Waals interaction between bilayers should account for the variation of the dielectric constant inside the bilayer. Here, instead of considering (as usual) the whole bilayer as a hydrocarbon, we assume that the dielectric properties of a part of the dipolar headgroups are closer to those of water and the dielectric properties of the remaining part closer to those of the hydrocarbon chains. As in our previous paper,8 the region between the phosphate groups, of thickness tc (denoted DHH in ref 13), is treated as a hydrocarbon and the rest of the bilayer, of thickness th ) t - tc, as water (where t is the thickness of the bilayer). Using eqs 6 and 8a, one obtains

[ { [

(

)

p AH 1 1 1 + + λ 2π (z + t )3 3λ zm + th m h 1 1 1 2 × 3 3λ z + t + 2t (zm + th + 2tc) (zm + th + tc)3 m h c

S0 ) 8 KC/

(

(

)

1 1 3λ zm + th + tc

)]}]

1/2

(11)

where zm is the solution of eq 8b and the distribution of the intermembrane distances is given by eq 7. Once the distribution is known, the average position 〈z〉 and the root-mean-square fluctuation σ can be calculated from

∫0

∫0

〈z〉 )

σ)

(

zF(z) dz )

z exp -

∫

(

)

S0(U(z) + pz) dz kT S0(U(z) + pz) ∞ exp dz 0 kT

∫0 (z - 〈z〉)2 exp ∞

( (

∫

(

) )

S0(U(z) + pz) dz kT S0(U(z) + pz) ∞ exp dz 0 kT (12a)

∞

∞

)

)

1/2

(12b)

The asymmetry of the distribution can be defined as the ratio r

r)

(

( (

) )

S0(U(z) + pz) (z - zm)2 exp dz zm kT S0(U(z) + pz) zm (z - zm)2 exp dz 0 kT

∫

∞

∫

)

1/2

(12c)

which is unity in the symmetrical case. Figure 1 presents the enthalpy and the distributions F1 and F2, calculated with the two procedures described (eqs 8 and 9, respectively), for typical values of the interaction parameters, namely, A ) 0.05 J/m2, λ ) 1.5 Å, AH ) 1.0 × 10-20 J, th ) 5.0 Å, tc ) 40 Å, KC ) 1.0 × 10-19 J, T ) 300 K, and (a) p ) 1 × 104 N/m2 and (b) p ) 1 × 106 N/m2. For large pressures, both procedures lead to almost identical results, because most of the intermembrane distances are distributed in such cases in the vicinity of the minimum of the enthalpy. The asymmetry of the distribution function follows the asymmetry of the enthalpy, and, most importantly, is dependent on the pressure, being larger at low pressures. In Figure 2, the mean square fluctuation σ (Figure 2a), the asymmetry ratio r (Figure 2b), and the pressure p (Figure 2c) are plotted versus the average distance 〈z〉 , for various values of KC, using the first method. Figure 2d

Figure 1. The enthalpy H (thick line) and the distributions of the intermembrane separation F1 (continuous thin line) and F2 (dotted line) (calculated from eqs 8 and 9, respectively) vs the distance z for the interaction potential (eq 10) with A ) 0.05 J/m2, λ ) 1.5 Å, AH ) 1.0 × 10-20 J, th ) 5.0 Å, tc ) 40 Å, KC ) 1.0 × 10-19 J, and T ) 300 K, the applied pressure being: (a) p ) 1.0 × 104 N/m2; (b) p ) 1.0 × 106 N/m2. The subscripts 1 and 2 refer to quantities calculated using eqs 8 and 9, respectively.

represents the fluctuation pressure pfl, defined as

pfl(〈z〉) ) p(〈z〉) - p(〈z〉)|KC)∞

(13)

vs the average separation 〈z〉 . At low separation distances, pfl ∼ exp(-z/2λ), which is consistent with previous predictions.3,7,8 The average separation distance and its root-meansquare fluctuation were recently experimentally determined as functions of the applied pressure, for lipid bilayers/water multilayers,10 and a disagreement between the existing theories and experiment was noted.12 No comparable experimental results are yet available for two lipid bilayers, and it is difficult to extend the present theory to multilayers. For this reason, we will employ the present procedure to estimate the interaction parameters of eq 10 from the experimental results for multilayers.10 Figure 3 represents the root-mean-square fluctuations (Figure 3a) and the pressure (Figure 3b) as functions of the average separation distance, for (i) EPC (egg phosphatidylcholine, circles) and (ii) DMPC (1,2-dimyristoylsn-glycero-3-phosphatidylcholine, squares)/water multilayers. The values A ) 0.044 J/m2, λ ) 1.54 Å, AH ) 8.13 × 10-21 J, and KC ) 2.4 × 10-19 J (EPC, continuous line) (13) Petrache, H. I.; Tristram-Nagle S.; Nagle, J. F. Chem. Phys. Lipids 1998, 95, 83.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

4182

Langmuir, Vol. 18, No. 11, 2002

351

Letters

Figure 2. (a) The root-mean-square fluctuation σ, (b) the asymmetry ratio r, (c) the applied pressure p, and (d) the fluctuation pressure pfl vs average thickness 〈z〉 for (1) KC ) 0.3 × 10-19 J, (2) KC ) 1.0 × 10-19 J, and (3) KC ) 3.0 × 10-19 J (A ) 0.05 J/m2, λ ) 1.5 Å, AH ) 1.0 × 10-20 J, th ) 5.0 Å, tc ) 40 Å, and T ) 300 K).

Figure 3. Fit of the experimental data of ref 10 for (a) the root-mean-square fluctuation σ and (b) the applied pressure p as functions of the average separation distance 〈z〉, for EPC/ water (circles/continuous line) and DMPC/water (squares/dotted line) multilayers.

and A ) 0.043 J/m2, λ ) 1.62 Å, AH ) 1.04 × 10-20 J, and KC ) 3.4 × 10-19 J (DMPC, dotted line) were obtained from a simultaneous least-squares fit of the experimental data of Petrache et al.10 for both the pressure and the root-mean-square fluctuation. For the other parameters, we employed the values provided by Petrache et al.:10 th ) 7.6 Å, tc ) 45.4 Å for EPC; th ) 7.6 Å, tc ) 44.0 Å for DMPC. The agreement is good except for the root-meansquare fluctuation at low separation distances. There are several reasons for this deterioration of the fit. First, the experiments were carried out for stacks of lipid bilayers, where the correlation between the fluctuations of neighboring membranes is expected to play a role, especially at low separation distances. Second, we assumed that both

KC and tc were independent of the applied pressure, which is not accurate at large pressures.13 Third, while X-ray diffraction allowed an extremely precise determination of the total periodicity distance (one bilayer plus one water layer), the average thicknesses of the water and hydrocarbon layers were determined with an error of the order of 1 Å,10,13 which is clearly relevant at large applied pressures, for which the separation distance is small. It should be noted that the values obtained for the parameters are in the ranges characteristic for phospholipid bilayers. Since the model was developed for two bilayers in water and was compared to experimental results for lipid bilayer/ water multilayers, no complete accuracy is expected for the values obtained by fitting. Particularly, the values obtained for the bending moduli are somewhat larger than those provided by literature. This might be a consequence of the increased rigidity of a multilayer, when compared to that of two bilayers, due to the correlation between the fluctuations of adjacent bilayers. To conclude, we presented a new method to account for the effect of the thermal fluctuations on the interactions between elastic membranes, based on a predicted intermembrane separation distribution. It was shown that for a typical potential, the distribution function is asymmetric, with an asymmetry dependent on the applied pressure and on the interaction potential between membranes. Equations for the pressure, root-mean-square fluctuation, and asymmetry as functions of the average distance (and the parameters of the interacting membranes) were derived. While no experimental data are available for two interacting lipid bilayers, a comparison with experimental data for multilayers of lipid bilayer/water was provided. The values of the parameters, determined from the fit of experimental data, were found within the ranges determined from other experiments. LA0201568

Nanodispersions

352

Langmuir 2004, 20, 1775-1780

1775

Specific Ion Effects in Common Black Films: The Role of the Thermal Undulation of Surfaces Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received September 5, 2003. In Final Form: November 26, 2003 It has long been demonstrated that the traditional double layer theory does not provide an accurate description for the repulsion between colloidal particles at small separations and high ionic strengths. Recent experiments on black films revealed that the traditional theory seems to be inadequate even for large separation distances and moderate ionic strengths. These results raise doubts about the validity of the DLVO approximations. It is argued here that the results can be, however, understood within the framework of the traditional theory if one accounts for the thermal undulation of the film interfaces, and a new treatment to account for the thermal undulations is suggested.

1. Introduction When two charged particles immersed in an electrolyte approach each other, the overlap of their ionic atmospheres (the double layers) generates a repulsive force. The traditional Derjaguin-Landau-Verwey-Overbeek (DLVO) theory assumes that the stability of charged colloids is a consequence of a balance between this double layer repulsion and the attractive van der Waals interactions.1 The first theoretical description of the double layers assumed that the ions interact via a mean potential, which obeys the Poisson equation.2 Such a simple theory is clearly only approximate and sometimes predicts ionic concentrations in the vicinity of the surface that exceed the available volume.3 There were a number of attempts to improve the model, by accounting for the variation of the dielectric constant in the vicinity of the surface,4 for the volume-exclusion effects of the ions,5 or for additional interactions between ions and surfaces, due to the screened image force potential,6 to the van der Waals interactions of the ions7 with the system, or to the change in hydration energy when an ion approaches the interface.8 It is clear that the DLVO theory is incomplete, a simple example being the stability of neutral lipid bilayers,9 or of the water films involving nonionic surfactants,10 where there is no double layer to provide the required repulsion. Another example is provided by the specific ion effects, namely, the different behaviors of systems immersed in different electrolytes of the same valence. Various electrolytes have been classified long ago by Hofmeister in an * Corresponding author. E-mail address: [email protected]. edu. Phone: (716) 645-2911 ext. 2214. Fax: (716) 645-3822. (1) Deryagin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633. Verwey, E. J.; Overbeek, J. Th. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (2) Gouy, G. J. Phys. Radium 1910, 9, 457. Chapman, D. L. Phyl. Mag. 1913, 25, 475. (3) Stern, O. Z. Eletrochem. 1924, 30, 508. (4) Spaarnay, M. J. Recl. Trav. Chim. Pays-Bas 1958, 77, 872. (5) Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612. (6) Jonsson, B.; Wennerstrom, H. J. Chem. Soc., Faraday Trans. 2 1983, 79, 19. (7) Ninham, B. W.; Yaminsky, V. Langmuir 1997, 13, 2097. (8) Manciu, M.; Ruckenstein, E. Adv. Colloid Interface Sci. 2003, 105, 63. (9) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (10) Strey, R.; Schomacker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253.

empirical series, based on their efficiency in protein precipitation.11 More recently, it was shown that, in a certain pH range, amphoteric latex particles coagulated when the concentration of CsNO3 exceeded about 0.3 M but remained stable even at 3.0 M of KNO3.12 Since the DLVO theory of double layer forces accounts only for the ion valency and concentration, it cannot explain why different results were obtained when Cs was replaced by K. Experiments regarding the force between mica surfaces revealed that the traditional DLVO theory provides good agreement at large separations but fails at low separations and high electrolyte concentrations.13 For some electrolytes, even at relatively low ionic strengths (10-3 M) there is an additional strong repulsion at low separations between surfaces. All of the above experiments can be understood qualitatively if one assumes an additional short-range repulsion between surfaces. It was suggested that such a repulsion, the hydration force, is provided by the organization of water in the vicinity of the surface (the “hydration” of the surfaces).14 However, there is no agreement regarding the microscopic origin of the hydration repulsion. Some models assume that the hydrogen bonding of water near a surface is responsible for this repulsion and that the ions do not play any role.15 Another model considers that, when two surfaces approach each other, the electrolyte ions are losing their hydrating molecules and this unfavorable free energy process generates a repulsion, hence that the hydrated electrolyte ions are vital to the hydration force.12,16 The local interaction between dipoles (surfacewater and water-water dipoles) was also proposed to explain the hydration repulsion.17 Since the dipole interactions are electrostatic, this hydration and the double layer should be coupled into a single force.18 In this case, (11) Hofmeister, F. Naunin-Schmiedebergs Arch. Exp. Pathol. Pharmakol. 1888, 24, 247. (12) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156. (13) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (14) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Suface Forces; Plenum: New York, 1987. (15) Attard, P.; Batchelor M. T. Chem. Phys. Lett. 1988, 149, 206. (16) Paunov, V. N.; Kaler, E. W.; Sandler, S. I.; Petsev, D. N. J. Colloid Interface Sci. 2001, 240, 640. (17) Schiby, D.; Ruckenstein E. Chem. Phys. Lett. 1983, 95, 435. Manciu M.; Ruckenstein, E. Langmuir 2001, 17, 7061. (18) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584.

10.1021/la0356598 CCC: $27.50 © 2004 American Chemical Society Published on Web 01/20/2004

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

1776 Langmuir, Vol. 20, No. 5, 2004

Manciu and Ruckenstein

while a repulsion is present even in the absence of electrolyte ions, the force can depend on the electrolyte concentration via the Poisson equation,18 and particularly because of the charge recombination on the surface, which, while decreasing the surface charge, generates new surface dipoles.17 A common feature of the experiments discussed above is that the hydration repulsion is short ranged. In contrast, recent experiments with AOT-based black films (thin water films with surfactant monolayers adsorbed at the water/air interfaces) showed that the traditional double layer theory is not accurate even at large distances, of the order of 100 Å.19 Indeed, different cations (Cs and Li) led to different dependencies of the thickness of the film on the applied pressure, for electrolyte concentrations exceeding 0.05 M. For concentrations between 0.05 and 0.2 M CsCl, the results could be explained in terms of the DLVO interactions, when a surface potential of about 0.1 V was employed in the calculations. However, this implies that the surface charge density doubles when the electrolyte concentration increases from 0.05 to 0.2, whereas a decrease of the surface charge density because of charge recombination is expected to occur. When Li replaced Cs, the equilibrium thickness of the film increased by as much as 10 Å, and the slope of pressure vs distance plot decreased, suggesting an increase in the Debye decay length. Sentenac and Benattar19 pointed out that the difference is too large to be explained by different Stern layers.3 The difference is also too large to be accounted for by additional interactions between ions and surfaces, such as ion-dispersion7 or ion-hydration interactions.8 It was suggested19 that the hydration of ions might be responsible for these long-range effects, the Li ion, which is much strongly hydrated than Cs, losing some of its ability to screen the electrostatic repulsion. However, this effect should have been observed in the double layer repulsion between rigid surfaces, such as those reported by Pashley,13 and not only in the black films. In the present paper, we will argue that the experimental results of Sentenac and Benattar,19 which apparently contradict sharply the DLVO theory, can be explained in terms of the traditional theory, when the thermal undulations of the film interfaces are also taken into account. The interface will be treated as a membrane subjected to thermal undulations, which can take place at either constant or nonconstant surface area. In the first case the fluctuations generate bending, while in the second they can produce both bending and change of area. The bending free energy of a normal mode of wavelength λ, u(x,y) ) R sin(2π(x/λ)), where u(x,y) denotes the displacement of a point of coordinates (x,y) of the membrane, along the normal to the (x,y) plane and R the amplitude, is given by20

UBending )

∫0

λ

dx

∫ dy

353

( (

))

∂2u(x,y) ∂2u(x,y) 2 1 KC + ) 2 ∂x2 ∂2 KCR2 dy (1a) 4λ4

∫

where KC is the bending modulus, while the free energy due to surface tension is20 (19) Sentenac, D.; Benattar, J. J. Phys. Rev. Lett. 1998, 81, 160. (20) Sornette D.; Ostrowsky, N. In Micelles, Membranes, Microemulsions and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer-Verlag: Berlin, 1994.

UST )

∫0

λ

dx

∫ dy (12 γ[(

∂u(x,y) ∂x

) ( 2

+

∂u(x,y) ∂y

) ]) 2

) 2

∫ dy γR 4λ2

(1b)

where γ is the surface tension. While the exact value for the bending modulus of an AOT layer between air and water is not known, experiment provided a value of the order of 1 kT for AOT monolayers between water and oil.21 These values are not very accurate; indeed for water/decane, the neutron spin-echo experiments provided 5.0 kT, while the small-angle neutron scattering led to 0.5 kT.21 Assuming that KC ) 1 kT and γ ) 0.025 N/m,19 the bending energy (1a) and the surface tension energy (1b) become equal for a mode with λ ) (KC/γ)1/2 ∼ 4 Å. At smaller wavelengths, the free energy due to the change in area is smaller than that due to bending. However, for much larger wavelengths, the bending free energy becomes much smaller. Consequently, for λ . 4 Å, the bending at constant area becomes favorable thermodynamically. Since the wavelengths of the thermal undulation of AOT interface are in general much larger than 4 Å (the distance between two neighboring AOT molecules on the surface being about 7-8 Å),22 in what follows it will be assumed that the interfaces bends at constant area. It was shown that when the interfaces interact through a harmonic potential, the entropic contribution to the free energy due to the finite confinement is inversely proportional to the mean square fluctuation, σ2, of the distribution of distances between interfaces.23 In a seminal paper, Helfrich conjectured that for a hard-wall interaction between membranes, the entropic confinement contribution has the same functional form as for a harmonic potential and that σ is proportional to the average intersurface distance 〈d〉.23 This conjecture leads to a supplementary pressure, due to the confinement of the thermal undulations, Πu ∝ 〈d〉-3. This law is valid at large separations, for elastic membranes interacting via shortrange potentials, such as the neutral lipid bilayers;24 however, it is not accurate at small separations, where experiment25 and Monte Carlo simulations26 showed that σ depends exponentially on 〈d〉. The approach proposed here combines a Boltzmannlike distribution of the film thicknesses27 with the minimization of the total Gibbs free energy.28 One of the undulating interfaces is assumed flat and rigid (denoted 1 in Figure 1), while the other (2 in Figure 1) is considered to be composed of many flat pieces of finite area A, which behave independently of each other and are Boltzmannian distributed in the potential created by the rigid interface. As shown below, the intersurface distance distribution based on this model coincides, for a harmonic interaction, with the exact result obtained from the partition function of two undulating membranes. The model is extended to cases involving nonharmonic interactions for which analytic solutions are not available. (21) Kellay, H.; Binks, B. P.; Hendrikx, Y.; Lee, L. T.; Meunier, J. Adv. Colloid Interface Sci. 1994, 49, 85. (22) Bulavchenko, A. I.; Batischev, A. F.; Batischeva, E. K.; Torgov, V. G. J. Phys. Chem. B 2002, 106, 6381. (23) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (24) Larche, F. C.; Appell, J.; Porte, G.; Bassereau, P.; Marignan, J. Phys. Rev. Lett. 1986, 56, 1700. (25) Petrache, H. I.; Gouliaev, N.; Tristram-Nagle, S.; Zhang, R.; Suter, R. M.; Nagle, J. F. Phys. Rev. E 1998, 57, 7014. (26) Gouliaev N.; Nagle, J. F. Phys. Rev. Lett. 1998, 81, 2610. (27) Manciu M.; Ruckenstein, E. Langmuir 2002, 18, 4179. (28) Manciu M.; Ruckenstein, E. Langmuir 2001, 17, 2455.

Nanodispersions

354

Specific Ion Effects in Common Black Films

Langmuir, Vol. 20, No. 5, 2004

1777

Figure 1)

F(u) )

( ) AB 2πkT

(

1/2

exp -

)

ABu2 2kT

(6)

if the area A is selected as27

A) Figure 1. The distribution of distances between an undulating membrane and a rigid surface interacting via a harmonic potential is the same as the Boltzmann distribution of distances between a rigid surface and independent planar pieces of a suitable chosen area A, interacting via the same potential.

At constant external pressure the enthalpy is the relevant thermodynamic potential for the Boltzmann distribution. A large area A implies that the distribution of the thicknesses is confined near the minimum of the enthalpy, while a small value of A corresponds to large thickness fluctuations. The first case corresponds to a low enthalpy, but a large entropy, whereas the second to a large enthalpy, but a low entropy. The real distribution is provided by the minimization of the Gibbs free energy with respect to A at constant external pressure Π. 2. Elastic Interfaces Interacting via a Nonlinear Potential Let us first summarize the results obtained for a thermal undulating membrane interacting with a rigid, flat surface via a harmonic potential, Uh(u) ) (1/2)Bu2. The total energy, per unit area, is given by20

Uh )

∫ dx dy

( (

)

∂2u(x,y) ∂2u(x,y) 1 KC + 2 ∂x2 ∂y2

2

+

〈u˜ 2〉

(12 K q C

4

+

Z)

∫ ∏ du˜

)

σ2 ) 〈u2〉 )

∑ 〈u˜ 2〉 )

qx,qy

(2π)2

∫0

2πq dq

(3)

)

KCq4 + B

kT 8(KCB)1/2 (4)

and the distribution of distances is Gaussian

F(u) )

( )

1 u2 exp - 2 1/2 (2π) σ 2σ

( (

exp -β

))

1

∑ u˜ 2(KCq4 + B) q ,q 2 x

y

x

y

(

)

2πkT Kcq4 + B

)

1/2

(8)

The difference in the free energies of a free membrane (which formally corresponds to B f 0) and that interacting with the rigid wall via the harmonic potential (1/2)Bu2 is given by

∆F ) kT

1 (2π)2

∫0

∞

(

dq 2πq ln

)

KCq4 + B KCq4

1/2

)

()

kT B 8 KC

1/2

(9)

The entropic term of the free energy (per unit area) due to the confinement is obtained by subtracting the interaction energy per unit area from eq 9, a result which is essentially due to Helfrich23

∆Fent )

( )

kT B 8 KC

1/2

-

(kT)2 1 B〈u2〉 ) 2 128K σ2

(10)

C

Let us now assume that the functional form for the entropic contribution, eq 10, holds even when the membrane is subjected to an arbitrary potential, U(u), with a minimum at u ) 0. Therefore, the Gibbs free energy of the membrane per unit area at a constant external pressure Π is given by

G(A) )

(kT)2

∫u F(u)(U(u) + Πu) du + 128K σ2

(11a)

C

1 1 B ) kT 2 2

∞

(7)

∏ q ,q

)

Therefore for the mean square fluctuations one obtains

kT

1/2

An alternate procedure to calculate the distribution of an elastic membrane in a harmonic potential has as starting point the direct integration of the partition function of the canonical ensemble in Fourier space21

1 Bu2(x,y) (2) 2 where u(x,y) denotes the displacement of a point of coordinates (x,y) of the membrane, along the normal to the (x,y) plane, from the position of the potential minimum, u0 ≡ 0, KC is the bending modulus of the membrane and B is the constant of the harmonic potential of interaction between the undulating membrane and the rigid surface. Denoting by u˜ (qx,qy) the Fourier transform of u(x,y), the average energy of each undulation mode obtained from the equipartition principle is given by20,23

( )

KC kT )8 2 B Bσ

(5)

This distribution of thicknesses is formally identical with the Boltzmannian distribution of distances between the fixed, rigid membrane (1 in Figure 1) and a collection of planar pieces of membranes of area A, independent of each other, but free to move in the potential (1/2)Bu2 (see

where

F(u) )

(

A(U + Πu) 1 exp N kT

)

(11b)

with the normalization factor N given by

N)

∫u F(u) du

(11c)

The average distance between the membrane and the rigid surface is obtained from

〈d〉 ) d0 +

1 N

∫u uF(u) du

(11d)

where d0 is the distance between the position of the potential minimum (u ) 0) and the rigid surface. The mean square fluctuation of the membrane is given by

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

1778 Langmuir, Vol. 20, No. 5, 2004

σ2 )

1 N

Manciu and Ruckenstein

∫u (u - (〈d〉 - d0))2F(u) du

(11e)

The above integrations are performed over the entire range allowed for u. By employing eqs 11b-e, the Gibbs free energy eq 11a becomes a function of A alone for any value of the external pressure Π, and consequently the value of A can be obtained from the minimization of G. For the harmonic potential Uh(u) ) (1/2)Bu2, eq 11a can be integrated

G(A) )

∫-∞

+∞

F(u)

(12 Bu

2

)

+ Πu du +

(kT) 128KC

∫-∞

+∞

2

) 2

u F(u) du

355

kT kTBA + (12) 2A 128KC

where the Boltzmannian distribution of the independent pieces of area A (eq 5) was taken into account. Minimization of G with respect to A recovers the relation between the area A and the elastic parameters of the membrane, eq 7, obtained by employing the equipartition principle (eq 4). The main assumption in this approach is that the functional form of the entropic term is given by eq 10. This constitutes a very good approximation when most of the membrane is confined near the minimum of the enthalpy (the equilibrium position). This assumption is probably acceptable even when parts of the membrane wander far from the equilibrium position, since the Helfrich law (based on eq 10) seems to be obeyed at large separations.10,24 3. Long-Range Interactions in Black Films Let us now return to the long-range interactions between AOT-based black films at moderately high electrolyte concentrations (0.05-0.2 M CsCl and LiCl).19 It has been long known that the DLVO theory is not accurate for uniunivalent electrolytes above about 0.05 M.14 It is also clear that the differences between the hydration behavior of the Li and Cs ions should lead to different interactions of the ions with the interfaces and to different rates for their recombination with the surface charges. Furthermore, the different polarizabilities of the ions should lead to different van der Waals interactions between ions and the rest of the system. In addition, the excluded volume effects are more important for the much strongly hydrated Li+ than for Cs+. There is no doubt that all of the above corrections to the DLVO theory, as well as many others, should be included in a complete theory. However, the purpose of this article is not to provide a set of parameters which lead to an excellent agreement with experiment but to show that the traditional DLVO theory can still provide a reasonable accurate description of the experiments, when the thermal undulations of the interfaces are taken into account. Therefore, the specific ion effects will be taken into account in what follows only via the ability of different ions to modify the bending modulus of the interface. There are no reliable data for the bending modulus of AOT interfaces between air and water, and it is not known how the bending modulus depends on the electrolyte concentration. However, there is experimental evidence that the addition of salt (NaCl) can decrease drastically the bending modulus of the AOT film, possibly because of the decrease of the surface charge density at high ionic strength.29 (29) Alexandridis, P.; Holzwarth J. F.; Hatton, T. A. Langmuir 1993, 9, 2045.

The concentration of AOT in all experiments was 2.5 × 10-3 M, which roughly corresponds to its critical micelle concentration (cmc) in the absence of an electrolyte. When an electrolyte is added, the cmc concentration is lowered. For the ionic strengths employed in the experiments of Sentenac and Bennatar,19 the surface tension reached saturation well before 2.5 × 10-3 M.19 It will be therefore assumed that in all those experiments, the amount of AOT adsorbed did not change and that each surfactant molecule occupies in average an area of 60 Å2, which is in agreement with the values (40-70 Å2) reported in the literature.22 The concentration of Na cation, resulting from the dissociation of AOT, will be neglected, since it is at least a few times lower than the concentration of the electrolyte cation. The reassociation of the AOT adsorbed on the surface with the electrolytes cations is taken into account via the association-dissociation equilibrium, which leads to a surface charge density σS given by30

σS ) -

eΓ CE eψS 1+ exp KD kT

(

)

(13)

where e is the protonic charge, Γ is the surfactant surface density (1 molecule/60 Å2 ≡ 2.77 × 10-6 mol/m2), CE is the bulk electrolyte concentration (assumed completely dissociated), KD is the dissociation constant, and ψs the surface potential. While specific ion effects can be introduced by employing different dissociation constants for AOT-Li and AOT-Cs, here the same value will be used for both, KD ) 0.5 M, implying that the most important contribution to specificity is due to the different bending moduli of the interfaces. Let us first calculate the free energy of interactions between two planar, nonundulating interfaces. The potential obeys the Poisson-Boltzmann equation

∇2ψ )

2eCE eψ sinh 0 kT

( )

(14a)

with the boundary conditions

|

σS dψ )dz z)d/2 0

(14b)

and

|

dψ )0 dz z)0

(14c)

where z is measured from the middle distance between the planar (nonundulating) interfaces separated by a distance d,  is the dielectric constant, and 0 is the vacuum permittivity. After the solution of the Poisson-Boltzmann equation is obtained, the total double layer free energy per unit area is obtained by adding the electrostatic

Fel(d) )

1 2

d/2 0(∇ψ)2 dz ∫-d/2

(15a)

entropic (30) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205.

Nanodispersions

356

Specific Ion Effects in Common Black Films

Langmuir, Vol. 20, No. 5, 2004

1779

Figure 2. The enthalpy H and its harmonic approximation Uh in the vicinity of the minimum, for planar, rigid surfaces separated by a distance d (CE ) 0.1 M, Π ) 1 × 104 N/m2, Γ ) 3.3 × 10-6 mol/m2, KD ) 0.5 M, b1 ) -3.08 × 10-22 J, b2 ) -6.28 × 10-14 J/m, b3 ) 8.28 × 107 J/m, b4 ) 6.13 × 1015 J/m2, b5 ) -9.00 × 10-23 J, and T ) 300 K). F1 is the distribution of the intersurface distances for interfaces with bending modulus KC ) 2 kT interacting via the potential Uh. This distribution coincides with the Boltzmann distribution of finite pieces of area A. F2 is the Boltzmann distribution of the pieces of area A, but now the enthalpy H and not its harmonic approximation Uh is the thermodynamic potential. F3 is calculated using for A the value obtained from the minimization of the Gibbs free energy (eq 11a).

Fent,ions(d) ) 2CEkT

eψ eψ ∫-d/2 [(kT ) sinh(kT )+ d/2

eψ (kT )] dz (15b)

1 - cosh and chemical contributions31

∆Fch(d) ) -2

∫σσ(∞)(d) ψS(σS) dσS S

(15c)

S

where ψS is the surface potential and σS(∞) is the surface charge at infinite separation. For the van der Waals interaction between nonundulating AOT interfaces, whose hydrocarbon chains are about 7 Å long,22 the Pade aproximant, calculated on the basis of the Lifshitz theory by Donners et al., will be used32

UvdW(d) )

(

b1 + b2d 1 + b5 2 d 1 + b3d + b4d2

)

(16)

where d is the thickness of the water layer and the empirical parameters bi are assumed equal to those provided by Donners et al. for dodecane films with thickness of 9 Å32 (in SI units): b1 ) -3.08 × 10-22, b2 ) -6.28 × 10-14, b3 ) 8.28 × 107, b4 ) 6.13 × 1015, and b5 ) -9.00 × 10-23. The enthalpy per unit area, H, between the surface 1 and one of the flat surfaces of area A is given by

H(u) ) [Fel(d) - Fel(∞) + Fent,ions(d) - Fent,ions(∞) + ∆Fch(d) + Uvdw(d)] + Πu (17) where d ) d0 + u, d0 being the distance between the rigid membrane and the position of the local minimum of the enthalpy. The enthalpy defined by eq 17 has an absolute minimum (H ) -∞) for d ) 0, due to the divergence of the (31) Manciu, M.; Ruckenstein, E. Langmuir 2003, 19, 1114. (32) Donners, W. A. B.; Rijnbout, J. B.; Vrij, A. J. Colloids Interface Sci. 1977, 60, 540.

van der Waals attraction. However, the black film (before rupture) is in a metastable state, with the intersurface distances distributed around the local minimum of the enthalpy and all the intersurface distances larger than dmin (see Figure 2). The integration over u in eqs 11 are therefore performed in the range corresponding to dmin < d < ∞ (dmin - d0 < u < ∞). When, because of thermal undulations, one region of one of the interfaces is separated from the corresponding region of the other interface by distances smaller than dmin, the black film ruptures. Details of this process are provided elsewhere.33 In Figure 2 the enthalpy H of the system, per unit area, for a nonundulating interface is plotted as a function of the separation distance d for CE ) 0.1 M and Π ) 1 × 104 N/m2, the other parameters having the values given above. When both interfaces are rigid, their separation distance d0 corresponds to the local minimum of the enthalpy. If the bending modulus is large, the separation between the interfaces is distributed in the vicinity of this minimum, where the enthalpy can be well approximated by the harmonic potential Uh. The distribution of the separation distances F1 was calculated using this harmonic approximation and eq 5, for a bending modulus KC ) 2 kT (T ) 300 K). As already noted, the distribution is identical to a Boltzmann distribution of independent pieces of interfaces of area A, given by eq 7. A further improvement in calculating the distribution of the intersurface distances, proposed previously,27 consisted in using the value of A provided by the harmonic approximant of the enthalpy, Uh, but employing the real enthalpy H for the Boltzmann distribution (F2 in Figure 2). A new better approximation is to consider the value of A as a parameter which minimizes the Gibbs free energy, eq 11a (curve F3 in Figure 2). The differences between F2 and F3 are in general significant only for small values of KC (flexible interfaces) and large separation distances. The bending moduli of the AOT films in the presence of various electrolytes are not known. Experiment showed that19 at low AOT concentrations, the surface tension (33) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 2727.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

1780 Langmuir, Vol. 20, No. 5, 2004

357

Manciu and Ruckenstein

(which are clearly non-negligible) were completely disregarded. It is of interest to note that the calculations at low ionic strengths seems to be better explained by a larger bending modulus, which is expected on intuitive grounds because the higher charged interface should be also more rigid. Consequently, the specific ion effects could be explained in terms of a change in interface flexibility within the traditional DLVO theory. 4. Conclusions

Figure 3. The experimental data for the external pressure Π vs average thickness 〈d〉 are compared to the results of the present theory, which accounts for the undulation of the interfaces combined with the traditional DLVO treatment of interactions. Different bending moduli affect not only the strength of the interactions but also its slope (traditionally related to the Debye screening length).

increased by about 5 × 10-3 N/m when Cs was replaced by Li. This indicates that the more strongly hydrated Li ion somehow diminishes the adsorption of AOT on the interface and also that the (negative) surface adsorption of Li is larger (in absolute value) than the (negative) surface adsorption of Cs. Since the Li ions cannot approach the water/AOT interface (a simple theory for surface tension showed that the distance of closest approach might be a few times larger than the ion’s diameter),8 the water in the vicinity of the surface is not bound to the hydrated ions; hence the interface can more easily undulate in the “free” water. On the other hand, the Cs ions can approach closer to the interface binding of some water molecules. The presence of these more rigid structures (hydrated ions) in the vicinity of the interface attenuates the undulations. On this ground, one expects the AOT interfaces to be more flexible in the presence of Li ions. Another possibility is that different recombination constants for Cs and Li lead to different surface charging and hence to different bending moduli. However, in this article the same dissociation constant was employed in both cases, to emphasize the importance of membrane flexibility. In Figure 3, the experimental results of ref 19 are compared with the calculations involving the traditional DLVO theory for the interactions between parallel (nonundulating) interfaces, combined with the present approach to account for the thermal undulations. The only parameter modified was the unknown bending modulus, whose values were changed from 0.5 to 5 kT, which are of the right order of magnitude for the water/oil AOT interfaces.21 A reasonable agreement with experiment was obtained for KC in the range 0.5-5 kT even though the other corrections to the traditional DLVO interactions

It is well-known that the traditional interactions of the DLVO theory are not accurate at high electrolyte interactions, particularly at small separation distances. Recent experiments on AOT-based black films apparently suggested that the traditional theory is also not valid at moderate ionic strengths and large separations. In this paper, it was argued that the experiments can be understood in terms of the traditional theory, when the thermal undulations of the interfaces are taken into account. A new approach to account for the thermal undulations of elastic interfaces interacting via arbitrary potentials was suggested. For interfaces interacting via a harmonic potential, the distribution of distances between interfaces is formally identical to a Boltzmann distribution of many independent pieces of area A, where A is related to the bending modulus and to the constant of the interaction potential. Assuming that the distribution remains Boltzmannian in a nonharmonic potential, and that the entropic term due to undulation confinement remains proportional to σ-2, as for a harmonic potential, the value of A can be obtained via the minimization of the Gibbs free energy of the system. On the basis of this approach, it was shown that the experiments on AOT-based black films, which apparently disagree drastically with the DLVO theory, can still be understood in the traditional framework, if the bending modulus of the AOT-based interfaces depends on the kind of electrolyte used Li or Cs. One of the reasons for this explanation is that such a long-range departure from the classical theory was observed in thin films with flexible charged interfaces34 and not between rigid interfaces.13 It should be emphasized that, without any doubt, there are many differences between the double layers formed in LiCl and CsCl electrolytes, particularly at high ionic strengths, which are completely disregarded here. The dissociation constant of Li-AOT is probably different from that of Cs-AOT. In addition, the dependence of the bending modulus on the electrolyte concentration for the two types of electrolytes is unknown. The purpose of this paper was not to provide a set of parameters, which can describe the experiment, but merely to suggest that the DLVO theory, the traditional workhorse of colloid science, still provides reasonable results, when the undulations of the interfaces are taken into account. LA0356598 (34) Kolarov, T.; Cohen, R.; Exerowa, D. Colloids Surf. 1989, 42, 49.

Nanodispersions

358

5412

Langmuir 2004, 20, 5412-5417

Interaction Force between Two Charged Plates Immersed in a Solution of Charged Particles. Coupling between Double Layer and Depletion Forces Haohao Huang and Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, Amherst, New York 14260 Received February 3, 2004. In Final Form: April 9, 2004 When two parallel plates are immersed in a solution of small charged particles, the center of the particles is excluded from a region of thickness D/2 near the plate, where D is their diameter. The approach which Langmuir developed for the double layer repulsion in the presence of an electrolyte with ions of negligible size is extended to the case in which one of the “ions” is a charged particle of finite, relatively small size. A general expression for the force generated between the two charged plates immersed in an electrolyte solution containing relatively small charged particles is derived. In this expression, only the electrical potential at the middle distance between the plates is required to calculate the force. A Poisson-Boltzmann equation which accounts for the volume exclusion of the charged particles in the vicinity of the surface is solved to obtain the electrical potential at the middle between the two plates. Starting from this expression, some results obtained previously for the depletion force acting between two plates or two spheres are rederived. For charged plates immersed in a solution of an electrolyte and charged small particles, the effects of the particle charge, particle charge sign, particle size, and volume fraction of the particles on the force acting between the two plates are examined.

1. Introduction The presence of nonadsorbing polymer particles in a colloidal dispersion affects the colloidal stability.1 The depletion force has been the subject of many experimental and theoretical studies. A depletion force between two colloidal particles immersed in a solution of small particles, such as polymer particles or micelles, arises when the gap between the two particles becomes smaller than the diameter of the small particles. At small gaps, the small particles are expelled from the gap and an attraction between the two large particles is generated. Asakura and Oosawa were the first to provide an explanation for this attractive force.2,3 They considered the cases of two plates or two relatively large particles immersed in a solution containing rigid, spherical, relatively small macromolecules. When the gap between the two large particles becomes smaller than the diameter of the macromolecules, the depletion of macromolecules in the gap leads to a concentration difference of the macromolecules between the bulk and the gap. As a result, an attraction between the two large particles is generated due to the difference between the osmotic pressures in the gap and in the bulk. The interactions between two parallel plates or spheres in a solution of nonadsorbing polymers assumed to be spherical have been extensively investigated experimentally,4-6 theoretically,7-13 and by simulations.14,15 The * To whom correspondence should be addressed. Phone: (716) 645-2911 ext. 2214. Fax: (716) 645-3822. E-mail: feaeliru@ acsu.buffalo.edu. (1) Napper, D. H. Polymeric stabilization of colloidal dispersions; Academic Press: New York, 1983. (2) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (3) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (4) Verma, R.; Crocker, J. C.; Lubensky, T. C.; Yodh, A. G. Phys. Rev. Lett. 1998, 81, 4004. (5) Liang, W.; Tadros, T. F. J. Colloid Interface Sci. 1993, 158, 152. (6) Ohshima, Y. N.; Sakagami, H.; Okumoto, K.; Tokoyada, A.; Igarashi, T.; Shintaku, K. B.; Toride, S.; Sekino, H.; Kabuto, K.; Nishio, I. Phys. Rev. Lett. 1997, 78, 3963.

second virial coefficients of the polymer particles were introduced into the theory by Sieglaff and Vrij.16-18 Walz and Sharma calculated the force acting between charged plates or particles immersed in an electrolyte solution also containing small but finite size particles using a “force balance” approach.19 The effect of the polydispersity of the small particles was examined by Chu et al.20 and by Walz.21 In this paper, the approach suggested by Langmuir22 for the double layer repulsion is employed to calculate the force between two charged parallel plates immersed in an electrolyte solution containing also relatively small charged particles. Langmuir’s approach has the advantage that only the electrical potential at the middle between the two plates appears in the expression for the force. Consequently, the force can be directly obtained from the electrical potential distribution provided by a PoissonBoltzmann equation which accounts for the excluded volume due to the finite size of the small charged particles. This paper is organized as follows: first, a general (7) Vrji, A. Pure Appl. Chem. 1976, 48, 471. (8) Joanny, J. F.; Leibler, L.; de Gennes, P. G. J. Polym. Sci., Polym. Phys. Ed. 1979, 17, 1073. (9) Attard, P. J. Chem. Phys. 1989, 91, 3083. (10) Henderson, D. J. Colloid Interface Sci. 1988, 121, 486. (11) Gast, A. P.; Hall, C. K.; Russel, W. B. J. Colloid Interface Sci. 1983, 96, 251. (12) Rao, I. V.; Ruckenstein, E. J. Colloid Interface Sci. 1985, 108, 389. (13) Ruckenstein, E.; Rao, I. V. Colloids Surf. 1986, 17, 185. (14) Dickman, R.; Attard, P.; Simonian, V. J. Chem. Phys. 1997, 107, 205. (15) Feigin, R. I.; Napper, D. H. J. Colloid Interface Sci. 1980, 75, 525. (16) Sieglaff, C. L. J. Polym. Sci. 1959, 41, 319. (17) Vrij, A. Pure Appl. Chem. 1976, 48, 471. (18) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409. (19) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168, 485. (20) Chu, X. L.; Nikolov, A. D.; Wasan, D. T. Langmuir 1996, 12, 5004. (21) Walz, J. Y. J. Colloid Interface Sci. 1996, 178, 505. (22) Langmuir, I. J. Chem. Phys. 1938, 6, 893.

10.1021/la049703a CCC: $27.50 © 2004 American Chemical Society Published on Web 05/27/2004

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Coupling between Double Layer and Depletion Forces

Langmuir, Vol. 20, No. 13, 2004 5413

expression for the force acting between two parallel plates immersed in an electrolyte solution containing small charged particles of finite size is derived; second, the obtained expression is used to derive in a consistent manner expressions for the depletion force previously established; finally, the effects of the charge of the small finite size particles, their sign, their size, and the volume fraction as well as the electrolyte concentration on the force between the two charged plates are examined. 2. Theory Langmuir considered that the force acting per unit area between two charged plates is equal to the difference between the osmotic pressures at the middle between the plates and the bulk. On the basis of this idea, Langmuir derived the following expression for the force acting between two charged parallel plates (p) immersed in a symmetric v/v electrolyte solution22,23

[ (

p ) 2nkT cosh

veΨ0 -1 kT

)]

(1)

where n is the number density of the electrolyte molecules in the bulk, e is the protonic charge, Ψ0 is the electrical potential at the midway between the two plates, v is the valency of the ions, k is the Boltzmann constant, and T is the absolute temperature. This equation implies that the ion distributions are Boltzmannian and the solution is dilute. If small finite size charged particles are also present in the electrolyte solution, Langmuir’s approach can be generalized by writing the following expression for the force acting between the plates:

p ) Π0 - Πb )

∑i ni,0kT - ∑i ni,bkT

(2)

where Π0 and Πb are the osmotic pressures at the midway between the two plates and in the bulk, respectively, and ni,0 and ni,b are the number densities of ion i, including the small finite size charged particles, at the midway between the plates and in the bulk, respectively. The second equality in the above equation implies that the solution satisfies the van’t Hoff equation. Let us consider two parallel plates immersed in a solution of a 1:1 electrolyte and of small finite size particles which can dissociate monovalent counterions, thus acquiring a charge of ze per particle. The system contains two ions of the electrolyte which are assumed to be of negligible size, one kind of charged particles which have a finite size, and their counterions which are of negligible size. Assuming Boltzmann distributions for the ions and charged particles, eq 2 becomes

{ [ ( ) ] || [ (

eΨ0 -1 + kT eΨ0 z nm exp sgn(z) -1 + kT zeΨ0 nm exp -1 kT

p ) kT 2n cosh

) ] [ ( ) ]}

359

(3)

(23) Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, The Netherlands, 1948.

for distances L between the plates that are larger than the diameter D of the charged small particles and

{ [ ( ) ] || [ (

p ) kT 2n cosh

eΨ0 -1 + kT z nm exp sgn(z)

) ] }

eΨ0 - 1 - nm kT

(4)

for distances L between the plates that are smaller than the diameter D, because the gap is in this case depleted of particles. In eqs 3 and 4, ze is the charge of the particles, sgn is the sign function, and nm is the number density of the charged particles in the bulk. The main assumption in eq 3 is that the small particles can be treated in the framework of the Poisson-Boltzmann approximation. However, being of finite size and having a relatively large charge, the interactions among particles as well as their van der Waals interactions with the plates should also be involved in the distribution of their concentration. The latter treatment24 will allow identification of the conditions under which the present simple treatment is valid. The electrical potential distribution can be calculated using the Poisson equation

d2Ψ F )2  dx 0

(5)

where F is the charge density in solution,  is the dielectric constant, 0 is the vacuum permittivity, and x is the distance measured from the middle between the plates. Let us consider that the distance L between the plates is larger than D. The center of the charged particle cannot be located in a layer of thickness D/2 near the plate. Consequently, the charge density in region I of thickness D/2 close to the plate is given by

eΨ - zn e exp(sgn(z) ) (eΨ kT ) kT

F ) -2ne sinh

m

-L/2 < x < -(L - D)/2, L > D (6)

and the charge density outside this region, region II, is given by

(eΨ kT ) eΨ zeΨ zn e exp(sgn(z) ) + zn e exp(kT kT )

F ) -2ne sinh m

m

-(L - D)/2 < x < 0, L > D (7)

When the distance between the two plates becomes smaller than the diameter D of the small particles, eq 6 for the charge density can be employed in the whole gap between the plates. To obtain the electrical potential at the midway between the plates, boundary conditions are required. At the plate surface with a charge density of σ,

dΨ σ )for x ) -L/2 dx 0 (24) Huang, H.; Ruckenstein, E. Manuscript in preparation.

(8)

Nanodispersions

360

5414 Langmuir, Vol. 20, No. 13, 2004

Huang and Ruckenstein

At the boundary between the two regions, I and II, the electrical potential and its derivative with respect to x should be continuous.

Ψ|Ix)(D-L)/2 ) Ψ|II x)(D-L)/2

|

|

dΨ I dΨ II ) dx x)(D-L)/2 dx x)(D-L)/2

(9) (10)

At the midway between the plates, the symmetry of the system provides another boundary condition.

dΨ )0 dx

(x ) 0)

(11)

Combined with the boundary conditions (eqs 8-11) and the charge density distributions (eqs 6 and 7), eq 5 can be solved numerically to obtain the electrical potential distribution. Introducing the value obtained for the electrical potential at the middle between the plates in eq 3 or 4, one can obtain the force that is acting between the two plates. The force can also be calculated starting from an expression for the free energy. The free energy (F) of a colloidal system composed of two charged parallel plates immersed in an electrolyte solution containing charged particles is composed of three contributions: electrical (Fele), entropic (Fent), and chemical (Fche).25,26

F ) Fele + Fent + Fche

(12)

The electrical contribution, the electrostatic energy of the system, can be expressed as

1 Fele ) 0 2

∫-L/2(dΨ dx ) L/2

2

dx

(13)

The entropic contribution with respect to that in the bulk is given by

∫-L/2[∑(ni ln xi - ni,b ln xi,b) +

Fent ) kT

L/2

nw ln xw - nw,b ln xw,b] dx (14)

where nw is the number density of the water molecules located in the gap, xi and xw are the mole fractions of ion i and the water molecules located in the gap, respectively, and the subscript b indicates the bulk. The chemical free energy with respect to the bulk can be calculated using the expression26

∆Fche ) -2

∫

σ(L)

Ψs dσ σ(∞)

dF dL

is the depletion force. By the substitution of Ψ0 ) 0 into eqs 3 and 4, the equations acquire the form

p)

{

-nmkT for L < D 0

for L g D

(17)

where the negative sign indicates an attractive force. Consequently, the depletion free energy (E) is given by

∫∞ p dx )

E)-

L

{

nmkT(D - L) for L < D 0

for L g D

(18)

Equation 12 for the free energy also leads to the above expression for the depletion force. Indeed, because there is no charge on the plates, the electrical and chemical contributions become 0 and only the entropic contribution remains. For distances smaller than the particle diameter, no particle is located between the plates and the entropic contribution is provided by the difference of the mixing entropies of the water molecules in the gap and the bulk. Because the number density of the small particles (nm) of diameter D in the bulk is much smaller than that of the water molecules (nw,b), the free energy can be approximated by

F ) 2kT

0 (0 - nw,b ln xw,b) dx ) ∫-L/2

-nw,bkTL ln xw,b = nwkTLxm = nmkTL (19) Therefore, the depletion force for a distance of L < D is

p)-

dF ) -nmkT dL

(20)

For distances larger than the diameter D,

(16)

3. Depletion Force Acting between Uncharged Plates or Spheres If the two plates are uncharged, the electrical potential in the middle is 0, and the force acting between the plates (25) Overbeek, J. Th. G. Colloids Surf. 1990, 51, 61. (26) Manciu, M.; Ruckenstein, E. Langmuir 2003, 19, 1114.

spheres, is 1/2x(a+D)2-l2.

(15)

where σ(L) is the surface charge density for a distance between the plates equal to L and Ψs is the surface potential. The force is provided by the derivative of the free energy with respect to the distance L

p)-

Figure 1. Schematic of two large spheres immersed in a solution of small particles. When the center-to-center distance between the two spheres of diameter a is l, the distance from the connecting line of the centers of the large particles to the center of the particle of diameter D, which contacts the two

F ) 2kT

-(L-D)/2 (0 - nw,b ln xw,b) dx ) ∫-L/2

-nw,bkTD ln xw,b = nwkTDxm = nmkTD (21) and the entropic contribution becomes independent of the distance L. Therefore, the force is 0 for L > D. Using Derjaguin’s approximation, the force between two uncharged spheres (ps) of diameter a separated by a centerto-center distance of l can be calculated from the force between two parallel plates (p) using the expression23

ps )

∫0∞2πhp(h) dh

where h is defined in Figure 1.

(22)

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Coupling between Double Layer and Depletion Forces

361

Langmuir, Vol. 20, No. 13, 2004 5415

Since, for D > l - a, the center of the small particles cannot be located in the region where h < 1/2x(a+D)2-l2 (see Figure 1), eq 17 becomes

p)

{

-nmkT for h < (x(a + D)2 - l2)/2

(23)

for h g (x(a + D)2 - l2)/2

0

For D e l - a, p ) 0. Equations 22 and 23 lead to

ps )

{

1 - nmπkT[(a + D)2 - l2] for a < l < a + D 4 (24) 0 for l g a + D

Equations 17, 18, and 24 have been obtained previously by different methods.2,19 4. Effect of Polydispersity of the Particle Size on the Depletion Force Let us now consider the case of two uncharged plates immersed in a solution of small polydisperse particles. An expression for the force between the plates can be obtained directly from eq 2. Only the particles with sizes smaller than the distance between the plates can be present in the gap. However, the particles with larger sizes cannot be located in the gap. Therefore, the depletion force can be expressed as

p)

∫0 nmkTf(D) dD - ∫0 nmkTf(D) dD ) ∞ -nmkT∫L f(D) dD ∞

L

(25)

where f(D) is the probability density of the particle size distribution. If the size distribution of the particles is Gaussian, f(D) has the form

f(D) )

1 1D-D h exp 2 R Rx2π

[ (

2

)]

(26)

where D h is the mean particle diameter and R is the standard deviation. Substituting eq 26 into eq 25, one can readily obtain for the depletion force between two plates immersed in a solution of normally distributed particles the expression

1 L-D h 1 L-D h p ) - nmkT erfc ) nmkT erf -1 2 R 2 R (27)

(

[ (

)

) ]

where erfc and erf are the complementary and the error functions, respectively. The depletion energy is

E)

{

) ]}

nmkTR L - D h L-D h 1 L-D h 2 erfc exp 2 R R R xπ (28)

(

)

[(

The force changes from -nmkT to 0 as the distance increases. For a standard deviation equal to 0, the force reduces to eq 7.

Figure 2. Effect of the charge of the small particles on the force. The force was calculated using eqs 3 and 4. The parameters used in the calculation are σ ) -0.01 C/m2, electrolyte concentration c ) 0.1 M, volume fraction of the small particles ) 0.01, and D ) 100 Å: (1) z ) 0; (2) z ) -10; (3) z ) -20.

Using eq 27 and Derjaguin’s approximation, one obtains for the force between two spherical particles immersed in a solution of particles of polydisperse sizes the expression

ps )

{

anmkTπR L - D h L-D h erfc 4 R R 1 L-D h 2 exp R xπ

(

)

[(

) ]} (29)

Equations 27 and 29 were first derived by Walz who employed a force balance procedure.21 5. Force between Two Charged Plates Immersed in an Electrolyte Solution Containing Small Charged Particles When the two plates are charged, an electrical potential is generated between them and the ions and the charged small particles are no longer uniformly distributed between the plates. 5.1. Effect of the Charge of the Small Particles on the Force. Let us assume that both the plates and the particles are negatively charged. As the charge of the small particles becomes more negative, the concentration of the small particles between the two plates decreases because they have the same sign of the charge as the plates. Therefore, the contribution of the small particles to the osmotic pressure at the middle distance also decreases. Consequently, the force decreases (becomes less positive) as the charge of the small particles becomes more negative (Figure 2). The interaction between the two plates is repulsive at small distances and becomes attractive at large distances. At small distances, the electrostatic interactions between the two plates dominate and the net force is repulsive. As the distance increases but remains smaller than the diameter of the particles, the electrostatic repulsion between the plates decreases but, because the attractive depletion force remains almost constant, the net force can become attractive. As soon as the distance between the two plates becomes larger than the particle diameter, the electrostatic and depletion contributions become comparable and the net force becomes small. The above considerations are valid when D is sufficiently large for the double layer repulsion at the distance L ) D to become

362

5416 Langmuir, Vol. 20, No. 13, 2004

Figure 3. Effect of the sign of the charge of the small particles on the force. The force was calculated using eqs 3 and 4. The charge of the small particles is positive. The parameters used in the calculation are σ ) -0.01 C/m2, electrolyte concentration c ) 0.1 M, volume fraction of the small particles ) 0.01, and D ) 100 Å: (1) z ) 10; (2) z ) 20.

Figure 4. Effect of the concentration of the small particles on the force. The force was calculated using eqs 3 and 4. The parameters used in the calculation are σ ) -0.01 C/m2, electrolyte concentration c ) 0.1 M, D ) 100 Å, and z ) -10. The volume fractions of the small particles with respect to the entire volume are noted in the figure.

sufficiently small. Cases in which D is small are examined later in the paper. 5.2. Effect of the Sign of the Charge of the Particle on the Force. If the particle possesses a charge with a sign opposite to that of the plates (the latter are assumed negatively charged in all calculations), a larger number of particles will be located between the plates. Therefore, the contribution of the positively charged particles to the osmotic pressure is larger than that of the negatively charged particles. As a result, the net force decreases (becomes less positive) as the charge of the particles becomes more positive (Figure 3). As the distance increases, the force decreases, being repulsive at short distances and attractive at large ones, until the distance L becomes equal to the diameter D. For values of L > D, the force becomes again repulsive and tends to 0 for large distances. 5.3. Effect of the Concentration of the Small Particles on the Force. Figure 4 shows that, for the same sign of charges of the plates and the particles, the force decreases as the volume fraction of the particles in

Nanodispersions

Huang and Ruckenstein

Figure 5. Effect of the size of the small particles on the force. The force was calculated using eqs 3 and 4. The parameters used in the calculation are σ ) -0.01 C/m2, electrolyte concentration c ) 0.1 M, volume fraction of the small particles ) 0.01, and z ) -10: (1) D ) 60 Å; (2) D ) 80 Å; (3) D ) 100 Å.

Figure 6. Effect of the charge of the small particles on the force for an electrolyte concentration of c ) 0.01 M. The force was calculated using eqs 3 and 4. The parameters used in the calculation are σ ) -0.01 C/m2, volume fraction of the small particles ) 0.01, D ) 100 Å: (1) z ) 0; (2) z ) -10; (3) z ) -20.

the solution increases. Obviously, for distances between plates smaller than D, the particle concentration difference between the bulk and the gap is larger for higher volume fractions. Therefore, the net force is smaller (less positive) for higher volume fractions. 5.4. Effect of the Size of the Small Particles on the Force. Three values for the particle diameter are considered in Figure 5, and the force was calculated for negative charges of the plates and the particles. The force is the same for all three sizes for distances between the plates smaller than the smallest particle diameter. However, when the distance between the plates becomes equal to the diameter of the smallest particle, the force acting on the plates first increases and then becomes positive. At this distance, the electrostatic interaction is dominant and the net force repulsive. For the other two sizes, the force increases as soon as the distance between the plates becomes equal to their diameters. These two sizes of particles can enter in the gap between the plates for relatively large distances between the plates at which the electrostatic repulsion is relatively weak. For this reason, the net forces for particles of these two sizes are smaller than that for the particle of the smallest size.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

Coupling between Double Layer and Depletion Forces

The calculation also showed that, as expected, the net force for very small charged particles approached the double layer repulsion between two plates immersed in a solution of two electrolytes. 5.5. Effect of Electrolyte Concentration on the Force. As the electrolyte concentration increases, the double layer interaction decreases. Comparing Figures 2 and 6, one can see that the net force for the 0.01 M electrolyte solution (Figure 6) is much larger than that for the 0.1 M electrolyte solution (Figure 2). For the 0.1 M electrolyte solution, there is a sharp increase from attraction to small repulsion when the distance between the plates becomes equal to the diameter of the small particle. However, for the 0.01 M electrolyte solution, no such increase occurs. Therefore, in the latter case, the depletion force was negligible compared to the strong electrostatic interactions.

363

Langmuir, Vol. 20, No. 13, 2004 5417

6. Conclusions A general expression for the net force acting between two charged plates immersed in an electrolyte solution containing charged particles was obtained. The derivation of equations for the force based on this expression is simpler than those available in the literature. The same results as those in previous papers for the depletion force acting between two plates or two spheres were obtained. An expression for the interaction between two plates or two spheres immersed in a solution of particles of polydisperse size could also be derived. In addition, the present approach could be used to treat the case of the interaction between two charged plates immersed in an electrolyte solution containing charged particles of finite size, a case which has not been previously treated. LA049703A

Nanodispersions

364

Journal of Colloid and Interface Science 290 (2005) 336–342 www.elsevier.com/locate/jcis

Thermodynamically stable dispersions induced by depletion interactions Haohao Huang, Eli Ruckenstein ∗ Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 16 December 2004; accepted 20 April 2005 Available online 28 June 2005

Abstract When small particles are added to a colloidal dispersion of large particles, a depletion interaction between large particles occurs because the small ones are depleted from the gaps between the former particles. In the present paper, a cell model is employed to examine the behavior of a dispersion of large particles immersed in an electrolyte solution containing small particles. In this model, each cell consists of one large particle in its center and an associated atmosphere. Double-layer, van der Waals, and depletion interactions, as well as entropic effects, have been taken into account. When the change of the free energy with respect to that of the electrolyte solution is negative (and this happens in most cases), the dispersions of large particles are stable from a thermodynamic point of view. With increasing volume fraction of the small particles, the free energy change becomes more negative. The formation of gels observed experimentally in concentrated emulsions is explained through the formation of a thermodynamically stable dispersion.  2005 Elsevier Inc. All rights reserved. Keywords: Depletion force; Electrostatic interaction; Van der Waals interactions; Thermodynamic stability

1. Introduction During recent years, the behavior of colloidal particles immersed in an electrolyte solution containing small particles has been extensively studied experimentally [1–3], theoretically [4–8], and by simulations [9–11]. In addition to van der Waals and double-layer interactions, a depletion interaction was present in such systems. By changing the volume fraction of the small particles and the ratio of the diameters of the small and large ones, one can control the strength of the depletion interactions and hence the behavior of the system. Experiment has shown that colloidal dispersions of large particles can form a gel or a glass in the presence of depletion interactions caused by polymer molecules or surfactant micelles present in the dispersion. It was observed that with increasing small particle (polymer) concentration a solidlike gel was generated when the ratio of the radius of gyration of the polymer to the large particle radius became sufficiently small [3]. * Corresponding author. Fax: +1 (716) 645 3822.

E-mail address: [email protected] (E. Ruckenstein). 0021-9797/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.04.059

A first explanation for the origin of the depletion interaction between two large particles present in a solution containing small particles was suggested by Asakura and Oosawa [12]. In their theory, the hard small particles are excluded from the gaps between the large particles, and for this reason, there is a difference between the osmotic pressures in the gap and the bulk solution. For uncharged small particles, this interaction is caused by an entropic change due to the depletion of small particles. However, if the small particles that induce the depletion force are charged, the latter macroions affect the electrical potential distribution in the colloidal system and hence the depletion and the double-layer interactions [13]. The phase behavior of nonaqueous colloidal suspensions containing nonadsorbing polymer was investigated by Gast et al. [3] on the basis of statistical mechanics. In their theory, a second-order perturbation approach was used to calculate the free energy. Rao and Ruckenstein [4,5] examined the phase behavior of systems involving steric, depletion, and van der Waals interactions. If a depletion interaction is present in a colloidal system, it generates a negative free energy change, and, as a result,

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 290 (2005) 336–342

the system can become thermodynamically stable. If the system contains a high volume fraction of dispersed phase, it can become a gel. The goal of the present paper is to investigate the behavior of a dispersion of large particles in an electrolyte solution containing small uncharged or charged particles. A cell model is employed to calculate the free energy of the system. In this model, the charged small particles of radius RS are depleted from layers of thickness RS around the large particles of radius RL due to hard-sphere and electrostatic interactions. The small particles also introduce a charge into the Poisson–Boltzmann equation, which affects the electrical potential distribution. In addition to the above interactions, the van der Waals interactions between the large particles and an entropic contribution are also taken into account.

2. The model The colloidal system is divided into a number of Wigner– Seitz cells. A large particle of radius RL is located at the center of each cell, and a liquid atmosphere is associated with the particle. The atmosphere contains an electrolyte solution and small particles of radius RS (Fig. 1). Each of the small particles has a charge ze, and the large particles have a surface electrical potential Ψ0 , which is assumed to be negative in the calculations. In the present model, the Wigner–Seitz cell is approximated by a sphere. The radius of the sphere RC can be obtained from the number density nL of the large particles, RC3

3 = . 4πnL

(1)

The small particles cannot be located in a layer of thickness RS around the large particles due to hard-sphere interactions. Outside this region, the distribution of the small particle is assumed to be Boltzmannian. To calculate the free energy of the system, one must solve the Poisson–Boltzmann equation to obtain the electrical potential Ψ and the ionic distribution. Because the Wigner–

365 337

Seitz cell is approximated by a sphere, the electrical potential depends only on the radial coordinate r, ∇ 2Ψ = −

ρ , εε0

(2)

where ρ is the charge density in the solution (which includes the ions of the electrolyte, the charged small particles, and the counterions of the large and small particles), ε is the dielectric constant, and ε0 is the vacuum permittivity. The concentrations of ions can be expressed in the form
 e(Ψ − Ψ0 ) c+ = c+0 exp − , (3) kT
 e(Ψ − Ψ0 ) c− = c−0 exp , (4) kT   −ze(Ψ −Ψc )  for RL + RS < r < RC , kT cs = csc exp (5) 0 for RL < r < RL + RS , where k is the Boltzmann constant, T is the absolute temperature in K, c+ and c− are the concentrations of cations and anions, respectively, c+0 and c−0 are the corresponding concentrations at the surface of the large particles, cs is the concentration of small particles, and csc and Ψc are the concentration of small particles and the electrical potential at the periphery of the spherical cell, respectively. The average electrolyte and small particle concentrations in each cell are equal to their overall concentrations in the whole system, c¯+ = 3

RC  c+ r 2 dr RC3 − RL3 = c + ce ,

(6)

RL

RC  c¯− = 3 c− r 2 dr RC3 − RL3 = c,

(7)

RL

and RC c¯s = 3

cs r 2 dr



RC3 − (RL + RS )3 ,

(8)

RL +RS

where c is the electrolyte concentration and ce is the concentration of the counterions of the particles. The overall concentration of cations also includes the counterions dissociated from all particles, which are assumed to be negatively charged. To solve the Poisson–Boltzmann equation, boundary conditions are required. For a constant surface potential Ψ0 on the surface of the large particles, Ψ (r = RL ) = Ψ0 .

(9)

Due to the periodicity of the cells, one can write that at the periphery of the spherical cells, Fig. 1. The cell model. The small particles are depleted from the layer between the radii RL and RL + RS .

dΨ = 0 for r = RC . dr

(10)

Nanodispersions

366 338

H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 290 (2005) 336–342

Equation (2) and the boundary conditions (9) and (10) can be solved numerically to obtain the electrical potential distribution for selected overall concentrations of electrolyte and small particles. The change of the free energy Fcell of each cell with respect to that of the electrolyte solution involves three contributions: electrical Fele , entropic Fent , and chemical Fche [14,15], Fcell = Fele + Fent + Fche .

(11)

The electrical contribution, the electrostatic free energy of the system, can be calculated using the expression

 RC 1 dΨ 2 Fele = εε0 4πr 2 dr. 2 dr

(12)

RL

The probability of finding one particle per unit area of the surface of radius R1 is 12/4πR12 . Therefore, the van der Waals interaction between one particle located at a distance q from the center of the sphere of radius R1 and the particles on the surface element R12 sin α dα dθ is given by

The entropic contribution is given by

 RC

2 Fent = kT 4πr ci ln xi + ns ln xs + cw ln xw i

RL

−



cib ln xib + cwb ln xwb

 dr,

(13)

i

where ci is the concentration of ion i in the cell, cw is the concentration of water in the cell, x is the molar fraction, subscript i refers to ions i, subscript s to the small particles, subscript b indicates the overall average concentration of the ion, and subscript w refers to water. Obviously, the concentration of small particles is zero for RL < r < RL + RS . The calculated free energy change is relative to that of a uniform electrolyte solution. The chemical contribution Fche is 0 at the constant surface charge density of the large particles and −Ψ0 σ at the constant surface potential Ψ0 of the large particles, where σ is the surface charge per large particle. The van der Waals interaction among the large particles of the system was calculated on the basis of a model that was previously employed to examine the stability of emulsions [16] and microemulsions [17]. In this model, a particular particle is considered to be surrounded by 12 nearest neighbor particles randomly located on a sphere of radius R1 , and all the other particles of the system are assumed to be homogeneously distributed outside a sphere of radius R2 (Fig. 2). The radius R1 is selected as the average distance between large particles, and is therefore given by RL3 , (14) (R1 /2)3 where ϕ1 is the volume fraction of the large particles, and the constant 0.74 represents the volume fraction of the most compact arrangement of large particles. The number of large particles in the sphere of radius R2 is 13. Assuming that they are arranged in a compact manner, one can write ϕ1 = 0.74

4π 3 13 4π R = (R1 /2)3 . 3 2 0.74 3

Fig. 2. Interaction between one particle and the other particles. The neighboring 12 particles are located on a sphere of radius R1 and the other are located outside a sphere of radius R2 .

(15)

3 U sin α dα dθ, (16) π where U , the van der Waals interaction between two identical  particles with the center-to-center distance x = dV =

R12 + q 2 − 2qR1 cos α, is provided by the expression AH RL 12 (x − 2RL )3 AH RL  =− , 12 (−2R + R 2 + q 2 − 2qR cos α)3 L 1 1

U =−

(17)

where AH is the Hamaker constant. The overall van der Waals interaction between one particle located at the distance q from the center of the sphere of radius R1 and the other particles can be expressed in the form  3 V= U sin α dα dθ π 0<α<π, 0<θ<2π

 AH RL R1 + q − 2RL =− RL ln +q . qR1 R1 − q − 2RL

(18)

If one assumes that the particle can be located in a sphere of radius R1 − 2RL at any place with equal probability, the following average van der Waals interaction between a particle and the neighboring 12 is obtained: V¯ =

R1−2RL

0

4πq 2 V dq AH RL (R1 + RL ) =− . (19) R1 (R1 − 2RL ) 4π(R1 − 2RL )3 /3

Because the van der Waals interaction is a short-range one, the van der Waals interaction between one large particle and the large particles other than the neighboring ones can be neglected.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 290 (2005) 336–342

367 339

The free energy change as calculated above refers to one cell. However, the total free energy change Fsys of the system also contains an entropic contribution due to the mixing of the large particles with the dispersing medium containing an electrolyte and small particles. The entropic contribution for the dispersion of the large particles in a medium containing small particles is given in Appendix A. An upper bound for the entropy of mixing is given by  n1 Nt − n2 nS Nt − n2 nS Su = k nL ln + ln nL n1 n1



 Nt − n 2 n S Nt − n2 nS − − nL ln − nL , (20) n1 n1 where Nt denotes the total number of sites per unit volume of the system (each site is considered to have the volume of a molecule of the dispersing medium, a water molecule for instance), n1 the number of sites occupied by each large particle, and n2 is the number of sites occupied by each small particle. Although Eq. (20) provides an upper bound for the entropic contribution, the calculations revealed that this contribution to the free energy is negligible. Adding the entropy of mixing to the double-layer depletion and the van der Waals interactions, one obtains the following expression for the free energy change of the system, Fsys = nL (Fcell + V /2) − T Su ,

Fig. 3. Effect of the volume fraction of uncharged small particles. RL = 200 Å, RS = 10 Å, c = 0.01 M, Ψ0 = −0.05 V, AH = 0.1 × 10−20 J.

(21)

when the upper bound of the entropy of mixing is employed.

3. Numerical calculations 3.1. The free energy change of the system for uncharged small particles The small particles are depleted from the gaps between the large particles when RC < RL + RS . As a result, an attractive force is induced between the large particles. The effect of the volume fraction ϕ2 of the small particles on the free energy change is presented in Fig. 3. The dispersion is thermodynamically stable because the free energy change with respect to that of the electrolyte solution is negative. As expected, the free energy change becomes more negative as the concentration of uncharged small particles increases. This happens because with the introduction of a larger number of small particles, the entropic contribution provided by Eq. (13) increases and the free energy change becomes more negative. As the size of the small particles increases, the thickness of the depletion zone becomes larger, and at constant ϕ2 , the free energy change becomes less negative (Fig. 4). The free energy change becomes less negative with increasing ϕ1 and the van der Waals interaction between the large particles becomes significant at large ϕ1 .

Fig. 4. Effect of the size of the uncharged small particles. RL = 200 Å, c = 0.01 M, Ψ0 = −0.05 V, ϕ2 = 0.1, AH = 0.1 × 10−20 J.

As the electrolyte concentration increases at constant ϕ1 and ϕ2 , the free energy change becomes less negative (Fig. 5). Two effects play a role in this case: (i) the electrical free energy (Eq. (12)), which becomes less positive with increasing electrolyte solution, and (ii) the entropic contribution (Eq. (13)), which becomes more negative with increasing electrolyte concentration. It is clear that under the conditions examined the entropic contribution is dominant (Fig. 6) and the free energy change increases as the electrolyte concentration increases because of the entropic contribution. 3.2. Distributions of electrical potential and concentrations of ions in the system with charged small particles Fig. 7 shows that the electrical potential first increases with the distance from the surface of a large particle and then decreases. The magnitude of the change of the electrical potential is, however, small, less than 10% of the surface electrical potential in the case examined.

Nanodispersions

368 340

H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 290 (2005) 336–342

Fig. 5. Effect of the electrolyte concentration. RL = 200 Å, RS = 30 Å, Ψ0 = −0.05 V, ϕ2 = 0.1, AH = 0.1 × 10−20 J.

Fig. 8. Distributions of cation and anion concentrations in the radial direction. (1) c = 0.15 M, RL = 200 Å; (2) c = 0.005 M, RL = 200 Å; (3) c = 0.15 M, RL = 250 Å. (a) Anion; (b) cation. The other parameter values are as in Fig. 7.

The concentrations of the cations and the anions are plotted in Fig. 8. Because both the large and the small particles are negatively charged, their counterions make the average cation concentration higher than the average anion concentration. With increasing distance from the surface of a large particle, the cation concentration first decreases because the electrical potential increases. However, at a distance corresponding to the maximum in the electrical potential, the cation concentration begins to increase until it reaches the boundary of the cell. The highest difference between the concentrations of the cations and anions is at the surface of the particles. The concentration difference at the surface is 8.7 mM for an electrolyte concentration of 150 mM and 2.3 mM for an electrolyte concentration of 5 mM, both for ϕ1 = 0.1 and ϕ2 = 0.05. Fig. 6. Contributions to the free energy change: (1) c = 0.02 M; (2) c = 0.01 M; (3) c = 0.005 M. The values of the other parameters are as in Fig. 5.

Fig. 7. Electrical potential distribution in the radial direction. Ψ0 = −0.05 V, z = −3, ϕ1 = 0.1, ϕ2 = 0.05, RS = 30 Å. (1) c = 0.15 M, RL = 200 Å; (2) c = 0.005 M, RL = 200 Å; (3) c = 0.15 M, RL = 250 Å.

3.3. The free energy change of the system containing charged small particles If the small particles are charged, they affect the electrical potential and the depletion interactions. The small particles are repelled by the large ones if they have the same sign of charge and the depletion interaction becomes stronger. As the volume fraction of small particles increases, the free energy change becomes more negative (Fig. 9). As for the uncharged small particles, the free energy change of the system containing charged small particles becomes less negative at high electrolyte concentrations (Fig. 10). As the charge of the small particles increases, the free energy change becomes more negative (Fig. 11). This occurs because (i) the ionic strength increases very much and this decreases the repulsion between the large particles; (ii) the depletion interaction is stronger for the charged small particles than the uncharged small particles. The free energy change of the system becomes less negative with increasing volume fraction of large particles, be-

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 290 (2005) 336–342

Fig. 9. Effect of the volume fraction of small particles. RL = 200 Å, Ψ0 = −0.05 V, RS = 30 Å, z = −3, AH = 1 × 10−21 J, c = 0.01 M.

369 341

Fig. 12. Prediction of the behavior of an experimental system. Ψ0 = −40 mV. T = 293 K, RL = 180 nm, RS = 10 nm, z = −5, AH = 1 × 10−21 J, and ε = 68.9. (1) ϕ2 = 0.0336; (2) ϕ2 = 0.0206.

cause the separation between the large particles becomes smaller and the repulsion between them stronger. 3.4. Prediction of the phase behavior of an experimental system

Fig. 10. Effect of salt concentration. RL = 200 Å, Ψ0 = −0.05 V, RS = 30 Å, z = −3, ϕ2 = 0.1, AH = 1 × 10−21 J.

Berli et al. have investigated the rheological behavior of protein-stabilized oil-in-water emulsions [18]. They measured the viscosity of an oil-in-water emulsion as a function of the shear stress at various concentrations of sodium caseinate. The sodium caseinate molecules aggregate in watergenerating micelles, which can be considered as the small particles. The oil droplets are considered as the large particles. For large values of ϕ1 , as the concentration of sodium caseinate increased, the viscosity of the emulsion exhibited a sharp increase because of the formation of a gel. The gel was generated because of the depletion interactions induced by the large number of sodium caseinate micelles. Using the values Ψ0 = −40 mV, T = 293 K, RL = 180 nm, and RS = 10 nm provided in Ref. [18], and selecting the charge of the small particle z = −5, AH = 1 × 10−20 J, ε = 68.9 for the mixture of water and alcohol, the free energy of the system is plotted in Fig. 12. As the volume fraction of oil increases, the free energy decreases. For a volume fraction ϕ2 = 0.0206, the free energy is positive for ϕ1 < 0.60, but becomes negative for ϕ1 > 0.60. A thermodynamically stable gel is formed at large volume fractions of the oil droplets and this explains the sharp increase in the viscosity of the system for ϕ1 ≈ 0.60.

4. Conclusions

Fig. 11. Effect of the charge of the small particles. RL = 200 Å, Ψ0 = −0.05 V, RS = 30 Å, AH = 1 × 10−21 J, c = 0.01 M, ϕ2 = 0.1.

A spherical cell model was employed to describe the behavior of large particles immersed in a solution of electrolyte containing small particles. An attractive depletion

370

Nanodispersions

342

H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 290 (2005) 336–342

force was included in the free energy change of the system. The free energy change becomes more negative as the volume fraction of small uncharged particles increases because of a stronger depletion force. As the size of the small uncharged particles increases, the free energy change becomes less negative. As the charge of the small particles with the same sign of charge as the large particles increases, the free energy change becomes more negative than for uncharged small particles because of a stronger depletion force induced by the charged small particles. Some experimental facts regarding the behavior of the viscosity of an oil-inwater emulsion with increasing concentration of a surfactant are explained via the formation of a gel induced by depletion forces. Appendix A Let us derive an expression for the entropy of mixing of the large particles with a fluid containing small particles. The procedure employed is similar to that of Ref. [17]. The entropy of mixing is given by the Boltzmann relation S = k ln Ω,

(A.1)

where Ω is the number of configurations of a mixture formed from N1 molecules of liquid, nL identical large particles, and nS identical small particles, all per unit volume. If Nt is the total number of sites per unit volume and the number of sites occupied by each small particle is n2 , the center of the first large particle can be located at any of the Nt − nS n2 remaining sites, the center of the second large particle at any of the Nt − nS n2 − α2 sites, the center of the j th large particle at any of Nt − nS n2 − αj , and so on. Therefore, the number of configurations Ω can be calculated using the following expression: Ω = (Nt − nS n2 )(Nt − nS n2 − α2 ) × (Nt − nS n2 − α3 ) . . . (Nt − nS n2 − αnL ) × nL !−1 .

(A.2)

An upper bound of Ω can be obtained if αj is selected in the form αj = (j − 1)n1 ,

(A.3)

where n1 is the number of sites occupied by each large particle. Consequently,

Ω = (Nt − nS n2 )(Nt − nS n2 − α2 ) × (Nt − nS n2 − α3 ) . . . (Nt − nS n2 − αnL ) × nL !−1 =

nn1 L [(Nt − nS n2 )/n1 ]! . [((Nt − nS n2 )/n1 ) − nL ]!nL !

(A.4)

Substituting Eq. (A.4) into the Boltzmann relation, and using the Stirling approximation for the factorials, one obtains  n1 Nt − n2 nS Nt − n2 nS Su = k nL ln + ln nL n1 n1



 Nt − n2 nS Nt − n 2 n S − − nL ln − nL . n1 n1 (A.5) One can show that the contribution of the entropy of mixing to the total system free energy change is in the present case negligible.

References [1] E.K. Hobbie, Phys. Rev. Lett. 81 (1998) 3996. [2] E.H.A. de Hoog, W.K. Kegel, A. van Blaaderen, H.N.W. Lekkerkerker, Phys. Rev. E 64 (2001) 021407. [3] A.P. Gast, C.K. Hall, W.B. Russel, J. Colloid Interface Sci. 96 (1983) 251. [4] I.V. Rao, E. Ruckenstein, J. Colloid Interface Sci. 108 (1985) 389. [5] E. Ruckenstein, I.V. Rao, Colloids Surf. 17 (1986) 185. [6] S. Ramakrishnan, M. Fuchs, K.S. Schweizer, C.F. Zukoski, J. Chem. Phys. 116 (2002) 2201. [7] Y.L. Chen, K.S. Schweizer, M. Fuchs, J. Chem. Phys. 118 (2003) 3880. [8] Y.L. Chen, K.S. Schweizer, J. Chem. Phys. 120 (2004) 7212. [9] J.J. Cerda, T. Sintes, C.M. Sorensen, A. Chakrabarti, Phys. Rev. E 70 (2004) 011405. [10] K.G. Soga, J.R. Melrose, R.C. Ball, J. Chem. Phys. 110 (1999) 2280. [11] D. Costa, P. Ballone, C. Caccamo, J. Chem. Phys. 116 (2002) 3327. [12] S. Asakura, F. Oosawa, J. Chem. Phys. 22 (1954) 1255. [13] H.H. Huang, E. Ruckenstein, Langmuir 20 (2004) 5412. [14] J.Th.G. Overbeek, Colloids Surf. 51 (1990) 61. [15] M. Manciu, E. Ruckenstein, Langmuir 19 (2003) 1114. [16] W. Albers, J.Th.G. Overbeek, J. Colloid Sci. 14 (1959) 519. [17] E. Ruckenstein, J.C. Chi, J. Chem. Soc. Faraday Trans. II 71 (1975) 1690. [18] C.L. Berli, D. Quemada, A. Parker, Colloids Surf. A 215 (2003) 201.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects

371

Journal of Colloid and Interface Science 272 (2004) 430–437 www.elsevier.com/locate/jcis

Attractive interactions in dispersions of identical charged colloidal particles: a Monte Carlo simulation Jie Feng and Eli Ruckenstein ∗ Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260-4200, USA Received 30 June 2003; accepted 12 November 2003

Abstract Using the Debye–Hückel pair potential, the collective interactions between identical charged particles were examined via Monte Carlo simulations. The results have shown that when the number of charges per particle and the particle volume fraction were sufficiently large, the pair long-range electrostatic repulsion generated an effective attractive interaction between identical charged particles because of manybody effects. Disordered liquid-like structures, ordered crystal-like structures, ordered structures dispersed in disordered ones, and disordered structures dispersed in ordered ones have been found. The structures are dependent on the volume fraction and charge of the particles, as well as on the electrolyte concentration.  2003 Elsevier Inc. All rights reserved. Keywords: Effective attractive interaction; Colloidal dispersions; Monte Carlo simulation

1. Introduction Experiment has revealed that concentrated dispersions of monodispersed latices exhibit iridescence [1,2]. This phenomenon occurs because of Bragg diffraction of the visible light by ordered arrangements of spheres and the equality of their lattice parameter with the optical wavelength [3,4]. Because of the large distances involved, the van der Waals interactions are negligible and the only interactions that can explain the ordered lattice arrangement of the colloidal particles are the electrostatic ones. Microscopy experiments carried out by Ise and co-workers [5–7] with colloidal dispersions of latex particles indicated that the distance between particles for low particle charges was approximately equal to that corresponding to a uniform distribution. However, for highly charged particles, the distance between some of them became smaller than that for a uniform distribution. To explain this observation, Ise and co-workers suggested that the smaller distances occur in ordered regions generated by the long-range attractive interactions caused by the electrostatic interactions between identical highly charged particles. A theory was formulated by Sogami and Ise [8,9] that starting from the Poisson–Boltzmann equation, predicted that * Corresponding author.

E-mail address: [email protected] (E. Ruckenstein). 0021-9797/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.11.059

the electrostatic interaction between two identical charged colloidal particles immersed in an electrolyte solution is repulsive at short distances but becomes attractive at long ones. This conclusion contradicts the DLVO theory [10], where the electrostatic interactions between two colloidal particles are repulsive at all distances. This issue has been discussed in several papers, some in favor of the Sogami–Ise theory [10–12] but most against [13–18]. Thermodynamics rigorously demonstrates that in the framework of the Poisson– Boltzmann equation, the interaction between two identical charged particles is repulsive at all distances [18]. The fact that interactions between ions of the same kind can become attractive was noted a long time ago. Using the integral equation approach and considering higher order correlations between ions [19], Kirkwood predicted attractive interactions between identical charged ions. Outhwaite and Hutson arrived at a similar conclusion, starting from the symmetrized Poisson–Boltzmann equation [20]. Schmidt and Ruckenstein [21,22], considering the colloidal particles a component of a particles–electrolyte–liquid mixture, and assuming coulombic interactions among the colloidal particles, anions and cations, found conditions under which electrostatic interaction generated attraction between particles. They attributed this phenomenon to a collective Coulomb interaction among all the charged species. The Ornstein–Zernike equation in its zero-order approximation,

372

Nanodispersions J. Feng, E. Ruckenstein / Journal of Colloid and Interface Science 272 (2004) 430–437

which constitutes a symmetrized Poisson–Boltzmann equation, was employed. The calculations revealed that the pair correlation function of the particles became larger than unity for particle concentrations larger than a critical value. This result indicates the presence of a potential of mean force representing an effective attraction between particles. Chu and Wasan [23] solved the Ornstein–Zernike equation with the mean spherical approximation. They found an oscillatory attractive effective interaction potential between two identical charged particles at high particle charges and concentrations and a repulsive interaction at low particle charges and concentrations. The scope of this paper is to demonstrate via Monte Carlo simulations that the conventional Debye–Hückel screened pair potential, which is based on the DLVO theory and is repulsive at all distances, can explain the experimental observations of Ise et al. Consequently, the pair potential does not have to become attractive at large distances to explain the Ise et al. observations. The simulations have been carried out for a wide range of charges and volume fractions of the particles as well as a few electrolyte concentrations. The paper is organized as follows: In Section 2, the model and the Monte Carlo algorithm will be described. In Section 3, the main results will be presented. Section 4 will emphasize the conclusions.

2. Model and algorithm N identical charged hard spherical particles with a radius of 670 Å located in a three-dimensional cubic box were considered in the simulations. The pair interaction between two particles was represented by the Debye–Hückel screened potential (ED−H (rij )),
∞, for rij < 2r, ED−H (rij ) = (1) (Ze)2 exp(−κ(rij −2r)) , for rij
2r, 4πε0 εd rij (1+κr)2

431

constant, and T is the temperature in K. In what follows zi and zc will be considered equal to unity. The effective potential U was calculated from the radial distribution function (g(r)) using the expression U = −kB T ln g(r).

(3)

The effects of the number of charges Z per particle and of the volume fraction f of the particles on the effective interaction potential between identical charged particles were examined. The number of charges per particle was varied between 150 and 1200 and the particle volume fraction f was varied between 0.005 and 0.12. The electrolyte concentration was assumed to be 10−5 or 10−4 M. In the simulations, the traditional algorithm was applied to an off-lattice at each step, and periodic boundary conditions were employed. The interactions were truncated using the minimum image convention, and the information was recorded every 50 cycles, after the system reached equilibrium; 1000 samples were averaged in each simulation.

3. Results and discussion As shown in Fig. 1, for the low particle volume fraction f of 0.005 and an electrolyte concentration of 10−5 M, there was little attraction between identical charged particles and very small oscillations occurred when the number of charges per particle Z was 300. With number of charges per particle Z increasing to 600 and 1200, the curves of the effective interaction between identical charged particles versus distance between particles became oscillatory, but the amplitudes of oscillations were relatively small because of the low particle volume fraction. However, it is clear that the larger the charge per particle the larger the oscillations. Fig. 2 presents the structure generated at a large charge per particle (Z = 1200) for f = 0.005.

where rij is the distance between the centers of two particles, r is the radius of the particles, e is the protonic charge, ε0 is the vacuum permittivity, εd is the dielectric constant, Z represents the number of charges per particle, and κ is the reciprocal Debye length, which is given by the expression    4πe2 2 2 2 κ = nc zc + ni zi . (2a) ε0 εd kB T i

Because of electroneutrality, np Z = nc zc and Eq. (2a) becomes    4πe2 2 2 κ = np Zzc + ni zi . (2b) ε0 εd kB T i

In Eqs. (2), np is the particle concentration, ni is the concentration of ions of species i and valence zi , nc is the concentration of counterions of the particles, zc is the valence of counterions of the particles, kB is the Boltzmann

Fig. 1. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.005 and electrolyte concentration of 10−5 M. (1) Z = 300; (2) Z = 600; (3) Z = 1200.

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects 432

373

J. Feng, E. Ruckenstein / Journal of Colloid and Interface Science 272 (2004) 430–437

(a)

Fig. 2. Structure of charged colloidal particles with Z = 1200, particle volume fraction of 0.005, and electrolyte concentration of 10−5 M.

(b)

Fig. 3. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.015, and electrolyte concentration of 10−5 M. (1) Z = 300; (2) Z = 600; (3) Z = 1200.

When the particle volume fraction f was increased to 0.015, the oscillations of the effective interaction between identical charged particles became larger than those for f = 0.005 at an electrolyte concentration of 10−5 M (Fig. 3). The effective interaction between identical charged particles versus the distance between particles is plotted in Fig. 3 for various values of Z (Z = 300, Z = 600, Z = 1200). As shown in Fig. 4, the colloidal dispersion has a disordered liquidlike structure for Z = 600, but a more ordered structure for Z = 1200. When the electrolyte concentration was increased to 10−4 M, the interaction between identical charged particles became completely screened. As shown in Fig. 5, no oscillations of the effective interaction potential were present for Z = 300, 600, and 1200.

Fig. 4. Structure of charged colloidal particles with particle volume fraction of 0.015 and electrolyte concentration of 10−5 M. (a) Z = 600; (b) Z = 1200.

Fig. 5. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.015, and electrolyte concentration of 10−4 M. (1) Z = 300; (2) Z = 600; (3) Z = 1200.

374

Nanodispersions J. Feng, E. Ruckenstein / Journal of Colloid and Interface Science 272 (2004) 430–437

Fig. 6. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.03 and electrolyte concentration of 10−5 M. (1) Z = 300; (2) Z = 600; (3) Z = 1200.

433

Fig. 8. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.03, and electrolyte concentration of 10−4 M. (1) Z = 300; (2) Z = 600; (3) Z = 1200.

(a)

Fig. 9. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.06, and electrolyte concentration of 10−5 M. (1) Z = 150; (2) Z = 300; (3) Z = 600; (4) Z = 1200.

(b) Fig. 7. Structure of charged colloidal particles with particle volume fraction of 0.03 and electrolyte concentration of 10−5 M. (a) Z = 600; (b) Z = 1200.

When the particle volume fraction f was increased to 0.03 (Fig. 6), oscillations could be observed at the electrolyte concentration of 10−5 M. Fig. 7a shows that a disordered structure surrounded by an ordered one occurred for

Z = 600. For Z = 1200 (Fig. 7b), the structure became more ordered. When the electrolyte concentration was increased to 10−4 M, the effective interaction became almost completely screened and no oscillations could be found (Fig. 8). When the particle volume fraction f was further increased to 0.06, oscillations of the effective interaction potential between identical charged particles occurred under a wide range of charges per particle (from Z = 150 to 1200) at an electrolyte concentration of 10−5 M (Fig. 9). The effective interaction potential curves between identical charged particles versus distance between particles can be divided into two groups: the curves for Z = 600 and Z = 1200, and those for Z = 150 and Z = 300. Disordered structures characterize the second group and very ordered structures the first one (Fig. 10). When the electrolyte concentration was

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects 434

375

J. Feng, E. Ruckenstein / Journal of Colloid and Interface Science 272 (2004) 430–437

(a)

(b)

(c)

(d)

Fig. 10. Structure of charged colloidal particles with particle volume fraction of 0.06 and electrolyte concentration of 10−5 M. (a) Z = 150; (b) Z = 300; (c) Z = 600; (d) Z = 1200.

increased to 10−4 M, the oscillations of the effective interactions remained, but because of increased screening, their amplitudes were smaller than for an electrolyte concentration of 10−5 M (Fig. 11). Fig. 12 shows that the structure was no longer ordered even for Z = 1200. As shown in Fig. 13, when the particle volume fraction f was further increased to 0.12 and the electrolyte concentration was 10−5 M, oscillations of the effective interaction occurred from Z = 150 to Z = 1200. As shown in Fig. 14, for Z = 150 the system had a disordered structure, but for Z = 300, 600, and 1200 the systems became very ordered. When the electrolyte concentration was increased to 10−4 M, the effective interaction exhibited large oscillations for Z = 1200, but the oscillations for Z = 150, 300, and 600 became much smaller (Fig. 15). As shown in Fig. 16, the structure was very ordered for Z = 1200, but disordered for Z = 150, 300, and 600. The effective interaction potential is a result of collective behavior. Let us select two neighboring particles immersed in a dispersion of particles of the same kind. Two kinds of

Fig. 11. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.06, and electrolyte concentration of 10−4 M. (1) Z = 150; (2) Z = 300; (3) Z = 600; (4) Z = 1200.

Nanodispersions

376 J. Feng, E. Ruckenstein / Journal of Colloid and Interface Science 272 (2004) 430–437

Fig. 12. Structure of charged colloidal particles with particle volume fraction of 0.06, electrolyte concentration of 10−4 M, and charge per particle Z = 1200.

435

Fig. 13. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.12, and electrolyte concentration of 10−5 M. (1) Z = 150; (2) Z = 300; (3) Z = 600; (4) Z = 1200.

(a)

(b)

(c)

(d)

Fig. 14. Structure of charged colloidal particles with particle volume fraction of 0.12 and electrolyte concentration of 10−5 M. (a) Z = 150; (b) Z = 300; (c) Z = 600; (d) Z = 1200.

interactions are involved: (1) a repulsive interaction between the two selected particles, and (2) a collective interaction to which each of the two selected particles is subjected by all the other particles of the system. If the latter interaction is a repulsive one which can overcome the former, then the two selected particles will be subjected to an effective attraction.

At equilibrium, the repulsion between the two neighboring particles becomes equal to the repulsion to which they are subjected by all the other particles of the system. As shown in Fig. 17, for Z = 300 and an electrolyte concentration of 10−5 M, the oscillations of the effective interaction potential were weak for low particle-volume frac-

Non-DLVO colloidal interactions: excluded volumes, undulation interactions, depletion forces and many-body effects 436

377

J. Feng, E. Ruckenstein / Journal of Colloid and Interface Science 272 (2004) 430–437

Fig. 15. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.12, and electrolyte concentration of 10−4 M. (1) Z = 150; (2) Z = 300; (3) Z = 600; (4) Z = 1200.

Fig. 17. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various particle volume fractions (f ), for charges per particle Z = 300 and electrolyte concentration 10−5 M. (1) f = 0.005; (2) f = 0.015; (3) f = 0.03; (4) f = 0.06; (5) f = 0.12.

tions, but became increasingly large as the volume fraction increased. The strengthening of the oscillations is caused by the increased collective interaction. As shown in Fig. 18, for Z = 1200 and an electrolyte concentration of 10−5 M, large oscillations of the effective interaction were present for most particle volume fractions because of the large number of charges per particle.

4. Conclusion Using Monte Carlo simulations and a Debye–Hückel pair repulsive potential, effective interaction potentials between identical charged particles were calculated and the phase structures were obtained for various charges per particle and various particle volume fractions. The simulation re-

(a)

(b)

(c)

(d)

Fig. 16. Structure of charged colloidal particles with particle volume fraction of 0.12 and electrolyte concentration of 10−4 M. (a) Z = 150; (b) Z = 300; (c) Z = 600; (d) Z = 1200.

378

Nanodispersions J. Feng, E. Ruckenstein / Journal of Colloid and Interface Science 272 (2004) 430–437

437

Table 1 The effective interaction and the phase structure for identical charged particles Z = 150 f = 0.005

Z = 300

f = 0.030

f = 0.1200

f = 0.015 f = 0.030 f = 0.060 f = 0.1200

Z = 1200

Diameter of particle d = 670 Å and electrolyte concentration ni = 10−5 M

f = 0.015

f = 0.060

Z = 600

Oscillations; disordered structure Oscillations; disordered structure

Weak oscillations; disordered structure Oscillations; disordered structure Oscillations; disordered structure Oscillations; disordered structure Large oscillations; ordered structure

Weak oscillations; disordered structure Oscillations; disordered structure Oscillations; local ordered structures Large oscillations; ordered structure Large oscillations; ordered structure

Diameter of particle d = 670 Å and electrolyte concentration ni = 10−4 M No oscillations No oscillations No oscillations No oscillations Oscillations; Oscillations; Oscillations; disordered structure disordered structure disordered structure Oscillations; Oscillations; Oscillations; disordered structure disordered structure disordered structure

Weak oscillations; disordered structure Strong oscillations; partially ordered structure Oscillations; local ordered structures Large oscillations; ordered structure Large oscillations; ordered structure No oscillations No oscillations Oscillations; disordered structure Large oscillations; ordered structure

fractions lead to very ordered structures, while smaller ones lead to disordered structures.

References

Fig. 18. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various particle volume fractions (f ), for charges per particle Z = 1200 and electrolyte concentration 10−5 M. (1) f = 0.005; (2) f = 0.015; (3) f = 0.03; (4) f = 0.06; (5) f = 0.12.

sults have shown that an effective attractive interaction can be generated even though there is no attractive component in the pair potential employed. This effective attraction is a result of many-body effects. The general equilibrium properties of the identical charged particles systems are summarized in Table 1. Generally, when the number of charges per particle is small and/or the particle volume fraction is low, no oscillatory effective interactions between identical charged particles are present. As the number of charges per particle and the particle volume fraction increased, the effective interaction between identical charged particles became oscillatory. Large charges per particle and particle volume

[1] T. Alfrey Jr., E.B. Bradford, J.W. Vanderhoff, G. Oster, J. Opt. Soc. Am. 44 (1954) 603. [2] W. Luck, M. Klier, H. Wesslau, Ber. Bunsenges. Phys. Chem. 67 (1963) 75. [3] S. Hachisu, Y. Kobayashi, A. Kose, J. Colloid Interface Sci. 42 (1973) 342. [4] A. Kose, M. Ozaki, K. Takano, K. Kobayashi, S. Hashisu, J. Colloid Interface Sci. 44 (1973) 330. [5] N. Ise, T. Okubo, M. Sugimura, K. Ito, H.J. Nolte, J. Chem. Phys. 78 (1982) 536. [6] Y. Ishii, H. Matusuoka, N. Ise, Ber. Bunsenges. Phys. Chem. 90 (1986) 50. [7] N. Ise, Angew. Chem. Int. Ed. 25 (1986) 323. [8] I. Sogami, N. Ise, Phys. Lett. A 96 (1983) 199. [9] I. Sogami, N. Ise, J. Chem. Phys. 81 (1984) 6320. [10] M.V. Smalley, Mol. Phys. 71 (1990) 1251. [11] I.S. Sogami, T. Shinohara, M.V. Smalley, Mol. Phys. 74 (1991) 599. [12] M.V. Smalley, Langmuir 11 (1995) 1813. [13] J.T.G. Overbeek, J. Chem. Phys. 87 (1987) 4406. [14] C.E. Woodward, J. Chem. Phys. 89 (1988) 5140. [15] S. Levine, D.G. Hall, Langmuir 8 (1992) 1090. [16] R. Ettelare, Langmuir 9 (1993) 1888. [17] J.T.G. Overbeek, Mol. Phys. 80 (1993) 685. [18] E. Ruckenstein, Adv. Colloid Interface Sci. 75 (1998) 169. [19] J.G. Kirkwood, J. Chem. Phys. 3 (1935) 300. [20] C.W. Outhwaite, V.C.L. Hutson, Mol. Phys. 29 (1975) 1521. [21] A.B. Schmidt, E. Ruckenstein, J. Colloid Interface Sci. 150 (1992) 169. [22] E. Ruckenstein, A.B. Schmidt, J. Chem. Soc. Faraday Trans. 88 (1992) 2365. [23] X. Chu, D.T. Wasan, J. Colloid Interface Sci. 184 (1996) 268.

Introduction to CHAPTER 5 Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

5.1

5.2

5.3

5.4

5.5

M. Manciu, E. Ruckenstein: “Specific ion effects via ion hydration: I. Surface tension” ADVANCES IN COLLOID AND INTERFACE SCIENCE 105 (2003) 63–101. E. Ruckenstein, M. Manciu: “Specific ion effects via ion hydration: II. Double layer interaction” ADVANCES IN COLLOID AND INTERFACE SCIENCE 105 (2003) 177–200. M. Manciu, E. Ruckenstein: Comments on the “Osmotic Coefficients and Surface Tensions of Aqueous Electrolyte Solutions: Role of the Dispersion Forces” JOURNAL OF PHYSICAL CHEMISTRY B 108 (2004) 20479–20481. M. Manciu, E. Ruckenstein: “On the Interactions of Ions with the Air/Water Interface,” LANGMUIR 21(2005) 11312–11319. M. Manciu, E. Ruckenstein: “Ions at the air/water interface,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 304 (2006) 541–544.

One of the main shortcomings of the traditional DLVO theory is its inability to predict specific ionic effects. These effects are sometimes much stronger than what the excluded volume effects (examined in the previous chapter) can explain. It was recently suggested by Ninham and his group that the van der Waals interactions between ions and the rest of the system might be responsible for the ionic specificity [M. Boström, D.M.R. Williams, B.W. Ninham, Phys. Rev. Lett. 87 (2001) 168103]. However, these ion-dispersion forces lead to qualitatively incorrect results, namely that the cations are attracted toward the air/water interface and the anions are repelled by that interface. Experiment shows that the opposite is actually true. A simple model that accounts for the large specific interactions between ions and surfaces is based on the longknown role of ions in structuring the water around them; the structure-making ions interact preferentially with the bulk water, hence they are effectively repelled by the surface, while the structure-breaking ions disturb the

order of bulk water and therefore they are expelled toward the interface [5.1]. This simple Structure Making/Structure Breaking (SM/SB) model can explain quantitatively many effects related to ion specificity, such as the dependence of the surface potential and surface tension on pH and type and concentration of electrolyte; the existence of a minimum (dependent on the nature of the ions) of the surface tension at low electrolyte concentrations; and an almost linear dependence (with a positive or sometimes negative slope, again dependent on the nature of the ions) of the surface tension on the electrolyte concentration at sufficiently high ionic strengths [5.1]. The SM/SB model for ion-hydration forces is used to calculate the force between two parallel plates, and it is shown that specific ion effects emerge through changes in the surface charge and potential induced by ion hydration forces [5.2]. There are difficulties in accounting for the iondispersion forces at short separation distances, particularly when other strong interactions (such as ion-hydration or Born hard-sphere repulsion) are also present. It is demonstrated that the dispersion force between ions is negligible at distances larger than about 5 Ǻ. At smaller distances it is not appropriate to base its calculation on a continuous dielectric model, because a small change in the cut-off parameter changes the value of the dispersion interaction from negligible into one that exceeds the mutual electrostatic repulsion between ions of the same kind. If real, the latter value can lead to the clustering of ions of the same kind, an interesting effect which was not yet observed [5.3]. The SM / SB model for the ion hydration force is compared with the traditional model for ion hydration (based on the Born energy of ions) and with a recent model due to Bostrom, Kunz and Ninham [Bostrom, M.; Kunz, W.; Ninham, B. W. Langmuir 2005, 21, 2619], which accounts for the decrease in hydration energy caused by the water density decrease in the vicinity of the air/water interface. Even when the ion-dispersion force is included in the latter model, it leads to an interfacial depletion of both

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_5, © Springer Science+Business Media, LLC 2010

380

anions and cations, which contradicts the Molecular Dynamics simulations results of the Jungwirth and Tobias group [Mucha, M.; Frigato, T.; Levering, L. M.; Allen, H. C.; Tobias, D.J.; Dang, L. X.; Jungwirth, P. J. Phys. Chem. B 2005, 109, 7617]. In contrast, the SM / SB model reproduces more accurately the results of the simulations [5.4].

Nanodispersions

There are an increasingly large number of experiments which demonstrate that the cations (except H+) are repelled by the air-water interface whereas the anions (except F-) accumulate there. The simple SM/SB model can account, both qualitatively and quantitatively, for these experimental results [5.5].

381

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

Advances in Colloid and Interface Science 105 (2003) 63–101

Specific ion effects via ion hydration: I. Surface tension Marian Manciu, Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA

Abstract A simple modality is suggested to include, in the framework of a modified Poisson– Boltzmann approach, specific ion effects via the change in the ion hydration between the bulk and the vicinity of the surface. This approach can account for both the depletion of the interfacial region of structure-making ions as well as for the accumulation of structurebreaking ions near the interface. Expressions for the change in interfacial tension as a function of electrolyte concentrations are derived. On the basis of this theory, one explains the dependence of the surface potential on pH and electrolyte concentration, the existence of a minimum in the surface tension at low electrolyte concentrations and the linear dependence, with a positive or sometimes negative slope, of the surface tension on the electrolyte concentration at sufficiently high ionic strengths. 䊚 2003 Elsevier Science B.V. All rights reserved. Keywords: Specific ion effect; Ion hydration; Structure-making ion; Structure-breaking ion

Contents 1. Introduction ............................................................................................. 64 2. The 2.1. 2.2. 2.3. 2.4.

mean-field formalism (the Poisson–Boltzmann approach) .............................. The Poisson–Boltzmann equation ........................................................... The image forces (Wagner–Onsager–Samaras theory) .................................. Ion hydration effects via the dielectric constant........................................... Ion size effects ....................................................................................

*Corresponding author. Tel.: q1-716-645-2911y2214; fax: q1-716-645-3822. E-mail address: [email protected] (E. Ruckenstein). 0001-8686/03/$ - see front matter 䊚 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0001-8686Ž03.00018-6

65 66 67 68 69

382

Nanodispersions 64

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

2.5. The van der Waals interactions between ions and media ............................... 70 3. A simple treatment of ion hydration. General behavior........................................ 73 3.1. d1sd2sdE ......................................................................................... 74 3.2. d1-d2 ............................................................................................... 77 3.3. The anions are adsorbed on and the cations are repelled by the interface ........... 83 3.3.1. d1sd2sdE. Linear approximation ..................................................... 83 3.3.2. d1sd2sdE. Approximate solution of the nonlinear equation .................... 85 4. Applications .............................................................................................. 89 4.1. Zeta potential ...................................................................................... 89 4.2. Surface tension at low ionic strength. Jones–Ray effect ................................ 92 4.3. Surface tension at high ionic strength ....................................................... 96 5. Conclusions .............................................................................................. 98 References................................................................................................ 101

1. Introduction The distribution of electrolyte ions around a charged surface was first calculated by Gouy w1x and Chapman w2x. Their theory assumed that the ions are point charges embedded in a continuum of constant dielectric constant, which are distributed according to Boltzmann statistics and interact with all the other charges via a mean field obeying the Poisson equation. The limitations of a theory with such drastic approximations was obvious from the beginning, and Stern pointed out w3x that this Poisson–Boltzmann approach can predict, in the vicinity of the surface, an ion concentration larger than that corresponding to close compaction. Since then, the theory was corrected in many ways to account for image forces w4x, finite sizes of the hydrated ions w5x, ion correlations w6,7x, dependence of dielectric constant on the field w8x and electrolyte concentration w9x and many other effects. The Poisson–Boltzmann approach w1,2x has the advantage of simplicity and is surprisingly accurate, at least for univalent ions in a certain range of electrolyte concentrations (1.0=10y3 –5=10y2 M) and not too close to the interface. It was later employed w10x to explain the repulsion due to the overlap of two double layers, and the stability of colloids thereafter w11,12x. It is instructive to compare the Poisson–Boltzmann approach as applied to colloid stability to a related example regarding the surface tension of electrolyte solutions. For most electrolytes, the surface tension of the wateryair interface increases with increasing ionic strength. This increase is caused (via the Gibbs equation) by the negative adsorption of electrolyte ions on the surface w13x. The depletions of such ions near the interface were attributed for a long time to a more favorable free energy when the ion is fully hydrated w13x, which occurs when the ion is sufficiently far from the interface. However, the Poisson–Boltzmann approach always predicts, for a charged interface and a symmetric electrolyte, an increase in the total number of ions in the vicinity of the interface, as compared to the bulk, because the counterions increase always exceeds the coions depletion. Hence, to calculate the distribution of ions in the vicinity of a surface and their surface adsorption, additional interactions have to be included in the Poisson–Boltzmann treatment. A

383

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

65

charge embedded in a dielectric is repelled by the charge density it induces on the dielectric boundary (the ‘image force’). The use of the image force to calculate the (negative) surface adsorption of ions led to a diverging increase in surface tension w14x. However, Wagner w14x, inspired by the then-recent Debye–Huckel ¨ theory of electrolytes, realized that the other ions screen the image force. The screening length around an ion depends on the local concentration of ions and is, therefore, a function of the distance from the surface. Onsager and Samaras w15x simplified the calculations of Wagner, by assuming a constant screening length through electrolyte, and obtained an analytic ‘limiting law’ for the dependence of the surface tension on electrolyte concentration. However, the theory failed at high electrolyte concentrations (above f0.2 M), and the authors pointed out various sources of error and suggested various modalities to account for them into a more complete theory. In contrast, the bare-bone Poisson–Boltzmann approach has had an unusual success in explaining quantitatively the double layer repulsion and the colloid stability. A few reasons for this success might be the partial cancellation of some effects decorating the bare-bone equations w5x, or the fact that the corrections are important particularly within a few molecular diameters from the interface. The model inaccuracies could be reasonably well accounted for by a simple theoretical construction (the Stern layer) or by using the surface charge, surface potential or the dissociation constant of surface groups as a fitting parameter. The series of decorations of the Poisson–Boltzmann equation is long, and in spite of the rich literature available in the field, the calculations of the interactions are still mostly performed in the traditional DLVO manner (the bare-bone Poisson–Boltzmann approach). It was recently shown that such a decoration of the theory with van der Waals interactions between individual ions and the interface might account in a simple manner for the inaccuracies of the Poisson–Boltzmann approach regarding the ion specific effects w16x and also for those of the Wagner–Onsager–Samaras theory w17,18x. In this article, we will first review the general framework of the Poisson– Boltzmann approach and will discuss the various kinds of approximations it involves. Some of the decorations will be briefly reviewed. The emphasis will be, however, on the recent van der Waals correction and arguments will be brought that the specific ion effects can be explained through ion hydration. A simple manner to account quantitatively for ion hydration and its relevance to ion specific effects will be suggested, and a comparison with the approach involving the van der Waals interactions made. 2. The mean-field formalism (the Poisson–Boltzmann approach) A fundamental hypothesis of the Gouy–Chapman theory is that the interaction of an ion with all the other charges can be described by a ‘mean potential’ c(x), where x is the distance from the surface. The chemical potential of an ion of species ‘i’ with charge qi in the liquid is then given by: mi(x)sm0iqkTlnŽci(x).qqic(x)qWi(x),

(1)

384

Nanodispersions 66

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

where m0i represents the standard chemical potential of the ion of species ‘i’, of concentration ci(x) and charge qi. The interaction free energy Wi (x) includes all the other interactions of the ion with the medium (not accounted for by the mean field). Some examples are examined below. The distribution of ions of species ‘i’, ci(x), is related to the bulk concentrations c0i of ions by the Boltzmann equation: B

cisc0iexpCy D

qic(x)qDWi(x) E F, G kT

(2)

where DWi(x) is calculated with respect to the bulk and can depend also on c, ci and other parameters, k is the Boltzmann constant and T is the absolute temperature. The mean field is assumed to satisfy the Poisson equation:

2

= c(x)sy

8ici(x)qi ´0

q

≠P(x) , ´0≠x

(3)

where the polarization P(x) is typically assumed proportional to the macroscopic electric field: P(x)sy´0(´y1)=c(x),

(4)

´0 being the vacuum permittivity and ´ the dielectric constant of the medium. 2.1. The Poisson–Boltzmann equation The most simple approximation is DWi(x)s0; for an uni-univalent electrolyte of concentration cE the potential in the vicinity of a planar surface can be calculated from Eqs. (2)–(4), which lead to, B ec(x) E d2c(x) 2ecE F s sinhC 2 D kT G dx ´´0

(5)

the well-known Poisson–Boltzmann equation, where e is the proton charge. It can be solved for suitable boundary conditions, such as constant surface charge density, constant surface potential, dissociation or adsorption equilibrium at the surface. Once the ion distribution is known, the other characteristics of the system, such as the free energy, can be calculated. Several kinds of approximations are involved in the derivation of Eq. (5). First, it implies that the interactions between charges can be described by an ‘average’ potential, which is a mean field type of approximation. A theory devoid of this assumption can be developed in the general framework of statistical mechanics, using pair correlation functions. One of the successful approaches, based on the

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

385 67

inhomogeneous hypernetted chain approximation, was presented by Kjellander and Marcelja w7x. However, the results provided by this method depend strongly on the ion pair potentials employed and it was shown w19x that minor changes in the ionpair potentials can lead to qualitatively different results. It is not yet clear if the ion-correlation effects can be described to a good approximation by a suitablechosen function DWi(x). Another limitation of the Poisson–Boltzmann approach is that the interaction between two surfaces immersed in water might not be exclusively due to the electrolyte ions. For instance, water has a different structure in the vicinity of the surface than in the bulk and the overlapping of such structures generates a repulsion even in the absence of electrolyte w20x. In this traditional picture, the ‘hydration’ repulsion is not related to ion hydration; actually it is not related at all to electrolyte ions. However, as recently suggested w21x, this ‘hydration’ interaction can still be accounted for within the Poisson–Boltzmann framework, assuming that the polarization is not proportional to the macroscopic electric field, but depends also on the field generated by the neighboring water dipoles and by the surface dipoles. There are also other effects, which cannot be incorporated easily in the Poisson– Boltzmann formalism, such as the non-planarity of the interface, its thermal fluctuations, the charge fluctuations and so on. The other kinds of approximations are due to the oversimplification DWis0. Many effects can be accounted for by decorating the Poisson–Boltzmann equation with more suitable expressions for DWi(x). The rest of the article is devoted to the examination of a few possible choices. 2.2. The image forces (Wagner–Onsager–Samaras theory) As already noted, the Poisson–Boltzmann equation fails to describe the ion depletion in the vicinity of the surface, which is basically due to the preference of the ions for the bulk liquid, where they can be fully hydrated. One of the first corrections to the formalism was the addition of the interaction between the ions of charge qi and the surface via the (screened) ‘image force’ w14x:

B ´y´9 E

B q2i 2x E F, expCy D ´q´9 G 16p´0´x D l(x) G

DWW,i(x)sC

F

(6a)

where ´9 is the dielectric constant of the medium which has an interface with water and l(x) is the screening length. This interaction is the same for both kinds of ions of a symmetric electrolyte, hence the ion depletions are in this case the same for both ions and, in the absence of surface charges or other interactions, the electric potential is vanishing everywhere. To account for different ion sizes (and hence to obtain, to some extent, ion-specific effects), Onsager and Samaras w15x proposed an

Nanodispersions

386 68

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

¨ interaction potential modified in a Debye–Huckel spirit: B 2a E

B ´y´9 E

DWOS,i(x)sC

F

D ´q´9 G

expC D

i

l

2ai 1q l

F G

B q2i 2x E expCy F, D 16p´0´x lG

(6b)

¨ where ai are the ion radii and lsyŽ´´0kTySic0iqi2. is the Debye–Huckel screening length. The Onsager–Samaras theory fails at electrolyte concentrations above approximately 0.2 M. One of the reasons is the assumption that the screening length l does not depend on the distance to the interface. In reality, the screening in the iondepleted region near the surface is smaller than in the bulk. Only when l is large compared to the thickness of the depleted layer, the above approximation is accurate. Other reasons for the inaccuracy (additional interactions and surface charge) will be examined later in the paper. The image force formalism was intended primarily to relate, via the Gibbs adsorption equation, the increase of the surface tension of water to the negative adsorption of the electrolyte ions. Its effect on the double layer interactions was ¨ ¨ w 4x . examined by Jonsson and Wennerstrom 2.3. Ion hydration effects via the dielectric constant Another modality to include the ion hydration effects in the Poisson–Boltzmann formalism is via the dielectric constant. If the surface charge is sufficiently large, then there is a large electric field and a high concentration of counterions near the surface. Both of these factors lead to a decrease of the dielectric constant. The main contribution to the polarizability of a water molecule is a result of the alignment of its dipole by the electric field, the polarization being proportional with the field when the latter is sufficiently low, but becoming saturated at high fields. In addition, the presence of an electrolyte hinders the polarizability of the water molecules, thus decreasing the dielectric constant. To account for these effects, one should first account for the dependence of the dielectric constant on the magnitude of the field w9x and on the counterions concentration w8x in Eq. (4). Second, one should account, via the Born energy, for the change of the ion hydration free energy due to the change in the dielectric constant of the medium in Eq. (2). The Born hydration energy is the electrostatic energy of an ion in a dielectric and depends on the field generated by the ion in the entire space. When there is a boundary between two media with different dielectric constants, the change in the electrostatic energy when the ion approaches the boundary generates an image force, which induces a diverging negative surface adsorption. One can avoid this behavior by accounting for the screening of the interaction by the electrolyte w14x. This treatment does not include the non-uniformity of the dielectric constant.

387

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

69

Another possible treatment w9x is to define the Born energy in term of a local dielectric constant ´(x):

DWB,i(x)s

q2i B 1 1E C y F, 8p´0ai D ´(x) ´ G

(7)

where ´s´(`) is the bulk dielectric constant. In this case, the ions are repelled by the interface only because the dielectric constant of water in its vicinity, ´(x), is lower than in the bulk. If the dielectric constant would be larger near the interface, the ions would be attracted there. When the interface is charged, the Poisson–Boltzmann equation predicts stronger fields and higher ionic concentrations in the vicinity of the surface, therefore a dielectric constant smaller than in the bulk, and the ‘local’ Born energy indicates that the ions are repelled by the interface. However, a neutral surface depleted of ions should have a dielectric constant larger than the bulk and the ions should be attracted, not repelled by such an interface. Therefore, other interactions should be included to explain the ion depletion near a neutral surface. 2.4. Ion size effects It has been long accepted that the Poisson–Boltzmann equation can lead to ion concentrations that exceed the volume available in the vicinity of a surface. Stern included the effect of the ion size on its distribution around a charged surface by assuming the existence of a layer in the vicinity of the surface containing only condensed counterions w3x. The effect of the ion size on the double layer interaction was later incorporated in the modified Poisson–Boltzmann formalism in a manner similar to that described here w22x. The difference in the free energy of ions between bulk and interface can be written as a function of ion concentrations, either in a lattice model w22x, in terms of the occupation probabilities or in a continuum treatment w23x, by assuming that the hydrated ions behave as rigid spheres and employing the excess chemical potential with respect to an ideal gas. By accounting for the size of the ions one can explain the depletion of ions in the vicinity of the surface and obtain in most cases an increase in the double layer repulsion as compared to the Poisson–Boltzmann equation (in some special cases, it might lead to a decrease of the repulsion w24x). It also can account, at least partially, for some specific ion effects. However, the differences in hydrated radii between various ions are small and the specific effects become important in this treatment only at high ionic strengths and high surface potentials. While the hydrated ion radius hinders the ions to approach to close the interface, the ‘depletion region’, which will be estimated later in this paper from the change in the interfacial tension, might either exceed or be much smaller (sometimes, even negative) than the hydrated radius of the ions.

388

Nanodispersions 70

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

2.5. The van der Waals interactions between ions and media Onsager and Samaras were interested in a qualitative explanation of the experimentally determined increase in surface tension with increasing electrolyte concentrations. The screened image force provided a good agreement with experiment, up to approximately 0.2 M. They suggested that better agreement could be obtained if higher multipolar correction would be included in the treatment. Onsager and Samaras also remarked that these interactions should be important only in the close vicinity of the surface, since they should decrease with at least the third power of the distance. It was later conjectured by Sienko w25x that the multipolar corrections to the Onsager–Samaras theory might account for specific ion effects, because of different ion polarizabilities. A quantitative attempt to investigate this conjecture was made recently by Stairs w25x, by including the interaction between a dipole and its image, using the (screened) Onsager–Samaras potential. However, because of screening, the effect is small at high ionic strengths. Ninham and Yaminsky w26x, and Karraker and Radke w18x realized that the van der Waals interactions between the ions and interface are not screened by the electrolyte and hence might become more important than the image force, at large electrolyte concentrations. Recognizing that the hydration of ions might also play a ¨ et al. w16,17x showed that the van der Waals interactions alone (with role, Bostrom suitable values selected for the interaction parameters) might account for the ion specific effects. The van der Waals interaction between an ion and the interface between two dielectric media, calculated in the Lifshitz approach, has as leading term w26x: DWNY(x)sq

B . x3

(8)

Let us first address some quantitative issues, using the more simple Hamaker approach. If one assumes pairwise additivity, the interaction between a sphere of kind (1) immersed in a fluid of kind (3) and a planar half-space of kind (2) is given by w27x: DWH(x)sy

A132a w xya xya B xya Ez 2A a3 4A a5 F|sy 1323 y 1325 y«, q lnC x1q D xqa G~ 6(xya) y xqa a 9x 15x

(9)

where x is the distance from the interface to the center of the ion of radius a and A132 is the effective Hamaker constant between ion (1) embedded in a fluid (3) and a medium (2) on the other side of the interface, given by w27x: A132sA12qA33yA13yA23,

(10)

389

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

71

where Aij are the Hamaker constants of the interactions between the media ‘i’ and ‘j’ in vacuum. Eq. (8), with Bsy(2A132 a 3 )y9, is a good approximation of Eq. (9) for a<x (the difference is less than 10% for x)4a). Since three media are involved in these van der Waals interactions, the effective Hamaker constant can become negative. For the wateryair interface, A12sA23s0, and hence: A132sqA33yA13.

(11)

If the Hamaker constant between ion and water A13 (in vacuum) is larger than that between water and water A33, the effective Hamaker constant is negative and the ions are repelled by the interface. On the other hand, if A13-A33, the ions are attracted by the interface. ¨ et al. w16,17x showed that the positive ions are less polarizable and the Bostrom negative ions more polarizable than water. Therefore, the former should be attracted to and the latter repelled by the interface. The values of Bi have been, however, obtained by fitting the experimental interfacial tensions. Karraker and Radke w18x, accounting only for the interactions between the individual ions and the water and air continua (thus ignoring the term A33 in Eq. (11)), concluded that the dispersion interactions always repel the ions from the wateryair interface, with the exception of the non-polarizable Hq, for which BHs 0. ˚ for the Cly ion embedded in a water continuum, By assuming a radius as1.8 A ¨ et al. w16,17x if one obtains the value BCls31=10y50 J m3 employed by Bostrom A132s23=10y20 J, which is typical for metal–metal interactions through vacuum (10–30=10y20 J w27x). If the ‘Hamaker constants’ A13 of ion–water interactions would be of this order of magnitude, then the water–water interaction (A33s 3.7=10y20 J) can be neglected in Eq. (11). However, the derivation of Eq. (10) is based on the (macroscopic) hypothesis that the ‘ion sphere’ replaces a ‘water sphere’ of the same size. If the ions are thought as occupying interstitial places in a (rigid) water lattice, the displacement of ions is not followed by that of water. In such a case, the effect of the ‘intervening medium’ has to be discarded and the Hamaker constants calculated (in the hypothesis of pairwise additivity) as through vacuum. Furthermore, these van der Waals interactions are important only near the interface, where it is unlikely that either Lifshitz or Hamaker approaches are accurate for spheres of molecular sizes. For example, the magnitude of the interaction for Naq ˚ from the interface is only approximately 0.02kT (the values of B used ions at 5 A ¨ et al. in the calculation, BNasy1=10y50 J m3 was obtained from fit by Bostrom w17x and BNasq0.8=10y50 J m3 was calculated by Karraker and Radke w18x). Eq. (8) might provide a convenient way to account for the interfacial interactions, if suitable values for Bi (not related to the ‘macroscopic’ Hamaker constants) would be selected. There are, however, some more important qualitative issues, which should be discussed. If the cations would be attracted by the interface, they will condense

390

Nanodispersions 72

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

there (since this attraction diverges for x™0) and the accumulation of ions on the ¨ et al. interface will lead to an infinite decrease of the interfacial tension. Bostrom ˚ in the calculations w17x avoided this ‘cation catastrophe’ by selecting a cut-off of 2 A and obtained good agreement with experiment by fitting the values of B. As will ˚ is used, good be shown in the next section, if a cut-off of approximately 4 A agreement with experiment can be obtained by completely disregarding the van der Waals interactions of ions. More dramatic are the long-range consequences of the van der Waals attraction, ¨ et al. w16x, the double layer even when a cut-off is included. As shown by Bostrom interactions in electrolytes of physiological concentrations become in this case ˚ This unexpected theoretical prediction has yet attractive at distances less than 30 A. to be verified experimentally. Recent molecular dynamics simulations w28x showed that the ‘commonly held opinion’ that the interfacial region is depleted of electrolyte ions might not be true, and that some ions might approach the interface. Their accumulation on the interface can be predicted by a suitable negative value for the parameter Bi in the van der Waals interactions of the ions (coupled with an interaction cut-off). However, there are the large negative ions (Cly, Bry) which prefer to accumulate in the vicinity of the interface w28x, and not the less-polarizable small and positive ions, as expected from the van der Waals interactions. In the treatment of Karraker and Radke w18x, the interaction coefficients Bi are always positive, and the cation catastrophe is avoided. However, the van der Waals repulsions are still larger for the negative ions than for the positive ones; therefore, the accounting for the van der Waals interaction alone would imply that all airy ¨ et al. w17x electrolyte interfaces should have a positive surface potential. Bostrom argued that this might be indeed the case, and that the negative values of the potentials typically obtained experimentally might be caused by the double layers generated around the electrodes used in the experiments, and do not represent intrinsic properties of the wateryair interface. Karraker and Radke w18x suggested that the negative potentials of the surfaces are due to the preference of OHy for the surface, thus including another interaction (in their treatment, the van der Waals interactions still repel the OHy ions from the interface, stronger than they repel the cations). The increase of the surface tension of water by the addition of an electrolyte was traditionally related (via Gibbs adsorption equation) to the negative adsorption of ions on the interface. However, some electrolytes decrease the interfacial tension w29x, hence should be positively adsorbed. Therefore, if the van der Waals interactions would repel all the ions from the interface, some additional interactions have to be included to explain the positive adsorption. The accounting for the van der Waals interactions of ions has the advantage of simplicity, since the determination of the values of the Bi coefficients for all the ions could lead to a quantitative treatment of ion specific effects within the Poisson– Boltzmann framework. However, as mentioned above, this treatment provides the debatable prediction that the cations would approach the interface closer than the anions.

391

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

73

The change in ion hydration between bulk and interface was suspected since a long time w13x to be responsible for the distribution of ions in the vicinity of the interface; whereas this effect can be easily understood qualitatively, a consistent quantitative theory has not yet been developed. The reason is that accurate information about the structure of water in the vicinity of an interface and about the corresponding changes in the ion hydration with the distance are yet beyond our reach. In the next section, a simple modality to account in a semi-quantitative manner for the ion hydration effects will be suggested. The effects of the van der Waals interactions and image forces will be compared with those of hydration. 3. A simple treatment of ion hydration. General behavior It is well known that, in general, there is a strong correlation between the standard enthalpy of hydration of an electrolyte and the modification of the surface tension by that electrolyte w30x. The effect of ion hydration on interfacial tension can be easily understood qualitatively. For the structure-making ions, their approach to the interface is unfavorable, because they can better organize the water dipoles in bulk water than at the interface. The opposite is true for the structure-breaking ions; in such cases, the total free energy of the system is minimized by pushing the structurebreaking ions toward the interface, because the bulk water molecules can better organize their hydrogen bonding network without these ions. The surface tension can either increase or decrease with electrolyte concentration, depending whether the depletion of the structure-making ions in the vicinity of the surface or the adsorption of the structure-breaking ions dominates the process. The distribution of ions in the vicinity of the interface could be calculated if accurate expressions for DWi(x) would be available. There is, however, a simple manner to obtain reasonable estimations of ion distribution, even when the expressions DWi (x) are not known. The hydration free energy per ion, in bulk water, is of the order of y102kT w31x. An increase in the total free energy by 5kT implies (via the Boltzmann distribution) a concentration decrease by more than two orders of magnitude. Therefore, if at a particular distance di from the interface, the hydration free energy of a structure-making ion is increased by only a few percent from the bulk hydration free energy, one can consider, for calculation purposes, that the region from the interface to the distance di is totally depleted of those ions. Hence, one can describe the hydration interaction of a structure-making ion by a parameter, representing a distance of closest approach to the surface. It is plausible to consider di related to the hydrated radii of the ions of species ‘i’; however, they are not the same because the structure of water is different in the vicinity of the interface from that in the bulk. This approach might seem to be rather crude; however, it should be noted that recent molecular simulations of simple electrolytes interfaces w28x show that the variation of the electrolyte concentrations is about as steep as the variation of water density at the interface. Therefore, if one assumes a sharp wateryair interface, one can also assume step distributions for the ions of the electrolyte.

392

Nanodispersions 74

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

A similar simple treatment is less accurate for the structure-breaking ions. The change in the total free energy between surface and bulk cannot exceed a few kT, because otherwise the ions would be trapped at the interface and generate a large surface potential, possibility ruled out by experiment. Since DWi (x) is not very large, it cannot be well approximated, in general, by a step function. However, in the spirit of the above approximation and in the absence of more accurate information, we will assume that it can be described by a potential well and will examine the consequences. In this section, we will investigate the effect of the changes in the free energies of ions between bulk and interfacial region, on the ion distributions in the vicinity of the interface, and consequently on the surface tension, for an uni-univalent electrolyte in the following cases: 1. Both kinds of ions are structure-making, and d1sd2; 2. Both kinds of ions are structure-making, but the cations (2) are repelled more strongly than the anions (1): d2)d1; 3. The cations are structure-making (d2 )0) but the anions are structure-breaking, hence positively adsorbed on the interface. 3.1. d1sd2sdE In this case (in the absence of a surface charge) the interface is neutral. The dependence of the interfacial tension g on the uni-univalent electrolyte concentration cE can be calculated using the Gibbs adsorption equation: dgsy 8 Gidmi,

(12)

is1,2

where mi is the (total) electrochemical potential and Gi is the surface excess of ion i:

|

`

Žci(x)ycE.dx

GiŽcE.s

(13)

0

with the wateryair interface located at xs0. The total electrochemical potential, which is independent of the distance from the interface, can be related to the bulk electrolyte concentration cEsci(`) via: dmisdŽkTlnŽci(`)fi..,

(14)

where f i is the activity coefficient of the electrolyte in the bulk. Using Eqs. (12)–(14) and assuming f i'1, the change in the interfacial tension due to the addition of an electrolyte is given by: 1 ≠g G1qG2 sy '2D, kT ≠cE cE

(15)

393

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

75

where the ratio y(G1 qG2)y2cE has the dimension of a length which is denoted by D, the depletion thickness. The surface tension as a function of electrolyte concentration can be obtained by integrating Eq. (15):

|

gŽcE.yg(0)s

cE

≠g dcsykT cs0 ≠c

|

cE 0

G1(c)qG2(c) dc. c

(16)

In the present case, there is no surface charge and d1sd2sdE, therefore ci(x)sc for x)dE and ci(x)s0 for 0-x-dE, which provides Gi(ci)'ycDsycdE and (17)

DgŽcE.sŽ2kTdE.cEAcE.

This expression was employed before to account for the dependence of g on the electrolyte concentration w30x, which is roughly linear up to large electrolyte concentrations. It should be noted that this simple model is in better agreement with experiment up to large ionic concentrations w29x than the Onsager–Samaras theory w15x, which assumes that the image forces are screened by the electrolyte and hence are small at large electrolyte concentrations. However, dE has to be regarded as a parameter, which depends on both kinds of ions of the electrolyte; therefore, the ‘specific ion effects’ are transformed into ‘specific electrolyte effects’. For some electrolytes, the value of dE determined through fitting is reasonable and seems to ˚ for NaCl and dEs3.8 be close to the hydrated ion radius, for example dEs3.9 A ˚ A for KCl w30x. For these electrolytes, the selection of a cut-off distance equal to ˚ in the Poisson–Boltzmann equation would lead to an excellent agreement dE(f4 A) with surface tension experiments, even if the van der Waals interactions and other interactions that decorate the Poisson–Boltzmann approach are disregarded w30x. It is instructive to compare how the ion-interface interactions (image force, van der Waals interactions, ion hydration) affect the depletion thickness at various electrolyte concentrations. To estimate the magnitudes of these effects, it will be assumed that the interactions are non ion-specific (which implies to use Eq. (6a) for the potential of the image force, and B1sB2sB in Eq. (8) for the van der Waals interactions). Consequently, the interface is neutral and the surface adsorption can be calculated from the simple expression: G1 G2 s syDs cE cE

|

`B

B

0 D

D

CexpCy

DW(x) E E Fy1Fdx. kT G G

(18)

As that generated by ion hydration for d1sd2, the depletion thickness generated by the van der Waals interactions, when B1 sB2 , does not depend on concentration. When only the van der Waals interactions are accounted for, Eqs. (8) and (18) lead

394

Nanodispersions M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

76

Fig. 1. Comparison between the depletion distances generated by several symmetric (the same for both ions) interactions: the image forces (Eq. (6a), ´s80, ´9s1, Ts300 K), the van der Waals interaction ˚ The interactions (Eq. (8), B1sB2s31=10y50 or 1=10y50 J m3 ) and ion hydration (d1 sd2'dEs4 A). accounted for on each curve are: (1) the ion hydration and the image forces; (2) the image forces alone; (3) the ion hydration and the van der Waals interactions, Bs31=10y50 J m3; (4) the van der Waals interactions alone, Bs31=10y50 J m3; (5) the ion hydration and the van der Waals interactions, Bs 1=10y50 J m3; (6) the ion hydration alone; (7) the van der Waals interactions alone, Bs1=10y50 J m 3.

to the following depletion distance w17x:

|

D'

`B

B

0 D

D

C1yexpCy

B B E1y3 B EE FF C F dxs D kT G kTx3 GG

|

`B

B

0 D

D

C1yexpCy

B B E1y3 1 EE FF C F . dy(1.354 D kT G y3 GG (19a)

while when both the ion hydration and van der Waals are accounted, the effect of the van der Waals interactions is to increase (decrease) the cut-off distance:

|

dE

D'

|

dxq 0

`B

B

dED

D

C1yexpCy

B EE B FFdx(dEq . kTx3 GG 2kTd2E

(19b)

The correction of the cut-off distance in Eq. (19b) is significant only for large values of B.

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

395

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

77

There is no reason to assume that B1sB2 or that d1sd2. However, when the surface potential is low (hence in the absence of a surface charge and at sufficiently low electrolyte concentrations), Eq. (18) can be used to calculate separately the depletion thicknesses for each kind of ions. For comparison purposes, the depletion thickness is plotted in Fig. 1 as a function of electrolyte concentration (´s80, ´9s1, Ts300 K) for a number of cases listed ˚ was selected to obtain agreement there. The value of the cut-off distance dEs4 A with the surface tension data for the aqueous solutions of some common electrolytes (NaCl, KCl) at high ionic strengths. The effects of the van der Waals interactions or of ion hydration on the depletion thickness are qualitatively the same. When B is large (Bs31=10y50 Jm3), the additional accounting of ion hydration provides only a small correction to the result obtained with van der Waals interactions alone. When B is small, the additional accounting of the van der Waals interactions does not affect much the result obtained with ion hydration interactions alone. It should be noted than neither B nor dE are precisely known; what can be inferred from the data on surface tension at high ionic strengths is that both effects together should ˚ Therefore, the rather large value lead to a depletion thickness of approximately 4 A. y50 3 BCls31=10 J m , evaluated from fitting in Ref. w17x, might have been obtained because the ion hydration was neglected. ˚ at high Assuming an electrolyte with a depletion thickness of approximately 4 A ionic strengths, Fig. 1 suggests that the image forces play a dominant role at very low electrolyte concentrations (cE-1.0=10y4 M), but are negligible at relatively large concentrations (cE)0.1 M). For some electrolytes, the ‘characteristic distance’ dE required to fit the experimental data on surface tension with Eq. (17) is smaller than the hydrated radius of the ions and might even become negative (e.g. HCl, HNO3, HClO4 w29x). This means that at least one component of the electrolyte is preferentially adsorbed on the interface. Therefore, dE can be regarded only as a parameter in an extremely simplified theory. 3.2. d1-d2 The differences in the hydration energies of the ions and in their hydrated radii are expected to lead to d1/d2. As indicated by simulations w28x, the negative ions are less repelled, therefore they can approach the interface closer than the positive ions. The asymmetric ion depletions generate a surface potential, even in the absence of an ‘external’ surface charge. The charge that generates the diffuse double layer ˚ is located a few Angstroms from the interface, and is due to the anions. The corresponding modified Poisson–Boltzmann equations for the planar interface located at xs0 are: d2c s0 dx2

0-x-d1,

(20a)

Nanodispersions

396 78

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

B ec E d2c ecE s expC F d1-x-d2, 2 D kT G dx ´´0

(20b)

B ec E d2c 2ecE s sin hC F d2-x-`, 2 D kT G dx ´´0

(20c)

where e is the protonic charge. Eqs. (20a), (20b) and (20c) can be solved for the boundary conditions: dc(x) s0 dx

(no surface charge density) xs0,

(21a) (21b)

x™`.

c(x)™0

The surface potential generated by the asymmetric distribution of the ions in the vicinity of the surface is small when the electrolyte concentration is sufficiently low. It is, therefore, worthwhile to obtain an analytical solution in the linear approximation. The general solution of the linear approximation of Eqs. (20a), (20b) and (20c) is: cI(x)sC1xqC2 0-x-d1, cII(x)sy

(22a)

B xyd E B kT xyd1 E 1 qC3expC FqC4expCy F d1-x-d2, e y2l G D y2l G D B xyd E

B

l

D

cIII(x)sC5expC D

2

FqC6expCy G

xyd2 E F d2-x-`, l G

(22b)

(22c)

¨ where l is the Debye–Huckel length, ls6(´´0kTy2e 2cE). The boundary condition Eq. (21a) leads to C1s0 and the boundary condition Eq. (21b) to C5s0. The continuity of c and dcydx at xsd1 and xsd2 leads finally to the solution: cI(x)sy

kT B 1y e

C

D

C

2

D

ly2

cos hC

yd1

B

kT cII(x)sy 1y e D

E

1 Bd

E

1

G

y2

Fq

D

Bd

cos hC

D

2

D

ly2

sin hC

B xyd E

cos hC

Bd

1

ly2

F G

yd1

FF E

(23a)

d1-x-d2,

(23b)

GG

F

E

B d yd E yd1 E 1 Fq sin hC 2 1 F D ly2 G G ly2 G y2

2

0-x-d1,

397

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

79

˚ d2s Fig. 2. (a) The potential generated by the asymmetric distribution of the electrolyte ions (d1 s2 A, ˚ ´s80, Ts300 K) provided by Eq. (23a), Eq. (23b) and Eq. (23c) is compared with the numerical 5 A, solution of Eqs. (20a), (20b) and (20c) for various electrolyte concentrations. (b) The distribution of ˚ d2 s5 A, ˚ cE s1 M, ´s80, Ts300 K). (c) The electrolyte ions in the vicinity of the interface (d1 s2 A, ˚ and d1s0, 1, 2, 3, 4 and 5 A, ˚ as a function of electrolyte concentration depletion thickness for d2 s5 A (´s80, Ts300 K). (d) The change in surface tension as a function of electrolyte concentration, for ˚ and d1s0, 1, 2, 3, 4 and 5 A. ˚ The value g0 corresponds to cE'0. d2s5 A

cIII(x)sy

kT e

B

1 B

Bd

2

D

D

ly2

1qy2Ctan hC

yd1

EEy1

FF

expCy D

xyd2 E F d2-x-`. l G

(23c)

GG

The magnitude of the potential is larger for larger values of (d2yd1)yl, but is everywhere smaller than kTyes0.026 V. The potential is generated by the asymmetric distributions of the two kinds of ions (it vanishes for d1sd2) and increases with increasing cE. At sufficiently low electrolyte concentrations, the linear approximation is always accurate; however, at sufficiently high ionic strengths, the linear approximation is no longer valid and the magnitude of the surface potential can exceed kTye. The potential obtained via the numerical integration of Eqs. (20a), (20b) and (20c) is compared with Eqs. (23a), (23b) and (23c) in Fig. 2a, for d1s

398

Nanodispersions M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

80

˚ d2s5 A ˚ and various electrolyte concentrations. Up to cEs1 M (ls3 A), ˚ the 2 A, numerical integration of the nonlinear Eqs. (20a), (20b) and (20c) provides a solution c(x) within a few percents from the analytical solution of the linear approximation. The distributions of electrolyte ions for cEs1.0 M are plotted in Fig. 2b. The change in the interfacial tension due to the electrolyte ions can be calculated using the Gibbs adsorption equation (Eqs. (12)–(14)). The surface adsorptions are given by:

|

`

|

Žc1(x)ycE.dxs

G1s

0

|

d1

|

ycEdxq 0

`

|

Žc1(x)ycE.dxq

sycEd2q

d2

d2

d2

|

`

Žc1(x)ycE.dxq

d1

Žc1(x)ycE.dx d2

c1(x)dx

(24a)

d1

and

|

`

|

`

Žc2(x)ycE.dxsycEd2q

G2s

0

Žc2(x)ycE.dx.

(24b)

d2

For small values of cII, the last integral in Eq. (24a) can be evaluated using Eq. (23b): B xyd E

|

d2 d1

B ec

cEexpC D

E II(x) F(cE kT G

cos hC

|

D

d2

Bd

1

ly2

F G

B d yd E yd1 E 1 cos hC Fq sin hC 2 1 F y2 D ly2 G D ly2 G 1 s2cEl . B d yd E 2 1 1qy2tan hy1C F D ly2 G d1

dx

2

(25)

Since in the linear approximation the second integrals in Eqs. (24a) and (24b) cancel each other, Eq. (15) becomes: B 1 ≠g G1qG2 sy '2Ds2 d2y kT ≠cE cE

C D

E

l 1qy2tan h

Bd

2

D

ly2

C

y1

FF

yd1 E

.

(26)

GG

The slope of the surface tension as a function of electrolyte concentration is no longer constant, since l depends on cE. For low electrolyte concentrations,

399

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

2DsŽd1qd2.q

Žd2yd1.2 2l2

y

Žd2yd1.3 12l3

q∆

81

(27a)

For cE™0, the surface potential generated by the asymmetry of the ion concentrations vanishes, the anions are uniformly distributed from d1 to infinity, and the total depletion thickness is given by the sum of the depletion thicknesses of each kind of ions, calculated independently. For sufficiently high ionic strengths, but low enough for the linear approximation to remain still valid, Eq. (26) leads to:

2D(2d2y

l 1qy2

.

(27b)

The latter equation shows that the cation hydration (described by the cut-off distance d2) dominates the process. Because of the increase in potential with electrolyte concentration, the region between d1 and d2 is practically depleted of anions. The depletion distance 2D (given by Eq. (26)) is plotted in Fig. 2c as a function ˚ and various values of d1. The slope of electrolyte concentrations for d2s5 A increases with ionic strength because the surface potential increases. The change in interfacial tension (obtained by integrating Eq. (26)) is represented for the same cases in Fig. 2d. The dependence is well approximated in all cases by a straight line, with a slope dependent on both d2 and d1. As already noted in Section 3.1, for neutral systems with d1sd2 and B1sB2, the accounting of ion hydration through a cut-off distance provides similar results as the ion-surface van der Waals interactions. The surface potential generated by the asymmetric distribution of ions, due to van der Waals interaction alone (with B1/B2) is plotted in Fig. 3a. We employed for curve (1) the values B1sq 15=10y50 J m3 and B2sq0.8=10y50 J m3, which were calculated for Cly and Naq in Ref. w18x. The values B1sq31=10y50 J m3 and B2sy1.0=10y50 J m3, employed for the curves (2–4), were obtained in Ref. w17x by fitting the experimental surface tension data. In the latter case, it was necessary to use a cut-off, Dcut-off, in the cation interactions, to avoid their infinite accumulation on the interface. The ˚ was suggested in Ref. w17x; as shown in Fig. 3, the results are value Dcut-offs2 A strongly dependent upon this choice. The corresponding dependence of surface tension on electrolyte concentration is presented in Fig. 3b. The only significant difference between the two models (one accounting for hydration alone and the other for van der Waals interactions alone) is that the surface potential is always positive in the latter method, because always B1)B2. At sufficiently large electrolyte concentrations, both models predict an almost linear increase of the surface tension, and agreement with experiment can be obtained by using suitable choices for the parameters (di or Bi).

400

Nanodispersions 82

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

Fig. 3. (a) The surface potential and (b) the surface tension vs. electrolyte concentration, calculated by accounting for the van der Waals interactions alone in the Poisson–Boltzmann formalism (´s80, Ts 300 K), for (1) B1s15=10y50 J m3 and B2s0.8=10y50 J m3; (2) B1s31=10y50 J m3, B2sy ˚ (3) B1s31=10y50 J m3, B2sy1=10y50 J m3 and Dcut-offs2 A; ˚ (4) 1=10y50 J m3 and Dcut-offs1 A; ˚ B1s31=10-50 J m3, B2sy1=10y50 J m3 and Dcut-offs4 A.

401

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

83

3.3. The anions are adsorbed on and the cations are repelled by the interface The change in hydration energy for the cations will be assumed, as before, given by: DW2(x)s` 0-x-d2

(28a)

DW2(x)s0

(28b)

d2-x-`

The interactions between anions and the interface will be, however, approximated by a potential of the type: DW1(x)syW1 0-x-d1

(29a)

DW1(x)s0

(29b)

d1-x-`

In this case, two parameters (W1, d1) are used to describe the interfacial interactions of the anions. The modified Poisson–Boltzmann equation is (for d1d2): BW E B ec E d2cI ecE 1 C F C I F 0-x-d1, s exp exp 2 D kT G D kT G dx ´´0

(30a)

B ec E d2cII ecE II F d1-x-d2, s expC 2 D kT G dx ´´0

(30b)

B ec E d2cIII 2ecE III F d2-x-` s sin hC 2 D kT G dx ´´0

(30c)

with the same boundary conditions as before: dc s0 dx c™0

xs0,

(31a)

x™`.

(31b)

To evaluate the effect of the potential well on the ion distributions, it will be assumed that d1sd2sdE. 3.3.1. d1sd2sdE. Linear approximation For sufficiently small electrolyte concentrations, the surface potential generated is small and the linear approximation is valid. The solution of Eqs. (30a), (30b) and (30c) for the boundary conditions Eqs. (31a) and (31b), is given in this

402

Nanodispersions 84

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

Fig. 4. The anion (1) and cation (2) distributions and the potential (3) in the vicinity of an interface, ˚ W1s1kT, cEs0.1 M, ´s80, Ts300 K. d1sd2s4 A,

approximation by:

C

B

cI(x)sy

B

kT 1y e D

cII(x)sy

kT e

x Dl

cos hC

AE F 2G

y d A A d A cos hC y Fqy sin hC y F l 2 2 l 2 B

E

D

E

B

G

D

1

y

1q

B 2 dE tan hy1C D l A

y

E

E

F

E

0-x-dE,

(32a)

GG

B xE C F dE-x-`, exp y D lG AE F 2G

(32b)

where A'exp(W1 ykT). The potential in the diffuse layer and the corresponding ion ˚ cEs0.1 M and W1s1kT. distributions are presented in Fig. 4 for dEs4 A, For the depletion thickness, one obtains in the linear approximation: y

BW E G1qG2 1 '2Ds2dEyexpC F D kT G cE

|

dE

B ec E

expC

0

D

y

1q

B 2 dE tan hy1C D A l

I

B

2l

s2dEy

Fdx kT G

y

C

(dE 2y E

A F 2G

D

A E . AdE 1q 2l G

F

(33)

403

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

85

The last approximate expression shows that, at low electrolyte concentrations (l™`), the total adsorption is positive when the potential well is sufficiently deep (A'exp(W1 ykT))2). In this case, the surface tension decreases almost linearly with increasing electrolyte concentration. However, the slope does not remain constant, its magnitude decreasing with increasing electrolyte concentration, because of the surface potential generated by the asymmetric distribution of electrolyte ions. The slope ≠gy≠c becomes positive at concentrations higher than a critical electrolyte concentration, obtained from: 2y

A dE 1 s0´lcs . AdE 2 1 1 1q y 2lc 2 A

(34)

3.3.2. d1sd2sdE. Approximate solution of the nonlinear equation At high electrolyte concentrations, the linear approximation fails and Eqs. (31a), (31b), (32a), (32b) and (33) are no longer valid. A simple solution, even approximate, of the non-linear Poisson–Boltzmann equation is more difficult to obtain; however, the general behavior of the system can be understood from the following semi-quantitative analysis. The diffuse double layer is generated by the anions trapped in the potential well. The ‘surface charge density’, due to their distribution between 0 and dE, is given by:

|

dE

BW E

0

D kT G

r(x)dxsyecEexpC

ss

1

F

|

dE

B ec(x) E

0

D

expC

kT

B

W1qec¯ E F, D kT G

Fdx'yecEdEexpC G

(35)

where c¯ is a suitable average value between c(0) and c(dE). Starting from Poisson equation, one can show that, because of the electroneutrality, the potential gradient at distance dE is related to the ‘surface charge density’ via: dc Z sy Z dx ZxsdE Z Z

|

d2c dxs 2 dE dx `

|

r(x) dxsy dE ´´0 `

|

dE 0

r(x) s dx'y . ´´0 ´´0

(36)

Outside the well, the integration of the Poisson–Boltzmann equation (with the boundary condition Eq. (31a)) leads to w27x: B dc(x) 2kT ec(x) E F dE-x-`. s sin hCy D dx el 2kT G

(37)

Ž Ž

. .

86

Fig. 5. Graphical solution of Eq. (38). If W1 )0, there is a positive adsorption of anions in the potential well if the electrolyte concentration is sufficiently low. The increase of the electrolyte concentration displaces Nc¯ N toward larger values. If the electrolyte concentration is sufficiently high, the anion adsorption ¯ .ykT -1 . in the well becomes negative exp ŽW1qec

404 Nanodispersions

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

405 87

Combining Eqs. (35)–(37), yields: B

B E W1qec¯ E 4l Fs sin hCy ec(dE) F. D D kT G dE 2kT G

expC

(38)

If the well is not very broad and the potential is sufficiently large, then the ¯ potential varies slowly between 0 and dE and one can approximate cfc(d E). Under such conditions, Eq. (38) allows to calculate c(dE ) at any electrolyte concentration. ¯ In Fig. 5, the left-hand-side term of Eq. (38) is plotted against yecykT (c¯ is negative) for W1s0, 1kT and 3kT. The right-hand-side term is plotted for 4lydEs 0.1, 1, 10. The intersection of the two curves provides c¯ for the value selected for 4lydE. The cations cannot approach the interface closer than dE. When W1s0, the region 0-x-dE, which is depleted of cations, is also partially depleted of anions, since expŽŽW1qec¯ . ykT.-1, because of the potential generated by the asymmetric distributions of ions. At low electrolyte concentrations (curve (1) in Fig. 5), the potential is small and the anion depletion negligible. The increase of the electrolyte concentration displaces the intersection toward larger magnitudes of c¯ , and at high electrolyte concentrations (curve (3) in Fig. 5) the potential well can be considered as being completely depleted of anions, because expŽŽW1qec¯ . ykT.f0. When W1)0, there is a positive adsorption of anions at the interface if the electrolyte concentration is sufficiently low, but the unbounded increase of the surface potential with increasing ionic strength generates a negative adsorption of the anions in the potential well at large electrolyte concentrations. Consequently, the surface tension can first decrease (if the potential well is deep enough) and then increase with increasing electrolyte concentration. When the surface potential is large, the hyperbolic sinus in the right term can be approximated by its largest exponential term and Eq. (38) becomes: 2 B W1 E B dE E ¯ C FlnC F, csy 3 D e G D 2l G

(39)

which shows that the surface potential increases without bound with the electrolyte concentration. Therefore, the potential well will be always depleted by the anions at sufficiently high electrolyte concentrations, in spite of their favorable interaction with the interface. To illustrate the above discussions, the surface potential and the surface tension are plotted vs. electrolyte concentration in Fig. 6a and b, respectively, for d1sd2s ˚ and various values for W1. The Poisson–Boltzmann Eq. (30b) was dEs5 A integrated numerically and its solution was employed in Eq. (15) to calculate the surface tension. If the potential well for anions is shallow, the total depletion thickness is positive at any electrolyte concentration and the surface tension always

Nanodispersions

406 88

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

Fig. 6. (a) The surface potential and (b) the surface tension vs. electrolyte concentration, calculated for ˚ ´s80, Ts300 K. a potential well, with d1sd2s5 A,

increases with increasing ionic strength. If the potential well is sufficiently deep (A'exp(W1 ykT))2), the total depletion thickness is negative at low ionic strength (because of the anion adsorption in the potential well) and the surface tension is decreased by the addition of electrolyte. However, the increase in surface potential with ionic strength expels the anions from the potential well and, at large electrolyte concentrations, the depletion thickness becomes positive. Consequently, the surface tension passes through a minimum. For W1s2kT, the minimum is at approximately

407

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

89

cEs0.8 M; the position of the minimum depends on the parameters of the interactions and increases with the depth of the well. 4. Applications 4.1. Zeta potential As noted above, the surface potential generated by the asymmetric distributions of electrolyte ions increases with increasing ionic strength. In contrast, electrophoretic experiments on air bubbles indicated that the surface potential decreases with the addition of electrolyte and depends strongly on the pH of the solution. For this reason, the effect of pH was included in the model, by assuming that the OHy ions are adsorbed on the interface. The adsorption of the Hq will be neglected in comparison to that of OHy. Assuming an equilibrium constant KOH for the adsorption of OHy on the interface sites, one can write: KOHs

(1yx)wOHyxS , x

(40)

where (1yx) is the fraction of free sites and wOHy xS the concentration of OHy in the vicinity of the interface sites. Consequently, the surface charge density is given by: sOHsyexNsy

eN KOH 1q wOHyxS

(41a)

where N is the number of adsorption sites per unit area of the interface. It will be assumed that the pH value was attained by adding an acid (or a base) with the same anion (cation) as the electrolyte. The salt and the acid or base will be assumed totally dissociated. The surface charge is therefore screened by monovalent ions with a total bulk concentration c9scEqcAqcB, where cA(cB) is the concentration of the added acid (base). The concentration of OHy ions in the vicinity of the surface, wOHy xS , is related to the bulk concentration wOHyx through: B

wOHyxSswOHyxexpCy D

DWOH(0) E B ecS E FexpC F, G D kT G kT

(42)

where cSsc(xs0) is the surface potential, which is negative, wOHy x is the bulk concentration, and DWOH includes all the ion interactions which are not contained in the mean field c. The latter interactions can be accounted for by using a new

408

Nanodispersions M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

90

apparent dissociation constant K9OH. With this choice, Eq. (41a) becomes: sOHsy

eN B K9OH ecS E F, 1q expCy y wOH x D kT G

(42b)

A more complete treatment should include in the Poisson–Boltzmann formalism all the interactions between the electrolyte ions and the interface, summarized in Section 2. Here it will be shown that, by accounting only for the hydration interaction of the electrolyte ions, one can capture the experimental dependence of the surface potential on pH and electrolyte concentration. It will be assumed that the cations are structure-making, with a cut-off distance d2sdE and the anions structure-breaking, with interactions given by Eq. (29a) and Eq. (29b). For simplicity, it will be assumed that d1 sd2 'dE , and that the interactions of Hq and OHy with the interface (DWH(x), DWOH(x)) are negligible. The latter approximation affects the results only at very low or very high pH values. Consequently, the potential satisfies the equations: E e BW E B ec E d2c B 1 C F C F expC F s c qc exp qc Ž . E A B 2 D kT G D kT G dx D G ´´0

qcA

B e ec E expCy F 0-x-dE D ´´0 kT G

B ec E d2c 2eŽcEqcAqcB. s sin hC F dE-x-` 2 D kT G dx ´´0

(43a)

(43b)

where cE is the electrolyte concentration and cA(cB ) is the concentration of the added acid (base). The boundary conditions are: dc Z sOH eN s Z sy dx Zxs0 ´´0 ´´0 Z Z

1 B K9OH ecS E F 1q expCy y wOH x D kT G

(44a)

and: c(x)Zx™`s0

(44b)

A boundary condition, which is equivalent to Eq. (44b), but is more convenient for calculations, is provided by Eq. (37), which is valid because for x)dE the potential satisfies the traditional Poisson–Boltzmann equation. In this case, it is enough to solve Eq. (43a) for 0-x-dE, with the boundary conditions Eq. (44a) and Eq. (37).

409

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

91

Fig. 7. The experimental values of the zeta potential for wateryair interface in presence of NaCl, as a function of pH, from Ref. w32x (triangles: 0.1 M; circles: 0.01 M; squares: 0.00001 M) are compared ˚ W1s0.5kT, K9OHs with the potential values (cS and cd ) predicted by the present model: d1 sd2 s4 A, 10y10 M, Ns5.0=1016 sitesym2, ´s80 and Ts300 K.

The numerical solution for the surface potential as a function of pH is compared in Fig. 7, for various NaCl concentrations, with the experimental results provided by Li and Somasundaran w32x. The potentials ycSsyc(0) and ycdsyc(dE) are plotted as functions of distance, since the zeta potential determined by electrophoresis is not defined at the surface, but at an unknown location, the ‘plane of slip’. The magnitude of cS is always larger than that of cd, since the potential ˚ which is provided by the dependence decays with the distance. The value dEs4 A, of the surface tension of water on the NaCl concentration at high ionic strengths was employed. For the equilibrium constant, the value K9OHs10y10 M, which is consistent with the experimental data for pH values between 3 and 6, was selected. A reasonable agreement with the data (which have a rather large error) was obtained by selecting Ns5.0=1016 sitesym2 and W1s0.5kT.. One important consequence of the present hydration model is that the predicted surface potential does not vanish at high ionic strength, because of the cut-off distance for cations. Since there is no screening in the cation depletion region, the surface potential at large electrolyte concentrations is roughly given by: cSZcE™`(

sdE ´´0

(45)

Another consequence is that the asymmetric distributions of electrolyte ions, due to their different hydration interactions, generate themselves a surface potential, which increases with increasing electrolyte concentration. At sufficiently high pH values (larger than f3) the surface charge generated by the adsorption of OHy on

410

Nanodispersions 92

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

the interface provides the dominant contribution to the surface potential. In this case, the increase of the electrolyte concentration enhances the screening of the surface charge and hence the absolute value of the surface potential decreases. However, at low pH values, the adsorption of the OHy ion is negligible and the surface potential is generated by the asymmetric distributions of the two kinds of ions. In this case, the absolute value of the surface potential is increased by the addition of electrolyte. The decrease of the absolute value of the surface potential with increasing electrolyte concentration at most pH values and its increase at very low pH values (Fig. 7) appears to be supported by experiment w32x. Karraker and Radke w18x compared the same experimental data with a model in which the surface potential was generated by the charge of the wOHy x adsorbed on the interface, which is screened by the ions of the electrolyte. While good agreement for the surface potential was obtained for cEs1=10y5 M, the predicted potential was much lower than the calculated one for 10y2 and particularly for 10y1 M. They have taken into account the image and van der Waals forces acting on the ions while we accounted only for the ion hydration instead. As noted in Section 3.2, the two approaches can lead to similar results, for suitable choices of the values of the parameters. However, for the values of B1 and B2 calculated from the polarizabilities of ions w18x, the surface potential is positive al low pH values, and not negative, as indicated by experiment. Its magnitude at 0.1 M, (f0.003 V, Fig. 3a) is also smaller than that provided by experiment. At large pH values, the adsorption of the OHy ions on the interface is responsible for the negative surface potential. A possibility to generate large surface potentials, at low pH values, would be to employ values for B1 and B2 of different signs. If one would select B1-0 and B2) 0 and a suitable cut-off distance (to avoid the divergent accumulation of anions at the interface), results similar to those obtained here would be obtained. However, this implies that the van der Waals interactions attract the anions toward and repel the cations from the interface, which contradicts the general theory of the van der Waals interactions. 4.2. Surface tension at low ionic strength. Jones–Ray effect The dependence of interfacial tension on electrolyte concentration can be related, via the Gibbs adsorption equation, to the depletion of the interfacial region of ions. At relatively large concentrations (between 0.1 and 1 M) of simple electrolytes (NaCl, KCl), the increase in surface tension with increasing ionic strength has an almost constant slope w29x. In the absence of accurate experimental data, it was reasonable to expect that this trend can be extended to low ionic strengths. The Wagner–Onsager–Samaras theory supported this expectation by predicting a positive slope at all electrolyte concentrations, which decreases with increasing ionic strength. However, a few years after the Onsager–Samaras theory, experiments performed by Jones and Ray w33x indicated that the surface tension at the wateryair interface is decreased by the addition of small amounts of KCl and exhibits a minimum at approximately cEs0.001 M. Because the relative changes in surface tension are of

411

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

93

the order of 0.01% at that electrolyte concentration, hence very susceptible to experimental error, the existence of this minimum was doubted. The negative slope of the surface tension at low ionic strength contradicted the Onsager–Samaras theory, which was considered valid in that range. Dole and Swartout w34x confirmed, however, the experimental results of Jones and Ray. Dole w35x explained the Jones–Ray effect using a modified Langmuir adsorption equation. He assumed that the interface contains ‘active spots’ which attract anions and that at low electrolyte concentration, the adsorption of the negative ions on these spots provides the main contribution in the Gibbs adsorption equation. By adding electrolyte, all the ‘active spots’ become occupied, no further adsorption of negative ions occurs and the change in surface tension is controlled by the overall depletion of ions due to image forces. Karraker and Radke w18x proposed another explanation of this minimum. At low electrolyte concentrations, the positive adsorption of OHy leads to a decrease of surface tension, while at sufficiently high ionic strengths, the depletion of Cly, repelled by the negative surface charge as well as by the van der Waals and image forces, becomes dominant and the surface tension increases. A simpler mechanism is suggested here. The adsorption of OHy ions does not affect, in a first approximation, the surface tension via the Gibbs adsorption equation. However, it generates a surface charge, which is responsible for the accumulation of cations in the vicinity of the surface. Consequently, the total surface adsorption of ions is positive and the slope of the surface tension, as a function of electrolyte concentration, is negative. The surface potential is, however, decreased by the addition of electrolyte, and the accumulation of cations is compensated by the anion depletion. Hence, by adding electrolyte, the ions depletions of the interfacial region, due to other interactions (ion hydration, image or van der Waals forces) dominate the process, and the slope becomes positive. The surface adsorption of OHy is obviously not negligible. However, the chemical potential of OHy in the bulk is given by: mOHsm0OHqkTlnŽwOHyxfOH.

(46)

and at constant pH, mOH depends on cE only via the activity coefficient, which was taken unity. Therefore, Eq. (15) yields: 1 ≠g GClyqGKq sy . kT ≠cE cE

(47)

It should be noted that the same result can be obtained by using an activity coefficient of 1 in Eq. (11) of Ref. w18x. At concentrations of approximately 0.001 M of uni-univalent electrolytes, (relevant for the Jones–Ray effect) fs1 is, in general, a good approximation. Let us evaluate how much the surface potential affects the surface adsorption of electrolyte ions. When only the short-range hydration interactions of ions with the

412

Nanodispersions M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

94

interface are taken into account, the depletion thickness is given by: y

GClyqGKq s cE

| s|

`B

2cEyc1(x)yc2(x) E Fdx D G cE

C

0

dEB

C2y

0 D

c1(x) c2(x) E Fdxy2 y cE cE G

|

`B

B ec(x) E

E

dED

D

kT

G

Ccos hC

Fy1Fdx, G

(48)

where the potential satisfies Eqs. (43a) and (43b). The first term of the right-hand˚ At cEs0.001 M and assuming side is less than 2dE, which is approximately 10 A. yec(dE)s2kT, the negative depletion thickness (the adsorption thickness) provided ˚ by the second term of the right-hand-side of Eq. (48) is approximately 200 A, largely exceeding the magnitude of the depletion thickness due to the cut-off distance. This value is so large, that the total surface adsorption remains positive and large, even when other long-range repulsions (such as the image or the van der Waals force) are taken into account. The inclusion of the image potential decreases ˚ and a similar effect is obtained by the adsorption thickness by approximately 10 A, including the van der Waals interactions. When the electrolyte concentration is increased, the range of the double layer ¨ decreases dramatically (the Debye–Huckel length decreases) and the magnitude of the surface potential also decreases. In the linear approximation, c(x)sc(dE)exp(y (xydE)yl) (for x)dE) and the second right-hand-side term of Eq. (48) becomes:

2

|

`B

B ec(x) E

E

dED

D

kT

G

Ccos hC

Fy1Fdx( G

|

B

E2

ecŽdE. l ec(x) E2 F F dxsC D kT G 2 dED kT G

`B

C

(49)

In summary, at very low electrolyte concentrations (below f0.001 M) and in the absence of a surface charge, the image forces provide the most important contribution to the (negative) ion adsorptions, and the other repulsions (ion hydration, van der Waals interactions) can be neglected. However, if there is a surface charge, the overall accumulation of ions in the vicinity of the surface (the positive overall adsorption) due to the mean field c (the Poisson–Boltzmann effect) becomes important. Furthermore, at electrolyte concentrations smaller than 0.001 M and if ec(dE) is at least of the order of kT, all the other interactions, including the image forces, become negligible. The overall ion adsorption, which is positive, is well approximated in this case by the traditional (bare-bone) Poisson–Boltzmann equation. At very large electrolyte concentrations, the range of the mean field Poisson– Boltzmann interactions as well as the image forces decrease drastically, since their decay lengths are roughly proportional to l and ly2, respectively. In this case, other interactions, such as ion hydration or van der Waals forces, become dominant. In Fig. 8, curve (1) represents the dependence of surface tension on electrolyte concentration, based on the present model. It is assumed that OHy ions are adsorbed on the surface, with an apparent equilibrium constant K9OHs1=10y10 M, and that ˚ for the cations, and a potential well of depth W1s0.5kT there are a cut-off dEs4 A

413

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

95

Fig. 8. The change in surface tension at low electrolyte concentrations (the Jones–Ray effect). Circles: Experimental data from Ref. w33x. Curve (1): predicted surface tension for the simple model which accounts for OHy adsorption and only ion hydration effects for electrolyte ions, within the Poisson– Boltzmann approach. Curve (2): predicted surface tension when image forces are also included in the model. As noted in Section 3, the image forces cannot be neglected at concentration lower than 0.01 M.

˚ for the anions. These values of the parameters are those used in and width dEs4 A the previous section, which was concerned with the zeta potential of air bubbles determined from electrophoretic experiments. However, to recover the Jones–Ray minimum approximately 0.001 M (for pH 7), a much smaller value for the number of adsorption sites on the interface N than the one used there had to be employed, namely Ns3=1015 sitesym2. At the minimum of the surface tension, the positive adsorption due to the mean field (the Poisson–Boltzmann effect) is compensated by the negative adsorption due to the ion hydration, image forces and van der Waals interactions. As already noted (Fig. 1), at the low ionic strengths where the Jones–Ray minimum occurs (f0.001 M), the image force contribution to the depletion thickness cannot be neglected. By including the image potential in the calculation of the surface adsorption, good agreement with experiment was obtained for Ns6=1015 sitesy m2 (Fig. 8, curve (2)). From the experimental values of the zeta potentials for air bubbles w32x, one could infer that the surface potential should be approximately 0.060 V for 0.001 M NaCl at pH 7. However, this value is much too large to be in agreement with the experiments of Jones and Ray w33x. The depletion thickness vanishes at the minimum of the surface tension, and at this surface potential, the adsorption thickness, due to ˚ and could not be compensated the Poisson–Boltzmann effect, would be over 200 A by the other interactions discussed here. The value which had to be selected for N

414

Nanodispersions 96

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

Fig. 9. Approximation of the interactions between ions and the interface: (1) square well for anions; (2) triangular well for anions; (3) hard wall for cations.

(Ns6=1015 sitesym2 ) to provide a good agreement with Jones–Ray minimum, led to a surface potential of only approximately 0.014 V. 4.3. Surface tension at high ionic strength At high ionic strength, the adsorption of OHy no longer plays a central role. Because the potential at the surface and its decay length are both small, the behavior of the surface tension is determined by the ion distributions within the first few ˚ Angstroms from the interface. The almost linear increase of the surface tension with the concentration of the common uni-univalent electrolytes can be easily described by employing a cut-off distance for the cations. However, experiment w29x indicated that for other electrolytes (HCl, HNO3, HClO4) the surface tension decreases with increasing electrolyte concentration up to 1 M. The adsorption thickness due to the Poisson–Boltzmann effect, which is responsible for the Jones–Ray effect at low ionic strength, cannot explain the linear decrease of surface tension at high ionic strength, since it decays too strongly with the electrolyte concentration. The image forces and the repulsive van der Waals interactions (Bi)0) generate a depletion thickness (a negative adsorption), and, therefore, also cannot explain this behavior. Can the change in cation hydration, between bulk and interface explain this effect? If the cations (such as Naq or Kq) cannot approach the interface, and the potential well for anions is not very deep, the overall adsorption of ions is negative at any electrolyte concentration. However, if the cations (such as Hq ) can penetrate at least a part of the interfacial potential well of the anions (d2-d1 ), they are

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

415 97

Fig. 10. The depletion thickness as a function of electrolyte concentration for (a) a square well interaction ˚ and W1s1kT; (b) a triangular well interaction for anions, with d1s5 A ˚ and for anions, with d1s5 A ˚ ´s80 and Ts300 K. DW(0)sy2kT; and a cut-off distance for cations d2s0, 1, 2, 3, 4 and 5 A;

attracted there by the accumulated anions and the overall adsorption can become positive. Two cases have been investigated. First, the solution of the Poisson–Boltzmann equation was obtained for an interfacial square-well potential for the anions, with ˚ Second, a triangular well was employed for the potential W1s1kT and d1s5 A. (the interaction potential varying linearly form DW1 sy2kT at xs0 to DW1s0 at

Nanodispersions

416 98

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

˚ (Fig. 9). For cations, the distance of closest approach to the interface was d1s5 A) ˚ The dependence of the slope of the interfacial selected d1s5, 4, 3, 2, 1 and 0 A. tension, (1ykT)(≠gy≠cE )sy(G1 qG2 )ycE on the electrolyte concentration is presented in Fig. 10a (for a square-well) and Fig. 10b (for a triangular well). The results are qualitatively the same: if the cations are allowed to penetrate most of the well, the surface tension decreases with increasing electrolyte concentration. If the cations are not allowed to penetrate the well, the potential is large in the well and the total surface adsorption is negative. At intermediate situations, the total surface adsorption is positive at low electrolyte concentrations (corresponding to a decrease of the surface tension with increasing electrolyte concentration) and negative at high ionic strengths (corresponding to an increase of the surface tension with increasing electrolyte concentration). The corresponding changes in surface tension as a function of electrolyte concentration are presented in Fig. 11a and b, for the square and the triangular well, respectively. While in the intermediate cases the surface tension passes through a minimum, at the extreme cases the dependence of the surface tension on electrolyte concentration is almost linear. It should be noted that a decrease of the interfacial tension with increasing electrolyte concentration was observed w29x only when the cation was Hq. The major differences in the hydration of Hq as compared to those of Naq or Kq, are probably responsible for the fact that the Hq can approach closer the interface than any other cation. Consequently, the simple model suggested here can predict a negative slope for the surface tension when Hq is the cation and a positive slope for the other cations. 5. Conclusions It has been long known that the over-simplified Poisson–Boltzmann equation is accurate in predicting the double layer interaction only in a relatively narrow range of electrolyte concentrations. One obvious weakness of the treatment is the prediction that the ions of the same valence produce the same results, regardless of their nature. In contrast, experiment shows marked differences when different kinds of ions are used. The ion-specific effects can be typically ordered in series (the Hofmeister series w36x), and the placement of ions in this series correlates well with the hydration properties of the ions in bulk water. It was suggested recently w18,26x that the van der Waals interactions between ions and two media separated by an interface can account, at least partially, for the order in the Hofmeister series and therefore can explain ion specific effects in the distribution of electrolyte ions in the vicinity of an interface. However, there is no consensus regarding the values of the interaction constants which should be used in the van der Waals interactions. It was expected that, due to the van der Waals interactions, the negative ions should be more strongly repelled by the wateryair interface than the positive ions. This seems to be, however, contradicted by experiment and simulations. In this paper, the old idea that the ion hydration should be mainly responsible for the ion-specific effects is reiterated. After a brief review of the Poisson–Boltzmann

417

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

99

Fig. 11. The change in surface tension as a function of electrolyte concentration for (a) a square well ˚ and W1 s1kT; (b) a triangular well interaction for anions, with d1s interaction for anions, with d1s5 A ˚ and DW(0)sy2kT; and a cut-off distance for cations d2s0, 1, 2, 3, 4 and 5 A; ˚ ´s80 and Ts 5A 300 K.

formalism and of a few approaches that describe ion-specific effects, a simple approach is proposed to account for ion hydration. Assuming that most of the hydration free energy is due to the interaction between ion and its water neighbors, the approach of a structure-making ion to the interface is unfavorable, because it

418

Nanodispersions 100 M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101

cannot be fully hydrated there. Since the hydration energy is large, only a minute percentage change of its value can lead to a complete depletion of ions. Therefore, a cut-off distance from the interface can describe well the interaction between ion and the interface. The long-range Wagner–Onsager–Samaras image force is important for neutral interfaces at very low electrolyte concentration, but is negligible when the interface is charged or above the physiological concentrations (f0.1 M). The van der Waals interactions can be approximately taken into account by a small change in the cut-off distance. More difficult is to treat the case of structure-breaking ions, which are pushed toward the interface, because they have there more favorable interactions with water. The consequence of a potential well for the anions in the vicinity of the interface is therefore investigated. However, the depth of the well should not be larger than a few kT, otherwise huge surface potentials would be generated at high ionic strengths, and this was not observed experimentally. It is not yet clear whether the total interaction of ions with the interface can be accounted by a simple function of distance; however, the simple treatment presented here leads to qualitatively appealing predictions. The assumptions that d1 sd2 or the simple functional form chosen for the anion interaction with the interface were made only to simplify the calculations. Better agreement with experiment can be obtained if other details are included in the model (dependence of the activity coefficient on the electrolyte concentration, additional interactions). The dependence of the surface potential on the electrolyte concentration and on the pH, determined experimentally, was reasonably well described by the traditional Poisson–Boltzmann equation, using a model involving the OHy adsorption on the interface and the treatment suggested for ion hydration. The same simple model can also explain the Jones–Ray effect (a minimum of the surface tension at a concentration of approximately 1.0=10y3 KCl w33x). At low electrolyte concentrations, the accumulation of the electrolyte ions due to the surface charge (as predicted by the Poisson–Boltzmann equation) overwhelms all the other contribution to the surface adsorptions. Consequently, the interfacial tension first decreases with increasing electrolyte concentration. However, the Poisson–Boltzmann adsorption thickness decays strongly with increasing electrolyte concentration, and the repulsion of ions by the interface (due to image forces, change in ion hydration and van der Waals interactions) becomes dominant, and the slope of the interfacial tension becomes positive. The behavior of surface tension at high ionic strength also can be understood on the basis of changes in ion hydration changes between bulk and interface. The tendency of the structure-breaking ions to accumulate at the surface can lead to a positive surface adsorption. However, if the cations cannot approach the interface, the asymmetry of the ion distributions generates a potential, which repels the anions from the interface, and the total adsorption becomes negative. Consequently, the surface tension increases with electrolyte concentration; this occurs for simple salts (NaCl, KCl). If the cations can approach the interface, the accumulation of anions in the vicinity of the interface is also followed by an accumulation of cations,

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63–101 101

without generating large surface potentials. Consequently, the total adsorption becomes positive and the surface tension decreases with electrolyte concentrations, up to high ionic strengths. This occurs, for example, for acids (HCl, HNO3, HClO4), and can be attributed to the ability of the Hq ion to approach the interface. References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x w35x w36x

G. Gouy, J. Phys. Radium 9 (1910) 457. D.L. Chapman, Phyl. Mag. 25 (1913) 475. O. Stern, Z. Eletrochem. 30 (1924) 508. ¨ ¨ J. Chem. Soc. Faraday Trans. II 79 (1983) 19. B. Jonsson, H. Wennerstrom, S. Levine, G.M. Bell, Discuss. Faraday Soc. 42 (1966) 69. ¨ B. Jonsson, ¨ H. Wennerstrom, P. Linse, J. Chem. Phys. 76 (1982) 4665. R. Kjellander, S. Marcelja, J. Chem. Phys. 82 (1985) 2122. M.J. Spaarnay, Recl. Trav. Chim. Pays-Bas 77 (1958) 872. Y. Gur, I. Ravina, A.J. Babchin, J. Colloid Interface Sci. 64 (1978) 333. I. Langmuir, J. Chem. Phys. 6 (1938) 893. B.V. Deryagin, L. Landau, Acta Physicochim. USSR 14 (1941) 633. E.J. Verwey, J.Th.G. Overbeek, Theory of Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. N.K. Adam, Physics and Chemistry of Surfaces, Oxford University Press, London, 1941. C. Wagner, Phys. Zeit. 25 (1924) 474. L. Onsager, N.N.T. Samaras, J. Chem. Phys. 2 (1934) 628. ¨ D.M.R. Williams, B.W. Ninham, Phys. Rev. Lett. 87 (2001) 168103. M. Bostrom, ¨ D.M.R. Williams, B.W. Ninham, Langmuir 17 (2001) 4475. M. Bostrom, K.A. Karraker, C.J. Radke, Adv. Colloid Interface Sci. 96 (2002) 231. F. Otto, G.N. Patey, J. Chem. Phys. 113 (2000) 2851. B.V. Deryagin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum, New York, 1987. E. Ruckenstein, M. Manciu, Langmuir 18 (2002) 7584. E. Ruckenstein, D. Schiby, Langmuir 1 (1985) 612. L. Lue, N. Zoeller, D. Blankschtein, Langmuir 15 (1999) 3726. M. Manciu, E. Ruckenstein, Langmuir 18 (2002) 5178. R.A. Stairs, Can. J. Chem. 73 (1995) 781. B.W. Ninham, V. Yaminsky, Langmuir 13 (1997) 2097. R.J. Hunter, Foundations of Colloid Science, Oxford Science Publications, 1987. P. Jungwirth, D.J. Tobias, J. Phys. Chem. 105 (2001) 10468. P.K. Weissenborn, R.J. Pugh, J. Colloid Interface Sci. 184 (1996) 550. K. Johansson, J.C. Eriksson, J. Colloid Interface Sci. 49 (1974) 469. B.E. Conway, Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam, 1981. C. Li, P. Somasundaran, J. Colloid Interface Sci. 146 (1991) 215. G. Jones, W.A. Ray, J. Am. Chem. Soc. 59 (1937) 187. M. Dole, J.A. Swartout, J. Am. Chem. Soc. 62 (1940) 3039. M. Dole, J. Am. Chem. Soc. 60 (1939) 904. K.D. Collinns, M.W. Washabaugh, Q. Rev. Biophys. 18 (1985) 323.

419

420

Nanodispersions

Advances in Colloid and Interface Science 105 (2003) 177–200

Specific ion effects via ion hydration: II. Double layer interaction Eli Ruckenstein*, Marian Manciu Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA

Abstract A simple modified Poisson–Boltzmann formalism, which accounts also for those interactions between electrolyte ions and colloidal particles not included in the mean potential, is used to calculate the force between two parallel plates. It is shown that the short-range interactions between ions and plates, such as those due to the change in the hydration free energy of a structure-makingybreaking ion that approaches the interface, affect the double layer interaction at large separations through the modification of the surface potential and surface charge density. While at short separations (below the range of the short-range ionhydration forces) the interaction can be attractive, at larger separations the interaction is always repulsive, as in the traditional theory. When the long-range van der Waals interactions between the ions and the system (ion-dispersion interactions) are accounted for in the modified Poisson–Boltzmann approach, an attractive force between plates can be generated. At sufficiently large separations, this attraction can become even stronger than the traditional van der Waals attraction between plates of finite thickness, thus generating a dominant longrange ‘double layer attraction’. At small plate separations, the attraction generated by the ion-dispersion forces combined with the electrostatic repulsion due to the double layers overlap can lead to a variety of interactions, from a weak attraction (which is typically by at least one order of magnitude smaller than the traditional van der Waals attraction between plates) to a strong double layer repulsion (for sufficiently large surface charges). Both types of ion interactions (long-range van der Waals or short-range ionic hydration) strongly affect the magnitude of the double layer interaction, and can account for the specific ion effects observed experimentally. However, they do not affect qualitatively the traditional theory of the colloid stability, which predicts that the colloid is stable when there is a sufficiently large charge on the surface, and coagulates when the van der Waals interactions between two colloidal particles dominate. The only qualitative difference found when the ion-dispersion *Corresponding author. Tel.: q1-716-645-2911, ext. 2214; fax: q1-716-645-3822. E-mail address: [email protected] (E. Ruckenstein). 0001-8686/03/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0001-8686Ž03.00068-X

421

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 178 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

interactions were incorporated into the traditional double layer theory was the emergence of a ‘double layer attraction’ at very large separations, which, however, does not affect much the stability of colloids. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Colloidal particles; Double layer interaction; Ion-hydration forces

Contents 1. Introduction ............................................................................................. 178 2. Ion distribution and interaction free energy. General formalism ............................ 3. Interaction between parallel plates immersed in a gas of neutral particles ............... 3.1. Free energy of the system and the Gibbs adsorption equation........................ 3.2. Long-range interactions between neutral particles and plates ......................... 3.3. Short-range interactions between neutral particles and plates ......................... 4. The interaction between parallel plates immersed in an electrolyte ........................ 4.1. Short-range interactions (ion-hydration forces)........................................... 4.2. Long-range interactions (ion-dispersion forces) .......................................... 5. Conclusions ............................................................................................. References ...................................................................................................

180 182 182 184 186 188 188 194 198 199

1. Introduction Langmuir demonstrated long ago w1x that the overlap of the ionic atmospheres generated near two charged parallel plates immersed in an electrolyte, calculated within the Poisson–Boltzmann formalism w2,3x, leads always to a repulsive force between the plates. This repulsion (the double layer force), combined with the van der Waals attraction, was later successfully employed to explain the stability of colloids (the DLVO theory) w4,5x. The original Poisson–Boltzmann treatment (which assumes that the ions obey the Boltzmann equation and interact only via a ‘mean potential’, which satisfies the Poisson equation) is clearly a drastic simplification. However, it provided in many cases a good qualitative agreement with experimental data, and sometimes—in a certain range of electrolyte concentrations—even a good quantitative agreement. The limitations of the traditional Poisson–Boltzmann treatment are numerous. A particularly important one refers to specific ion effects. The amphoteric latex particles coagulate when the concentration of CsNO3 is approximately 1 M, but remain stable at high pH values even when the concentration of KNO3 exceeds 3 M w6x. Nevertheless, in the traditional Poisson–Boltzmann approach the double layer repulsions for Cs and K are the same. A number of attempts were made to include additional interactions in the formalism, some of them being briefly discussed in the first part of this article w7x. With these corrections, many of them dependent on unknown parameters, the ‘modified Poisson–Boltzmann approach’ regained its explanatory power, and in general most of the experiments could be accounted for

422

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 179

by using suitable values for the unknown interaction parameters. However, both the simplicity and the predictability of the traditional treatment were lost, since in general, the choice of the parameter values was dependent on the particular problem investigated. In contrast, the ‘specific ion effects’ seems to be rather well organized in a ‘lyotropic series’, first obtained empirically by Hofmeister from data on protein salting out w8x. The ions seems to obey rather well the lyotropic series sequence in many different experiments (an exhaustive compilation being provided in Ref. w9x), and the specific ionic effects relevant in biophysics are particularly strong for anions at moderate ionic strengths (0.01–1 M w10x). It was also noted that different experiments might lead to slight alterations of the series w9x. An example is provided by the surface force experiments of Pashley w11x between crossed mica cylinders in various electrolytes, which indicated the occurrence of an additional repulsion in the vicinity of the surface in excess to the double layer interaction predicted by the traditional Poisson–Boltzmann equation. Such an interaction was observed at low ionic strengths (;10y4 –10y2 M) for NaCl, KCl and CsCl, but only at much larger electrolyte concentrations (;6=10y2 M) for LiCl. Therefore, the ‘specific ionic effect’ of Li seems to be quite different from those of Na, K and Cs. In contrast, the amphoteric latex colloids w6x were stable when the cation was K or Li, but coagulated for Cs. The relative regularity of the Hofmeister series suggested that the specific ion effects might be associated to the properties of the ions in the solvent, and it was noted long ago that the Hofmeister series is related to the hydration of ions w12x and to the ability of the ions to organize the neighboring water molecule w10x. More recently, it was proposed than the sequence of ions in the lyotropic series could be explained by the van der Waals interaction between the ions and the rest of the system w13x. When suitable constants for the van der Waals interactions were selected, the accounting for these ion-dispersion forces provided a quantitative agreement with the experiments regarding the surface tension of aqueous electrolytes w14,15x. More dramatic results were obtained when the ion-dispersion forces were accounted in the double layer interactions w16x. At low electrolyte concentrations (below 0.01 M), the specific ionic effects due to the van der Waals interactions of ions were negligible. However, under physiological conditions (cEs0.1 M NaCl, surface potential cSf0.010 V) the double layer repulsion between parallel plates ˚ The became attractive for separation distances smaller than approximately 20 A. maximum interaction free energy due to the double layer repulsion calculated in ˚ 2 , which is one order of magnitude Ref. w16x was only approximately 5=10y5 kTyA smaller than a typical van der Waals attraction between proteins at that separation distance. Therefore, the net interaction (namely, the double layer repulsion, which includes the van der Waals interactions of the electrolyte ions with the system, plus the van der Waals interactions between two parallel plates) is in general attractive and all the ‘physiological colloids’ should coagulate. It should be noted that there is an extensive literature about the existence of a ‘double layer attraction’ w17x, and there is no general agreement yet about its nature.

423

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 180 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

However, such an attraction occurs at large separation distances, and does not affect much the interactions at low separations, hence the coagulation of colloids. It is also well accepted that the classic Poisson–Boltzmann treatment is inaccurate at concentrations higher than approximately 0.05 M, and in general either additional interactions or unknown fitting parameters had to be employed for quantitative agreement. One of the major successes of the inclusion of the ion-dispersion forces into the Poisson–Boltzmann formalism was the explanation of the dependence of the surface tension of electrolytes on ionic strength w14,15x and pH w15x. In a previous article, it was shown that this dependence could be also explained by a simple accounting of the interactions between the ion and surface, through the changes in the ion hydration free energy between bulk and interface w7x. The latter treatment avoided some difficulties that occurred when only the van der Waals interactions of the ions with the system were taken into account (the accumulation of cations near the interface and the positive surface potential generated, in contrast to the accumulation of most anions and the negative surface potential provided by experiment and molecular simulations w18x). It also showed that the rather high values of the coefficients of the ion-dispersion interactions, obtained from the fit of the experimental data w14x, might have been caused by the fact that the ion-hydration interactions were neglected. In this article, a general formalism to calculate the interactions between colloidal particles, when additional interactions are included in the Poisson–Boltzmann formalism is presented. The effects of ion-hydration and ion-dispersion interaction potentials on the force between parallel plates are investigated. While quantitative effects strongly dependent on the ion specificity are obtained, the calculations indicate that the DLVO theory remains essentially valid, namely that the colloids are stable if their surface is sufficiently charged. The values of the surface charge density and surface potential, which provide stability, are of course different if additional interactions are accounted in the Poisson–Boltzmann formalism. It will be also shown that the ion-dispersion interactions might lead, at large separation distances, to an attraction between plates. This ‘double layer attraction’ might become at large separations, even stronger than the traditional van der Waals attraction between parallel plates of finite thickness. 2. Ion distribution and interaction free energy: general formalism As in the previous article w7x, the equality of the electrochemical potential through the system leads to the following equation for the distribution of the ions of an uniunivalent electrolyte of concentration cE: B

ciscEexpCy D

(y1)iecqDWi E F G kT

(1)

where ‘i’ stands for the kind of ions (1 for anions and 2 for cations), e is the protonic charge, k is the Boltzmann constant, T is the absolute temperature, DWi is

424

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 181

the change in the interaction energy of an ion ‘i’ (not included in the mean potential c) with respect to the bulk. The potential c obeys the Poisson equation: =2cs

eŽc1yc2.

(2)

´´0

where ´0 is the vacuum permittivity and ´ is the dielectric constant of the medium, which here is assumed to be constant through the solvent. The distribution of electrolyte ions can be calculated, for suitable boundary conditions, from Eqs. (1) and (2). The total free energy of the system is composed of several contributions: an ‘electrostatic energy’ due to the distribution of charges, an energy due to the ‘additional interactions of ions’ (not included in the mean potential), an ‘entropic’ term due to the mobile ions, and a ‘chemical free energy’ which is responsible for the formation of the double layer. The electrostatic energy is given by: Fels

1 2

| ´´ (=c) dV, 2

(3)

0

V

and the entropic contribution can be calculated from w19x: B

i

B

E

E

| Cc lnC cc Fyc qc FdV

FentsyTDSskT8

VD

i

i

D EG

i

E

G

(4)

where both integrals are calculated over the volume V of the solvent. The interaction free chemical energy (calculated with respect to the infinite separation of the components of the system) is given by w20x: B

| Cy|

DFchems

AD

E

s

cS(s)dsFdA G

s(`)

(5)

where s is the surface charge density, cS the surface potential, the surface integral is performed over the entire area A of the system (the interface between the solvent and the interacting surfaces), and the other integral is performed along the trajectory of the system from infinite separation distance between the surfaces to the final configuration of the system. To these components of the traditional double layer free energy one should add an energetic contribution due to the supplementary ion interactions:

| Žc DW qc DW .dV.

Fints

1

V

1

2

2

(6)

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

425

182 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

Consequently, the interaction free energy of the system (with respect to infinite separation distance between components) is given by: (7)

DFsDFelqDFentqDFchemqDFint

3. Interaction between parallel plates immersed in a gas of neutral particles 3.1. Free energy of the system and the Gibbs adsorption equation The formalism described in Section 2 can be applied to systems of any geometry. To simplify the calculations, it will be applied here to the interactions between two parallel plates. Let us first assume that two large parallel plates are immersed into a reservoir containing a neutral gas of concentration cN . If the gas interacts with the inner side of the plates (the interactions with the other sides are neglected), the free energy of the system (the plates and the adsorbed gas between them) per unit area is given by:

|

d

FN(2d)sFint(2d)qFent(2d)s B

d

| sc kT|

ydD

N

B c(x) E

E

cN

G

Cc(x) lnC

qkT

D

B

B

ydD

D

d

c(x)DW(x)dx yd

C1yexpCy

Fyc(x)qcNFdx G

DW(x) EE

FFdx

kT

GG

(8)

where 2d is the distance between plates, DW(x) is the change in the energy of a particle located at a distance x from the middle between the plates with respect to that at infinite distance and the gas particles are assumed to obey a Boltzmann distribution. The same result can be obtained using a Gibbs adsorption procedure, introduced by van der Waals and Makor w21x and generalized by Ash et al. w22x. The positive or negative surface excess adsorption of gas particles, GN, between plates is related, via the Gibbs adsorption equation, to the change in the surface tension g: B

dgsyGNdmNsyC

E

d

| Žc(x)yc .dxFdm

D

N

G

0

B

dB

B

E

E

E

D

0D

D

G

G

G

N

Fy1FdxFdm | CexpCy DW(x) kT

sycNC

N

(9)

where mN is the chemical potential of the neutral particles, which is constant through the system and can be related to the concentration of the neutral particles in the

Nanodispersions

426 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 183

reservoir, for a perfect gas, through: (10)

dmNsdŽkT lnŽcN...

Integrating Eqs. (9) and (10) for a concentration of particles varying from 0 to cN, one obtains:

|

cN

gŽcN.yg(0)sykT

0

GN(c) dcsycNkT c

dB

B

E

E

0D

D

G

G

Fy1Fdx | CexpCy DW(x) kT

(11)

and, as expected, the force between plates w21x, PŽ2d.sy

≠g ≠d

(12)

is identical to that obtained from the derivative of the free energy, Eq. (8), with respect to the separation distance 2d. For a neutral gas, the adsorption (which is proportional to the free energy of the system, see Eq. (8)) can be calculated easily. For charged particles one should account in the Gibbs adsorption equation for the adsorption of all particles of the system (including those responsible for the charging of the surface w23x). Therefore, one should first identify the mechanism of formation of the double layer. In this case, m(cG)

|

gŽcE.yg(0)sykT8

(11a)

Gidmi

y`

i

where i is taken over all the ionic species present in the system and Gi is the adsorption of ion i. The result is incomplete when one accounts only for the distributions of electrolyte ions between the plates. Indeed, for an uni-univalent electrolyte of concentration cE, the adsorption G9 of the two species of ions between two plates with constant surface charges s, is given at low surface potentials by: 2G9ŽcE. cE

|

s2

B

d

B

ydD

E

kT

G

D

esl E2 F D kT´´0 G

(C

B ec(x) E

CcoshC

|

B

e E2 F D kT G

Fy1Fdx(C G

BxE cosh2C F DlG

|

d

(c(x))2dx yd

B 2d E l sinhC Fqd D l G d B esl E2 2 C F dxs BdE BdE D kT´´0 G yd sinh2C F sinh2C F DlG DlG

(13)

where l is the Debye length. Since (≠y≠(2d))(G9(cE)ycE)-0 for any cE, using

427

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 184 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

Eqs. (9)–(12) would predict an attraction between the plates (P-0), which is incorrect. In what follows, the direct calculation of the free interaction energy by adding the energetic and entropic contributions (Eq. (7)) will be employed. 3.2. Long-range interactions between neutral particles and plates To have an idea about how important are the dispersion interactions of electrolyte ions, we will consider in this section the force between plates generated by a neutral gas, whose particles interact with the plates by long-range dispersion forces. An interaction of the type w16x: DWvdW(x)s

B B q (dyx)3 (dqx)3

(14)

will be assumed, where 2d is the separation distance between plates. For a molecule of gas, B is negative (the particles are attracted by the plates). However, if the particles are immersed in a medium, it is possible to have positive B values, depending on the Hamaker constants of the plates, particles and medium. Assuming a cut-off distance D for the interaction, the free energy per unit area (Eq. (8)) becomes, in the linear approximation (which is accurate if B is small or if the particle is sufficiently far from any interface):

|

FN(2d)(cNkT

d

B DW

C

ydD

(x) E 4d(dyD) FdxsBcN 2 G kT D (2dyD)2 vdW

(15)

and the corresponding interaction free energy, per unit area, is: F(2d)yF(`)sy

BcN (2dyD)2

(16)

This interaction, due to the interaction between particles and plates, has the same functional form at large separations (d4D) as the well-known van der Waals attraction between thick parallel plates w24x: FvdW,platessy

AH 12p(2d)2

(17)

where AH is the Hamaker constant. When the particles are attracted by the plates through dispersion forces (B-0), the net interaction between plates is repulsive, while when the interactions between particles and plates are repulsive (B)0), the plates are attracted to each other. The magnitude of the interaction generated by the dispersion forces between the neutral particles and the plates (Eq. (16)) is proportional to the concentration of

428

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 185

Fig. 1. The force between parallel plates for a neutral system, generated by ion-dispersion forces at cEs 0.1 M, for B1sB2s31=10y50 J m3 (curve (2)); B1sB2s15=10y50 J m3 (curve (3)); B1sB2sy 1=10y50 J m3 (curve (4)) are compared to the traditional van der Waals attraction between plates for AHs10y20 J (curve (1)).

particles in the reservoir (cN ) and is typically much smaller than the traditional van der Waals attraction between plates. The two interactions (Eqs. (16) and (17)) become comparable at large separation distances for: cN(

AH . 12pB

(18)

Since the Hamaker constant for proteins in a physiological solution is of the order of 10y20 J, Eq. (18) provides a concentration of approximately 2 M even for the large value of Bs31=10y50 J m3 corresponding to Cly ions w14x. This corresponds to an ideal gas pressure of approximately 50 atm. Consequently, the long-range repulsion or attraction between plates due to the van der Waals interactions between a gas of neutral particles and the plates is expected to be negligible in most cases. The interactions between neutral plates immersed in an electrolyte can be calculated from Eqs. (11) and (12), providing that the dispersion forces are the same for both kinds of ions and hence the potential vanishes everywhere. In Fig. 1,

429

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 186 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

the forces (per unit area) between parallel plates, calculated numerically for c1s ˚ and various values B1sB2 (curves 0.1 M, Ts300 K, a cut-off distance Ds2 A (2–4)) are compared with the traditional van der Waals attraction due to the polarizability of the plates, for a typical Hamaker constant of AHs1=10y20 J. The attraction generated by the ion-dispersion forces at this concentration is in general much smaller than the traditional van der Waals attraction (curve 1). When B1'B2-0 (curve 4), the interaction generated by ion-dispersion forces is a repulsion. In general, B1/B2 and a double layer is generated by the asymmetric distribution of the electrolyte ions even if there is no charge on the plate surface. A treatment based on the modified Poisson–Boltzmann formalism will be presented in Section 4.2. 3.3. Short-range interactions between neutral particles and plates In a previous article w7x, it was suggested that the changes in ion hydration between bulk and the vicinity of the surface might be approximated by a simple step function. In what follows the interaction between parallel plates will be investigated for an interaction between particles and plates of the type: DW(j)sW1 0-j-d1

(19a)

d1-j-`

(19b)

DW(j)s0

where j is the distance to one of the plates and W1 can be negative (corresponding to an attraction between particles and plates) or positive (repulsion). It is also assumed that the interactions between particles and plates are additive (Fig. 2): (19c)

DWsDW(j)qDW(2dyj),

2d being the separation between plates. Due to the short-range of this interaction, the force between plates vanishes for d)d1. For (d1 y2)-d-d1, the free energy per unit area of the system (Eq. (8)) is given by: F(2d)scNkT

|

B

B

ydD

D

d

C1yexpCy

DW(x) EE FFdx kT GG

B

B 2W EE

D

D

w

s2cNkTxŽd1yd.C1yexpC y

1

kT

B

B W EEz

D

D kT GG~

FFyŽ2dyd1.C1yexpC GG

1

FF|

w B

BW E

E2

B

BW E

B 2W EEz

y D

D kT G

G

D

D kT G

D

s2cNkTxdCexpC

1

Fy1F qd1CexpC

1

FyexpC

1

kT

FF| GG~

(20)

430

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 187

Fig. 2. An additive step-like interfacial interaction between neutral particles and plates. There is no force between plates for d)d1, attraction when (d1y2)-d-d1 and either attraction or repulsion for 0-d(d1y2).

and the force per unit area between plates by P(2d)sy

B BW E E2 ≠F(2d) 1 sycNkTCexpC Fy1F -0 D D kT G G ≠(2d)

d1 -d-d1 2

(21)

431

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 188 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

is always attractive in this region (P-0), regardless of whether the particles are attracted toward (W1-0) or repelled (W1)0) by the plates. For the range 0-d-(d1 y2), the force per unit area is given by: B

BW E

E

D

D kT G

G

P(2d)scNkTCexpC

1

Fy1F 0-d-

d1 2

(22)

which leads to a net repulsion between plates if the neutral particles are attracted by the plates and to an attraction when the neutral particles are repelled by the plates. In summary, short-ranged step-like interactions for ions generate only short-range interactions between plates, which are always attractive for (d1 y2)-d-d1 and become either attractive or repulsive, depending on the sign of W1, for 0-d-(d1 y 2). 4. The interaction between parallel plates immersed in an electrolyte 4.1. Short-range interactions (ion-hydration forces) The change in the ion hydration energy between the bulk and the water–air interface for structure-making ions is so large compared to kT, that these ions practically do not approach the interface. The change is not so steep when a solid interface is immersed in water, particularly when the surface has sites which can bound structure-making ions. In what follows, it will be assumed that the anions are structure breaking and hence interact with the surface via an attractive potential of the type (W1)0, see Fig. 3a): DW1(j)syW1 0-j-d1

(23a)

DW1(j)s0

(23b)

d1-j-`

and the cations are structure making and interact with the surface via a repulsive potential (W2)0): DW2(j)sW2 0-j-d2

(24a)

DW2(j)s0

(24b)

d2-j-`

The modified Poisson–Boltzmann equation for a uni-univalent electrolyte confined between parallel plates is: B d2c ecE B B DW1 E B ec E DW2 E B ec EE CexpCy FexpC FyexpCy FexpCy FF s 2 D d x ´´0 D D kT G D kT G kT G D kT GG

(25)

432

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 189

Fig. 3. (a) A typical ion-hydration potential: the anions are attracted toward the interface (W1s1 kT, ˚ while the cations are repelled by the surface (W2 s2 kT, d2 s3 A). ˚ (b) The potential and the d1s6 A) ˚ W2 s2 kT, d2 s3 A, ˚ ds15 A, ˚ cEs0.1 M, cSs corresponding ion distributions, for W1 s1 kT, d1 s6 A, y0.010 V, ´s80, Ts300 K.

which has to satisfy two boundary conditions, one due to the symmetry: dc Z Z s0 dx Zxs0 Z Z

(25a)

433

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 190 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

and the other providing the surface charge density, the surface potential or a relation between them. Since the selected DWi(j) is independent of j on each of the intervals 0-j-d2, d2-j-d1, d1-j-d (Fig. 3a), the solution in the linear approximation (valid at low potentials) in the interval ‘j’ is of the type: B

c(j)sy

kT AjyDj qC2jexpC e AjqDj D

y

E

B AjqDj j FqC2jq1expCy D 2 lG

E

j F y A qD 2 l j

j

G

(26)

where Ajsexp(y(DW1)j y(kT)), Dj sexp(y(DW2 )j y(kT)), ls6((´´0kT)y(2e2cE )) is the Debye length and in each interval j, C2j , C2jq1 are constants to be determined from the boundary conditions and the continuity of the electric potential and electric field. A typical result for the potential and the corresponding ion distribution is ˚ W2s2 kT, d2s3 A, ˚ ds15 A, ˚ cEs presented in Fig. 3b, for: W1s1 kT, d1s6 A, 0.1 M, cSsy0.010 V, ´s80, Ts300 K. Once the potential has been determined, the interaction free energy of the system can be calculated using Eq. (7). For separation distances d)d1, the Poisson–Boltzmann equation is obeyed within d1-j-2dyd1. The system is completely equivalent with a traditional double layer, but formed between the surfaces S9(jsd1 ) and not between the real surfaces S(js 0). The corresponding ‘surface potential’ is the potential at S9, cS'c(jsd1), and the corresponding surface charge density, s9, is given by:

|

d1

r(j)djsy´´0

s9ssq

0

dc Z . Z dj Zjsd1 Z Z

(27)

Therefore, the short-range interactions due to ion hydration do not modify qualitatively the double layer interaction, which remains repulsive at distances larger than the range of the interfacial interaction of the ions. The magnitude of the repulsion is, however, modified, since the relation between the ‘true’ surface charge density s and the ‘apparent’ surface charge density s9 depends on DWi and hence on the specificity of the ions. To illustrate this, the apparent charge density s9 is calculated in Fig. 4a and b as ˚ and constant surface charge a function of the separation distance, for d1'd2s5 A density. The apparent charge density is proportional to the derivative of the electric potential at the distance jsd1'd2. When the surface is negatively charged (ssy 0.010 Cym2, Fig. 4a), both the accumulation of anions (DW1-0) and the cation depletion (DW2)0) lead to an increase of the charge in the vicinity of the surface, therefore to an increase in the double layer repulsion. When the surface is positively charged (ss0.010 Cym2, Fig. 4b), the ion–hydration interaction can lead to a decrease in the apparent surface charge density (curve (4)), which can even change sign. The apparent charge density is plotted in Fig. 4c as a function of electrolyte ˚ and ssq0.010 Cym2. Depending on the potentials of concentration, for 2ds20 A

434

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 191

ion hydrations, s9, which is positive at sufficiently low concentrations, can become negative when the ionic strength is sufficiently high. The electrolyte concentration which correspond to s9s0 and hence, roughly, to coagulation depends strongly, via W1 and W2, on the type of electrolyte. Therefore, short-range interfacial interactions of ions, of the order of kT, can account for specific ion effects in the double layer interactions. The force between neutral plates (ss0) immersed in an electrolyte of concentration cEs0.1 M, calculated from the derivative of the free energy of the system (Eq. (7)) is plotted in Fig. 5a. The double layer, which is generated by the different distributions of the positive and negative ions of the electrolyte alone (it vanishes when DW1sDW2), is always repulsive for d)d1'd2, and can be either attractive or repulsive for d-d1.

Fig. 4. (a) The apparent surface charge density s9 for an interaction at ‘constant surface charge density’ (ssy0.01 Cym2, cEs0.1 M) as a function of the separation between plates. The long-range double layer interaction is controlled by s9, which in turn depends on the short-range potential of interaction between ions and surface. (b) The apparent surface charge density s9 for an interaction at a positive ‘constant surface charge density’ (ssq0.01 Cym2 , cE s0.1 M) as a function of the separation between plates. The apparent surface charge density can be decreased or even change its sign when the shortrange interaction between ions and surface is accounted for. (c) The apparent surface charge density s9 ˚ as a function of the electrolyte concentration. While the double layer for ssq0.01 Cym2 and 2ds20 A interaction is always repulsive at sufficiently large concentrations, the electrolyte concentration corresponding to s9s0 (hence, to a vanishing double layer repulsion) depends strongly on the short-range interactions between ions and surface.

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 192 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

Fig. 4 (Continued).

435

Nanodispersions

436 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 193

Fig. 5. (a) The force per unit area between neutral parallel plates immersed in an electrolyte with cEs 0.1 M. The (asymmetric) short-range interactions between ions and interface lead to an asymmetric distribution of electrolyte ions, which in turn generates a long-range double layer repulsion. In the vicinity ˚ the force can become attractive. (b) The force per unit area between negaof the surface (2d-10 A), tively charged parallel plates immersed in an electrolyte with cE s0.1 M. The interactions between ions and surfaces increase in this case the apparent surface charge density, hence increase the repulsion at ˚ large separation distances. The force can become attractive for 2d-10 A.

437

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 194 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

A more realistic calculation is presented in Fig. 5b, which implies that there is a surface charge density generated by the dissociation of surface groups. Assuming that the cations can be bound to N negative sites per unit area, the dissociation equilibrium provides the expression: KDs

B (hN)cq hcq h ecS W2 E E E s s cEexpCy y F D (1yh)N (1yh) 1yh kT kT G

(28)

where KD is the dissociation constant, cEq denotes the cation concentration in the liquid in the vicinity of the surface and h is the fraction of dissociated sites. The surface charge density is therefore given by: ssyehNsy

eN B

cE ecS W2 E 1q expCy y F D KD kT kT G

(29)

The ion–hydration interactions lead to an accumulation of anions and a depletion of cations near the interface, thus increasing the magnitude of the apparent surface charge density s9. Consequently, the double layer repulsion is enhanced at large distances. When the surface charge density is positive at low electrolyte concentrations (e.g. when the charge is generated by the dissociation of basic surface groups), the ion hydration can decrease the magnitude of the repulsion at large distances, but cannot transform it there into an attraction. The apparent charge density, which is ultimately responsible for the double layer interaction, depends strongly on the hydration potential of ions. Although the double layer interaction is always repulsive, for some electrolytes s9 can become sufficiently large to prevent coagulation, while for other electrolytes sufficiently low to allow coagulation. Therefore, the ion–hydration interaction can provide a suitable explanation for the specific ion effects observed in double layer interactions. 4.2. Long-range interactions (ion-dispersion forces) In this section, the effect of the ion-dispersion forces on the double layer interactions will be investigated, by assuming that they provide the only correction to the traditional Poisson–Boltzmann equation. When the dispersion interaction coefficients are different for anions and cations, the different distributions of the two kinds of ions generate a potential even when the plates are neutral. The potential is obtained from the solution of the correspondingly modified Poisson–Boltzmann equation (Eqs. (1) and (2)) and the interaction free energy can be calculated via the numerical integration of Eq. (7). The force between neutral plates immersed in an uni-univalent electrolyte of concentration cEs0.1 M, when B1/B2, is presented in Fig. 6(a), for a cut-off ˚ While the force was always attractive when B1sB2)0 (Fig. 1), distance Ds2 A. the double layer generated by the asymmetric distributions of electrolyte ions, when

438

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 195

B1/B2, can dominate the interaction (which becomes repulsive) at intermediate separations (B1s31=10y50 J m3, B2s1=10y50 J m3, curve (1) in Fig. 6a). If one type of the ions (the cation) is attracted by the interface (B2sy1=10y50 J m3, curve (2)), the repulsion induced by the ion-dispersion forces becomes, as expected, even stronger.

Fig. 6. (a) The force between neutral plates immersed in an electrolyte with cE s0.1 M, due to the iondispersion forces. At intermediate separations, the double layer generated by the asymmetric distribution of electrolyte ions leads to a net repulsion, although both types of ions are repelled by the interface (B1s31=10y50 J m3, B2s1=10y50 J m3). When the ions of one kind are attracted by the interface (B1sy1=10y50 J m3), the repulsion increases. The interactions at large separation distances (inset) are attractive. (b) The force between negatively charged plates immersed in an electrolyte with cEs0.1 M, (1) Ns3=1016 sitesym2, KDs0.1 M, B1sB2s0; (2) Ns3=1016 sitesym2, KDs0.1 M, B1s 31=10y50 J m3 and B2sy1=10y50 J m3; (3) ss0, B1s31=10y50 J m3 and B2sy1=10y50 J m3; (4) Ns1=1017 sitesym2, KDs0.1 M, B1s31=10y50 J m3 and B2 sy1=10y50 J m3 ; the ‘double layer attraction’ (inset) occurs at large separations, even in the presence of large surface charges. (c) The force between parallel plates, for a moderate surface charge generated by the dissociation of surface groups, for Ns1=1017 sitesym2, KDs0.1 M, ´s80, Ts300 K and various sets of ion-dispersion coefficients: (1) B1 s0, B2 s0; (2) B1s15=10y50 J m3, B2 s3=10y50 J m3 ; (3) B1s15=10y50 J m3, B2s1=10y50 J m3; (4) B1s15=10y50 J m3, B2sy1=10y50 J m3; (5) B1s15=10y50 J m3, B2sy 3=10y50 J m3; (6) B1s31=10y50 J m3, B2s3=10y50 J m3; (7) B1s31=10y50 J m3, B2s1=10y50 J m3; (8) B1s31=10y50 J m3, B2sy1=10y50 J m3; (9) B1s31=10y50 J m3, B2sy3=10y50 J m3; In the inset, the calculations are repeated for a strong double layer (Ns2=1017 sitesym2, KDs1 M).

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 196 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

Fig. 6 (Continued).

439

440

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 197

The maximum repulsion generated by the asymmetric distribution of ions is however, small when the surface charge is zero (approximately one order of magnitude smaller than a typical van der Waals attraction) and therefore the neutral physiological colloids will coagulate, in agreement with the DLVO theory. More interesting are the effects of ion-dispersion forces when the plates are charged. Let us first assume that a small negative surface charge is generated by the dissociation of surface groups (Eqs. (28) and (29), with cEs0.1 M, Ns 3=1016 sitesym2 and KDs0.1 M). In the absence of the van der Waals interactions of ions (B1sB2s0), a weak double layer repulsion is provided by the traditional theory (curve (1), Fig. 6b). By accounting in the modified Poisson–Boltzmann equation for the van der Waals interactions of ions with B1s31=10y50 J m3, B2s ˚ but assuming ss0, curve (3) was obtained. When y1=10y50 J m3 and Ds2 A, both the van der Waals interaction and the surface charge density were accounted for (curve (2)), the interaction became attractive at most distances. However, if the surface charge density is sufficiently large (curve (4) in Fig. 6b, with Ns1=1017 sitesym2) then the interaction in the vicinity of the surface is repulsive. Therefore, a strong double layer interaction is not modified much at small separations by ion-dispersion forces. The ion-dispersion forces (as the ion-hydration forces, discussed previously) might constitute a suitable candidate to account for the ion specific effects within the Poisson–Boltzmann formalism, provided that suitable values for the interaction parameters are chosen. In Fig. 6c, the force between plates was calculated for a moderate double layer, generated by the dissociation of surface groups (Ns 1=1017 sitesym2, KDs0.1 M), and for cEs0.1 M, ´s80 and Ts300 K. In the absence of ion–dispersion interactions (B1 sB2 s0), the surface potential is approximately y0.020 V. When the ion–dispersion interactions were included, the repulsion in the vicinity of the surface has always decreased. The decrease of the repulsion seems to be particularly affected by B1, which typically has larger magnitudes than B2 w14,15x (B1s15=10y50 J m3 for the curves (2)–(5) and B1s 31=10y50 J m3 for the curves (6)–(9)). The change of B2 from B2s1=10y50 to B2s3=10y50 J m3 (curves (2)–(3) and (6)–(7)) had a much smaller effect. The specific ionic effects of cations become more important when B2-0, because the accumulation of positive charges at the surface compensates for the negative charge generated through the dissociation and this attenuates the double layer. In the inset of Fig. 6c, the calculations were repeated for the same values of the parameters, but a stronger double layer (Ns2=1017 sitesym2, KDs1.0 M). The ion-dispersion forces have in this case only a minor effect, and the interaction can be well approximated by the traditional Poisson–Boltzmann approach, with slightly modified parameters (density of sites, dissociation constant). There is, however, a range in which the ion-dispersion forces generate qualitatively new results. The double layer between two plates immersed in an electrolyte decays exponentially with the distance and becomes vanishingly small at large separations. The interactions generated by the van der Waals interactions of the ions with the system decay slower and become dominant at those separations. In the inset of Fig.

441

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 198 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200

6b, the ‘double layer force’ between plates becomes attractive at distances larger ˚ despite the relatively large surface charge density, which than approximately 40 A, generates a repulsion in the vicinity of the surface. The ‘equivalent’ Hamaker constant of such an attraction f12pcE(B1 qB2 ) is small, for usual electrolyte concentrations (cE-1 M), when compared to the Hamaker constants of proteins (AHf10y20 J). However, if the plates have a finite thickness l, their van der Waals interactions at large separations is given by w24x: FvdW,platessy

E AH B 1 1 2 AHl2 1 C F q y fy Ay 2 2 2 4 12p D (2d) (2dq2l) (2dql) G 2p(2d) (2d)4

(30)

At very large separations (d4l), the attraction induced by the van der Waals interactions of ions, which decays only with the second power of the distance (Eq. (16)), becomes much larger than the traditional van der Waals attraction between plates, and its magnitude is proportional to the electrolyte concentration. 5. Conclusions The traditional Poisson–Boltzmann treatment of the electrolyte ions, while simple and relatively accurate, is incomplete, since the interactions are treated only via a mean field. It predicts that the double layer interactions depend only on the valence of the electrolyte ions, whereas it is well documented that different electrolytes of the same valence lead to qualitatively different results. It was recently suggested that the van der Waals interactions of the ions with the system can explain the ion specific effects w13x, and it was shown that by accounting for this interactions one can successfully predict the dependence of surface tension of aqueous solutions of electrolytes on their concentration w14,15x. In the first part of this article w7x, it was shown that the ion–hydration interaction can also explain the surface tension behaviour. In this article, the effect of the interfacial interaction of ions (ion-hydration and ion-dispersion forces) on the double layer interaction was investigated. When only short-ranged interactions were taken into account (such as the ion–hydration interactions), the magnitude of the repulsion between plates at large separations became dependent on the magnitude of an ‘apparent charge density’, which is the sum between the real surface charge density and the charge due to the accumulationy depletion of electrolyte ions in the vicinity of the interface. The accounting of the ion–hydration interactions can increase or decrease the apparent surface charge density, hence increases or decreases the interaction, which remained, however, always repulsive at large separations. The long-range double layer interactions are strongly dependent on the short-range interfacial interaction of ions. Consequently, specific ion effects can be accounted for in this manner. When the ion-dispersion forces are included in the Poisson–Boltzmann treatment, an attraction was obtained at most distances, in the absence of a surface charge density. The magnitude of the attraction was in general smaller than the traditional

442

Nanodispersions E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 199

van der Waals attraction between plates. When a small (negative) charge was added on the surface, the attraction was even increased. However, when a large charge density was present on the interface, the interaction became repulsive (and strong) at small separations. When either ion–hydration interaction or ion-dispersion forces were included in the treatment, the results were qualitatively identical to the traditional DLVO theory, which roughly predicts that ‘strongly charged’ colloidal particles are stable, and ‘ weakly charged’ particles coagulate. Significant quantitative differences, which can account for specific ion effects, could be introduced by either mechanism, when suitable interaction parameters were selected. There are some qualitative difficulties when the specific ion effects are explained via the dispersion forces of the ions. Particularly the anions, for which the dispersion coefficients B1 are large, affect the double layer interactions. However, experiments on colloid stability w6x or colloidal forces w11x revealed strong specific ion effects especially for cations. Furthermore, the ions which affect most strongly the solvating properties of the proteins are those from their vicinity, since they perturb mostly the structure of water near the proteins. However, the van der Waals interactions of ions predict that the cations remain in the vicinity of an interface, and the anions are strongly repelled, while Hofmeister concluded that anions are mainly responsible for the salting out of proteins. Accounting for ion hydration is more appealing qualitatively. The structuremaking ions (mostly cations) are repelled by the surface, which therefore becomes more negatively charged. Consequently, the cations can generate via this mechanism long-range specific effects. Furthermore, there are the chaotropes (structure-breaking) anions, which are expelled from the bulk water toward the interface, therefore they modify more strongly the structure of water near the proteins. The structuremaking cations prefer the bulk of the electrolyte, hence, they are expected to affect less the solvation properties of the proteins. It is noteworthy that the ion–dispersion interactions induce an attraction between plates, which prevails at sufficiently large separations, regardless of their surface charge density. At separations much larger than the thickness of the plates, this attraction can even exceed the traditional van der Waals attraction between plates. References w1x w2x w3x w4x w5x w6x w7x w8x w9 x

I. Langmuir, J. Chem. Phys. 6 (1938) 893. G. Gouy, J. Phys. Radium. 9 (1910) 457. D.L. Chapman, Phyl. Mag. 25 (1913) 475. B.V. Deryagin, L. Landau, Acta. Physicochim. URSS 14 (1941) 633. E.J. Verwey, J.Th.G. Overbeek, Theory of Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. T.W. Healy, A. Homola, R.O. James, R.J. Hunter, Farady Discuss. Chem. Soc. 65 (1978) 156. M. Manciu, E. Ruckenstein, Adv. Colloid Inter. Sci. 105 (2003) 63. ¨ Experimentelle Pathologie und Pharmakologie F. Hofmeister, Naunin-Schmiedebergs Archiv fur (Leipzig) 24 (1888) 247. K.D. Collinns, M.W. Washabaugh, Q. Rev. Biophys. 18 (1985) 323.

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces 200 E. Ruckenstein, M. Manciu / Advances in Colloid and Interface Science 105 (2003) 177–200 w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x

M.G. Cacace, E.M. Landau, J.J. Ramsden, Q. Rev. Biophys. 30 (1997) 241. R.M. Pashley, J. Colloid. Interf. Sci. 83 (1981) 531. N.K. Adam, The Physics and Chemistry of Surfaces, Oxford University Press, London, 1941. B.W. Ninham, V. Yaminsky, Langmuir 13 (1997) 2097. ¨ D.M.R. Williams, B.W. Ninham, Langmuir 17 (2001) 4475. M. Bostrom, K.A. Karraker, C.J. Radke, Adv. Colloid. Interf. Sci. 96 (2002) 231. ¨ D.M.R. Williams, B.W. Ninham, Phys. Rev. Lett. 87 (2001) 168103. M. Bostrom, N. Ise, T. Okubo, M. Sugimura, K. Ito, H.J. Nolte, J. Chem. Phys. 78 (1983) 1474. P. Jungwirth, D.J. Tobias, J. Phys. Chem. 105 (2001) 10468. J.Th.G. Overbeek, Colloid. Surf. 51 (1990) 61. M. Manciu, E. Ruckenstein, Langmuir 19 (2003) 1114. E.L. Makor, J.H. van der Waals, J. Colloid. Sci. 7 (1952) 535. S.G. Ash, D.H. Everett, C.J. Radke, J. Chem. Soc. Faraday Trans. 69 (1973) 1256. E. Ruckenstein, J. Colloid. Interf. Sci. 82 (1981) 490. R.J. Hunter, Foundations of Colloid Science, Oxford Science Publications, 1987.

443

444

Nanodispersions

J. Phys. Chem. B 2004, 108, 20479-20481

20479

COMMENTS Comments on the “Osmotic Coefficients and Surface Tensions of Aqueous Electrolyte Solutions: Role of the Dispersion Forces” Marian Manciu† and Eli Ruckenstein* Department of Chemical and Biological Engineering, State UniVersity of New York at Buffalo, Buffalo, New York 14260 ReceiVed: May 12, 2004 In the past decade or so, a series of papers have been published suggesting that the van der Waals interactions acting on ions near an interface can explain a plethora of phenomena, such as the ion specificity (the Hofmeister series),1 the failure of the DLVO theory in biological systems2 or the surface tension of electrolytes.3 More recently, the same idea was extended to the ion specificity in bulk solutions.4 However, the dispersion interactions (between ions and the whole system) generate repulsive forces between the water/air interface and the highly polarizable ions (Cl-, Br-, I-) and attractive forces between the interface and the less polarizable ions (Na+, Li+, K+).3 Some recent experimental results5 also challenged the traditional Langmuir picture of a surface layer depleted of ions; they revealed, however, the opposite, namely that the more polarizable anions are positively adsorbed on the interface.5 The conclusion of Hu et al.5 was supported by the numerical simulations of Jungwirth and Tobias,6 which demonstrated that the polarizability of halogen anions (Cl-, Br-, I-) is directly related to their propensity for the surface. The less-polarizable ions (Na+ and F-), however, preferred the bulk water.6 These results are opposite to the predictions of the ion dispersion theories.3 The failure of the ion dispersion forces to predict even the qualitative behavior of ions near an interface is probably due to the existence of other short-ranged forces that dominate the interactions in the vicinity of the interface. We suggested that such a role might be played by the interactions of the ions with their neighboring water molecules.7 The approach to the water/ air interface is unfavorable for the structure-making ions, because they have to lose a part of their hydration shell; however, it is favorable for the structure-breaking ions because the bulk water can organize better in their absence.7 A simple model based on short range ion-hydration forces explained fairly accurately the experiments on surface tension (namely, its increase for salt solutions but decrease for acidic solutions with increasing electrolyte concentration, its pH dependence, and the Jones-Ray effect).7 Kunz et al. reached a similar conclusion,4 namely, that the difficulties of the previous treatments based on ion dispersion forces were due to the fact that the incomplete hydration was not taken into account.4 The focus of the present Comment is on ion-ion dispersion forces, which in the opinion of Kunz et al.4 are important in bulk electrolyte solutions. One of the questions raised by them * Corresponding author. E-mail address: [email protected]. Phone: (716) 645 2911/2214. Fax: (716) 645 3822. † E-mail address: [email protected].

is why these forces have been neglected in previous theories. They answered by suggesting that the general belief is that the Coulomb interactions dominate the van der Waals ones, and hence the latter can be neglected. Kunz et al.4 also noted that the van der Waals interactions might be negligible at low ionic strength (cE < 0.02 M), but not at high ionic strength (cE ∼ 1 M), for which the Coulomb interactions are screened. As a simple calculation shows, at large separations the van der Waals interactions, though larger than the screened Coulomb interactions, are much smaller than kT, and hence negligible. To understand why the ion-ion dispersion interactions have been ignored by most theories of electrolytes, let us perform some simple calculations. The London theory provides for the ion-ion dispersion interactions:8,9

UL(r) = -

()

r0 6 CAB 3 h ω0 ≡- 6 4 2π r r

(1)

where r is the distance between the centers of two ions, h is the Planck constant, and CAB is a constant dependent on ω0 and r0, which are given by

ω0 )

4 ∫∞RA(iξ) RB(iξ) dξ πR0AR0B 0

(2a)

and

r0 )

( ) R0AR0B

1 6

(2b)

(4π02)

where RA is the polarizability of atom “A”, R0A is its zero frequency polarizability, and 0 is the vacuum permittivity. Typical values for r0 and ω0 are 2 Å and 1016 rad/s, respectively,9 which provide a typical value of about CAB ≈ 12 000 kT Å6. The screened Coulomb force is given by

UC(r) =

1 e2 r exp 4π0 r λ

( )

(3)

where e is the protonic charge,  is the dielectric constant in water, and λ is the Debye length. The van der Waals interactions (eq 1) dominate the screened Coulomb interactions (eq 3) at both very small and very large separations. In the latter case, for a concentration of 1 M of a uni-univalent electrolyte, λ = 3 Å, and by employing  ) 80, one finds that the ion-ion dispersion forces dominate the interactions only for distances larger than about 30 Å (see the inset in Figure 1). However, both interactions are negligible at those separations, being of the order of 10-5 kT. Because the typical value provided in ref 9, CAB ≈ 12 000 kT Å6, is rather large, it is likely that most ion-ion dispersion forces are in general negligible at large separations. The dispersion forces play an important role only for separations less than about 5 Å. The problem is that in this region, they seem to play an enormous role (curve 1 in Figure 1). A hard-sphere repulsion (Born interaction) is typically

10.1021/jp047959j CCC: $27.50 © 2004 American Chemical Society Published on Web 12/02/2004

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

20480 J. Phys. Chem. B, Vol. 108, No. 52, 2004

445

Comments

Figure 1. Ion-ion dispersion forces (curve 1) dominate the screened Coulomb interactions (curve 2,  ) 80, λ ) 3 Å) for separations smaller than about 7 Å and larger than about 30 Å (see the inset). However, the nonscreened Coulomb interactions (curve 4,  ) 5, λ ) ∞) are much larger than the ion-ion dispersion forces (1) for distances larger than about 3.5 Å.

Figure 2. Ion-ion dispersion interactions (curve 1) calculated using the Lifshitz approach in ref 4 dominate at small separations the waterscreened Coulomb interactions (curve 2,  ) 80, λ ) ∞); however, they are negligible compared to the nonscreened Coulomb interactions ( ) 10, λ ) ∞) (curve 3), in the presence of a hard wall repulsion at r ) 3.6 Å.

considered to be responsible for preventing the ion-ion collapse due to their (divergently large) van der Waals attractions. If the hard sphere repulsion would start at rHS ) 2 Å, the depth of the energy minimum would be about 200 kT, and all the ions would cluster, regardless of their charge. On the other hand, if the hard sphere repulsion would start at rHS ) 5 Å, the depth of the energy minimum would be less that 1 kT and the effect of ion-ion dispersion would be much less noticeable. The effect of the ion-ion interactions (from huge to negligible) is strongly dependent on the position selected for the hard sphere repulsion. Kirkwood considered long ago10 that a square well with a depth of the order of about 1 kT and with the width of about (1/4)r0 can represent the van der Waals interactions between molecules. This choice probably influenced the conclusion that the ionion dispersion is not important in electrolytes, drawn by Kirkwood in his theory of strong electrolytes.11 Later, Kirkwood and Poirier indicated a manner to include the short-ranged interactions (hard core repulsions and van der Waals attractions) and the long-ranged interactions (Coulomb force) into a unitary theory of strong electrolytes.12 They ignored, however, the van der Waals interactions at large separations, probably for the reasons outlined above. One might argue that the London theory for van der Waals interactions is inaccurate and the dispersion forces should be calculated using the Lifshitz approach (a macroscopic continuum theory) extrapolated to the limit of atomic distances.1,4 Though such a theory would not predict a divergence at r f 0,4 the attractive minimum suggested by Figure 2 of ref 4 would be probably much larger than 20 kT. Even for the ion-ion dispersions potential plotted in Figure 2 of ref 4, (reproduced here for convenience in Figure 2, curve 1), the location of the hard-sphere potential strongly affects the result. Na+ has a relatively low polarizability, and because rHS ) 4.10 Å, it seems that the ion-ion dispersions are not important for the Na+Na+ interactions.4 For NO3--NO3-, rHS ) 3.6 Å, and the depth of about 2 kT of the attractive minimum generates a significant clustering of these ions.4 In fact, the probability of the formation of NO3--NO3- pairs is even larger that the probability of the formation of Na+-NO3- pairs (see Figure 3 of ref 4), a surprising result. Whether the hard sphere repulsion would start at rHS = 2 Å instead of rHS = 3.6 Å (the rHS were adjusted in ref 4 to fit the osmotic coefficients data), an attractive minimum

of over 10 kT would predict a considerable clustering of ions of the same kind. This effect (if real) would be certainly impressive. There is, however, a more important issue. At separations of a few A° ngstroms, the Coulomb interactions are hardly screened either by other ions or by the water molecules. The large dielectric constant of bulk water is due to the ability of the water molecules to reorient their permanent dipole in an electric field. However, the molecules near ions are already oriented by the strong electric fields generated by ions; therefore the local dielectric constant of water (in the vicinity of the ions) is decreased. Using a lower value  ) 5 for the dielectric constant in the vicinity of ions, one finds only repulsive interactions between ions of the same type (curve 5 in Figure 1) for distances r larger than about 3.5 Å. Though at shorter distances divergently large attractions can be found, even in the absence of a hard-wall repulsion it is unlikely for the ions to be able to escalate a repulsive potential barrier larger than 20 kT to collapse on the attractive minimum (for  ) 1, the repulsive barrier would exceed 150 kT). For the ion-ion dispersion forces calculated via the Lifshitz approach,4 employing a dielectric constant  = 10 for separations distances of a few A° ngstroms (curve 5 in Figure 2) instead of the bulk value  ) 78.2 in the Coulomb interactions (curve 2 in Figure 2) would completely remove the attractive well induced by the ion-ion dispersion forces, the latter being much less important than the change in the local dielectric constant. In conclusion, we suggest that the ion dispersion forces were ignored by most (but by no means all) electrolyte theories mainly because they are important only for separations between ions smaller than about 5 Å, and the interactions at these distances are not well-known. It is hard to believe that at these distances the interactions can be accurately described by a sum between a hard-wall repulsion, a Coulomb interaction, and a London attraction. Even if the latter would be true, a correction in the local dielectric constant (because of incomplete screening by water molecules) would render again the van der Waals interactions negligible, up to distances of the order of ion diameters. It is true that, by adjusting the parameters of such interactions, a plethora of interesting phenomena can be predicted, such as the collapse of all the ions of the same kind of an electrolyte. We completely agree with Kunz et al.4 that the short-range

Nanodispersions

446

Comments interactions between ions are important for the behavior of the system; this was also emphasized in the work of Jungwirth and Tobias.6 In fact, the latter work showed that the local interactions between ions and neighboring water molecules are so important, that they control the behavior of ions near interfaces. Whereas the highly polarizable ions are drawn toward the interface in their discrete, microscopic treatment,6,7 a macroscopic, continuum van der Waals approach would predict exactly the opposite.3 Whether or not two ions of the same kind have an attractive minimum, possibly of many kT, when they are located at a few A° ngstroms apart in water it is quite an interesting issue. But this issue has probably to be resolved by ab initio microscopic theories that include all the interactions (Coulomb, dispersion, hydrogen bonding, etc.) of a representative system. Today’s computational capability (for up to about 20 water molecules) might be sufficient for such ab initio calculations.

J. Phys. Chem. B, Vol. 108, No. 52, 2004 20481 References and Notes (1) Ninham, B. W.; Yaminsky, V. Langmuir 1997, 13, 2097. (2) Bostro¨m, M.; Williams, D. M. R.; Ninham, B. W. Phys. ReV. Lett. 2001, 87, 168103. (3) Bostro¨m, M.; Williams, D. M. R.; Ninham, B. W. Langmuir 2001, 17, 4475. (4) Kunz, W.; Belloni, L.; Bernard, O.; Ninham, B. W. J. Phys. Chem. B 2004, 108, 2398. (5) Hu, J. H.; Shi, Q.; Davidovits, P.; Worsnop, D. R.; Zahniser, M. S.; Kolb, C. E. J. Phys. Chem. 1995, 99, 8768. (6) Jungwirth, P.; Tobias, D. J. J. Phys. Chem. 2001, 105, 10468. (7) Manciu, M.; Ruckenstein, E. AdV. Colloid Interface Sci. 2003, 105, 63. (8) London, F. Z. Phys. 1930, 63, 245. (9) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, U.K., 1987. (10) Kirkwood, J. G. J. Chem. Phys. 1935, 3, 300. (11) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 767. (12) Kirkwood, J. G.; Poirier, J. C. J. Phys. Chem. 1954, 58, 591.

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

11312

447

Langmuir 2005, 21, 11312-11319

On the Interactions of Ions with the Air/Water Interface Marian Manciu* Department of Physics, University of Texas at El Paso, El Paso, Texas 79968

Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received July 21, 2005. In Final Form: September 15, 2005

In the vicinity of a charged interface, the Poisson-Boltzmann approach considers that the ions obey Boltzmann distributions in a mean electrical field that satisfies the Poisson equation. However, the boundary between two dielectrics generates additional interactions between ions and the interface. The traditional models of ion hydration interactions, that assume that water is a homogeneous dielectric, predict that these interactions are repulsive for all kinds of ions, since all ions should prefer the medium with a larger dielectric constant, where they are better hydrated. In reality, the interactions between the ions and the neighboring water molecules can generate additional short-range ion-hydration interactions, which are either repulsive (for structure-making ions) or attractive (for structure-breaking ions). In the present paper, various models for the ion-hydration forces are examined and compared with the results of molecular dynamics simulations. At large ionic strengths, the latter results could be reproduced qualitatively only when short-ranged attractions between the structure-breaking ions and the interface were taken into account.

1. Introduction It has long been known that the addition of a salt increases the surface tension of the air/water interface, an experimental result that indicates, via the Gibbs adsorption equation, that the ions are repelled by the air/ water interface. The latter result can be intuitively understood in terms of simple electrostatics; the ions, being charged, prefer the medium with larger dielectric constant, in which their electrostatic energy is lower. When a charge approaches the interface between two dielectrics, the field generated by the charge induces a net charging of the interface. Because the interaction between the charge and the induced surface charge is mathematically equivalent to that between the charge and an imaginary charge placed on the other side of the interface (the image charge), the corresponding electrostatic force is commonly named the “image force”. Because of the long-range nature of Coulomb forces, the repelling of ions from the air/water interface due to the “image forces”, calculated as free of ion screening, is so large that the Gibbs adsorption equation predicts a divergently large surface tension. This difficulty has been solved by Wagner, who noted that the image force acting on an ion is screened by the cloud formed by the other ions.1 By assuming that the Debye-Hu¨ckel screening length λDH of the electrolyte solution does not depend on the distance from the interface, Onsager and Samaras obtained an analytical expression for the surface tension of dilute electrolyte solutions.2 * Corresponding authors. Phone: (915) 747 7531. Fax: (915) 747 5447. E-mail: [email protected]. (Marian Manciu); Phone: (716)6452911/2214.Fax: (716)6453822.E-mail: [email protected] (Eli Ruckenstein). (1) Wagner, C. Phys. Zeit. 1924, 25, 474. (2) Onsager, L.; Samaras, N. N. T. J. Chem. Phys. 1934, 2, 628.

At high salt concentrations, the surface tension of aqueous electrolyte solutions increases almost linearly with the salt concentration,3 and the Onsager-Samaras law, which is valid only for low electrolyte concentrations, is not applicable. The qualitative behavior of the surface tension at high ionic strengths can be, however, understood by using a simple model suggested by Langmuir, who noted that it could be explained by the complete depletion of ions from the first water layer of the interface, of a thickness of about 4 Å.4 Langmuir suggested that the specific ion effects (different slopes of the surface tension of salt aqueous solutions as functions of ionic strength) might be explained by different dissociation constants of the salts.4 By selecting suitable “depletion lengths” for all kinds of salts, the behavior of surface tension at large electrolyte concentrations could be explained even quantitatively.5 Since then, the depletion of ions from the air/water interface became the commonly accepted representation of the surface tension of electrolyte solutions. However, this common picture is not always true; near a charged interface, there is an accumulation of counterions, predicted by the Poisson-Boltzmann treatment, which at very low electrolyte concentrations (less than 10-3 M) can dominate the interfacial depletion of ions due to ion hydration forces,6 and consequently, the surface tension of aqueous electrolytes can decrease with increasing ionic strength (The Jones-Ray effect).7 Even more significant, the surface tension of acids decreases with increasing (3) Weissenborn, P. K.; Pugh, R. J. J. Colloid Interface Sci. 1996, 184, 550. (4) Langmuir, I J. Am. Chem. Soc. 1917, 39, 1848. (5) Aveyard, R.; Saleem, S. N. J. Chem Soc. Faraday I 1976, 72, 1609. (6) Manciu, M.; Ruckenstein, E. Adv. Colloid Interface Sci. 2003, 105, 63. (7) Jones, G.; Ray, G. W. A. J. Am. Chem. Soc. 1937, 59, 187.

10.1021/la051979a CCC: $30.25 © 2005 American Chemical Society Published on Web 10/27/2005

448

Nanodispersions

Interactions of Ions with the Air/Water Interface

electrolyte concentration,3 which clearly points to an overall accumulation of ions at the water/air interface. The possible accumulation of negative ions at the air/ water interface was first predicted by Perera and Berkowitz,8 who found out from molecular dynamics simulations, surprisingly, that the large anions (Cl -, Br -, and I -) are expelled from water clusters to their interface. Their predictions are supported by the recent large-scale molecular dynamics simulations for the air/water interface of various electrolyte solutions, which reveal that, when the polarization of ions and water molecules is explicitly taken into account, the large anions are accumulated near the interface.9 The traditional models, such as the Wagner-OnsagerSamaras “screened image force” or the Langmuir “interfacial depletion length” imply that the ions prefer the medium with a higher dielectric constant, where they are better hydrated (surrounded by water molecules that screen their electric field). It is clear that these models cannot explain the accumulation of ions at the air interface of aqueous acid solutions, and the main question is what kind of ion interactions might be responsible for this effect. It was suggested that the ion-dispersion forces might explain the interfacial distribution of ions.10 However, they predict that the cations are attracted by the interface and that the anions are repelled by the interface,6 in disagreement with experiment11 and simulations.8,9 Recently, two new models have been proposed to explain the interactions between ions and the air-water interface, based on changes of the hydration energy of ions with the distance from the interfaces.6,12 In the first one,6 the discrete nature of water molecule has been taken explicitly into account; when a structure-making ion approaches the interface, it loses some of its hydrating water molecules. Because the energetic penalty is much larger than kT, the interfacial region is practically forbidden to the structure-making ions, and the “Langmuir depletion length” approach can describe well the ion distribution. Because each ion has its particular hydrated structure, there is a particular “Langmuir depletion length” that can be associated with each kind of ion. In contrast, the structure-breaking ions are expelled from the bulk water toward the interface, and this implies an attraction between them and the interface, an interaction that can be roughly approximated by a square well described by two parameters. The other model12 suggests that, because there is a lower density of water in the interfacial region than in bulk water, the dielectric constant is lower than in the bulk, and this leads to repulsions between all kinds of ions and the air/water interface. The purpose of this article is to examine the effect of the ion-hydration forces, suggested by these models, on the distribution of ions near an interface, and to compare their predictions with those obtained from molecular dynamics simulations.9 2. Modified Poisson-Boltzmann Approach Under the assumption that the ions interact with a “mean field” ψ generated by all of the other charges of the (8) Perera, L.; Berkowitz, M. L. J. Chem. Phys. 1993, 99, 4222. (9) Mucha, M.; Frigato, T.; Levering, L. M.; Allen, H. C.; Tobias, D. J.; Dang, L. X.; Jungwirth, P. J. Phys. Chem. B 2005, 109, 7617. (10) Ninham, B. W.; Yaminsky, V. Langmuir 1997, 13, 2097. (11) Knipping, E. M.; Lakin M. J.; Foster K. L.; Jungwirth, P.; Tobias, D. J.; Gerber, R. B.; Dabdub, D.; Finlayson-Pitts, B. J. Science 2000, 288, 301. (12) Bostrom, M.; Kunz, W.; Ninham, B. W. Langmuir 2005, 21, 2619.

Langmuir, Vol. 21, No. 24, 2005 11313

system, the Boltzmann distribution of ions in the vicinity of an interface is provided by the expression

(

ci(x) ) cE exp -

)

qiψ(x) + ∆Wi(x) kT

(1a)

where ci is the concentration of ions of kind “i”, x is the distance from the interface (assumed planar), cE is the electrolyte concentration in a reservoir in contact with the system, qi is the charge of ions of kind “i”, k is the Boltzmann constant, T is the absolute temperature, and ∆Wi(x) is an additional free energy of interaction between the ion and surface, not included in the mean field ψ (note that ∆Wi(x) ) Wi(x) - Wi(∞) is calculated with respect to a reservoir far away from the surface6). The mean field "ψ" is assumed to obey the Poisson equation

∑i qici(x)

∆ψ(x) ) -

(1b)

0

where  is the dielectric constant and 0 the vacuum permittivity. Once the additional interactions are known, the distribution of the concentration of ions near the interface can be obtained from eqs 1a and 1b (for the appropriate boundary conditions), and consequently, the surface tension of electrolytes can be easily calculated, via the Gibbs adsorption equation. However, the additional interactions ∆Wi are not known, and there is no consensus about their microscopic origin. One natural candidate for this “additional interaction” is the screened image force (Wagner-Onsager-Samaras)2

( )

ai λ qi2  - ′ DH 2x ∆WOS,i(x) ) exp  + ′ ai 16π0x λDH 1+ λDH

( )

exp

(

) (2)

where ai is the radius of ion “i”,  is the bulk dielectric constant of water, ′ ) 1 is the dielectric constant of air and

λDH )

x

0kT

∑i cEqi2

is the Debye-Hu¨ckel screening length. Although the screened image forces are important at low electrolyte concentrations, simple calculations show that they become too weak at large ionic strengths and therefore cannot explain the surface tension of electrolytes at those concentrations.6 A simple treatment, which circumvents this difficulty, uses an alternative approximate treatment of the image forces. Because the Born energy of ions in bulk water is given by

WBorn,i(x) ) -

qi2 1 8π0ai 

(3)

one can suggest a distance-dependent interaction, based

449

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

11314

Langmuir, Vol. 21, No. 24, 2005

Manciu and Ruckenstein

on a local approximation of the Born energy13

∆WBorn,i(x) )

(

qi2 1 1 8π0ai (x) 

)

3. Distribution of Simple Ions near the Water/Air Interface

(4)

where (x) is the variable dielectric constant near the interface. In the vicinity of the air/water interface, the dielectric constant is expected to be lower that in the bulk, because the rotation of the water molecules is there somewhat obstructed. As a result, eq 4 indicates a repulsion between ions and the interface. This approximate treatment of electrostatic interactions has the advantages that it does not lead to a diverging surface tension, and that the dependence of the dielectric constant on distance can be selected “a posteriori”, such as to match the results provided by the experiment. However, the local approximation of the Born energy has two shortcomings. First, it predicts that all ions are repelled by the interface, which is not true. Second, eq 4 is not an accurate approximation: the correct Born energy has to be calculated by integrating the energy of the electric field over the entire space, including both dielectrics (note that eq 3 implies only one infinite dielectric). Consequently, the derivative of the Born energy with respect to the distance between the charge and the interface should provide rigorously the “image force”. It is clear that the derivative of the approximate expression, eq 4, does not lead to the image force. However, it should be noted that both the “screened image force” and the “local approximation of the Born energy” are approximations of the same physical process, namely the change in the electrostatic energy of the system when a charge approaches a boundary between two continuous dielectrics. Therefore, they do not represent different contributions to the free energy. Both the “image force” and the “local approximation for the Born energy” approaches account for the ion hydration, through the macroscopic dielectric constant of water, assumed a continuous dielectric. Because none of them could explain satisfactorily the distribution of ions near a water-air interface, it was suggested that another effect plays the main role, namely the van der Waals interactions between ions and the rest of the system. These iondispersion forces can be described by an equation of the type10

∆WvdW,i )

Bi x3

(5)

The theory based on ion-dispersion forces predicts, however, that the cations are attracted toward the interface and that the anions are repelled toward the bulk,6 in contrast with the experimental11 and computational8,9 findings, which indicate the opposite. It was therefore suggested6 that another short-range ion-hydration force, which involves changes in the hydration energy when an ion approaches the interface, due to interactions between the ion and the neighboring water molecules, should be employed to calculate the distribution of ions in the vicinity of the interfaces. In what follows, the predictions of the two models for the short-range ion-hydration force of refs 6 and 12 will be compared with the results of molecular dynamics simulations provided in ref 9 for various electrolyte solutions (NaCl, NaOH, HCl and HBr). (13) Spaarnay, M. J. Recl. Trav. Chim. Pays-Bas 1958, 77, 872.

3.1. Bostrom-Kunz-Ninham (BKN) Model for IonSolvation Forces. A new model for the short-ranged ionsolvation forces has been proposed recently.12 It is based on the observation that the air/water interface is not sharp, the water density increasing gradually, over a distance of a few angstroms, from zero (in air) to the density of bulk water. The water density profile was obtained by fitting the results of the molecular dynamics simulations with the empirical expression12

p(x) ) 1 - 1.0302 exp(3.5 - x)/(1 + exp(3.5 - x)) (6a) where p(x) ) F(x)/F0, F(x) being the density of water, F0 the density of bulk water, and x is measured in angstroms. In the region where the water density is lower than that of the bulk, the dielectric constant is expected to be smaller than in the bulk, and a Spaarnay-like local approximation of the Born energy of ions is given by12

1 1 1 + p(x)( - 1)  ∆Wsolvation(x) ) ∆Gsolvation 1 1

(7)

where ∆Gsolvation is the free energy of solvation in the bulk. The total additional interaction was considered to be the sum of the potentials of a screened image force (eq 2), an ion-dispersion force (eq 5), and an ion-solvation force (eq 7)12

∆Wi(x) ) ∆WOS,i + ∆WvdW,i + ∆Wsolvation,i

(8)

In Figure 1a, the interaction potentials at cE ) 1 M are plotted for Cl-, using in calculations the values ∆Gsolvation,Cl ) 361.79 kJ/mol and BCl ) 13.6 × 10-50 Jm3, provided in ref 12. The ion-solvation contribution clearly dominates the interactions in the vicinity of the interface, for x > 1 Å. At lower distances (x < 1 Å), the total additional energy is larger than 20 kT, and practically no ions are allowed in that region; therefore, the fact that neither eqs 2 nor 5 are accurate for x < 1 Å is not relevant. It should also be noted that both the screened-image and ion-dispersion forces are longer-ranged than the ion-solvation force, and they dominate at large distances (see the inset of Figure 1a). However, at large distances and large ionic strengths, they are small compared to kT and hence negligible. In contrast, at low ionic strengths, the screened image force provides an important contribution at large distances, even when the short-range ion-solvation forces are strong: see the inset of Figure 1b, in which the additional interactions are calculated for Na+ at cE ) 0.001 M , using ∆Gsolvation,Na ) 407.65 kJ/mol and BNa ) 1.4 × 10-50 Jm3.12 From Figure 1b it is clear than the ion-dispersion forces are in this case negligible in the vicinity of the air/water interface. At large distances from the interface, they dominate the interactions (because of their 1/x3 dependence), but are too small compared to kT to affect the ion distribution. In summary, the recent BKN model12 supports the previous predictions6 that the long-ranged screened image forces govern the interfacial distribution of ions at low ionic strengths, that the short-range ion-hydration forces govern their distribution at high ionic strengths, and that the ion-dispersion forces provides only minor corrections to the ion distributions.

Nanodispersions

450

Interactions of Ions with the Air/Water Interface

Langmuir, Vol. 21, No. 24, 2005 11315

Figure 2. Distribution of ions predicted by BKN ion-hydration model, for ψ(x) = 0. The first curve is the water profile obtained from fitting MD simulations. The circles, triangles and stars represent the distribution of I-, Cl-, and Na+, respectively. Note that the interfacial region is depleted of all kind of ions, and therefore, the distribution of ions can be well approximated by a simple model using suitable chosen Langmuir depletion lengths (the distributions corresponding to dcut-off ) 1 Å and dcut-off ) 2 Å are represented in the figure).

Figure 1. Contributions of the screened image, ion dispersion and ion-solvation to the additional energy ∆Wi(x) predicted by BKN model for (a) Cl -, at cE ) 1 M and (b) Na+ at cE ) 0.001 M. In the inset are represented the long-range behavior of the energetic contributions.

Because the solvation energy of any kind of ion in the BKN model is extremely large near the air/water interface, the interface is practically depleted of any kind of ions. Let us assume, for the sake of simplicity, that the mean field electrical potential is negligible (because of the large ionic strength considered); therefore, the distribution of ions in the vicinity of the air/water interface is well approximated by

ci(x) =

(

cE exp -

)

∆Wsolvation,i(x) + ∆WOS,i + ∆WvdW,i(x) (9) kT

The distributions of the ions Na+, Cl-, and I-, for which both the solvation energies and the ion-dispersion constants Bi have been provided in ref 12 (∆Gsolvation,i ) 407.65, 361.79, and 209.49 kJ/mol; Bi ) 1.4, 13.6, and 15.1 × 10-50 J m3, respectively), are plotted in Figure 2, together with the density profile of the air/water interface. The ion distributions are also compared in Figure 2 with those predicted by a simple model, in which “Langmuir depletion lengths” (e.g., ∆Wi ) ∞ for x < dcut-off and ∆Wi ) 0, x > dcut-off), with dcut-off ) 1 and 2 Å, have been arbitrarily selected, and all of the other interactions have been ignored. The calculations for the latter distributions take into account the interfacial profile of the water using a statistical distribution of sharp interfaces, a procedure which is detailed in the next section.

Although providing comparable results with the simple model based on “Langmuir depletion lengths”, the main advantage of the BKN model is that it allows the direct calculation of the ion distributions without any fitting parameters, once the profile of the interface eq 6, the van der Waals parameters Bi in eq 5, and the solvation energies in eq 7 are known. There are, however, some difficulties associated with the model. The most important is that it predicts that all of the ions are repelled from the interface, in contrast with the experimental data11 and simulations,8,9 which indicate that some anions (e.g., Cl-, Br-, I-, but not F- ) are attracted by the interface. The second difficulty is that it predicts very small changes in the interfacial forces for different ions and, therefore, can hardly account for ion specific effects (Figure 2 shows only a minute difference between the distributions of Na+ and I- ions). In Figure 3a, the predictions of the BKN model for cE = 1.2 M are compared with the distributions of Na+ and Cl- ions near the air/water interface, obtained via molecular dynamics simulations in ref 9; note that in the molecular dynamics simulations the water molecules occupy the region from x ) 0 to the water/air interface, located at x0 ) 13.9 Å; therefore, the expression

p(x) ) 1 - 1.0302 exp(x - 13.9)/(1 + exp(x - 13.9)) (6b) has been employed instead of eq 6a. The distributions of Na+ and Cl- predicted by the BKN model are very close to each other, both kinds of ions being repelled from the air/water interface. Furthermore, the Na+ cations more closely approach the interface than the Cl- anions, in qualitative disagreement with the simulations. Whereas the average interfacial depletion of ions might be close to the value required to explain the behavior of the surface tension of NaCl solution, the distributions of ions does not reproduce even qualitatively the Molecular Dynamics simulations. The situation is particularly dramatic for acids: the hydrogen ion has a Gibbs free

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

11316

451

Langmuir, Vol. 21, No. 24, 2005

Figure 3. (a) Distributions of Na+ and Cl- ions near the air/ water interface of a NaCl solution, predicted by the BKN model (note that the Na+ ions approaches the interface closer than the Cl- ions), are compared with the molecular dynamics simulations of ref 9. (b) The distributions of H+ and Cl- ions near the air/water interface of a HCl solution, predicted by the BKN model (note that the interface is depleted of both H+ and Cl- ions ), are compared with the molecular dynamics simulations of ref 9, that show an accumulation of both H+ and Clions at the interface.

energy of hydration of about ∆Gsolvation,H = 1100 kJ/mol14 and consequently should be repelled from the interface by ion-solvation forces much stronger than the simple cations (Na+, K+, and Li+). Assuming a negligible iondispersion interaction for H+, the ions distributions in the vicinity of the air/water interface of the HCl solution at cE ) 1.2 M, predicted by the BKN model, are compared to molecular dynamics simulations9 in Figure 3b. In this case, the BKN model leads to an average interfacial depletion of ions larger than that predicted for the NaCl solution (Figure 3a), which implies that the surface tension of the HCl solution should increase with ionic strength stronger than the surface tension of the NaCl solution. This is not true; in reality, the surface tension of acid solutions is not even increasing but decreases with the ionic strength.3 The reason for this behavior can be clearly seen from the results of the molecular dynamics simulations9 reproduced in Figure 3b: both H+ and Cl- prefer the air/water interface, where they accumulate. 3.2. Structure Making/Structure Breaking (SM/ SB) Model for the Short-Range Ion-Hydration (14) Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. H.; Earhart, A. D.; Coe, J. V.; Tuttle, T. R. J. Phys. Chem. A 1998, 102, 7787.

Manciu and Ruckenstein

Forces. A simple model has been suggested recently for the ion-hydration forces,6 based on the old common intuition that some ions are able to structure the water (the structure making ions), and for this reason they prefer the bulk, while other ions generate disorder in water (the structure-breaking ions), and for this reason, they are expelled toward the interface. When a structure making ion approaches an interface, it loses some of its hydration molecules, a process which is thermodynamically unfavorable. The hydration energies are of the order of 102 kT, and even a change of a few percents of this energy affects drastically the ion distribution; an increase in the total free energy by 5 kT implies (via the Boltzmann distribution) a decrease in concentration by more than 2 orders of magnitude. For this reason, one can assume that this region is practically depleted of structure-making ions, in agreement with the Langmuir suggestion.4 The main difference between the traditional ionhydration models and the SM/SB model is that the previous ones assumed that water is a continuous dielectric, characterized by a macroscopic dielectric constant, whereas the latter takes into account that, microscopically, water is constituted of discrete molecules. The change in hydration occurs when the hydrated ion loses some of its hydrating water molecules, a process that starts from a certain distance dcut-off from the interface. Therefore, for x > dcut-off, there are no “ion-hydration” interactions, whereas the region with x < dcut-off can be considered completely depleted of ions. Furthermore, because different structure-making ions have different cutoff thicknesses in which they lose hydrating water molecules, dcut-off is a parameter that describes the specificity of the structure-making ions (in contrast to the suggestion of Aveyard and Saleem, who suggested that a dcut-off should be associated with the specificity of salts, not of the corresponding ions).5 For structure-breaking ions, it was suggested that their attraction toward the interface is governed by a simple surface potential well, described by two adjustable parameters (the width and the depth of the surface potential well).6 When this simple model was employed for the air/ water interface, it could explain the dependence of the surface potential on the electrolyte concentration and on pH, the behavior of surface tension of salt and acids at high ionic strengths and the Jones-Ray effect (the existence of a minimum in the surface tension of salts at a small electrolyte concentration).7 To compare the results with the existing simulations,9 one should note that the simulations were performed at a fixed number of ions, and not for a system in contact with a reservoir. Therefore, instead of eq 1 one should use

(

ci(x) ) c0,i exp -

qiψ(x) + ∆Wi(x) kT

)

(10a)

with the constant c0,i provided by the normalization condition

∫∫dydz∫0x

max

A

ci(x) dx )

N NA

(10b)

where the integral in the interface plane yz is on an area 30 × 30 Å2, the integral along the distance x, perpendicular to the interface, is up to xmax ) 25 Å, N ) 18 represents the number of ions (either cations or anions) from the box and NA is the Avogadro number. The total additional interaction ∆Wi is the sum of contributions from a short-range ion-hydration force, a long-range ion-hydration force (the screened image force)

Nanodispersions

452

Interactions of Ions with the Air/Water Interface

Langmuir, Vol. 21, No. 24, 2005 11317

and a ion-dispersion force. The long-range ion hydration forces are due to the behavior of water as a homogeneous dielectric, characterized by a macroscopic dielectric constant, whereas the short-range ion-hydration forces are due to the local, discrete structure of water. At low electrolyte concentrations, the screened image forces are dominant (leading to the Onsager-Samaras law),2 whereas at large ionic strengths, they are negligible and the behavior of ions near interfaces is determined by the shortranged ion-hydration forces.6 As noted earlier, the inclusion of the ion-dispersion interactions is equivalent to a slight modification of the parameters of the short-range ion-hydration forces.6 Explicit expressions for the additional interactions between ions and interfaces, ∆Wi(x), allow us to calculate the distribution of ions ci(x) (for a sharp water/air interface) eqs 10a and 10b, for the appropriate boundary conditions

σs dψ |x)0 ) dx 0

(11a)

ψ(x)|x)xmax = 0

(11b)

and

where σs is the surface charge density (assumed σs ) 0 in what follows) and with the last approximation holding because xmax of the numerical simulations is at about 5 Debye-Hu¨ckel lengths from the air/water interface. The traditional Modified Poisson-Boltzmann (MPB) approach (eqs 1a and 1b) assumes a sharp air/water interface. It is more reasonable to consider that our system is constituted from a statistical ensemble of sharp interfaces, with a statistical distribution

φ(ξ) )

1

x2πσ

(

exp -

(ξ - x0)2 2σ2

)

∫ξ∞) - ∞ φ(ξ)c′i(x - ξ) dξ

(12b)

p(x) )

∫ξ ) -∞ φ(ξ)Θ(x - ξ) dξ )

(

)

x - x0 σ 2

1 + erf

(13a)

∆WNa ) 0 for x >3 Å

(13b)

and

which involves a total depletion of ions in the first dcut-off ) 3 Å of water and

(12a)

where the prime denotes the concentration calculated for a sharp air/water interface. The water density profile for a sharp interface located at ξ is described by the Heaviside function Θ(x - ξ) ) 0 for x - ξ < 0, Θ(x - ξ) ) 1 for x ξ > 0; therefore, the water profile for the statistical ensemble is given by

∞

∆WNa ) ∞ for 0< x < 3 Å

∆WCl ) -1.7kT for 0 < x < 2.5 Å

(14a)

∆WCl ) 0 for x > 2.5 Å

(14b)

and

around the average position x)x0 of the interface, the variable ξ denoting the position of the sharp air/water interface in one component of the statistical ensemble. Once the ion distributions for sharp air/water interfaces, c′i(x), are calculated, the concentrations of ions for the statistical ensemble, ci(x), are obtained by summing up the corresponding contributions to concentrations provided by each element of the ensemble. For an element with a sharp interface located at ξ, the concentration of ions at the distance (x - ξ) from the sharp interface represents the contribution to the concentration of ions at location x in the statistical ensemble (because ξ + (x - ξ) ) x). Adding up all of these contributions (from the sharp interfaces located at all possible ξ values) multiplied with the probability of a sharp interface at the distance ξ in the statistical ensemble (eq 12a), one obtains the convolution

ci(x) )

The distributions of ions and the water density profile for a water/air interface (water molecules located in the region x < x0), as considered in the molecular dynamics simulations9 (instead of an air/water interface) are simply obtained by the transformation (x - ξ) f (ξ - x) for ci′(x - ξ) in eq 12b and Θ(x - ξ) in eq 12c. This treatment does not generate large repelling forces between ions and interfaces, as the BKN model does, because the structure-breaking ions expelled at the interface are still hydrated in each subsystem of the statistical ensemble. Intuitively, this means that the structure-breaking ions arrive at the interface surrounded by their hydration water molecules. Whereas, on average, the water density is lower near the air/water interface, so there are sufficient water molecules locally to hydrate each interfacial ion. In contrast, the BKN model suggests that the (average) low density of water leads to a low dielectric constant near the air/water interface, and consequently, no ion will approach the interface. The results of the numerical calculations for the distributions of ions for NaCl are presented in Figure 4a, in which we employed the following values for ∆Wi(x)

(12c)

which involves a simple square potential near the interface, with a width of 2.5 Å and a depth of 1.7 kT (the parameters selected in eqs 13 and 14 are assumed to include also the ion-dispersion interactions). The parameters of statistical distribution of the sharp air/water interfaces have been selected σ ) 1.4 Å and x0 ) 13.9 Å, in order to fit the profile of the interface provided by the molecular dynamics simulations.9 The simple SM/SB model clearly cannot explain the behavior of ions at distances larger than 10 Å from the interface, but it is not yet clear whether longer-ranged forces are involved or whether the strong variation of the ions distributions in that region are artifacts of the limited size of the system employed in simulations. To reproduce the simulations for the ion distributions in the aqueous solution of HCl (see Figure 1b), ∆WCl has been selected as above (eqs 14a and 14b) and an attractive interaction between hydrogen ions and the air/water interface has been employed

∆WH ) -1kT for 0 < x < 1 Å

(15a)

∆WH ) 0 for x > 1 Å

(15b)

and

Such an interaction has been suggested by experiment regarding the surface tension of aqueous acid solutions.3 The accumulation of H+ at the interface has been recently

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

11318

Langmuir, Vol. 21, No. 24, 2005

453

Manciu and Ruckenstein

Figure 4. (a-d) Distribution of cations (1), anions (2), and water (3) at the air/water interface. The continuous curves are calculations based on the SM/SB model for short-range ion-hydration forces. The squares (cations), circles (anions) and triangles (water) are the molecular dynamics results of ref 9. (a) NaCl; (b) HCl; (c) HBr; (d) NaOH.

confirmed directly by experiment.15 One should note that there is a minimum in the concentrations of both anions and cations around 10 Å. This minimum cannot be explained either by the simple potentials selected for the ion-hydration forces or by the traditional electrostatic mean field interactions (for which the depletion of one kind of ions is associated with the accumulation of ions of the other kind). Furthermore, the Poisson-Boltzmann treatment predicts a monotonic variation of charge, whereas in Figure 4b successive inversions of charges with the distance are apparent. One possible reason for this behavior might be another interaction between ion and the surface, perhaps due to the structuring of the water near the interface. The behavior of ions is similar for other aqueous acid solutions (HBr, Figure 4c) for which an attractive potential has been selected for the interaction between Br- and the interface, whereas for H+ the same potential as above (eqs 15a,b) was employed

∆WBr ) - 2.0kT for 0 < x < 1.5 Å

(16a)

∆WBr ) 0 for x > 1 Å

(16b)

and

It is of interest to note that, for a basic solution (NaOH, (15) Petersen, P. B.; Saykally, R. J. J. Phys. Chem. B 2005, 109, 7976.

Figure 4d), both the cations and the anions are repelled by the interface (in contrast with the simple acids, for which both H+ and Cl- or Br- are accumulated on the interface). To obtain a reasonable agreement with experiment, we used a repulsive potential for OH- with a cut off dOH ) 2.0 Å, whereas for Na+, we had to use dcut-off ) 3.5 Å, which is slightly different from the value selected in eqs 13. The agreement between the numerical results and those obtained via the simple SM/SB model for the short-range ion-hydration forces is satisfactory, if one takes into account that the interactions between ions and interfaces have been assumed to have a very simplified form, and the possible charging of the air/water interface by other mechanisms, other than the accumulation/depletion of ions, was ignored. 4. Conclusions The distributions of ions in the vicinity of an air/water interface are due not only to the electrostatic interactions among ions, but also to the additional interactions between ions and the interface. Whereas it is plausible to consider that the ions prefer the medium with a larger dielectric constant and hence that the corresponding interactions should repel the ions from the interface, the traditional treatments of the hydration of ions via screened image forces1,2 or local approximations of the Born energy,12,13 which consider water as a continuous dielectric, failed to reproduce the experiment at high ionic strengths. This

Nanodispersions

454

Interactions of Ions with the Air/Water Interface

failure led to a search for additional forces, such as the ion-dispersion forces, due to van der Waals interactions between ions and the rest of the system.10 Such interactions predicted, however, qualitatively wrong ions distributions (an accumulation of cations and a depletion of anions at the air/water interface, instead of the opposite). We proposed recently a simple model (SM/SB) for the short-range interactions between ions and interface, based on the fact that the water surrounding the ion is formed of discrete molecules.6 Consequently, the repelling ionhydration forces, acting on structure-making ions could be described by one parameter (a cutoff length), whereas the attractive ion-hydration forces, acting on structurebreaking ions, could be described by two parameters (a simple potential well at the interface). It is clear that the ion-hydration forces are more complex, but there is no theory yet that can provide an “ab initio” prediction of these interactions. The reason for this failure consists of the large number of molecules involved in these interactions. A natural way to calculate the interactions would be to simulate the air/water interface of an aqueous solution and to calculate “a posteriori” the potential that matches the distribution of ions. However, only systems with a limited number of molecules (for the time being, about 1000) could be treated in a reasonable time.9 It is apparent that in these cases, boundary effects are influencing the distribution of ions in all figures (Figure 4a-d), because the systems investigated were limited only to a slab of water of a thickness of about 15 Å from the air/water interface.

Langmuir, Vol. 21, No. 24, 2005 11319

We showed previously that a simple model for the ionhydration interactions, which separates the ion-hydration forces in a long-range term due to the behavior of water as a continuous dielectric (the screened image force) and a short-range term due to the discreetness of the water molecules (SM/SB), can explain almost quantitatively a number of phenomena related to the electrolyte interfaces.6 In this article, we examined the limitations of the model in predicting the distributions of ions near the air/water interface, by comparison with molecular dynamics simulations. It is clear that the real ion-hydration forces are more complicated than the simple model employed here; however, the interfacial phenomena (including specific ionic effects) can be understood, at least qualitatively, in terms of this simple approach. The situation is similar to the success of the traditional Poisson-Boltzmann approach: its ability in describing, at least qualitatively, and many times even quantitatively, the behavior of most colloidal systems probably resides in the use of at least one adjustable parameter (surface charge, surface potential, recombination constant and so on) in the fitting of the experimental results. If that parameter could be accurately measured, one would have to address the inaccuracies generated by the mean field treatment itself. Acknowledgment. We are indebted to Dr. Pavel Jungwirth for kindly providing us the files of the molecular dynamics simulations, reproduced in Figures 3a-b and 4a-d. LA051979A

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

455

Journal of Colloid and Interface Science 304 (2006) 541–544 www.elsevier.com/locate/jcis

Note

Ions at the air/water interface Marian Manciu a,∗ , Eli Ruckenstein b a Physics Department, University of Texas at El Paso, USA b Chemical and Biological Engineering Department, State University of New York at Buffalo, USA

Received 26 July 2006; accepted 15 September 2006 Available online 20 September 2006

Abstract In a recent review of this topic [B.C. Garett, Science 303 (2004) 1146] the emphasis was on some recent experiments, in which it was found that some anions accumulate at the air/water interface and not in the bulk, as usually happens to the cations, and on some simulations which explained those positive surface adsorption excesses. Because a large number of these experiments could be explained on the basis of some simple physical models proposed by the authors for the interaction between the ions and the air/water interface [M. Manciu, E. Ruckenstein, Adv. Colloid Interface Sci. 105 (2003) 63; Adv. Colloid Interface Sci. 112 (2004) 109; Langmuir 21 (2005) 11312], those models are reviewed in the present note, the goal being to draw attention to them. © 2006 Elsevier Inc. All rights reserved. Keywords: Interface; Ions; Double layer; Ion-hydration forces

Garrett [1] has briefly reviewed the recent progress in the understanding of the behavior of ions at the water/air interface, by emphasizing both some experiments [2,3], which have shown that some anions can accumulate at the interface and not in the bulk as usually considered, as well as some numerical simulations [4], which can explain such a behavior. His review stimulated us to draw attention to some physical models, which can provide additional clarifications and which have been considered recently by the authors [5–7]. First let us note that experiment revealed long ago that not all ions prefer the bulk to the interface [8]. Gibbs adsorption equation predicts that the surface tension increases with the electrolyte concentration when the total surface excess of ions is negative. The conventional picture, that the ions prefer the bulk, is probably due to Langmuir, who noted that the increase in the surface tension of aqueous solutions of simple salts with increasing concentration can be explained by assuming a surface layer of pure solvent with a thickness of about 4 Å [9]. However, because the aqueous solutions of some simple acids (such as HCl) possess surface tensions smaller than that of pure water [8], Gibbs adsorption equation indicates a positive total * Corresponding author. Fax: +1 915 747 5447.

E-mail address: [email protected] (M. Manciu). 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.09.034

surface excess adsorption of ions, hence at least one of the ion species must accumulate in this case at the interface. Therefore, experimental evidence for accumulation of at least one species of ions on the interface has been around as long as reliable measurements for the surface tension of acid solutions have been available. The first explanation for the negative ion adsorption was based on the “image force,” i.e., the electrostatic interactions between ions and the boundary between two dielectric media. However, the long range Coulomb force leads to a diverging large negative adsorption. The paradox was solved by Wagner [10] who, inspired by the then-developed Debye–Huckel theory of electrolytes, suggested that a screened image force could explain the surface tension behavior of salt solutions. A simplifying assumptions about the screening length provided later an analytic solution for the surface tension of electrolytes, the celebrated Onsager–Samaras theory [11]. Subsequently, the theory was challenged by the surface tension measurements of Jones and Ray [12], who observed an initial decrease, followed by a minimum at low concentrations, and a further increase with increasing ionic strength, which has been attributed to the charging of the interface [13]. The important conclusion was that there are other interactions between ions and interfaces in addition to image forces.

Nanodispersions

456 542

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 304 (2006) 541–544

Fig. 1. Change in surface tension at low electrolyte concentrations (the Jones–Ray effect). Circles: experimental data from Ref. [12]; line: predicted surface tension by the SM/SB model which accounts for OH− adsorption on the surface, ion hydration and image forces within the Poisson–Boltzmann approach (Ref. [5]).

Fig. 2. Possible dependence of the surface tension on electrolyte concentration, for various ion-hydration potentials for the anions (Ref. [5]). When the surface excess of the SB ions exceeds the surface depletion of the SM ions, the surface tension decreases with ionic strength.

In a mean field theory, the forces acting on each ion because of all the other species in the system can be described by a potential Ui , which can be separated into an electrical part due to the interaction between the charge qi of each ion “i” and the mean electrical potential ψ, and an additional term Wi , which accounts for all the other interactions between the ion and the rest of the system, not included in ψ, Ui = qi ψ + Wi .

(1)

Assuming a planar surface (hence, that the mean field depends only on the distance z from the surface) the chemical equilibrium for each species of ions implies that their concentration ci (z) is related to their bulk concentration c0i by   qi ψ(z) + Wi (z) ci (z) = c0i exp − , (2) kT where k is the Boltzmann constant, T is the absolute temperature, and z is the distance to the interface. Equation (2) also implies Wi (∞) ≡ 0 and that the electric potential obeys the Poisson equation: d2 ψ qA cA + qC cC =− , εε0 dz2

(3)

where ε is the dielectric constant of water, ε0 is the vacuum permittivity, and the subscripts A and C stand for anion and cations, respectively. The main difficulty consists in the determination of Wi , which should include (but is not limited to) image forces. It was recently suggested by Ninham and co-workers [14] that the different van der Waals interactions of anions and cations (due to their different polarizabilities) might explain the accumulation of ions near the interface (and not only the depletion provided by the image force alone). However, as emphasized in Ref. [5], the calculations based on ion-dispersion forces predicted that the cations should accumulate and the anions should

Fig. 3. Experimental values of the zeta potential for the air/water interface in the presence of NaCl, as a function of pH, from Ref. [15] (triangles: 0.1 M; circles: 0.01 M; squares: 0.00001 M) are compared with the potential values predicted by the SM/SB model.

be repelled by the interface, in contradiction with both experiment [2,3] and simulations [4], which indicate the opposite. Another possible explanation might be related to the ability of some ions to organize (structure-making ions) and others to disorganize (structure-breaking ions) the water structure. The approach to the interface of the structure-making ions is unfavorable, because they can better organize the water dipoles in bulk water than at the interface, thus decreasing the free energy of the system. The opposite is true for the structurebreaking ions: the total free energy of the system is minimized by pushing these ions toward the interface, because bulk water molecules can better organize their hydrogen bonding network without them. A recent attempt [5] to account for this structure-making/structure-breaking (SM/SB) interactions (ion-hydration forces) within the traditional mean field framework has shown that the surface excesses of ions can be either

Non-DLVO colloidal interactions: specific ion effects explained by ion-hydration forces

457

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 304 (2006) 541–544

543

(a)

(a)

(b)

(b)

Fig. 4. Distribution of cations (1), anions (2), and water (3) at the air/water interface. The continuous curves are calculations based on the short-range ion-hydration forces (the SM/SB model) (Ref. [7]). The squares (cations), circles (anions), and triangles (water) are the Molecular Dynamics results of Ref. [4]. (a) NaCl; (b) HCl.

Fig. 5. Distribution of cations (1), anions (2), and water (3) at the air/water interface. The continuous curves are calculations based on ion dispersion and ion-solvation forces (the BKN model, Ref. [16]). The squares (cations), circles (anions), and triangles (water) are the Molecular Dynamics results of Ref. [4]. (a) NaCl; (b) HCl.

positive or negative. In this model the potentials Wi have been taken as potential wells with depths of the order of kT and widths of a few Å for the structure breaking ions, and as generating depletion lengths of a few Å for the structuremaking ions. It was shown that this simple SM/SB model for the “ion-hydration forces” can explain the Jones–Ray effect (Fig. 1) [5], the increase of the surface tension of an aqueous salt solution with concentration and its decrease for aqueous acid solutions (Fig. 2) [5], as well as the dependence of the zeta potential on pH and electrolyte concentration (Fig. 3) [5]. The agreement is not surprising since the above simple structuremaking/breaking potentials can reproduce reasonably well the distributions of ions around the air/water interface, obtained by MD simulations [4], for various kinds of simple electrolytes, for an interfacial water profile assumed to be given by a sta-

tistical distribution of sharp interfaces [7] (see Figs. 4a and 4b). While the interactions potentials Wi used in the SM/SB model are qualitatively understandable [5], it is not yet possible to predict them a priori. It was recently suggested [16] that the short ranged ion-hydration interactions (denoted ion-solvation forces) might be calculated using the expression: Wi,solvation (z) = Gi,solvation

1 1+p(z)(ε−1) 1 − 1ε

−

1 ε

,

(4)

where Gi,solvation is the free energy of solvation of an ion “i” in the bulk [17], and p(z) is the profile of the water/air interface, which is reasonable well approximated by the empirical equation [16]:   p(z) = 1 − 1.0302 exp(3.5 − z)/ 1 + exp(3.5 − z) . (5)

Nanodispersions

458 544

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 304 (2006) 541–544

The latter treatment has the advantage of simplicity; however it predicts that the interface is depleted by all the kinds of ions, prediction contradicted by both experiment and simulations. In contrast, the earlier accounting of ion-dispersion forces [14] predicted that the cations are accumulated at the surface and the anions are depleted, whereas in reality the opposite occurs. In addition, the Bostrom–Kunz–Ninham (BKN) treatment of ion-solvation forces [16] predicts that the surface tension of the simple acid solutions increases with ionic strength more rapidly than the surface tension of simple salts (since the free energy of solvation of H+ is larger than those of the other cations, such as Na+ [17]). Actually, the former surface tension decreases with ionic strength, because both H+ and the anions are accumulated on the surface. As shown in Ref. [7], the inclusion of both ion-dispersion and ion-solvation forces (provided by the BKN model) in the interaction potential between ions and the air/water interface (see Figs. 5a and 5b) does not lead to good agreement with simulations. In contrast, the use of the ion interactions potentials employed in the SM/SB model [5] in conjunction with a statistical model for the profile of the air/water interface [7] provides a reasonable agreement with the simulated distribution of ions near the air/water interface [4]. In

summary, we can emphasize that our simple models provide insights regarding both the experimental and simulation results on the behavior of ions near the air/water interface. References [1] B.C. Garett, Science 303 (2004) 1146. [2] E.A. Raymond, G.L. Raymond, J. Phys. Chem. B 108 (2004) 5051. [3] D. Liu, G. Ma, L.M. Levering, H.C. Allen, J. Phys. Chem. B 108 (2004) 2252. [4] P. Jungwirth, D.J. Tobias, J. Phys. Chem. B 105 (2001) 10468. [5] M. Manciu, E. Ruckenstein, Adv. Colloid Interface Sci. 105 (2003) 63. [6] M. Manciu, E. Ruckenstein, Adv. Colloid Interface Sci. 112 (2004) 109. [7] M. Manciu, E. Ruckenstein, Langmuir 21 (2005) 11312. [8] J.W. Belton, Trans. Faraday Soc. 31 (1935) 1413. [9] I. Langmuir, J. Am. Chem. Soc. 39 (1917) 1848. [10] C. Wagner, Phys. Zeit. 25 (1924) 474. [11] L. Onsager, N.N.T. Samaras, J. Chem. Phys. 2 (1934) 628. [12] G. Jones, W.A. Ray, J. Am. Chem. Soc. 59 (1939) 187. [13] M. Dole, J. Am. Chem. Soc. 60 (1940) 904. [14] M. Bostrom, D.M.R. Williams, B.W. Ninham, Langmuir 17 (2001) 4475. [15] C. Li, P. Somasundaran, J. Colloid Interface Sci. 146 (1991) 215. [16] M. Bostrom, W. Kunz, B.W. Ninham, Langmuir 21 (2005) 2619. [17] B.E. Conway, Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam, 1981.

Introduction to CHAPTER 6 Polarization Model: a unified framework for hydration and double layer interactions

6.1

6.2

6.3

6.4

6.5

6.6

6.7

D. Schiby, E. Ruckenstein: “The role of the polarization layers in hydration forces,” CHEMICAL PHYSICS LETTERS 95 (1983) 435–438. E. Ruckenstein, D. Schiby: “On the origin of the repulsive hydration forces between mica plates,” CHEMICAL PHYSICS LETTERS 95 (1983) 439– 443. D. Schiby, E. Ruckenstein: “On the coupling between the double-layer and the solvent polarization-fields,” CHEMICAL PHYSICS LETTERS 100 (1983) 277– 281. M. Manciu, E. Ruckenstein: “Oscillatory and Monotonic Polarization. The Polarization Contribution to the Hydration Force,” LANGMUIR 17 (2001) 7582– 7592. M. Manciu, E. Ruckenstein: “Polarization of Water near Dipolar Surfaces: A Simple Model for Anomalous Dielectric Behavior,” LANGMUIR 21(2005) 11749– 11756. E. Ruckenstein M. Manciu: “The Coupling between the Hydration and Double Layer Interactions,” LANGMUIR 18 (2002) 7584–7593. M. Manciu, E. Ruckenstein: “On the Chemical Free Energy of the Electrical Double Layer,” LANGMUIR 19 (2003) 1114–1120.

Because the double layer force vanishes in the absence of surface charges, one expects the attractive van der Waals force to cause the coagulation of all neutral (or even weekly charged) colloids. The absence of such a behavior has been explained by the existence of an additional (non-DLVO) force, the hydration interaction, which is due to the structuring of water in the vicinity of hydrophilic surfaces. This chapter is devoted to the identification of the microscopic origin of the hydration force, and to the presentation of a unified treatment of the double layer and hydration forces, the Polarization Model. The basic idea is that the dipoles of the surface are responsible for the partial alignment of the dipoles of the neighboring water molecules via electrostatic dipole-dipole

interactions. The dipoles of the partially aligned water molecules induce partial alignment in the next layer of water molecules and so on; consequently a decaying polarization field is generated in the vicinity of a dipolar surface even in the absence of a surface charge [6.1]. When two such surfaces are brought close to each other, the overlap of the polarization fields (which have opposite orientations) increases the free energy of the system, thus generating repulsion. This repulsive force is similar (both in magnitude and distance dependence) to the experimentally determined hydration force [6.2]. Since both hydration and double layer forces are based on electrostatic interactions (dipole-dipole and charge-electric field, respectively), a unified theory for hydration and double layer force is proposed [6.3]. Whereas in the traditional electrodynamics a uniform dipolar field on a surface does not induce an electric field outside the surface, it is suggested that the discrete nature of the water molecules (which cannot be ignored at molecular distances) is responsible for a local, non vanishing electric field. This field is calculated by assuming that water is organized in ice-like layers in the vicinity of the interface, and leads to interactions compatible with the experimentally observed hydration force [6.4]. Two consequences emerge from this model: (i) in water, the polarization is no longer proportional to the electric field in the vicinity of dipolar surfaces (hence, an anomalous dielectric constant) , and (ii) the polarization oscillates near interfaces (consequently the corresponding hydration force is non-monotonic). The experimentally observed monotonic hydration force is most likely a result of the statistical roughness of the interacting surfaces: the oscillations are averaged out and thus a monotonically decreasing repulsion results [6.4]. Experiments to measure the electric field and water polarization within 10 Ǻ of the surface are difficult to perform. However, recent Molecular Dynamics simulations carried out by Faraudo and Bresme for water between two sodium dodecyl sulfate layers revealed oscillatory behaviors for both the polarization and the electric fields near the surface, and non-proportionality between them [Faraudo, J.; Bresme, F. Phys. Rev. Lett. 2004, 92, 236102]. Our polari-

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_6, © Springer Science+Business Media, LLC 2010

460

zation model is able to account for such unexpected behaviors [6.5] When all the electrical interactions (between surface charges and dipoles, ions and water molecule dipoles are taken into account, the electric "mean field" potential and the polarization become coupled via two nonlinear differential equations of second order (in which the polarization is no longer proportional to the electric field). The same constitutive equations are also derived using a variational approach [6.6]. In the linear approximation (valid for small potentials), the equations can be solved analytically and the interaction is the result of the sum of two free energy terms with two decay lengths, which (in the limit of vanishing small electrolyte concentrations) are related to the large Debye-Hückel length, and to a small hydration decay length

Nanodispersions

of about 2 Å which is independent of ionic strength [6.6]. The latter decay length is comparable to that measured experimentally for neutral lipid bilayers. Therefore, if the "coupling" between the interactions is small, they can be perceived as two different interactions with two different decay lengths - the double layer and hydration forces [6.6]. In order to extend the polarization model to systems of arbitrary geometries and arbitrary surface conditions, the corresponding chemical part of the free energy has to be accounted for. It is shown that the change in the chemical free energy depends not only on the surface charge density and potential in the final state, but also on their values at each distance between infinity and the final state. An exact expression for the chemical free energy has been derived and a simple approximation suggested [6.7].

Polarization Model: a unified framework for hydration and double layer interactions

461

462

Nanodispersions

Polarization Model: a unified framework for hydration and double layer interactions

463

464

Nanodispersions

Polarization Model: a unified framework for hydration and double layer interactions

465

466

Nanodispersions

Polarization Model: a unified framework for hydration and double layer interactions

467

468

Nanodispersions

Polarization Model: a unified framework for hydration and double layer interactions

469

470

Nanodispersions

Polarization Model: a unified framework for hydration and double layer interactions

471

472

Nanodispersions

Polarization Model: a unified framework for hydration and double layer interactions

473

474

Nanodispersions

Polarization Model: a unified framework for hydration and double layer interactions

7582

475

Langmuir 2001, 17, 7582-7592

Oscillatory and Monotonic Polarization. The Polarization Contribution to the Hydration Force Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received June 25, 2001. In Final Form: September 20, 2001 The propagation of polarization in water, in the vicinity of a planar surface containing dipoles, is considered to be a consequence of the nonuniformity of the dielectric constant near the dipoles. Discrete finite difference equations have been derived by assuming a layered icelike structure. These equations predicted a polarization that oscillates with the distance from the surface. When the polarizing surface was considered rough or fluctuating, the oscillations were smoothed out and a monotonically decaying average polarization was obtained. Assuming that there are local icelike structures around any of the water molecules, a secondorder differential equation for the polarization and expressions for its decay length were derived.

I. Introduction The direct measurement of the interaction force between two mica surfaces1 indicated a large repulsion at relatively short distances, which could not be accounted for by the DLVO theory. This force was associated with the structuring of water in the vicinity of the surface.2 Theoretical work and computer simulations3-5 indicated that, in the vicinity of a planar surface, the density of the liquid oscillates with the distance, with a periodicity of the order of molecular size. This reveals that, near the surface, the liquid is ordered in quasi-discrete layers. When two planar surfaces approach each other at sufficiently short distances, the molecules of the liquid reorder in discrete layers, generating an oscillatory force.6 When the surfaces were not rigid, as in the case of lipid bilayers, the oscillations of the force were smoothed out and the interactions became monotonic. The short-range repulsion between neutral7 or weakly charged8 bilayers, often called hydration force, was exhaustively investigated experimentally and was found to have an exponential decay, with a decay length between 1.5 and 3 Å, while the preexponential factor varied by more than an order of magnitude. A phenomenological treatment of the hydration repulsion, based on a Landau expansion of the free energy density, was proposed by Marcelya and Radic.9 They showed that, if the free energy density is a function of an “order parameter” that varies continuously from the surface, and if only the quadratic terms in this parameter and its derivative are nonnegligible, an exponential decay * Corresponding author. E-mail address: feaeliru@ acsu.buffalo.edu. Phone: (716) 645-2911/2214. Fax: (716) 645-3822. (1) Horn, R. G.; Israelachvili, J. N. J. Chem. Phys. 1981, 75, 1400. (2) Derjaguin, B. V.; Churaev, N. V. J. Colloid Interface Sci. 1974, 49, 249. (3) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: New York, 1982. (4) Ciccotti, G.; Frenkel, D.; MacDonald, I. R. Simulation of Liquids and Solids; North-Holland: Amsterdam, 1987. (5) Evans, R.; Parry, A. O. J. Phys.: Condens. Matter 1990, 2, SA15SA32. (6) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1992. (7) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (8) Cowley, A. C.; Fuller, N. L.; Rand, R. P.; Parsegian, V. A. Biochemistry 1978, 17, 3163. (9) Marcelja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129.

of the repulsion could be obtained for not too short distances. They also suggested that the correlation (decay) length of the order parameter should be of the order of the molecular size. However, the microscopic origin of the order parameter, and more importantly, the dependence of the free energy density on the order parameter, remained obscure. The interaction was later associated with the propagation of the polarization in water, and two models were proposed. Schiby and Ruckenstein10 suggested that the polarization propagates because of interaction at the molecular level (mutual orientation of neighboring dipoles); we will return later to this model. Gruen and Marcelja11 introduced a continuum model, in which equations for both the polarization and the electric field, considered independent quantities, were derived via the minimization of the total free energy of the system. The spatial variation of the polarization was considered dependent on the concentration of the Bjerrum defects, which constitute the source of the polarization field, while the spatial variation of the electric field was considered dependent on the concentration of ions, which constitute the source of the electric field. They showed that the polarization varies smoothly from the interface and that the electrostatic double-layer repulsion is enhanced by a contribution due to the surface polarization. At low separation distances, the effect of polarization becomes dominant and the repulsion decays exponentially, with a decay length inversely proportional to the square root of the concentration of the Bjerrum defects. A simplified version of the theory was presented by Cevc and Marsh.12 They started from the Marcelja-Radic phenomenological treatment, assumed that the polarization constitutes the order parameter, and used the Gruen-Marcelja model to explain the various contributions to the free energy density. The subsequent molecular dynamics simulations13,14 indicated that the polarization in water does not vary smoothly near the surface, but oscillates, and hence is not (10) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (11) Gruen, D. W. R.; Marcelja, S. Faraday Trans. 2 1983, 211 and 225. (12) Cevc, G.; Marsh, D. Biophys. J. 1985, 47, 21. (13) Kjellander, R.; Marcelja, S. Chem. Scr. 1985, 25, 73; Chem. Phys. Lett. 1985, 120, 393. (14) Berkowitz, M. L.; Raghavan, K. Langmuir 1991, 7, 1042.

10.1021/la010979h CCC: $20.00 © 2001 American Chemical Society Published on Web 11/01/2001

Nanodispersions

476

Oscillatory and Monotonic Polarization

accurately described by the Gruen-Marcelja theory.11 It was therefore concluded that the polarization is not a suitable order parameter. It was also noted15 that the exponential decay of the hydration force in the GruenMarcelja model was a consequence of an inconsistent choice for the functional form of the “non-local dielectric function”. The molecular dynamics simulation indicated that, unlike polarization (which oscillates), the disruption of hydrogen bonding varies smoothly in the vicinity of the surface and constitutes therefore a better choice as an order parameter. There were a number of computational approaches13,14,16 and lattice model calculations17 which related the hydration force to the disruption of the hydrogen bonding networks when two surfaces approach each other. An essentially different point of view was suggested by Israelachvili and Wennerstrom, who consider that any interaction that may arise from water structuring effects should be either oscillatory or attractive but not repulsive, and hence that the microscopic origin of the hydration repulsion should be sought elsewhere than in hydrogen bonding or polarizability.18 They proposed that the protrusion of the molecules belonging to the lipid bilayers could explain the hydration repulsion.19 It was however argued that the interactions due to protrusion are much smaller than the experimentally measured hydration repulsion,20 and experiment21 indicated that the polymerization of the bilayers does not modify the hydration repulsion. Another mechanism for the hydration repulsion between lipid bilayers was more recently proposed by Marcelja.22 It is based on the fact that in water the ions are hydrated and hence occupy a larger volume. The volume exclusion effects of the ions are important corrections to the PoissonBoltzmann equation and modify substantially the doublelayer interaction at low separation distances. The same conclusion was reached earlier by Ruckenstein and Schiby,23 and there is little doubt that the hydration of individual ions modifies the double-layer interaction, providing an excess repulsion force.23 However, while the hydration of ions affects the double-layer interactions, the hydration repulsion is strong even in the absence of an electrolyte, or double-layer repulsion. One can therefore conclude that there is no commonly accepted explanation for the microscopic origin of the hydration repulsion. The main purpose of this paper is to show that a suitable model for the polarization of water layers, based on the earlier work of Schiby and Ruckenstein,10 is compatible with both simulations and experiments on hydration repulsion. The treatment of polarization based on the assumption that water has a uniform dielectric constant involves a fundamental difficulty. Indeed, a uniform, continuous distribution of dipoles on a planar surface does not generate a field in a homogeneous medium and hence is not able to polarize the water. If the dipoles are distributed in the sites of a 2D planar square lattice, the field is (15) Attard, P.; Wei, D. Q.; Patey, G. N. Chem. Phys. Lett. 1990, 172, 69. (16) Besseling, N. A. M. Langmuir 1997, 13, 2113. (17) Attard, P.; Batchelor, M. T. Chem. Phys. Lett. 1988, 149, 206. (18) Israelachvili, J. N.; Wennerstrom, H. Nature 1996, 379, 219. (19) Israelachvili, J. N.; Wennerstrom, H. Langmuir 1990, 6, 873. (20) Parsegian, V. A.; Rand, R. P. Langmuir 1991, 7, 1299. (21) Binder, H.; Dietrich, U.; Schalke, M.; Pfeiffer, H. Langmuir 1999, 15, 4857. (22) Marcelja, S. Nature 1997, 385, 689. (23) Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612.

Langmuir, Vol. 17, No. 24, 2001

7583

oscillating above the lattice, with an amplitude decaying exponentially with the normal distance from the lattice. Hence, the average field is vanishing, and the polarization does not propagate in water. In the model of Gruen and Marcelja the spatial variation of polarization was associated with the existence of the Bjerrum defects. Consequently, the absence of such defects implies that the decay length is infinite and the polarization is constant in the medium, instead of being zero, as expected on the above grounds. Cevc and Marsh suggested that the polarization of water near the interface is proportional to the normal component of the dipolar density,12 which is a plausible assumption. However, the proportionality constant was provided without explanation, and it is not clear why the surface dipoles polarize the continuous medium. While one cannot rule out that there are contributions of different origins to the hydration repulsion, the polarization contribution might be the dominant one, at least for not too small separations, and this can explain, as shown later in the paper, the quadratic dependence, determined experimentally by Simon and McIntosh,24 of the hydration repulsion on the surface dipolar potential. The model proposed in the present paper for the polarization is based on a simple idea. In a real medium, the interactions between remote dipoles are screened by the intervening solvent molecules and hence are decreased by a factor equal to the inverse of the dielectric constant of the medium. On the other hand, the interactions between adjacent dipoles are much less screened and hence are more important. In a medium with homogeneous dielectric properties, a planar layer of dipoles produces a vanishing average electric field above it. However, because the medium is not homogeneous at the molecular level, the field of a nearby dipole is screened less and an electric field is generated above the plane. To calculate the net field in water, when the dielectric constant is inhomogeneous, one should know the distribution of molecules around a given site. In the following, we will consider that a water molecule belongs locally to an icelike structure. This was suggested by Nemethy and Scheraga,25 who treated the bulk water as an aggregate of icelike clusters of various sizes. In the vicinity of a surface, it is reasonable to assume that the clusters (which have a layered structure) are aligned by the surface. The alignment is expected to gradually decrease away from the surface. It will be shown that local inhomogeneities in the dielectric constant of the medium are responsible for the propagation of polarization inside the medium. A discrete representation of the water layers leads to equations with finite differences whose solution allows one to relate the total electrostatic energy (due to polarization) to the density and location of the dipole moments on the surface. We will show that a long-range order in the water structure, normal to the interface, can lead to oscillations of the polarization and that in this case the repulsive force cannot be approximated by an exponentially decaying function. However, the roughness or the undulations of the polarizing surface can smooth out the oscillations and lead to an exponentially decaying repulsion. II. Hydration Interaction II.1. Net Field above a Plane of Dipoles. In a homogeneous dielectric described by a constant , the field (24) Simon, S. A.; McIntosh, T. J. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 9263. (25) Nemethy, G.; Scheraga, H. A. J. Chem. Phys. 1962, 36, 3382.

477

Polarization Model: a unified framework for hydration and double layer interactions

7584

Langmuir, Vol. 17, No. 24, 2001

Manciu and Ruckenstein

generated by an infinite planar, uniform distribution of dipoles vanishes. The dipole distribution can be thought of as two planar surface charge distributions, of equal magnitude and opposite sign, separated by the infinitesimal distance δ (δ f 0 while the surface dipole density δσ ) constant, where σ is the surface density of the charges on the two planes). Both planar charge densities generate constant electric fields, which cancel in the region outside the planes. The field, normal to the surface, generated by the surface dipoles assumed to be distributed on the sites of a square lattice can be calculated as follows. The field along the z direction, normal to the surface, generated by the positive charges is given by6

Ez+ )

σ 2πx 2πy 2πz 1 + 2 cos + cos exp + ... 20 d d d (1a)

(

(

) (

)

)

where  is the dielectric constant, 0 is the permittivity constant, x and y are the coordinates in the plane of the surface, and the charges are located at the positions (x, y, z) ) (nd, md, 0) with n and m integers and d the length of the unit cell of the lattice. The field generated by the negative charges located at (x, y, z) ) (nd, md, -δ) is given by

Ez- ) -

( (

σ 2πx 2πy 1 + 2 cos + cos 20 d d 2π(z + δ) exp + ... (1b) d

)

(

) )

Hence, for the total field generated by the dipoles normal to the surface one obtains

Ezd ) Ez+ + Ez- ) 2πσδ 2πx 2πy 2πz cos + cos exp (2) 0d d d d

(

) (

)

where the terms of higher order in δ were neglected. The field generated by the surface dipoles, averaged on a planar surface equal to a unit cell of the lattice, vanishes. However, the medium is not homogeneous at molecular distances, and this can be taken into account by assuming that the field caused by the nearest dipole is screened by a different dielectric constant, 1 < . If one denotes by Ez,id the field along the z direction, generated in a vacuum by the dipole i of the surface, the field above the lattice is given by

Ezd )

1

Ez,nearestd +

1

1

( ) 1

1

-

1 

Ez,id ) ∑  all others

Ez,nearestd + ( - 1) 1

1

∑Ez,id )

 all

1 Ez,nearestd ≡ Ez,nearestd (3) ′

where the sum of the fields generated by all the dipoles in a medium of constant  vanishes on average. Therefore, the net contribution to the field can be calculated by considering only the nearest dipole in a medium with an effective dielectric constant ′.26 A correction to eq 3 could

be obtained if the dependence of  on distance would be known. While the effective dielectric constant ′ is expected to be small near the surface dipoles, it is expected to acquire the bulk value after a few molecular distances. When calculating the net field generated by the surface dipoles in water, it will be assumed that effective dielectric constants can be used only for the interactions with the water molecules close to the surface (the first water layer). The interactions with more distant water molecules will be considered as in a continuum medium; hence, the net field will vanish on average. To calculate the net field produced by a surface dipole, we will assume that one dipole of the surface polarizes the water molecules of the first layer located nearby above it. For the sake of simplicity, the area polarized by one dipole will be consider circular of radius R and equal to the area corresponding to a surface dipole (the inverse of the number of dipoles per unit area). The surface dipole b p has the components pxy in the plane and pz normal to the surface. Because of symmetry, the field produced by pxy vanishes, and so does the electric field parallel to the surface. The electric field, normal to the surface, generated by a dipole in a medium with an effective constant ′, at a point whose position vector makes an angle θ with the z direction, is given by

Ezd )

pz 4π′0σ3

(3 cos2 θ - 1)

(4)

where σ ) ∆′/cos θ, ∆′ being the distance between the center of the dipole and the center of the first water layer. For the average field produced by a surface dipole one obtains26

E h )

∫0 4π′ 2πFdF (∆′2 + F2)3/2

1 πR2

R

0

{(

pz 3

∆′2 -1 (∆′ + F2) 2

)}

)

pz 1 (5) ′ 2π (R2 + ∆′2)3/2 0 II.2. Polarization of an Ordered, Icelike Water Structure. While the water molecules in the vicinity of the surface are polarized by the dipoles of the surface and by the neighboring water molecules, the other water molecules are polarized only by the field generated by the neighboring water molecules. To calculate this field, the structure of water between the external surfaces must be known. Here we will assume that icelike clusters which, as suggested by Nemethy and Scheraga,25 constitute the liquid water are aligned by the external surfaces and hence that a layered, icelike structure is formed between the surfaces. The water layers, parallel to the surface, are made of out-of-plane hexagonal rings of water molecules. Each layer is composed of two planar sublayers; a water molecule from a sublayer has three hydrogen bonds with three molecules of the other sublayer of the same layer, and one with a molecule belonging to a sublayer of one of the adjacent layers (a schematic picture is presented in Figure 1). All the molecules of a sublayer are assumed to have the same average dipole moment, and the local field, normal to the surface, acting on a site of a sublayer (I or II) of a layer j, can be formally written as (26) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7061.

478

Nanodispersions

Oscillatory and Monotonic Polarization

Langmuir, Vol. 17, No. 24, 2001

7585

where A, A h , and r are constants, eqs 6 and 7 lead to (j ) 2, ..., N - 1)

(D

-1

I,I

1 r + D+1I,Ir2 mj-1I + γ (D-1II,I + D0II,Ir + D+1II,Ir2)mj-1II ) 0 (9a)

(

+ D0I,I -

)

)

(D-1I,II + D0I,IIr + D+1I,IIr2)mj-1I + 1 D-1II,II + D0II,II - r + D+1II,IIr2 mj-1II ) 0 (9b) γ

(

(

)

)

The corresponding characteristic equation, which provides the values of r, has the form

(D Figure 1. Sketch of an icelike cluster around a selected water molecule. In this structure, a molecule from a sublayer (dotted lines) of one layer (delimited with full lines) is connected via hydrogen bonds with three molecules of the other sublayer of the same layer (only two bonds are drawn in the picture; it should be noted that the molecules drawn are not in the same plane) and with one water molecule from an adjacent layer. The first four neighbors are located at the vertexes of a tetrahedron. The projection of the position of the water molecules from one layer in the plane of the surface (denoted in the text as xy) is a hexagonal network. Each icelike cluster involves 26 molecules around the selected molecule (marked); they are located in three water layers: a central and the two adjacent ones.

EjI ) D0I,ImjI + D0II,ImjII + D-1I,Imj-1I +

-1

1 r + D+1I,Ir2 γ 1 II,II + D0II,II - r + D+1II,IIr2 -1 γ II,I II,I II,I 2 (D-1 + D0 r + D+1 r )(D-1I,II + D0I,IIr +

I,I

( (D

+ D0I,I -

D-1II,IImj-1II + D+1I,IImj+1I + D+1II,IImj+1II + ... (6b) where mjR is the average dipole moment of a molecule from the sublayer R (R ) I or II) of layer j and the interaction coefficients D(kR,β account for the contribution of the dipoles of the sublayer R of layer j ( k to the local field at a site of the sublayer β (R, β ) I or II) of layer j. The estimation of the coefficients D on the basis of the icelike model used is provided in Appendix A. It will be assumed that only the molecules within a radius 2l from the given site (where l is the distance between adjacent molecules) generate a nonnegligible field at a given site and hence that k ) -1, 0, and 1. The average polarization of a molecule is proportional to the local field:

mjR ) γEjR

(7)

where the proportionality constant is the molecular polarizability. Equations 6 and 7 constitute a system of (2N - 4) linear equations with constant coefficients for the average polarizations mjR, where N is the number of water layers. The other four equations are provided by the boundary conditions. Seeking solutions of the form27

mjI ) Arj and mjII ) A h rj

(8)

(27) Mickens R. E. Difference Equations; Van Nostrand Reinhold Company: New York, 1987.

)

)

)

D+1I,IIr2) ) 0 (9c) The general solution of the system in eqs 9a,b is

mjI ) A1r1j + A2r2j + A3r3j + A4r4j

(10a)

mjII ) A h 1r1j + A h 2r2j + A h 3r3j + A h 4r4j

(10b)

where the constants Ak (k ) 1, ..., 4) have to be determined using the four boundary conditions given below. The relation between the constants Ak and their conjugate A hk is obtained by substituting eq 8 in eq 9a or b:

D-1II,Imj-1II + D+1I,Imj+1I + D+1II,Imj+1II + ... (6a) EjII ) D0I,IImjI + D0II,IImjII + D-1I,IImj-1I +

) (

A h k ) -Ak

1 r + D+1I,Irk2 γ k ) ηkAk (11) D-1II,I + D0II,Irk + D+1II,Irk2

(

D-1I,I + D0I,I -

)

The boundary conditions should take into account the effect of both the field produced by the surface dipoles and the field produced by the molecules of the first two layers.

m1I ) γ(E h I + D0I,Im1I + D0II,Im1II + D+1I,Im2I + D+1II,Im2II) (12a) m1II ) γ(E h II + D0I,IIm1I + D0II,IIm1II + D+1I,IIm2I + D+1II,IIm2II) (12b) mNI ) γ(-E h II + D-1I,ImN-1I + D-1II,ImN-1II + D0I,ImNI + D0II,ImNII) (12c) mNII ) γ(-E h I + D-1I,IImN-1I + D-1II,IImN-1II + D0I,IImNI + D0II,IImNII) (12d) The fields generated by the surface dipoles on the sublayers of the first layer, E h I and E h II, are calculated using eq 5, for the corresponding distances from the center of the surface dipole to the centers of the sublayers, ∆I′ and ∆II′, respectively. The field in the first sublayer of the first water layer, generated by the surface dipoles, is calculated by considering an effective dielectric constant ′ < , while, for the field generated in the second sublayer of the same layer, a dielectric constant ξ′ is employed (ξ > 1) to account for the dependence of the screening on the distance.

479

Polarization Model: a unified framework for hydration and double layer interactions

7586

Langmuir, Vol. 17, No. 24, 2001

Manciu and Ruckenstein

Since the interaction coefficients D are invariant to the transformation I f II, k f -k, and, additionally, D-1I,I ) D+1I,I (see Appendix A), the characteristic equation is invariant to the transformation r f 1/r, which can be directly verified by substitution in eq 9c. Therefore, the solutions of the characteristic equation come in pairs,

r1 ) F1, r2 )

1 1 , r3 ) F2, r4 ) F1 F2

Because for identical surfaces the polarization should be antisymmetric with respect to the middle distance between the two, one can write

of their interaction with a strongly polarizable medium, the liquid water). The value predicted for the molecular polarizability is about four times larger than that provided by eq 16; however, this does not affect the qualitative features of the discussion which follows. For an icelike structure, the distance between the centers of two adjacent molecules is l ) 2.76 Å and v ) (8/3x3)l3 ) 32.37 Å3. Using  ) 80, the characteristic equation acquires the form

r4 + 0.575r3 - 25.78r2 + 0.575r + 1 ) 0

(17)

The solutions rk are

mjI ) -mN+1-jII for j ) 1, N

r1 ) Fa, r2 )

Using eqs 10 and 11, the latter equation leads to

1 1 , r3 ) -Fb, r4 ) Fa Fb

(18)

with Fa ) 4.783 and Fb ) 5.381. Consequently, the general antisymmetric solutions are

A1F1j + A2F1-j + A3F2j + A4F2-j ) -(η1A1F1N+1-j + η2A2F1-(N+1-j) + η3A3F2N+1-j + η4A4F2-(N+1-j)) (13) which can be rearranged as

+ η2A2F1-n)F1-k + (A2F1-n + η1A1F1n)F1k + (A3F2n + η4A4F2-n)F2-k + (A4F2-n + η3A3F2n)F2k

mjI ) A1(Faj - η1FaN+1-j) + A3(-1)j(Fbj - η3(-1)N+1FbN+1-j) (19a) mjII ) η1A1(Faj - η2FaN+1-j) +

(A1F1n

)0 (14)

where n ) (N + 1)/2 and k ) (N + 1 - 2j)/2. Since, for a given value of N, the equations should be valid for any value of j, the coefficients in parentheses should vanish. The four equations thus obtained reduce to two equations, because η1η2 ) η3η4 ≡ 1, and the constants A2 and A4 can be expressed in terms of A1 and A3, respectively. Consequently, the general solution has the form

mjI ) A1(F1j - η1F1N+1-j) + A3(F2j - η3F2N+1-j) (15a) mjII ) η1A1(F1j - η2F1N+1-j) + η3A3(F2j - η4F2N+1-j) (15b) For illustration purposes we will present a numerical example. The interaction coefficients D are evaluated in Appendix A, selecting for the effective dielectric constant for the interaction between neighboring water molecules ′′ ) 1. This constitutes a lower bound for ′′. For the molecular polarizability, we will employ an expression26 based on the hypothesis of hindered molecular rotations introduced by Debye. This provides a lower bound for γ, which is given by26 γ)

η3A3(-1)j(Fbj - η4(-1)N+1FbN+1-j) (19b) which depend on two constants, A1 and A3 to be determined from the boundary conditions, eqs 12a and 12b. (The other two boundary conditions, eqs 12c and 12d, are equivalent to eqs 12b and 12a, respectively.) The above equations are defined only for integer values of j; however, since both Fa and Fb are real and positive, the above equations can be analytically extended to noninteger values of j. A continuous function of z, which provides the same values as eqs 19 at the discrete positions of the sublayers and interpolates between them can be obtained using the equality

(∆z ln F) ) exp(λz)

Fj ) exp(ln(Fj)) ) exp(j ln(F)) ) exp

where ∆ ) 4/3l is the distance between the centers of two adjacent water layers. In what follows, we will define the origin of z at one of the interfaces, located at the external boundary of the first water layer. The first sublayer of the first water layer is located at the distance z ) 1/2l ) 3/8∆ and corresponds to j ) 1 in eq 19a. Hence, in eq 19a, j ) z/∆ + 5/8 and N + 1 - j ) (H - z)/∆ + 3/8, where H ) N∆ is the separation distance between the two surfaces. From eqs 19, one obtains

mI(z) )

( ( ) () ( ) ( )) ( ( ))( ( ) ( ) ( ) ( ))

5∆ z 3∆ H-z exp - η1 exp exp + 8λa λa 8λa λa z 5 5∆ z A3 cos π + exp exp ∆ 8 8λb λb 3∆ H-z (-1)N+1η3 exp (20a) 8λb λb

A1 exp

0v +2 + 0v(D-1I,I + D-1II,I + D0I,I + D0II,I + D+1I,I + D+1II,I) 3( - 1)

(16) where  is the dielectric constant of water, 0 is the permittivity constant, and v is the volume occupied by a water molecule. The Onsager theory for the molecular polarizability28 implies that the water molecules can rotate freely in the liquid water and that their permanent dipole moments are higher in the liquid than in the vapor phase (because (28) Onsager, L. J. Am. Chem. Soc. 1936, 58, 261.

and

mII(z) ) -mI(H - z)

(20b)

where λa ) ∆/ln Fa ) 2.35 Å and λb ) ∆/ln Fb ) 2.18 Å are the decay lengths of the amplitudes of the monotonic and oscillatory terms, respectively.

480

Nanodispersions

Oscillatory and Monotonic Polarization

Langmuir, Vol. 17, No. 24, 2001

7587

Figure 2. Dipole moment as a function of the distance from one surface, calculated in the discrete approach for N ) 10 and various values of ξ. The values of the other parameters are l ) 2.76 Å, (pz/′) ) 1.0 D, ′′ ) 1,  ) 80, πR2 ) 50 Å2, ∆′I ) 2.88 Å, and ∆′II ) 3.8 Å.

The numerical solutions for N ) 10, pz/′ ) 1.0 D, A ) πR2 ) 50 Å2, ∆′I ) 2.88 Å, and ∆′II ) 3.8 Å are presented in Figure 2, for various values for ξ. The contributions of the monotonic and oscillatory parts of the polarizations depend strongly on the boundary conditions, which basically means the values of E h I and E h II. Both the monotonic and oscillatory parts decay from the surface, but with different decay lengths. The oscillations of the polarization are a consequence of the particular geometry of the system. The field generated by the average dipole moment of one sublayer tends to align in the same direction as the dipole moments of the molecules of the adjacent layers, but in the opposite direction as the dipole moments of the molecules of the other sublayer of the same layer. If the difference between E h I and E h II is large enough, this will produce a polarization which oscillates with the distance from the surface. The magnitude of the oscillations, which are attenuated with the distance, depends on the ratio E h II/E h I and on the strength of the electrostatic interaction between adjacent layers. The total free energy per unit area, due to the polarization of the water molecules, can be calculated using the expression

F(N) ) -

Nw 2

N

(mjIEjI + mjIIEjII) ) ∑ j)1 -

Nw

N

∑((mjI)2 + (mjII)2) 2γ j)1

(21)

where Nw is the number of water molecules in a sublayer, per unit area, and the factor 1/2 avoids double counting. The above expression can be computed only for integer values of N, that is, an integer number of water layers. The interaction free energy Fint(N), obtained by subtracting the free energy F(Nf∞), is represented in Figure 3 for various values of ξ. For higher values of ξ, the electrostatic interaction might become attractive for odd values of N (i.e. F(N) < F(∞)). II.3. Effect of Partial Disorder. The oscillations of the polarization are a consequence of the long-range order along the direction normal to the surface. For planar surfaces, some disorder is, however, induced by the gradual decrease of the cluster alignment away from the surface.

Figure 3. Interaction free energy in the discrete approach as a function of the separation distance between surfaces, calculated in the discrete model for various values of ξ: (a, top) linear scale (which shows better the oscillatory behavior near the surface; (b, bottom) logarithmic scale (which better reveals the behavior at large distances).

In addition, if the surfaces are not planar, but are sufficiently rough or fluctuating, the average polarization is no longer defined only for discrete values of z (the positions of the water sublayers with respect to one planar surface) but becomes a continuous function of distance, m(z). The system with rough or undulating surfaces will be approximated here by a statistical ensemble of systems with planar surfaces in which the water molecules are organized in long-range icelike structures. Assuming that in the statistical ensemble the surfaces are distributed Gaussian with the root-mean-square fluctuation σ, the average m at a position z from the average surface is obtained from

m(z) )

1

∫

x2πσ

∞

( )

exp -∞ 1

s2

2σ2

∑ ∑

m(z - s) ds )

x2πσ R)I,II j)1,N

(

mjR exp -

)

(z - zjR)2 2σ2

(22)

where zjR is the position of the sublayer R of the layer j. During the fluctuations, the number N of layers changes. However, the values of mjR in the layers near the surfaces are almost independent of N, for large N. Figure 4a presents the dependence of m on z for various values of

481

Polarization Model: a unified framework for hydration and double layer interactions

7588

Langmuir, Vol. 17, No. 24, 2001

Manciu and Ruckenstein

Fint(H) )

(

∞

1

∑(F(Hj) - F(∞)) exp -

x2πσ′j)0

)

(H - Hj)2 2σ′2

(23)

where Hj ) j∆ is the distance between surfaces in each of the elements of the statistical ensemble. The interaction free energy for various values of σ′ is presented in Figure 4b. II.4. Monotonic Polarization. Let us assume that it is possible to define an average m as a monotonic function of distance to the average surface and that locally an icelike order is preserved in each of the clusters with radius 2l, l being the distance between the centers of two adjacent water molecule, containing 26 water molecules around a selected water molecule (see Figure 1). If the cluster is aligned with its layers parallel to the external surface, and its center is located at the position z (measured from one of the surfaces), the distance between the centers of two adjacent water layers is ∆ ) 4/3l, the positions of the sublayers of the central layer are z ( 1/6l, and the positions of the other sublayers are z ( 7/6l and z ( 3/2l, respectively. The equations which describe the polarizations of the two sublayers of the central layer are

(

l 3l 7l ) γ D-1I,Im z + D-1II,Im z + 6 2 6 l l 7l D0I,Im z - + D0II,Im z + + D+1I,Im z + + 6 6 6 3l D+1II,Im z + 2

) ( ( )

mz-

(

)

(

(

)

the root-mean-square fluctuation and a large value of N (N ) 20). For σ f 0, the average dipole moment is nonvanishing only around z ) zjR, while, for σ > 5 Å, the average dipole moment becomes a smooth, monotonically decreasing function of the distance from the surface. Such a behavior was observed in computer simulations. Molecular dynamics studies of water between rigid surfaces13,14 indicated that the polarization is oscillating in the vicinity of the surface, with a spatial period of the order of molecular size. However, when the dynamic restrictions on bilayers’ headgroups were removed, the polarization became a smooth function of z, monotonically decaying.29 A similar procedure can be used to calculate the interaction free energy between two undulating (or rough) surfaces. Assuming that the separation distance between surfaces is normally distributed around H with a dispersion σ′, the interaction free energy is given by (29) Perera, L.; Essmann, U.; Berkowitz, M. L. Prog. Colloid Polym. Sci. 1997, 103, 107.

(

)

( )) l 3l 7l m (z + ) ) γ(D m (z - ) + D m (z - ) + 6 2 6 l l 7l D m(z - ) + D m(z + ) + D m (z + ) + 6 6 6 3l D m(z + )) (24a,b) 2 II,I +1

Figure 4. (a, top) Average polarization m as a function of the distance from one surface (the other surface is located at a large separation distance), for a rough or fluctuating surface, calculated using eq 22 for ξ ) 2 and various values of σ. The functions are multiplied with arbitrary factors for easier comparison of their shapes. (The factors have the following values: (1) 1; (2) 50; (3) 100; (4) 150; (5) 200.) (b, bottom) Interaction free energy as a function of separation distance, for a rough or fluctuating surface, calculated using eq 23 for ξ ) 2 and various values of σ′. The functions are multiplied with arbitrary factors for easier comparison of their shapes. (The factors have the following values: (1) 1; (2) 5; (3) 20; (4) 35; (5) 50.)

)

+1

II,I

I,I

0

I,I

II,I -1

0

I,I -1

where because of the symmetry of the system to the transformation I T II, k T -k only six of the interaction coefficients D were used (see Appendix A). When the cluster is not perfectly aligned parallel to the surface but makes a small angle R, the coefficients of interactions are slightly modified (see Appendix B). If the distribution of the tilt angles and their dependence on z would be known, it could be included in the calculations which follow. However, as shown later, the tilting of the local clusters by as much as 20° changes the decay length less than 10%. In addition, the tilting is expected to be lower near the surface. For these reasons, the solution with a constant decay length probably constitutes a good approximation. We will employ the procedure used by Schiby and Ruckenstein10 and will expand m around z. Adding the two eqs 24, one obtains (neglecting the terms of order 4 and higher)

2m(z) +

∂2m(z) l2 ) ∂z2 36

{

(

γ (D-1I,I + D+1II,I) 2m(z) +

(

2

)

∂2m(z) 9l2 + ∂z2 4 2

)

∂ m(z) l + ∂z2 36 ∂2m(z) 49l2 (D+1I,I + D-1II,I) 2m(z) + ∂z2 36

(D0I,I + D0II,I) 2m(z) +

(

)}

(25)

482

Nanodispersions

Oscillatory and Monotonic Polarization

Langmuir, Vol. 17, No. 24, 2001

which can be rewritten as

∂2m(z) ∂z

2

-

1 m(z) ) 0 λ12

(26)

λ1 ) l

x

-

9 49 1 1 (D I,I + D+1II,I) + (D-1II,I + D+1I,I) + D I,I + D0II,I 4 -1 36 36 0 γ 1 2 - (D-1I,I + D-1II,I + D0I,I + D0II,I + D+1I,I + D+1II,I) γ

(

(

)

)

(27)

A simpler procedure26 is to consider that the average dipole moment of the water molecules of one layer is obtained by averaging over the two sublayers, mj ) (mjI + mjII)/2. In this case, the quantities representing the interaction coefficients with the same (C0) and adjacent (C+1 and C-1) layers are given by

C0 )

The total electrostatic free energy per unit area, between the two parallel plates, due to the orientation of dipoles m(z) in the local field E ) m(z)/γ is given by26

F(H) ) -

with λ1 given by

D0I,I

+

D0II,I,

1 C1 ) C-1 ) C+1 ) (D-1I,I + D-1II,I + D+1I,I + D+1II,I) 2 (28) The average dipole moment is related in this case to the average dipole moment of the molecules from the same and adjacent layers via

m(z) ) γ(C-1m(z - ∆) + C0m(z) + C1m(z + ∆)) (29) where ∆ ) 4l/3 is the distance between adjacent layers. After a Taylor expansion and neglecting the terms of order 4 and higher, a differential equation of the same type as eq 26 is obtained, but with a decay length λ2 given by26

4l λ2 ) 3

x

C1

(30)

1 - (C0 + 2C1) γ

2z - H ( 2λ ) m(z) ) -m H-∆ sinh( 2λ ) sinh

(31)

1

where λ can be λ1 or λ2. The value of m1 is related to the local field, which can be obtained by adding to the field generated by the surface dipoles, E h , the field generated by the molecules of the first two layers from the surface

m1 ) γEe ) γ(E h + C0m1 + C1m2)

(32)

Using eq 31 for m2 ) m(3∆/2), one obtains

m1 )

(

γE h H - 3∆ ( 2λ ) 1 - γC - γC H-∆ sinh( 2λ ) sinh

0

1

1

∑

1

∫ 2S volume

((

)

(33)

(m(z))2

2S all molecules (m(z))2 γv

γE h

sinh

1 - γ C0 + C1

( (

dV ) -

) )

H - 3∆

sinh

)

γ

2

2λ

H-∆ 2λ

)

1

∫0 (m(z))2 dz ) H

2γv

()

H - λ sinh

2

2

4v sinh

(

H λ

)

H-∆ 2λ

(34)

where V ) SH is the volume occupied by the water molecules and S is the area of the planar interfaces. The interaction free energy is obtained by subtracting F(Hf∞) from F(H). Because E h is proportional to pz/′, the free energy is proportional to (pz/′)2. This result is in agreement with the experiments of Simon and McIntosh24 and consistent with the theory of Schiby and Ruckenstein.10 To verify the accuracy of the continuum approximation, we compared the average dipole moments along the z axis calculated using both the discrete and continuous procedures for integer values of N. The general solution of the finite differences equation (eq 29) is given by

mj ) A1r1j + A2r2j

(35)

where r1 and r2 are the solutions of the characteristic equation

2

The solution of eq 26 for two planar, parallel interfaces at a distance H apart, if the average dipole moment of the first water layers (located at z ) (∆/2) is (m1 (hence, for the boundary condition m(z)∆/2) ) m1, m(z)H-∆/2) ) -m1) has the form

7589

r +

C0 -

1 γ

C1

r+1)0

(36)

and the constants Ak are obtained from the boundary conditions. For the conditions specified in section II.2, C0 < 0 and -(C0 - 1/γ) > 2C1. Consequently, both solutions of the characteristic equation are real and positive:

1 - γC0 1 r1 ) F, r2 ) , F ) + F 2γC1

x(

1 - γC0 2γC1

)

2

-1 (37)

For a system with identical surfaces, the average dipole moment should be antisymmetric with respect to the middle distance; hence, mj ) -mN+1-j, j ) 1, N:

A1Fj + A2F-j ) -(A1FN+1-j + A2F-(N+1-j))

(38)

which can be rearranged as

(A1F(N+1)/2 + A2F-(N+1)/2)(F(N+1-2j)/2 + F-(N+1-2j)/2) ) 0 (39) and can be satisfied for any j (and a given N) only if A2 ) - A1FN+1. The corresponding (discrete) solution is

mj ) A1(Fj - FN+1-j)

(40)

483

Polarization Model: a unified framework for hydration and double layer interactions

7590

Langmuir, Vol. 17, No. 24, 2001

Manciu and Ruckenstein Table 1

4π0′′l3D-1I,I 4π0′′l3D-1II,I 4π0′′l3D0I,I 4π0′′l3D0II,I 4π0′′l3D+1I,I 4π0′′l3D+1II,I 4π0′′l3C0 4π0′′l3C1 λ1 (Å) λ2 (Å) λ3 (Å)

R ) 0°

R ) 5°

R ) 10°

R ) 15°

R ) 20°

0.688 92 1.844 62 -1.377 84 -2.388 44 0.688 92 0.432 00 -3.766 28 1.827 23 2.770 4 2.964 9 3.137 7

0.681 07 1.823 61 -1.362 14 -2.361 22 0.681 07 0.427 08 -3.723 36 1.806 41 2.755 0 2.948 6 3.122 2

0.657 76 1.761 19 -1.315 52 -2.280 41 0.657 76 0.412 46 -3.595 93 1.744 58 2.708 6 2.899 6 3.075 7

0.619 70 1.659 28 -1.239 39 -2.148 44 0.619 70 0.388 59 -3.387 84 1.643 63 2.630 8 2.817 5 2.997 8

0.568 04 1.520 95 -1.136 07 -1.969 35 0.568 04 0.356 20 -3.105 42 1.506 61 2.520 8 2.701 6 2.888 1

Since z is measured from the external boundary of the first water layer, the center of the first water layer is located at z ) ∆/2 and corresponds to j ) 1. Therefore j ) z/∆ + 1/2 and N + 1 - j ) (H - z)/∆ + 1/2, where H ) N∆ is the distance between surfaces. The analytical extension for m(z) (the continuous expression which provides the exact values at the discrete points and interpolates between them) is thus given by

(

m(z) ) 2A1 exp

) (

)

H+∆ 2z - H sinh 2λ3 2λ3

where A1 is obtained from the boundary condition m(z)∆/2) ) m1; hence,

( (

) )

2z - H 2λ3 m(z) ) -m1 H-∆ sinh 2λ3 sinh

(41)

This solution is of the same type as eq 31, but with a decay length λ3 given by

((

1 - γC0 ∆ λ3 ) ) ∆ ln + ln F 2γC1

x(

1 - γC0 2γC1

) )) 2

-1

-1

(42)

Figure 5a presents the values of m(z) calculated with the discrete and continuous approaches for N ) 10. The interaction free energies derived by the two approaches are compared in Figure 5b; the agreement is very good for distances larger than about the thicknesses of two water layers (7.36 Å). However, when the separation distance between surfaces becomes comparable to the molecular size, one expects the free energy associated with the restructuring of water to become dominant. Therefore, in this region neither the discrete nor its continuum approximation is likely to represent an accurate description of the hydration interaction. Let us now consider that the layers of the local clusters make an angle R with the surface. In this case, cos2 θ in eq A.1 of Appendix A should be replaced by (Appendix B, eq B.7)

cos2 θ′ ) cos2 θ cos2 R +

sin2 θ sin2 R 2

(B.7)

Assuming that the average dipole moment remains constant in a layer regardless of the tilt angle, one obtains for the interaction coefficients the values listed in Table 1. Additionally, we computed the decay lengths for the parameter values given in section II.4: λ1 from the continuous approximation, eq 27; λ2 from the continuous approximation which treats the polarization of one layer as an average over its two sublayers, eq 30; and λ3 from

Figure 5. (a, top) Polarization as a function of the distance from one surface. The solution of the discrete approach (eq 40, circles) and its analytical interpolation (eq 41, line 1) are compared to the solution obtained via the continuous approximation (eq 31, line 2). (b, bottom) Interaction energy, as a function of separation distance, for the discrete approach and for the continuous approximation.

the analytical extension of the solution of finite difference equation, eq 42. As already noted, if the average tilting angle, R, remains below 20°, the interaction coefficients and the decay lengths are only slightly modified. III. Summary and Conclusions We followed the model introduced by Schiby and Ruckenstein.10 Arguments are brought that the local inhomogeneities of the dielectric constant are responsible for the propagation of polarization in water, in the vicinity of a planar surface which has a dipole moment density. If the water has a long-range icelike order along the direction normal to the surface, the system of finite

484

Nanodispersions

Oscillatory and Monotonic Polarization

Langmuir, Vol. 17, No. 24, 2001

difference equations, corresponding to the layered structure, leads to discrete solutions, which can oscillate with the distance. However, if some disorder is present, due, for instance, to the roughness or fluctuations of the surfaces, the average polarization becomes a monotonically decaying function of the average distance from the surface. A continuous equation is derived for m(z), and expressions are obtained for the decay length.

(

D0II,I ) D0I,II ) 1 3 4π0′′l3

(( ) ) (x )

1 2 12 3 -1 -1 x33 3 +3 3 1 11 3

(() 3

)

Appendix A. Evaluation of the Coefficients Dkr,β The coefficients D(kR,β account for the contributions of the dipoles of the sublayer R of layer j ( k to the local field at a site of the sublayer β (R, β ) I or II) of layer j. While the field produced by remote dipoles can be treated as screened by a medium with a large dielectric constant ( = 80), the screening of the neighboring dipoles is much weaker. As noted in section II.1, the net contribution to the local field generated by a layer of dipoles is due to the neighboring dipoles, whose interactions take place in a medium with an effective dielectric constant smaller than . It is assumed that only the dipoles located within a radius 2l (where l denotes the distance between the centers of two adjacent water molecules) from the given site contribute to the local field and that the effective dielectric constant for them has a value ′′, smaller than . The electric field along z, caused by a neighboring molecule having an average dipole moment m along the z direction, is given by

Ez )

m(3 cos2 θ - 1) 4π′′0σ

(A.1)

3

where σ is the distance between a neighboring molecule and the selected site and θ is the angle between b σ and z. We will consider that the cluster of radius 2l has an icelike structure with the layers perpendicular to z (the effect of a tilting angle is evaluated in Appendix B). In an ice-I structure of perfect tetrahedrons, the distances between the sublayers of the same layer is l/3 and the distance between adjacent layers is 4l/3. The edges of the tetrahedron formed by the four first neighbors of a water molecule have the length l(x8/3), while the planar projection of the tilted hexagonal lattice has the side l(x8/9). The volume occupied in this structure by a water molecule is v ) [8/(3x3)]l3. Because the system is invariant at the transformation I f II, +k f -k (see Figure 1), only six interaction coefficients should be computed:

D-1I,I ) D+1II,II )

D-1II,I

) D+1

I,II

(

1 3 4π′′0l3

)

1 2+6 4π′′0l3

D0I,I ) D0II,II )

3

3

(x ) (x )

-1

8 3

(x ) (x ) 3 11

2

2 3

2

11 3

-1 3

( (x ) )

)

3

)

0.6889 4π′′0l3 (A.2a)

D+1I,I ) D-1II,II )

1 3 4π′′0l3

D+1II,I ) D-1I,II )

3

(x ) (x ) 2 3

)

)-

7591

2.3884 4π0′′l3 (A.2d)

2

-1

8 3

3

)

0.6889 4π′′0l3

1 2 0.432 ) 4π′′0l3 5 3 4π′′0l3 3

()

(A.2e)

(A.2f)

The averaged coefficients of interaction C between layers are obtained from

C0 ) D0I,I + D0II,I ) -

3.76628 4π′′0l3

(A.3a)

1 C1 ) C-1 ) (D-1I,I + D-1II,I + D+1I,I + D+1II,I) ) 2 1.82723 (A.3b) 4π′′0l3

Appendix B. Effect of a Tilt Angle of the Cluster The coefficients DkR,β were calculated in Appendix A by assuming that the layers of the cluster are perpendicular to z. When the cluster is tilted, the angles between z and the direction to the neighboring molecules is modified. To calculate the new angle, let us consider that the neighboring molecule, located at the point (x1, y1, z1) is rotated with an angle R (the tilt angle) around an axis in the xy plane, which makes the angle φ with the x axis. The final coordinate z3 of the molecule after rotation can be obtained by first calculating the coordinates (x2, y2, z2) in a system x′y′z whose axis x′, located in the xy plane, makes the angle φ with x

() (

)( )

x2 cos φ sin φ 0 x1 y2 ) -sin φ cos φ 0 y1 z2 0 0 1 z1

(B.1)

and then rotating the molecule around the x′ axis with an angle R:

)

1.8446 (A.2b) 4π′′0l3

(-1) 1 1.3778 6 )(A.2c) 3 3 4π0′′l 4π0′′l3 8 3

()(

)( )

x3 x2 1 0 0 y3 ) 0 cos R -sin R y2 z3 z2 0 sin R cos R

(B.2)

From eqs B.1 and B.2 one obtains

z3 ) z1 cos R - x1 sin R sin φ + y1 sin R cos φ (B.3)

Polarization Model: a unified framework for hydration and double layer interactions

7592

485

Langmuir, Vol. 17, No. 24, 2001

Manciu and Ruckenstein

The angles between the axis z and the direction of the molecule, θ and θ′, before and after rotation are given by

cos2 θ )

x12

+

z12 y12

+ z1

2

and cos2 θ′ )

z3 x32

2

Averaging over the angle φ

∫φ)0(z1 cos R - x1 sin R sin φ +

1 2π

2π

y1 sin R cos φ)2 dφ )

+ y32 + z32 (B.4)

Since the distance is not modified by rotation, x12 + y12 + z12 ) x32 + y32 + z32, and from eqs B.3 and B.4 one obtains

z12 cos2 R + (x12 + y22) one obtains from eq B.5 the result

〈cos2 θ′〉 ) cos2 θ cos2 R + 2

cos θ′ )

(z1 cos R - x1 sin R sin φ + y1 sin R cos φ)2 x12 + y12 + z12

(B.5)

sin2 R (B.6) 2

sin2 θ sin2 R 2

where 〈 〉 denotes the average over the angle φ. LA010979H

(B.7)

Nanodispersions

486

Langmuir 2005, 21, 11749-11756

11749

Polarization of Water near Dipolar Surfaces: A Simple Model for Anomalous Dielectric Behavior Marian Manciu* Department of Physics, University of Texas at El Paso, El Paso, Texas 79968

Eli Ruckenstein‡ Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received July 5, 2005. In Final Form: September 17, 2005 A model for the electrostatic interactions in water in the vicinity of a surface is suggested, which accounts, within the Poisson-Boltzmann mean field approach, for the screening of the charges and for the coupling interactions between neighboring dipoles. When the water molecules near a solid surface are assumed to be organized in icelike layers, the polarization is not a continuous function but exists only at the discrete positions of the water molecules. The particular positions of the water molecules in the icelike structure govern the manner in which the average water dipoles align with each other. On the basis of this model, one could explain the nonmonotonic behavior of the polarization and the electrical potential as well as the anomalous dielectric response of water (the nonproportionality of the polarization and the macroscopic electric field), which were obtained recently via molecular dynamics simulations.

1. Introduction The interactions between charged particles in water is described in the traditional Derjaguin-Landau-VerweyOverbeek (DLVO) model by a van der Waals attraction between particles, coupled with a double layer repulsion due to the overlapping of the ion clouds formed near charged surfaces.1 In the traditional approach, the ions are assumed to have Boltzmannian distributions in a mean electric field generated by charges, which in turn obeys the Poisson equation with a uniform dielectric constant (corresponding to bulk water).2 These assumptions are oversimplified, and many corrections to the model have been proposed to improve its accuracy (by accounting for image forces, finite sizes of the hydrated ions, ion correlations, dependence of dielectric constant on the field and electrolyte concentration, ion hydration and ion dispersion forces, etc.; for a recent review, see ref 3). Whereas the corrections to the traditional PoissonBoltzmann approach could explain many experimental results, there are systems, such as the vesicles formed by neutral lipid bilayers in water, for which an additional force is required to explain their stability.4 This force was related to the organization of water in the vicinity of hydrophilic surfaces; therefore it was called “hydration force”.5 The first physical models for the hydration force related the interactions to the polarization of water near a surface, * Corresponding author. Phone: (915) 747-7531; fax: (915) 7475447; e-mail: [email protected]. ‡ Phone: (716) 645-2911/2214; fax: (716) 645-3822; e-mail: [email protected]. (1) Deryagin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633; Verwey, E. J.; Overbeek, J. Th. G. Theory of Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (2) Gouy, G. J. Phys. Radium 1910, 9, 457; Chapman, D. L. Philos. Mag. 1913, 25, 475. (3) Manciu M.; Ruckenstein, E. Adv. Colloid Interface Science 2003, 105, 63. (4) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (5) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Suface Forces; Plenum Publishing: New York, 1987.

and two different approaches were proposed concomitantly.6,7 One of these models7 was based on the observation that the exponential decaying force, observed experimentally, is mathematically equivalent to a Landau order-parameter expansion if only the quadratic terms of the expansion are retained.8 While the physical grounds for such an expansion remained obscure,8 it was later conjectured that the Bjerrum defects might constitute the source of the polarization field in analogy with the ions, which provide the source of the electric field.7 The polarization and the electric fields are no longer considered proportional to each other, and coupled equations for them were obtained through a variational procedure. A suitable concentration of Bjerrum defects had to be selected to match the experimental results for the hydration force.7 The Gruen-Marcelja model could relate the hydration force to the physical properties of the surfaces by assuming that the polarization of water near the interface is proportional to the surface dipole density.9 This assumption led to the conclusion that the hydration force is proportional to the square of the surface dipolar potential of membranes (in agreement with the Schiby-Ruckenstein model),6 a result that was confirmed by experiment.10 However, subsequent molecular dynamics simulations revealed that the polarization of water oscillated in the vicinity of an interface, instead of being monotonic.11 Because the Gruen-Marcelja model was particularly built to explain the exponential decay of the polarization, it was clearly invalidated by the latter simulations. Other conceptual difficulties of this model have been also reported.12,13 (6) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (7) Gruen, D. W. R.; Marcelja, S. J. Chem. Soc., Faraday Trans. 2 1983, 79, 211. (8) Marcelja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (9) Cevc, G.; Marsh, D. Biophys. J. 1985, 47, 21. (10) Simon, S. A.; McIntosh, T. J. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 9263. (11) Berkowitz. M. L.; Raghavan, K. Langmuir 1991, 7, 1042. (12) Attard, P.; Wei, D. Q.; Patey, G. N. Chem. Phys. Lett. 1990, 172, 69.

10.1021/la051802g CCC: $30.25 © 2005 American Chemical Society Published on Web 10/29/2005

Polarization Model: a unified framework for hydration and double layer interactions

11750

Langmuir, Vol. 21, No. 25, 2005

Recent molecular dynamics simulations of water between two surfactant (sodium dodecyl sulfate) layers, reported by Faraudo and Bresme,14 revealed oscillatory behaviors for both the polarization and the electric fields near a surface and that the two fields are not proportional to each other. While the nonmonotonic behavior again invalidated the Gruen-Marcelja model for the polarization, the nonproportionality suggested that a more complex dielectric response of water might be at the origin of the hydration force. The latter conclusion was also supported by recent molecular dynamics simulations of Faraudo and Bresme, who reported interactions between surfactant surfaces with a nonmonotonic dependence on distance.15 This type of oscillatory hydration was previously observed experimentally in interactions between mica surfaces in water,16 and has been associated with the layering of water in the vicinity of a surface.16,17 The discrete nature of the water molecules, considered hard spheres, was suggested to be responsible for these nonmonotonic interactions;18,19 however, the high fluidity of the water confined in molecularly thin films20 seems to be inconsistent with the “crystallization” of water predicted by the hard-sphere model.18 It is commonly accepted that nonassociative liquids have the tendency to crystallize near surfaces.21 Whether or not water is layered near a surface, however, is still under debate. Experiments on phospholipid bilayers involving coherent anti-Stokes Raman scattering (CARS) microscopy22 or electron spin resonance (ESR)23 indicate that the water is highly ordered in the vicinity of a surface containing strong dipoles, with the permanent dipoles of the water molecules aligned parallel to the surface dipoles. The alignment is so strong that it weakens the hydrogen bonding in water near a strong dipolar interface.22 Even if one assumes that the water near a surface has the same structure as it does in bulk, the oscillations of the short-range interactions between surfaces could be explained by a nonlocal dielectric constant for water.24 This model assumes that the dielectric displacement field (D ) 0E + P) at a position r not only depends on the local electric field [D(r) ) (r)E(r)], but also depends on the electric field in the whole space: D(r) ) ∫(r,r′)E(r′)dr′. In this model, the oscillations of the interactions are due to charge overscreening25 and are analogous to the charge density waves in plasmas.24 Whereas in the Henderson and Lozada-Cassou model18 the oscillatory hydration interactions are not affected much by the electrolyte concentration, in the Cheperanov model24 the electrolyte concentration plays a major role (the oscillations increasing resonantly for concentrations larger than 0.01 M). Both models18,24 predict many oscillations in the hydration interactions with a periodicity of ∼2 Å, which is in excellent agreement with the (13) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582. (14) Faraudo, J.; Bresme, F. Phys. Rev. Lett. 2004, 92, 236102. (15) Faraudo, J.; Bresme, F. Phys. Rev. Lett. 2005, 94, 077802. (16) Israelachvilli, J. N.; Pashley, R. M. Nature 1983, 306, 249. (17) Cleveland, J. P.; Schaeffer, T. E.; Hansma, P. K. Phys. Rev. B 1995, 52, R8692. (18) Henderson, D.; Lozada-Cassou, M. J. Colloid Interface Sci. 1986, 114, 180. (19) Trokhymchuck, A.; Henderson, D.; Wasan, D. T. J. Colloid Interface Sci. 1999, 210, 320. (20) Zhu, Y. X.; Granick, S. Phys. Rev. Lett. 2001, 87, 096104. (21) Israelachvilli, J. N.; McGuigan, P. M.; Homola, A. M. Science 1988, 240, 189. (22) Cheng, J.-X.; Pautot, S.; Weitz, D. A.; Xie, X. S. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 9826. (23) Ge, M.; Freed, J. H. Biophys. J. 2003, 85, 4023. (24) Cherepanov, D. A. Phys. Rev. Lett. 2004, 93, 266104. (25) Kornyshev, A. A.; Leikin, S.; Sutmann, G. Electrochim. Acta 1997, 42, 849.

487

Manciu and Ruckenstein

experiment for the force between mica surfaces at 10-3 M. However, they are inconsistent with the polarization data obtained via molecular dynamics simulations of electrolyte solutions of higher ionic strengths confined between flat surfaces.11,14 Because the above models do not relate the magnitude of the hydration force to the physical properties of the surfaces, they cannot explain why in most systems the hydration interactions are monotonic functions of the separation distance (such as for neutral lipid bilayers),4 why their magnitude depends on the surface dipole density,4,10 and why the hydration decay length itself depends on the nature of the surface (being ∼2 Å for neutral lipid bilayers, but on the order of 10 Å for the interactions between mica surfaces).26 It was recently shown that all of the above effects could be explained within the framework of Ruckenstein’s model for water polarization.27 Let us now briefly review the basics of the model for water polarization proposed by Schiby and Ruckenstein.6 The main idea was that the neighboring dipoles are able to orient each other, a hypothesis than seems to be validated by more recent experiments.22,23 A planar surface with a homogeneous, constant surface dipole density cannot generate any electric field in a continuous medium. At any point, the field generated by the neighboring dipoles is exactly compensated by the opposing field of the more remote dipoles. Consequently, in a continuum theory (such as the Gruen-Marcelja model),7 the surface dipoles are not able to polarize the nearby water molecules. However, at the molecular level water is hardly a continuum, and the interactions between neighboring dipoles are much stronger than the interactions between remote dipoles, which are screened by the intervening water molecules.13 Therefore, one can expect on intuitive grounds that the electrical interactions between neighboring dipoles take place in a medium with a much lower dielectric constant than that of bulk water. Because of the change in dielectric screening with distance, the fields generated by the neighboring and the remote dipoles do not cancel each other any longer, and a net electric field is generated in the medium. This field is able to polarize the neighboring dipoles, which in turn generate electric fields in their vicinity, polarizing their neighboring dipoles and so on, which is in agreement with the recent CARS microscopy results for the orientation of water molecules in the vicinity of a phospholipid bilayer surface.22 To calculate the dipole correlations, Schiby and Ruckenstein assumed a homogeneous distribution of water molecules, which simplified the calculations considerably and led to an exponential decay of the polarization of water from a dipolar surface. The overlap of the polarized regions, when two surfaces approach one another, increases the free energy of the system, and this generates a repulsion with a roughly exponential behavior, which has a magnitude and a decay length that are very similar to the hydration forces measured in neutral lipid bilayers systems.4 The monotonic decay of the polarization in the SchibyRuckenstein model (the main critique of the polarization models) is a consequence of the assumption of the homogeneous distribution of water molecules in the vicinity of the surface. However, when the water was assumed to be structured in icelike layers in the vicinity of the surface, the polarization became an oscillatory function of the distance from the interface.13 This result was due to the particular locations of the water molecules (26) Pashely, R. M. J. Colloid Interface Sci. 1981, 83, 531. (27) Manciu, M.; Ruckenstein, E. Adv. Colloid Interface Sci. 2004, 112, 109.

488

Polarization of Water Near Dipolar Surfaces

Figure 1. (a) The structure of ice-I. Each water molecule is connected by hydrogen bonding with three water molecules from sites of different kind of the same layer and one water molecule from one of the adjacent layers. The two different sites I and II of the same layer are marked on the Figure. (b) A schematic model: the dipoles p of the surface groups are embedded in a medium of dielectric constant ′, formed by the bound water molecules. The centers of the dipoles are at a distance ∆′ from the first water layer (of thickness ∆). The polarization of the water molecules is due to the interactions with neighboring dipoles as well as with all the other charges of the system (surface charges and electrolyte ions).

in the icelike structure (see Figure 1a). In each icelike layer, the water molecules can occupy two distinct sites. As it will be shown later, the dipoles generate an electric field in the same direction in neighboring water molecules from adjacent layers; however, they generate a field in the opposite direction in the neighboring water molecules from the same layer, thus polarizing them in opposite directions. Therefore, it is possible for the dipoles of the water molecules belonging to different sites of the same layer to become oriented in antiparallel directions, leading to an oscillatory behavior in the polarization. This picture is the electrostatic equivalent of the antiferromagnetic materials, in which neighboring spins align with each other in opposite directions. A schematic drawing is presented in Figure 1b in which the surface dipoles p align with the dipoles of the water molecules of sites “I” of the first layer “-S” in the same direction, generating the average dipolar moment mIS, whereas the average dipole moment mII S of the water molecules of sites “II” of the same layer are oriented in opposite direction. This picture is supported by the calculations presented in Section 3. It should be noted that molecular dynamics simulations of water between flat surfaces11,14,15 showed an oscillatory behavior in the polarization; however, when the dynamic restrictions on

Nanodispersions

Langmuir, Vol. 21, No. 25, 2005

11751

the surface headgroups were removed, the statistical fluctuations of the surfaces render the polarization a monotonic function with the distance from the surface.28 This explains the monotonic decay of the hydration force obtained in most experiments13 and justifies the approximation employed in the Schiby-Ruckenstein model.6 This “smearing out” of oscillations in disordered systems was suggested by Israelachvilli and Pashley16 and was analyzed more quantitatively elsewhere.13 It should also be noted that, if the surface dipoles are strong enough to orient all of the dipoles of the neighboring water molecules parallel to those of the surface by weakening the hydrogen bonding,22 the icelike layering of the water near the interface will be disrupted. Consequently, no oscillations in the polarization would occur. A simple model that illustrated the behavior of the polarization when the water molecules are organized in water layers between perfectly flat surfaces was previously suggested.13 That model took into account the nearestneighbor dipole interactions, but ignored the surface charges and the electrolyte ions. The model is extended here to cases in which an electrolyte as well as surface charges are also present. It will be shown that a treatment of all electrostatic interactions, in the assumption of an icelike structuring of water near interfaces, can predict an oscillatory behavior for both the polarization and the electric potential as well as a nonproportionality between the polarization and the electric fields. It will be shown in what follows that the oscillations of the polarization are due to the structuring of water in a particular form, the coupling interactions between neighboring dipoles, the electrolyte concentration, and the boundary conditions (surface charge and surface dipole density). 2. Basic Equations The average polarization of a water molecule is due to the electric field generated by all of the other charges and dipoles and obeys the Poisson equation, which in the vicinity of a planar surface has the form29

d2ψ(x) dx2

)-

∑i ci(x)qi 0

+

1 dP(x) 0 dx

(1)

in which ψ is the electrical “mean field” potential, considered to depend only on the distance x from the planar surface; qi is the charge of an ion of kind i and concentration ci(x); P(x) is the polarization of the medium; and 0 is the vacuum permittivity. The local concentrations of the ions in the “mean” electric field are assumed to have Boltzmannian distributions:

(

ci ) cE exp -

)

qiψ(x) kT

(2)

in which cE represents the electrolyte concentration in the reservoir far from the plate. By assuming that the polarization is proportional to the macroscopic field, P ) -0( - 1)dψ/dx, the Poisson-Boltzmann equation is obtained from eqs 1 and 2. However, the Schiby and Ruckenstein model suggested that the neighboring dipoles create a supplementary field, and the average polarization (28) Perera, L.; Essmann, U.; Berkowitz, M. L. Prog. Colloid Polym. Sci. 1997, 103, 107. (29) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584.

489

Polarization Model: a unified framework for hydration and double layer interactions

11752

Langmuir, Vol. 21, No. 25, 2005

Manciu and Ruckenstein

moments of the molecules in the two sites, I and II, of layer j are therefore provided by

(

mIj ) γ E + EP +

[(

)

mIj dψ ) 0v0( - 1) 30v0 dx

)

I j

+

I II,I II I,I I II,I II DI,I -1 mj-1 + D-1 mj-1 + D0 mj + D0 mj +

]

I II,I II DI,I +1 mj+1 + D+1 mj+1 (5a)

(

mII j ) γ E + EP +

[(

)

mII j dψ ) 0v0( - 1) 30v0 dx

)

II j

+

I II,II II I,II I II,II DI,II mII -1 mj-1 + D-1 mj-1 + D0 mj + D0 j +

]

I II,II II DI,II +1 mj+1 + D+1 mj+1 (5b)

Figure 2. Sketch of the structuring of water in icelike layers in the vicinity of surfaces and the points at which the discrete variables are defined. The water molecules (circles) for the two sites of each icelike layer j have the average polarization mIj I and mII j , respectively. The average value of the potential ψj is defined at the position between two adjacent layers j-1 and j, whereas ψII j is defined at the middle of layer j.

of a water molecule is proportional to the total field acting at its location. The molecular field acting at the location of a water molecule is composed of a macroscopic field, E ) -(dψ/ dx), a local field (the Lorentz field EL) that occurs because each molecule is surrounded by a dielectric medium (constituted of all the other water molecules), EL ) P/30, plus an additional field, EP, due to the neighboring dipoles.13,27 Assuming that the additional field is generated only by the neighboring molecules from the same and adjacent icelike water layers (see Figure 2), at each of the sites (I or II) of layer “j” the field is provided by13 I,I I II,I II I,I I II,I II EI,j P ) D0 mj + D0 mj + D-1 mj-1 + D-1 mj-1 + I II,I II DI,I +1 mj+1 + D+1 mj+1 (3a) I,II I II,II I,II I II,II II EII,j mII P ) D0 mj + D0 j + D-1 mj-1 + D-1 mj-1 + I II,II II DI,II +1 mj+1 + D+1 mj+1 (3b)

in which the m’s and the D’s are average dipole moments and coupling coefficients, respectively (mIj represents the average dipole moment of a molecule at site I of layer j, and the coupling coefficient DI,II represents the ratio 0 between the electric field generated by all of the molecules of sites I of layer j at the positions of sites II of layer j and the average dipole moment of the former molecules, mIj ). The molecular polarizability of a water molecule, γ, can be related to macroscopic quantities by taking into account that, in an uniformly polarized medium, the fields generated by the near neighbors cancel each other. Consequently, one obtains29

(

m)γ E+

)

m ) Pv0 ) 0v0( - 1)E 30v0

(4)

in which v0 represent the average volume occupied by a water molecule, which leads to the Claussius-Mossotti equation, γ ) 30v0 ( - 1)/( + 2). The average dipole

in which the derivatives of the potentials of eqs 5a and 5b should be taken at the location of sites I and II, respectively, of layer j. 3. General Behavior of the Solutions of the System Although the nonlinear system of equations can be solved numerically, here we focus on the linear approximation of the Poisson-Boltzmann equation (which is accurate for small values of the potentials ψ, qψ/kT , 1). Because the average polarization of water is P(x) ) m(x)/v0, in this approximation eqs 1 and 2 become

d2ψ  1 dm(x) ) ψ+ 0v0 dx dx2 λDH2

(6)

in which  is the dielectric constant of water, and λDH ) (0kT/2e2cE)1/2 is the Debye-Hu¨ckel length for a uniunivalent electrolyte of concentration cE, where e is the elementary charge, k is the Boltzmann constant, and T is the absolute temperature. The polarizations of the water molecules are discrete quantities, being defined only at the locations of the water molecules (sites I and II in an icelike layer). To combine the discrete eq 5a,b with the continuous PoissonBoltzmann eq 6, the latter equation must also be discretized. To do this, we define for each layer j the average discrete potentials ψIj and ψII j , which are defined at the boundary and the middle of the layer, respectively (see Figure 2). These positions were selected because the potential undergoes a steep change at the crossing of a dipolar surface (at the locations of dipoles mIj and mII j ), whereas, in the remainder of the space, it varies slowly (only because of the distribution of electrolyte ions). By integrating eq 6 between site II of layer j-1 and site I of layer j, and between site I of layer j and site II of the same layer j one obtains

(

)

I ψII dψ I dψ II j - ψj |j| j-1 = dx dx ∆/2 II I II ψIj - ψj-1 ψII j - 2ψj + ψj-1  I ≡2 ) ψj (∆ - δ) + ∆/2 ∆ λ 2

(

)

DH

1 (mI - mII j-1); (7a) 0v0 j

490

Nanodispersions

Polarization of Water Near Dipolar Surfaces

(

Langmuir, Vol. 21, No. 25, 2005 11753

)

ψIj+1 - ψII dψ II dψ I j |j |j= dx dx ∆/2 I I I ψII ψj+1 - 2ψII j - ψj j + ψj  II ≡2 ) ψj δ + ∆/2 ∆ λ 2

(

)

DH

1 (mII - mIj ) (7b) 0v0 j in which ∆ is the distance between the centers of two adjacent icelike layers, δ is the distance between the sites I and II of the same layer, and dψ/dx | Ij represents the derivative of the potential with respect to distance, calculated at site I of layer j, which is approximated using finite differences. The same approximations in eqs 5a and 5b lead to

(

mIj ) 0v0( - 1) 2

ψIj - ψII j I II,I II + DI,I -1mj-1 + D-1 mj-1 + ∆

vanishes. This condition leads to an equation of 6 degrees in r, which provides 6 values for r. The physical symmetry of the system (the invariance at a simultaneous exchange of layers j-1 and j+1 and of sites I and II) requires that for each solution rk, its reciprocal 1/rk should also be a solution of the system; hence, only three values of rk are independent. There is a simple correspondence between the solutions rk of a linear system with finite differences and the characteristic lengths of a linear system of differential equations. By denoting x ) j∆ as a continuous position variable, one obtains

I ψII j - ψj+1 I + DI,II -1 mj-1 + ∆ II I,II I II,II II I,II I DII,II -1 mj-1 + D0 mj + D0 mj + D+1 mj+1 +

mII j ) 0v0( - 1) 2

)

II DII,II +1 mj+1 (7d)

Consequently, there are 4 independent variables mIj , ψIj , and ψII j for each icelike layer j, and a system of 4 linear equations with finite differences (eqs 7a-7d) is obtained. We try solutions of the type30

mII j ,

ψIj ) arj j ψII j ) br

mIj ) crj (8)

which, when introduced in the system of eqs 7a-7d, lead to

[

] ( )

4  2 1 1 1 1 + (∆ - δ) a - 1 + b + cd) ∆ λ 2 ∆ r 0v0 0v0 r DH

0 (9a) 2 4  1 1 - (1 + r)a + + (∆ - δ) b c+ d) ∆ ∆ λ 2 0v0 0v0 DH 0 (9b)

[

λk ≡

]

DI,I 2 2 -1 1 I,I mIj ) a - b + + DI,I c+ 0 + D+1r ∆ ∆ r 0v0( - 1)

mII j ) -

[

2r 2 a+ b+ ∆ ∆

(

(

) ) ]

DII,I -1 II,I + DII,I 0 + D+1 r d ) 0 (9c) r

DI,II -1 r

(10b)

as a characteristic length associated to the solution rk. The associate solution 1/rk corresponds to the characteristic length -λk. While a description in terms of characteristic lengths is more appealing intuitively, one should note that those solutions are defined only at the discrete points x ) j∆ (with j integer). Because the determinant of the system (eq 9) vanishes for any rk, three of the constants for each set of ak, bk, ck, and dk can be determined as functions of the forth one. This means that the solution depends on six constants, which have to be determined from the boundary conditions. When the surfaces are identical, the symmetry implies that there are only three independent boundary conditions (For an even number of layers, ψI is a symmetric function with respect to the middle distance between surfaces and mIj ) mII -j-1, whereas, for an odd number of layers, ψII is symmetric with respect to the middle distance and mIj ) mII -j). Assuming a surface charge density σ and a surface dipole density p⊥/A, in which p⊥ is the normal component of the dipole moment of a polar surface group and A is the average area occupied by each dipole located at a distance ∆′ from the boundary of the first icelike water layer, the Poisson equation provides the first boundary condition:

]

[

∆ ln(rk)

with

)

j mII j ) dr

(10a)

k

k

I II,I II I,I I II,I II DI,I 0 mj + D0 mj + D+1mj+1 + D+1 mj+1 (7c)

(

(∆x ln r ) ) expλx

rjk ) rx/∆ ) exp k

I,II + DI,II 0 + D+1 r c +

dψ-SI dx

)-

σ 0′

(11a)

in which ′ is the dielectric constant of the medium formed by the surface polar groups and the disorganized water molecules between them (see Figure 1b). The remaining two boundary conditions are provided by the polarization of the water molecules located on the two sites of the first water layers (denoted by j ) (S):

(

mI-S ) 0v0( - 1) 2

ψI-S - ψII -S I + DI,I 0 m-S + ∆

)

DII,II -1 1 + DII,II + DII,II d ) 0 (9d) 0 +1 r r 0v0( - 1)

II I,I I II,I II I DII,I 0 m-S + D+1 m-S+1 + D+1 m-S+1 + E-S (11b)

The linear system of eqs 9a-9d has nontrivial solutions for the constants a, b, c, and d only if its determinant

(30) Mickens, R. E. Difference Equations; Van Nostrand Reinhold Company: New York, 1987.

Polarization Model: a unified framework for hydration and double layer interactions

11754

Langmuir, Vol. 21, No. 25, 2005

(

mII -S ) 0v0( - 1) 2

491

Manciu and Ruckenstein

I ψII -S - ψ-S+1 I + DI,II 0 m-S + ∆

)

I,II I II,II II II DII,II mII 0 -S + D+1 m-S+1 + D+1 m-S+1 + E-S (11c)

The main differences between eqs 11a-11c and eqs 5a and 5b exist in the absence of water dipoles in the layer -S-1 (because the layer -S is the first icelike layer) and the presence of the fields EI-S and EII -S, generated at sites I and II of layer -S, respectively, by the surface dipoles. Because the dielectric constant for the latter interaction, ′, is much smaller than the dielectric constant of bulk water, only the nearest surface dipole has a significant contribution to these fields, which can be approximated by13

EI-S )

p⊥ ′

1 A ∆-δ 2π0 + ∆′ + π 2

E-SII )

p⊥ ′

1 A ∆+δ 2π0 + ∆′ + π 2

[ (

[ (

2 3/2

)]

2 3/2

)]

(12a)

(12b)

The linear system of eqs 9a-9d with finite differences, together with the three boundary conditions (for identical surfaces), can be solved analytically to provide solutions of the type30 6

ψIj )

akrjk ∑ k)1

ψII j )

bkrjk ∑ k)1

6

6

mIj )

∑ ckrjk

k)1 6

mII j )

dkrjk ∑ k)1

(13)

In the following section, a model solution of the system is presented for surfaces separated by seven icelike layers (corresponding to a separation distance of ∼25 Å). This will clarify the general behavior of the system and will demonstrate the nonproportionality between the polarization and the electric field. 4. A Model Solution for Polarization and Electric Potential In what follows, we examine the general behavior of the solutions of the system of eqs 9a-9d for the boundary conditions given by eqs 11a-11c by assuming perfect icelike layers. It is taken into account that the distance between the centers of two adjacent water molecules is l ) 2.76 Å, the distance between the centers of two adjacent icelike water layers is ∆ ) 4/3l ) 3.68 Å, the distance between the two distinct sites in the same layer is δ ) l/3 ) 0.92 Å (see Figure 2), and the volume occupied by a water molecule is v0 ) 8l3/3x3 ) 32.37 Å3. A legitimate first question is related to the magnitude of errors generated by the discretization of the PoissonBoltzmann equation (the replacement of eq 6 by eqs 7a-

Figure 3. (a) The ratio between the Debye-Hu¨ckel length λDH and the characteristic length of the system (eqs 9a-9d), in the absence of the coupling interactions between dipoles (all D coefficients vanish), as a function of λDH. The results obtained via the discretization of the Poisson-Boltzmann equation are accurate, except for very small values of λDH. (b) The characteristic lengths of the linear system with finite differences as a function of the Debye-Hu¨ckel length, for the coupling coefficients given by eqs 14a-14f.

7d). Considering that all the coupling coefficients D vanish (therefore, neglecting the coupling between neighboring dipoles), one should recover the traditional PoissonBoltzmann equation. Indeed, the condition of vanishing of the determinant of the system (eqs 9a-9d) generates in this case only two solutions, r1 and 1/r1, and the corresponding decay lengths (provided by eq 10b) are almost identical to (λDH, at least for distances larger than ∆ (see Figure 3a). To calculate the coupling coefficients D, we employ the approximations used in ref 13, which assume that only the dipoles of the neighboring water molecules, within a distance 2l from a particular point, generate a significant electric field at that point and that the fields generated by all the other dipoles are negligible. This means that the field Ep is generated at a site of a water molecule only by the dipoles of its first 26 neighbors, which are located either in the same icelike layer or in one of the adjacent layers. By assuming an effective dielectric constant, ′′ ) 1, for this interaction, one obtains13

492

Nanodispersions

Polarization of Water Near Dipolar Surfaces

Langmuir, Vol. 21, No. 25, 2005 11755

II,II 0.689 DI,I -1 ≡ D1 4π0l3

(14a)

0.689 4π0l3

(14b)

I,I DII,II -1 ≡ D1

II,I DI,II -1 ) D1 )

0.432 4π0l3

(14c)

I,II DII,I -1 ) D1 )

1.845 4π0l3

(14d)

II,II DI,I )0 ) D0

1.378 4π0l3

(14e)

I,II DI,II 0 ) D0 ) -

2.388 4π0l3

(14f)

The condition of vanishing of the determinant provides three pairs of solutions r1,1/r1; r2,1/r2; and r3,1/r3, with the last one being negative. The 3 decay lengths corresponding to the 6 solutions are plotted in Figure 3b as functions of the decay length λDH. The length λ1 is reminiscent of the Debye-Hu¨ckel length (at least at low electrolyte concentrations), whereas λ2 is comparable to the hydration decay length, which was determined experimentally for the interactions between neutral lipid bilayers.4 The negative sign of r3 implies that the terms proportional to rj3 and 1/rj3 have different signs in adjacent layers; hence, λ3 corresponds to a decay length of an oscillatory contribution to the polarization and potential (with an oscillation length 2∆). These types of oscillations are provided by the system of equations with finite differences (eqs 9a-9d) and are generated by the coupling coefficients between neighboring dipoles. The physical origin of this behavior is related to the particular structure of the icelike layers: the field generated by a dipole b p, at a distance b r from the dipole, is given by

E B)

1 [3(p bb r )r b - r2 b p] 4π0r5

(15a)

Because we are interested only in the polarization along the x axis, the field along this axis, at a position that makes an angle θ with the x axis, becomes

Ex )

px 4π0r3

(3 cos2 θ - 1)

(15b)

Consequently, the neighboring water molecules belonging to different layers (with small θ) tend to align with each other in the same direction, whereas the neighboring dipoles from the different sites of the same layer (for which the angle θ ) cos-1(1/3) = 71°) tend to align with each other in opposite directions (these interactions dominate the coupling coefficient DI,II 0 , which is negative). The relatively low magnitude of the decay length of the “oscillatory” part of the polarization and electrical potential indicates that the oscillations are attenuated rapidly with the distance from the surface. In a continuous model, in which the water molecules are not constrained to particular positions corresponding to an icelike structure, the water dipoles tend to align themselves in the same directions. Under these conditions, the oscillations vanish,

Figure 4. (a) The electric potential (circles) as a function of the distance from the surface for the system described in the text. The continuous line represents a Spline interpolation. (b) The average polarization of a water molecule (squares) as a function of the distance from surface for the system described in text; the macroscopic electric field (triangles) obtained through the numerical derivative of the potential is not proportional to the average polarization.

and only the characteristic lengths λ1 and λ2 remain.29 The vanishing of the oscillations of polarization due to structural disorder (the absence of the icelike structuring of water near interfaces) has also been observed in molecular dynamics simulations. In the latter case, when surface disorder was generated by allowing the motion of the surface groups, the polarization decayed monotonically from the surface.28 Let us investigate the electrical potential and the polarization of a typical system. For two surfaces separated by 7 icelike water layers (S ) 3, j ) 0 corresponding to the middle layer), with σ ) 1.9 × 10-4 C/m2, p⊥/′ ) 0.1 D, A ) 50 Å2 and λDH ) 50 Å, the solutions of the system (eqs 9a-9d) with boundary conditions given in eqs 11a-11c for the potential and polarization as functions of the distance between surfaces are plotted in Figure 4, panels a and b, respectively. Although the solutions are defined only at particular positions in each layer (dots), for clarity they have been numerically interpolated using splines. Both the electrical potential (Figure 4a, circles) and the polarization (Figure 4b, squares) have an oscillatory behavior with respect to the distance. Furthermore, the numerical derivative of the electrical potential (Figure 4b, triangles) shows that the polarization is not proportional to the macroscopic electric field. The result is

Polarization Model: a unified framework for hydration and double layer interactions

11756

Langmuir, Vol. 21, No. 25, 2005

particularly clear in the middle region between the surfaces, where the polarization varies slowly (mainly because of the coupling between dipoles), whereas the electric potential is almost constant (because the electrolyte ions can attenuate the electric field). Near the surfaces, the rapid variation of ψ generates a strong electric field, which has a large contribution to the polarization of the water molecules. Accounting for the facts that, in a real system, the water molecules are not perfectly structured in icelike layers (the consequences of this disorder have been investigated elsewhere),13 and that the charges and surface dipoles were selected here to be sufficiently small for the linear approximation to hold, the behavior of the calculated electrical potential and polarization are in excellent qualitative agreement with the results of molecular dynamics simulations (see Figures 1 and 2 of ref 14). 5. Conclusions It was recently shown via molecular dynamics simulations14 that, in the close vicinity of a surface, water molecules exhibit an anomalous dielectric response, in which the local polarization is not proportional to the local electric field. The recent findings are also in agreement with earlier molecular dynamics simulations, which showed that the polarization of water oscillates in the vicinity of a dipolar surface,11,14 leading therefore to a nonmonotonic hydration force.15 Previous models for oscillatory hydration forces, based either on volumeexcluded effects,18,19 or on a nonlocal dielectric constant,24 predicted many oscillations with a periodicity of ∼2 Å, which is inconsistent with these molecular dynamics simulations,11,13,14 in which the polarization exhibits only a few oscillations in the vicinity of the surface, with a larger periodicity. It is suggested here that this behavior of water in the vicinity of a surface is a result of a coupling between neighboring dipoles when the water is organized in an icelike layer in the vicinity of the surface. This coupling constituted the basis of a model for water polarization, which was proposed earlier to explain the monotonic hydration interactions6 (either in the absence of water structuring, or when statistical disorder has been taken into account).13,27 The interactions between neighboring dipoles are the most important because they are much

493

Manciu and Ruckenstein

less screened than those between more remote dipoles. Whereas in the latter case, there are many intervening water molecules that screen the interactions, in the former case there are no intervening water molecules. This behavior, due to the discrete (molecular) nature of water, is in sharp contrast with the classical results obtained by treating water as a continuum with a constant bulk dielectric constant. The continuum treatment assumes that a homogeneous surface dipolar density cannot polarize the water, whereas, in this model, the water is polarized by the surface dipoles.6,13 As shown earlier,13 if one takes into account the coupling between neighboring dipoles and one also assumes a structuring of the water in icelike layers near the surfaces, one can explain the oscillation of the water polarization in the vicinity of the surfaces. The oscillations are essentially due to the particular positions of water molecules in an icelike structure. Whereas the water molecules from different icelike layers tend to align with each other in the same direction, the water molecules from two sites of the same layer tend to align with each other in opposite directions (other crystalline structures may not necessarily lead to an oscillatory polarization). That model, which involved a vanishing electrolyte concentration, was extended here to the general case that takes into account all of the electrostatic interactions between charges (electrolyte ions and surface charges) and dipoles (water as well as surface dipoles). A general formalism, which employs an analytical solution in the linear approximation, is suggested, and the general behavior of the system is investigated. It was shown that, in the linear approximation, the polarization is a superposition of three exponentially decreasing functions, two of them being monotonic and one oscillating with the periodicity of two icelike water layers (∼7Å). The decay lengths of the functions depend on the electrolyte concentration and on the coupling interactions between neighboring dipoles. A model calculation predicted an anomalous dielectric behavior of water in the vicinity of a surface and an oscillatory dependence on distance of the water polarization and the electric potential in the close vicinity of a surface, which are very similar to the results obtained via molecular dynamics simulations.14 LA051802G

494

Nanodispersions

7584

Langmuir 2002, 18, 7584-7593

The Coupling between the Hydration and Double Layer Interactions Eli Ruckenstein* and Marian Manciu Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received May 8, 2002. In Final Form: July 31, 2002 The electric potential and the polarization between two charged, flat surfaces immersed in water are calculated without the usual assumption that the polarization is proportional to the electric field. The new constitutive equation accounts for an additional interaction, due to the orientational correlation of the water dipoles, which is a result of the mutual interaction between neighboring dipoles. The surfaces should be characterized not only by their charges or potentials but also by their dipole densities. Because both the double layer and the hydration forces are dependent in the present model on the polarization, the repulsion cannot be separated into two additive terms, one being the traditional “double layer” repulsion (DLVO theory) and the other the “structural” repulsion (hydration). In the absence of surface dipoles, the repulsion between charged surfaces becomes stronger than that predicted by the DLVO theory, particularly at high ionic strengths. The total repulsion can be increased or even decreased by the presence of dipoles on the surfaces, which contradicts the additivity of the repulsions. The repulsion between uncharged surfaces that possess dipoles was found to depend on the electrolyte concentration, and to be extended over a much longer distance than the conventional exponential decay, particularly at high ionic strengths. As a consequence of the coupling between the double layer and hydration, the decay length of the repulsive force becomes larger than those of the two conventional repulsions and at high ionic strength the difference becomes increasingly larger.

I. Introduction 1

2

Gouy and Chapman, who were the first to predict the distribution of electrolyte ions in water around a charged flat surface, demonstrated that the ions form a diffuse layer (the electric double layer) in the liquid near the interface. The interaction between two charged surfaces, due to the overlapping of the double layers, was calculated much later by Deryaguin and Landau3 and Verwey and Overbeek.4 The stability of the colloids was successfully explained by them in terms of a balance between the double layer and van der Waals interactions (the DLVO theory).3,4 However, it is well-known that the DLVO theory is reliable only for a limited range of electrolyte concentrations. Deryaguin himself has noted that this theory is valid between approximately 10-3 and 5.0 × 10-2 M. There have been numerous attempts to improve the DLVO theory, by accounting for the saturation of the molecular polarizability at high fields,5 the image forces,6 finite ion sizes,7 or the correlation between ions,8 to cite only a few. Experiments with uncharged lipid bilayers in water9 and the restabilization of some colloids at high ionic strengths10 indicated that another, non-DLVO repulsion is also * Corresponding author: e-mail address, feaeliru@ acsu.buffalo.edu; phone, (716) 645-2911 (ext 2214); fax, (716) 6453822. (1) Gouy, G. J. Phys. Radium 1910, 9, 457. (2) Chapman, D. L. Philos. Mag. 1913, 25, 475. (3) Deryaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633. (4) Verwey, E. J.; Overbeek, J. Th. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (5) Henderson, D.; Lozada-Cassou, M. J. Colloid Interface Sci. 1986, 114, 180. (6) Jo¨nnson, B.; Wennerstro¨m, H. J. Chem. Soc., Faraday Trans. 1983, 79, 19. (7) Levine, S.; Bell, G. M. Discuss. Faraday Soc. 1966, 42, 69. Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612. (8) Wennerstro¨m, H.; Jo¨nnson, B.; Linse, P. J. Chem. Phys. 1982, 76, 4665. (9) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351.

present. This repulsion was related to the structuring of the solvent around the interface and is known (when water is the solvent) as the hydration force. While the existence of the hydration force is undisputed, its origin is still a matter of debate. Marcelja and Radic showed that the exponential repulsion observed experimentally can be obtained if a suitable Landau free energy density, dependent on an unknown “order parameter”, is associated with the correlation of the water molecules in the vicinity of the surface.11 Later, Schiby and Ruckenstein12 and Gruen and Marcelja13 presented two different models, both involving the polarization of the water molecules. Gruen and Marcelja considered that the electric and polarization fields are not proportional in the vicinity of a surface and that while the electric field has the ion concentrations as its source, the source of the polarization field is provided by the Bjerrum defects. The coupled equations for the electric and polarization fields were derived through a variational method. Attard et al.14 contested the Gruen-Marcelja model because, to obtain an exponential decay of the repulsion, the nonlocal dielectric function was assumed to have a simple monotonic dependence upon the wavelength (eq 33 in ref 13). This was found to be inconsistent with the exact expression for multipolar models.14 In addition, the characteristic decay length for polarization (denoted ξ in eq 18, ref 13) is inversely proportional to the square of the (unknown) concentration of Bjerrum defects in ice. While at large concentrations of Bjerrum defects the disordered ice becomes similar to water and the traditional Poisson(10) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156. (11) Marcelja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (12) (a) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (b) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 100, 277. (13) Gruen, D. W. R.; Marcelja, S. J. Chem. Soc., Faraday Trans. 2 1983, 79, 211, and 225. (14) Attard, P.; Wei, D. Q.; Patey, G. N. Chem. Phys. Lett. 1990, 172, 69.

10.1021/la020435v CCC: $22.00 © 2002 American Chemical Society Published on Web 09/06/2002

Polarization Model: a unified framework for hydration and double layer interactions

Coupling between Hydration and Double Layer Interactions

Boltzmann equation is recovered, a low concentration of Bjerrum defects provided a much too large polarization decay length (ref 13 indicated as an example the value ξ ) 332 Å), which is by more than 2 orders of magnitude larger than the decay length of the hydration force. The model of Schiby and Ruckenstein12a predicted that the surface dipoles induced a polarization, even in the absence of a double layer, which decayed exponentially from the surface. The electrodynamics of continuous media predicts that the field generated by a planar surface with an uniform dipole density immersed in a fluid of uniform dielectric constant vanishes outside the surface. However, at a molecular scale, the fluid is not uniform. The interactions between remote charges or dipoles can be considered “screened” by the intervening medium which has a large dielectric constant ( ∼ 80 for water); however, the interaction between adjacent dipoles is much less screened ( ) 1 represents no screening).16 Consequently, a net electric field is generated by the surface dipoles which polarizes the nearby water molecules. These dipoles generate in turn electric fields in the neighboring molecules and so on. The decay length for polarization, calculated from this model, was found in good agreement with the values determined from experiment on neutral lipid bilayers.9 Schiby and Ruckenstein also suggested a new constitutive equation that related the polarization to a local electric field, which included the interaction between neighboring dipoles.12b Both models predicted a monotonic decay of the polarization from the surface. Because the two models could not explain the oscillatory profile of the average polarization obtained by Monte Carlo simulations, it was suggested that the density of hydrogen bonding is a more suitable order parameter.15 Recently, it was, however, shown17 that the Schiby-Ruckenstein model can lead to an oscillatory polarization, if the water in the vicinity of a flat surface is assumed organized in icelike layers. The oscillations are smoothed out when the surface is rough (or fluctuating) or if some disorder is assumed to exist in the icelike structure. It was also shown that the continuum approximation, based on a second order differential equation, can describe well the average interaction.17 One cannot yet rule out that other interactions contribute to the hydration, such as the disruption of the hydrogen bond networks when two surfaces approach each other. However, at least a part of this disruption is already contained in the dipole-dipole interactions included in the polarization model. In addition, the polarization model of hydration can relate the magnitude of the hydration force to the density of dipoles on the surface. This can explain the dependence of the hydration repulsion on the surface dipolar potential18 or the restabilization of some colloids at high ionic strength16 observed experimentally.10 It is usually assumed that the total repulsion is the sum between a “double layer” repulsion, due to the charges on the interface, and a “hydration” repulsion, due to the structuring of water in the vicinity of the interface, and that the two effects are independent of each other. This is, however, not accurate when the hydration is induced by the orientational correlation of neighboring dipoles, because both forces depend on polarization. The presence of dipoles on a surface free of charge generates an electric field and a polarization, dependent (15) Kjellander, R.; Marcelja, S. Chem. Scr. 1985, 25, 73. Attard, P.; Batchelor, M. T. Chem. Phys. Lett. 1988, 149, 206. (16) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7061. (17) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582. (18) Simon, S. A.; McIntosh, T. J. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 9263.

495

Langmuir, Vol. 18, No. 20, 2002 7585

on the distance from the surface. The gradient of polarization affects, via the Poisson equation, the macroscopic electric field and, hence, the double layer, even in the absence of a surface charge. If ions are present in the system (because of the addition of an electrolyte), they interact with the electric field and hence affect both the electric field and the polarization. Therefore, the total repulsion should depend on the electrolyte concentration, even for uncharged surfaces. On the other hand, in the absence of surface dipoles, the correlation between the water dipoles induced by a surface charge is expected to increase the repulsion as compared to the DLVO theory. This is due to the tendency of the water dipoles to orient in the same direction the adjacent water dipoles, thus increasing the decay length of the polarization. Consequently, even when only a surface charge is present (and the surface dipole density is zero), the repulsion is not accurately described by the DLVO theory (particularly at high ionic strength, as will be shown later) because it disregards the orientational correlation of neighboring dipoles. In addition, even for uncharged surfaces, the repulsion induced by the surface dipoles is not accurately described by the traditional “hydration” force, independent of the electrolyte concentration, because the gradient of polarization generates an electric field which is affected by the ionic strength. The problem is further complicated when surface charges as well as surface dipoles are present. Can the total repulsive interaction still be described by a superposition between two independent interactions, the “double layer” and the “hydration” repulsions? The purpose of this paper is to present a model that accounts in an unitary manner for both the double layer and hydration repulsions. The model is an extension of an earlier treatment of Schiby and Ruckenstein.12 The equation coupling the polarization and the electric potential are derived here using an analysis based on the Lorentz model for the polarization and also a variational treatment (see Appendix). It will be shown that if the mutual interaction between neighboring water dipoles is taken into account, the double layer repulsion increases in the absence of surface dipoles, when compared to the DLVO theory, particularly at high ionic strengths. Similarly, in the absence of a surface charge, but the presence of a surface dipole density, the repulsion is increased by the addition of an electrolyte, an effect that is important particularly at high ionic strengths. For a surface whose charge is generated by the dissociation of a surfactant, and the surface dipoles are provided by the nondissociated surfactant molecules, it will be shown that the “hydration” and the “double layer” repulsions are not only nonadditive but that the presence of surface dipoles can decrease the repulsion. II.1. The Basic Equations We will employ here the polarization model developed in refs 12 and 17. The water molecules are considered dipoles localized at the site of a lattice; however, the fields generated by the adjacent dipoles are treated differently from those of the remote dipoles, because the latter are screened by the intervening water molecules. Starting from the dipoles of the surface, the polarization propagates through water because of the interactions between the neighboring dipoles. When two external surfaces approach each other, the overlap of the polarization layers decreases the dipole moments and hence increases the free energy of the system, thus generating a repulsion.

Nanodispersions

496

7586

Langmuir, Vol. 18, No. 20, 2002

Ruckenstein and Manciu

The electric field EP at a site of the layer “i” of a water lattice, situated at zi, generated by all the other dipoles of magnitude mi of the same layer and those of magnitude mi(1 of the adjacent layers (the contribution of the other layers is neglected because of the screening) can be expressed as17

EP(zi) ) C1mi-1 + C0mi + C1mi+1 ) C0m(zi) + C1(m(zi-1) + m(zi+1)) ≈ (C0 + 2C1)m(zi) + 2

C1∆2

∂ m(z) ∂z2

|

2

z)zi

≈ C1∆2

∂ m(z) ∂z2

|

3.766 4π0′′l3

C1 )

1.827 4π0′′l3

(1)

z)zi

P 30

(3)

the macroscopic field being generated by all the charges and dipoles present, while the local field is generated by all the charges and dipoles, except the selected dipole. Extending the Lorentz relation when the additional field EP is present, one obtains

Elocal ) E +

P + EP 30

(4)

The average dipole moment m(z) ) P(z)v0, where v0 is the volume of a water molecule, is related to Elocal via

(

m(z) ) γ E + EP +

)

m(z) w 30v0 γ 1-

γ 30v0

)

m 30v0

(7)

∂z2

(6)

The macroscopic relation between the polarization and (19) Frankl, D. R. Electromagnetic Theory; Prentice Hall: Englewood Cliffs, NJ, 1986.

(8)

where F is the local charge density, given, as in the traditional theory, by

( ( )

(

eψ(z) eψ(z) - exp kT kT

F(z) ) -ecE exp

))

)

-2ecE sinh

( )

eψ(z) (9) kT

where cE is the bulk electrolyte concentration, e is the protonic charge, k is the Boltzmann constant, and T is the absolute temperature, can be rewritten as

∂2ψ(z) ∂z

2

)

2ecE eψ 1 ∂m(z) sinh + 0 kT 0v0 ∂z

( )

(10)

Equations 7 and 10 constitute a complete system of equations for ψ(z) and m(z), which replaces the traditional equations of the double layer. II.2. The Linear Approximation For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson-Boltzmann equation (the Debye-Hu¨ckel approximation). The extension to the nonlinear cases is (relatively) straightforward; however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by

2e2cE 0 eψ =ψ ≡ - 2 ψ (11) kT kT λ

( )

F ) -2ecE sinh

D

where λD ) (0kT/2e2cE)1/2 is the Debye-Hu¨ckel length. In the linear approximation, the system of eqs 7 and 10 becomes

(E + EP) (5)

where γ is the molecular polarizability. At constant polarization, EP ) (C0 + 2C1)m = 0. Hence eq 5 becomes

(

∂2m(z)

∂E(z) ∂2ψ(z) F 1 ∂m(z) ≡ )- + 2 ∂z 0 0v0 ∂z ∂z

∂2ψ(z)

m(z) )

m)γ E+

-

(2)

with l the distance between the centers of two adjacent water molecules, 0 the vacuum permittivitty, and ′′ the dielectric constant for the interaction between neighboring molecules, which is expected to be nearer to unity than to the dielectric constant of water,  ) 80. In addition to the field generated by the adjacent dipoles, there is a macroscopic field E due to the presence of charges and of the “average” polarization P of the medium. In the Lorentz treatment of polarization, for a constant macroscopic field in a linear and homogeneous medium of dielectric constant  (hence satisfying P ) 0( - 1)E), the local field Elocal at a site of a selected dipole is related to the macroscopic field E via19

Elocal ) E +

m(z) ) 0v0( - 1)E(z) + 0v0( - 1)C1∆2 The Poisson equation

where ∆ is the distance between the water layers, m(z) is a continuous function which is equal in every layer (located at zi ) i∆) to the average polarization of the water molecules in that layer (m(z)|z)zi ) mi), and Cj (j ) 0, 1) are interaction coefficients (see ref 17)

C0 ) -

the electric field, m ) 0v0( - 1)E can be employed in eq 6 to determine the value of γ. Assuming that the molecular polarizability remains the same in a nonconstant field, eqs 1 and 5 lead to

∂z2 λm2

∂2m(z) 2

∂z

)

 1 ∂m(z) ψ+ 0v0 ∂z λD2

) m(z) + 0v0( - 1)

∂ψ(z) ∂z

(12a)

(12b)

where λm2 ≡ 0v0( - 1)C1∆2. In the absence of an electrolyte (cE ) 0, λD ) ∞), eq 12a leads to

m(z) ) 0v0

∂ψ(z) + constant ∂z

(13)

The constant is zero since the average polarization should

497

Polarization Model: a unified framework for hydration and double layer interactions

Coupling between Hydration and Double Layer Interactions

Langmuir, Vol. 18, No. 20, 2002 7587

vanish when the electric potential is constant (this can be seen better from eq 12b, since a constant polarization and constant electric potential imply a vanishing polarization). Introducing the result (with constant ) 0) in eq 12b yields

m(z) )

2 λm2 ∂2m 2 ∂ m(z) ≡ λ H  ∂z2 ∂z2

(14)

which predicts a monotonic decay of the polarization from the surfaces, with a decay length λH ) (0v0C1∆2( - 1)/ )1/2. The solution of eq 14, which accounts for the antisymmetry of the polarization (m(-d) ) m0, m(d) ) -m0), is

m(z) ) -m0

sinh(z/λH) sinh(d/λH)

(15)

where z is measured from the middle distance and 2d is the distance between the plates. Employing the values l ) 2.76 Å for the distance between the centers of two adjacent water molecules in an icelike structure, ′′ ) 1 for the dielectric constant for the interaction between neighboring molecules (selected as in refs 16 and 17), ∆ ) 4/3l ) 3.68 Å for the distance between two adjacent water layers in an icelike structure, v0 ) 30 Å3 for the volume occupied by one water molecule, and  ) 80, one obtains λm ) 14.9 Å and λH ) λm/1/2 ) 1.67 Å, which is in good agreement with the hydration length experimentally determined for neutral lipid bilayers.9 This exponential dependence of the polarization was found also (in the continuum approximation) by earlier analysis.12,16,17 It should also be noted that using eq 14 (which is valid only in the absence of electrolyte), the free energy of the system (discussed in detail in section II.5) reduces to the form employed in previous polarization-based treatments of hydration forces.16,17 However, as will be shown below, the addition of electrolyte affects the hydration even in the absence of surface charges.

Figure 1. The dependence of the characteristic decay lengths λ1, λ2 of the system on the Debye-Hu¨ckel length λD (λm ) 14.9 Å, λH ) λm/1/2 ) 1.67 Å).

II.3. The Characteristic Decay Lengths

The remaining two independent constants, a1 and a2 can be determined using the boundary conditions for the electrical potential and polarization at the surfaces. For constant surface potential ψ0, the boundary condition is

In the presence of an electrolyte (λD * ∞), we will seek solutions of the type ψ ) ψ0 exp(z/λ) and m ) m0 exp(z/λ) for the homogeneous system of the two linear equations (12a,b). The condition for existence of nontrivial solutions leads to the characteristic equation

(

)(

)

2 1  λm -1 -1 ) 0 S λ4 2 2 2 λ λD λ λ2

λD2 λm2 (λD2 + λm2)λ2 + ) 0 (16)  The solutions of eq 16, (λ1 and (λ2 are always real, since the dielectric constant of the medium is higher than the vacuum dielectric constant (the discriminant λD4 + λm4 + 2λD2 λm2(1 - 2/) > 0 for  > 1). The dependence of the decay lengths λ1 and λ2 on the electrolyte concentration (λD) for  ) 80 and λm ) 14.9 Å are presented in Figure 1. At low electrolyte concentrations (λD . λm), the decay lengths λ1 and λ2 are well approximated by λD and λH ) λm/1/2; however, when λD becomes comparable to λm, λ1, and λ2 differ markedly from λD and λ H. II.4. Boundary Conditions for Two Identical Surfaces Immersed in Water The symmetry of the system implies that the potential is symmetric and the average polarization is antisym-

metric with respect to the middle distance; hence, the general solutions of the system (12a,b) are

() ()

() ()

ψ(z) ) a1 cosh

z z + a2 cosh λ λ2

(17a)

m(z) ) a j 1 sinh

z z +a j 2 sinh λ1 λ2

(17b)

where the constants a j 1 and a j 2 are related to the constants a1 and a2 via eq 12a

a j 1 ) a10v0λ1 a j 2 ) a20v0λ2

a1 cosh

()

( (

1  λ12 λD2 1  - 2 2 λ2 λD

) )

()

d d + a2 cosh ) ψ0 λ1 λ2

(18a)

(18b)

(19a)

For constant surface charge density σ, the condition of overall electroneutrality 2σ ) -∫d-d F(z) dz leads to

a1λ1 sinh

()

()

λD2 d d + a2λ2 sinh ) σ λ1 λ2 0

(19b)

The double layer charge distribution (eq 11) was employed in the derivation of the last equation. When the value of the polarization at the boundary (m0 ) m(-d)) is known, the corresponding boundary condition is

( )

a j 1 sinh -

( )

d d +a j 2 sinh ) m0 λ1 λ2

(20a)

A more realistic approach17 is to assume that the average polarization of the water molecules of the first water layer near the surface is proportional to the local field, generated by the surface charges, surface dipoles, and the water dipoles of the first two water layers.

498

Nanodispersions

7588

Langmuir, Vol. 18, No. 20, 2002

Ruckenstein and Manciu

The average electric field generated by surface dipoles with a surface density of 1/A and with a dipole moment p normal to the surface, whose centers are located at a distance ∆′ from the center of the first water layer, is given by17

p 1 1 ES ) ′ 2π0 A + ∆′2 π

()

(

)

(21)

3/2

where ′ is the local dielectric constant for the field generated by the surface dipoles in the neighboring water molecules. The macroscopic field at the interface is related to the electric potential, via the expression E ) -∂ψ(z)/∂z, while the field generated by the water dipoles is17

ESP ) C0m1 + C1m2

(22)

where m1 and m2 are the average polarization of the water molecules of the first and second water layer near the surface, respectively, and C0 and C1 are given by eq 2. Hence, the boundary condition for polarization has the form

(

m1 ) γ E +

)

m1 + ES + ESP ) 30v0 0v0( - 1)(E + ES + ESP) (23)

where eq 6 was employed to derive the last equality. Assuming that the positions of the centers of the first water layers are at the distances (d, and using eqs 12a and 12b, eq 23 becomes

[ (

a1 λ10v0

)(

( )

1  d - 2 (1 - 0v0( - 1)C0) sinh 2 λ λ1 λd 1 d-∆ 0v0( - 1)C1 sinh + λ1 0v0( - 1) d sinh + λ1 λ1

[ (

)(

( )) ( )]

( )

1  d (1 - 0v0( - 1)C0) sinh λ2 λ22 λd2 d-∆ 0v0( - 1)C1 sinh + λ2 0v0( - 1) d p v0( - 1) 1 sinh ) 3/2 λ2 λ2 ′ 2π A + ∆′2 π (20b)

a2 λ20v0

( )]

(

()

))

(

)

II.5. The Free Energy of the System In the DLVO framework, the free energy of a system of two overlapped double layers is composed of an electrostatic energy, an entropic contribution due to the ions in the double layer, and a chemical term, where applicable.4 The electrostatic energy per unit area of the double layer is provided by the familiar expression19,20 (20) Schwinger, J.; DeRaad, L. L., Jr.; Milton, K. A.; Tsai, W.-Y. Classical Electrodynamics; Perseus Books: Reading, MA, 1998.

Fel )

1 2

∫-∞ F′(z)ψ(z) dz ) 21 ∫-∞ ∇D(z)ψ(z) dz ) 1 1 ∞ D(z)ψ(z)|z)∞ z)-∞ - ∫-∞ D(z)∇ψ(z) dz ) 2 2 1 ∞ 1 d D(z)E(z) dz ) ∫-d D(z)E(z) dz 2 ∫-∞ 2 ∞

∞

(24)

where F′ is the total charge density (which includes the charge distributed between surfaces, F, and the surface charge density, σ), ψ is the electric potential, E is the macroscopic electric field, z is the distance measured from the middle between the surfaces and D ) 0E + P is the displacement field. The above expression accounts for the fact that the field is nonvanishing only between the surfaces, located at (d; the Poisson equation ∇D ) F′ and the relation E ) -∇ψ were also employed. The excess entropy contribution (with respect to the bulk) per unit area due to the ions of an electrolyte, calculated as for an ideal solution, is given by21

-T∆S ) kT

∑i ∫-d d

( () ci ln

ci

ci0

)

- ci + ci0 dz (25)

where ci is the actual (double layer) concentration of ions of species “i”, ci0 is the concentration at large distances, and the subscript “i” runs over all ion species. For an 1-1 electrolyte of concentration cE, which obeys the Boltzmann distribution, the above expression becomes

-T∆S ) -cEkT

∑ ∫-d i)1,2 d

(

(-1)i eψ

(

(-1)i+1eψ

)

+ kT kT (-1)i+1eψ exp - 1 dz (26) kT exp

(

) )

The entropic contribution to the free energy (per unit area) becomes in the Debye-Hu¨ckel approximation

-T∆S )

cEe2 kT



∫-dd (ψ(z))2 dz ≡ 2λ 02 ∫-dd (ψ(z))2 dz

(27)

D

The chemical contribution to the free energy, per unit area, due to the adsorption of n molecules (per unit area) of charge q on each surface of potential ψS is4

Fch ) 2n∆µ ) -2nqψS ) -2σψS

(28)

at constant surface potential, ∆µ being the change in the chemical part of the electrochemical potential of a molecule at its adsorption from the bulk on the surface, and σ the surface charge density. At equilibrium, the electrochemical potential is constant through the system (∆µ ) -qψS). At constant surface charge the chemical contribution to the free energy is zero.4 In addition to these well-known free energy contributions, one has to consider another one, which accounts for the mutual interactions between neighboring dipoles

Fm ) -

C1∆2 ∂2m 1 mEP ) m 2 2 2 ∂z

(29)

where eq 1 was employed. It should be noted that the above expression contains both the energy and the entropy of the dipole, in the weak field (linear) approximation. (21) Overbeek, J. T. G. Colloids Surf. 1990, 51, 61.

499

Polarization Model: a unified framework for hydration and double layer interactions

Coupling between Hydration and Double Layer Interactions

Consequently, the total electrostatic free energy per unit area (accounting for all the interactions between charges and dipoles) is provided by the expression

∫

[

]

1 1 Fel ) -d E(0E + P) - PEP dz ) 2 2 m(z) m(z) 1 d 0E(z) + E(z) EP(z) dz (30) 2 -d v0 v0 d

∫

[(

)

]

The free energy of the surface layer formed by surface dipoles and the water molecules between them is assumed to be independent of the distance 2d. At constant surface potential, the free energy per unit area of a system of two identical charged surfaces at distances (d from the middle distance can be written as the sum between a surface term (the chemical energy) and an integral over the density of the entropy of the mobile ions and all the electrostatic interactions between charges and dipoles

Fψ ) -2σψS +

1 2

∫-d d

[(

0

)

∂ψ m ∂ψ 0 + (ψ)2 ∂z v0 ∂z λD2

]

C1∆2 ∂2m m 2 dz (31a) v0 ∂z

The same results (eqs 12a,b and 31) were obtained using a variational method. The details are given in Appendix. At constant surface charge, the free energy is given by

Fσ )

1 2

∫-d d

[(

0

)

∂ψ m ∂ψ 0 + (ψ)2 ∂z v0 ∂z λD2

]

C1∆2 ∂2m m 2 dz (31b) v0 ∂z

III.1. The Influence of the Dipole-Dipole Interactions on the Double Layer Force At low electrolyte concentrations, because the DebyeHu¨ckel length is large, the polarization predicted by the DLVO theory is slowly varying in space. Therefore, when the dipole density on the surface is negligible, one expects the additional interaction, due to the mutual interaction between neighboring dipoles, to be also small, its density being proportional to m(∂2m/∂z2) ∝ (m2/λD2). Consequently, the contribution of the interaction between neighboring dipoles to the total free energy becomes negligible, and the DLVO theory is recovered. However, when λD is sufficiently small and becomes comparable to λm, a coupling between the two effects is expected to occur. As a consequence, a larger decay length (λ1 > λD) appears in the system (see Figure 1). There are two main reasons for the departure of the present model from the DLVO theory. First, the constitutive equations, which relate the polarization to the electric potential, are different. Second, the boundary conditions are different, since the average polarization in the DLVO theory is directly related to the surface charge, while in the present treatment it depends also on the surface dipole density. Let us first investigate the effect of the new equations alone, by using for both the DLVO theory and the present equations the same boundary conditions. For the surface charge density the constant value σ ) 5 × 10-4 C/m2 was employed (the value selected is low enough for the linear approximation to be accurate for all the electrolyte concentrations investigated here), while the polarization

Langmuir, Vol. 18, No. 20, 2002 7589

at the surface was considered induced by the surface charge only, as in the DLVO theory

m0 ) -0v0( - 1)

|

∂ψ(z) ∂z

z)-d

(32a)

In parts a and b of Figure 2, the electric potential and the polarization, respectively, calculated using eqs 12a,b are compared to those predicted by DLVO, with the boundary conditions (19b) and (32a), using for the parameters the values v0 ) 30 Å3,  ) 80, T ) 300 K, λm ≡ (0v0( - 1)C1∆2)1/2 ) 14.9 Å, and the values d ) 20 Å and λD ) 10 Å. The force per unit area between surfaces, Π ) -[∂F(2d)]/ [∂(2d)], with F(2d) given by eq 31, at constant surface charge σ ) 5 × 10-4 C/m2 and for a polarization m0 given by eq 32a, is compared to that predicted by the DLVO theory in parts c and d of Figure 2, for various electrolyte concentrations (λD ) 100, 30, 10, and 3 Å). As expected, at low electrolyte concentrations, the interaction is well described by the DLVO theory. At large electrolyte concentrations, the repulsion is, however, markedly larger than that provided by the DLVO theory, because the interactions between neighboring dipoles attenuate the decay of the polarization. To examine the effect of the surface dipole density, we consider that the polarization on the surface acquires the value

m0′ ) -0v0( - 1)

|

∂ψ(z) ∂z

z)-d

+ δm

(32b)

where the change δm of the average dipole moment of the water molecules at the interface is generated by the surface dipoles. It should be noted that the boundary condition (32b) affects the value of both ψ and m on the surface and hence m0′ * m0 + δm. Let us consider that the surface charge arises via the dissociation of surfactant molecules adsorbed on the interfaces and that the surface dipoles are due to the undissociated surfactant molecules adsorbed. In this case, the electric field induced by the surface dipoles is opposite to that generated by the surface charge (δm < 0). Hence, the presence of surface dipoles actually decreases the repulsion. The effect is illustrated in parts c and d of Figure 2, for δm ) -0.2 D. Therefore, in this case the total repulsion cannot be obtained (as usually assumed) by adding two independent terms, a “double layer force” due to the surface charges and a “hydration force” due to the surface dipoles. The second important difference with the DLVO theory arises from the boundary condition for the polarization; while the classical theory ignores the interactions between neighboring dipoles, eq 20b takes them into account. The electrical potential and the polarization calculated with eqs 12a,b and the boundary conditions eqs 19b and 20b with σ ) 5 × 10-4 C/m2 and p/′ ) 0 or p/′ ) -0.1 D are compared in parts a and b of Figure 3 with the DLVO predictions (the value A ) 100 Å2 was selected for the area occupied by a surface dipole). The repulsion forces, per unit area, at constant surface charge density are presented in Figure 3c and Figure 3d. The differences from the DLVO theory become again important at large electrolyte concentrations, and the presence of surface dipoles decreases the repulsion. At high ionic strengths, the range of the interaction is much longer than that predicted by the DLVO theory. At large separation distances, the first term of eqs 17a and 17b, with the decay length λ1 (λ1 . λ2), dominates both the electrical potential

Nanodispersions

500

7590

Langmuir, Vol. 18, No. 20, 2002

Ruckenstein and Manciu

Figure 2. (a) The electric potential and (b) the average polarization of a water molecule between two surfaces with σ ) 5 × 10-4 C/m2, separated by a distance 2d ) 40 Å, as a function of the position from the middle distance. The average polarization of the water molecules from interface, m0, is calculated from eq 32a. A perturbation δm ) -0.2 D illustrates the effect of surface dipoles. (c, d) The interaction force calculated, at various electrolyte concentrations, from eqs 12a,b with the boundary conditions (19b) and (32a), versus the separation distance, compared to the DLVO predictions. A perturbation δm ) -0.2 D illustrates the effect of surface dipoles.

and the polarization, respectively. Consequently, at large distances, the force depends mainly on the term containing the decay length λ1. For λD ) 0, one obtains from eq 16 the minimum of λ1

λ1(λD)0) ) lim

(12(λ

λDf0

(

λD4

2 D

4

+ λm +

+ λm2 + 2λD2

λm

(

2

2 1

1/2 1/2

)) ))

≡ λm (33)

with λm ) 14.9 Å for the values of the parameters employed here. Therefore, a long-ranged interaction, with a decay length larger than λm, is always present in the system, at any electrolyte concentration. The magnitude of this longrange interaction depends markedly on the ionic strength (see Figure 3d) and is much larger than the Debye and hydration decay lengths. When the interaction due to the surface charge is large, the presence of a low dipole density on the surface (generated by surface charge association with counterions, as discussed above) decreases the repulsion, since the electric field generated by the surface dipoles is opposite to the electric field generated by the charges. However, a sufficiently high surface dipole density, which still induces an electric field opposite to that generated by the charge, can eventually lead to an increase in the repulsion (the regime when the “hydration” dominates). This effect

is illustrated in Figure 3e, for weakly charged surfaces (σ ) -0.002 C/m2) and λD ) 3 Å. In this case, the increase of the dipole strength decreases initially the repulsion (when 0 < p/′ < 0.7 D); the repulsion is, however, increased for p/′ > 0.7 D. When p/′ > 1.7 D, the total repulsion becomes larger than that in the absence of surface dipoles. III.3. The Repulsion between Uncharged Surfaces The potential ψ(z) and the dipole moment m(z) provided by the system of eqs 12a,b are presented in parts a and b of Figure 4, for σ ) 0, p/′ ) 1 D and λD ) 3, 10, 30, and 100 Å. The forces between surfaces at various electrolyte concentrations (with the other parameters unchanged) are plotted in Figure 4c. At large separation distances, the first term in eqs 17a and 17b, which decays much slower, becomes dominant, with λ1 ∼ λD at low ionic strengths and λ1 ∼ λm at high ionic strengths. At short distances and low ionic strengths, the first term is, however, small, and the force is well described by an exponential “hydration force”, with constant preexponential factor and decay length (λ2 = λH ) λm/1/2). In the limiting cases σ ) 0 and λD f ∞, the free energy expression eq 31 leads (because of eq 13) to

FH ) -

1 2

∫-dd

[

]

C1∆2 ∂2m m 2 dz v0 ∂z

(34)

Polarization Model: a unified framework for hydration and double layer interactions

Coupling between Hydration and Double Layer Interactions

501

Langmuir, Vol. 18, No. 20, 2002 7591

Figure 3. (a) The electric potential and (b) the average polarization of a water molecule between two surfaces with σ ) 5 × 10-4 C/m2, separated by a distance 2d ) 40 Å, as a function of the position from the middle distance. The average polarization of the water molecules from interface, m1, was calculated using eq 20b, for p/′ ) 0 and p/′ ) -0.1 D, and A ) 100 Å2. (c, d) The interaction force calculated, at various electrolyte concentrations, from eqs 12a,b with the boundary conditions (19b) and (20b), versus separation distance, compared to the DLVO predictions. (e) The interaction force for λD ) 3 Å and σ ) -0.002 C/m2 for various dipole strengths (0 < p/′ < 3 D, A ) 100 Å2). The repulsion initially decreases and then increases with the increasing strength of the surface dipoles.

where m is given by eq 15 and the hydration free energy from the previous work16,17 is recovered. At high ionic strength, both terms, with decay length λ1 and λ2, are important and the force is smaller at short separation distances and larger at long separation distances when compared to the force for λD f ∞ (see Figure 4c). The long-range repulsion induced by the electrolyte concentration (see Figure 4c) can be explained as follows. At zero surface charge, the total charge of the electrolyte

ions between the surfaces (therefore the integral over potential (eq 11) between -d and d) vanishes. This can occur only if the potential changes sign between the surfaces, hence if the coefficients a1 and a2 in eq 17a have opposite signs, with the ratio of their magnitude determined by electroneutrality. The rapidly varying, positive potential near the surfaces (we assumed a positive surface charge) is compensated by a slowly varying, negative potential in the middle range (see Figure 4a. This generates a polarization, which decays with a decay length

Nanodispersions

502

7592

Langmuir, Vol. 18, No. 20, 2002

Ruckenstein and Manciu

Figure 4. (a) The electric potential and (b) the average polarization of a water molecule between two neutral surfaces (σ ) 0), separated by a distance 2d ) 40 Å and λD ) 3, 10, 30, and 100 Å, as a function of the position from the middle distance, calculated using eqs 12a,b. The average polarization of the water molecules at the interface, m1, was calculated from eq 20b, for p/′ ) 1 D and A ) 100 Å2. (c) The interaction force between two neutral surfaces calculated, at various electrolyte concentrations, using eqs 12a,b with the boundary conditions (19b) and (20b) for p/′ ) 1 D and A ) 100 Å2 versus the separation distance 2d.

>λm, which is much larger than λH (see Figure 4b). It should be, however, noted that the magnitude of the long-ranged repulsive force is small at low ionic strengths (see Figure 4c). IV. Conclusions The interaction between two charged surfaces immersed in a liquid was traditionally described by assuming that the polarization of the liquid is proportional to the electric field. However, an additional interaction, caused by the structuring of the liquid in the vicinity of a surface, should be also taken into account. This can be included, at least partially, via the mutual interactions between the water dipoles. This induces a polarization that propagates, from the surfaces, through water, being generated by both the surface charge and surface dipoles. The polarization gradient produces an electric field, which interacts both with the dipole moments of the water molecules and with the charges of the electrolyte. In this paper a model was presented, which allowed one to calculate both the electric potential and the polarization between two surfaces, without assuming, as in the traditional theory, that the polarization and the macroscopic electric field are proportional. An additional local field, due to the interaction between neighboring dipoles, was introduced in the constitutive equation which relates the polarization to the local field. The basic equations were also derived using a variational approach.

It was shown that the interaction between dipoles increases markedly the repulsion at high ionic strength and large separation distances, when compared to the DLVO theory. When both charges and dipoles are present on the surface, the repulsion is not provided by the sum of two independent repulsions, a “double layer” and a “hydration” repulsion. The presence of dipoles on the surface can even decrease the repulsion. It was also shown that the presence of an electrolyte generates a long-range repulsion, even at zero surface charge, if the interfaces carry a surface dipole density. At low ionic strength, this repulsion can be described, in the vicinity of the surface, by an exponential with both the decay length and the preexponential factor almost independent of electrolyte concentration, as usually considered for the hydration forces. The long-ranged interactions between the neutral surfaces is in this case small. However, at high ionic strengths, the repulsion between neutral surfaces differs markedly from this description, the force being smaller at short distances and larger at large distances than that at zero electrolyte concentration. Appendix. Derivation of the Equations for m and ψ from a Variational Principle The Maxwell equation of electrostatics in a vacuum for a system with planar xy symmetry (∂ψ/∂z ) - E, ∂E/∂z ) F′/0), F′ being the total charge density present in the system, can be derived as extremals of the electrostatic

Polarization Model: a unified framework for hydration and double layer interactions

503

Coupling between Hydration and Double Layer Interactions

free energy functional20

Fel )

∫V

(

at constant surface potential and the entropy contribution (per unit area, eq 27 is

)

∂ψ 0 2 + E dV ) ∂z 2

F′ψ + 0E

Langmuir, Vol. 18, No. 20, 2002 7593

∫V J dV

(A.1) -T∆S )

using the Euler-Lagrange equations

d

( ( )) ( ( ))

dz

∂J

∂

)

∂ψ

∂J

(A.2)

d

dz

∂J

∂

∂E

FDLVO,ψ ) )

∫

d

2λD2 -d

(ψ(z))2 dz

∂J

(

∫-dd

-

( )

0 ∂ψ(z) 2 ∂z

2

-

0 2λD2

∂z

Fel )

∂2ψ(z) ∂z

)

 1 ∂P(z) ψ+ 2  λD 0 ∂z

P(z) ) -0( - 1)

(A.3)

P2 20( - 1)

∫-d d

(

-

0 λD2

ψ2 -

-R

d

∂z

2

dz ) -RP

(A.6)

∂P ∂z

|

z)d

(A.10)

+

z)-d

∫-d R(

∂P(z) ∂z

)

2

dz (A.11)

Hence, the free energy at constant potential Fψ per unit area is given by

Fψ ) -RP

∂P ∂z

|

z)d

+

z)-d

ψ

∫-d d

(

-

0 ∂ψ 2 0 (ψ)2 2 ∂z 2λD2

( )

( ))

∂P P2 ∂P 2 + +R dz (A.12) ∂z 20( - 1) ∂z

and the extremals function ψ(z), P(z) satisfy the Euler equations

∂2ψ(z) ∂z2

)

∂2P(z) ∂z

)

∂ψ(z) ∂z

d

2R

where ψS is the surface potential. The chemical energy per unit area (eq 28) is given by

Fch ) -2σψS

∂2P(z)

∫-d P(z)

∂P ∂ψ ψ + 0E + ∂z ∂z

0 2 P2 E + dz (A.5) 2 20( - 1)

(A.9)

which represent the linear Poisson-Boltzmann equation and the proportionality relation between polarization and electric field, respectively. Until now, the classical DLVO results have been recovered. Let us suppose that another interaction, whose free energy density is -RP[∂2P(z)/∂z2] is also present. An integration by parts leads to

(A.4)

In this case, the conditions of extremum of the functional given by eq A.3 with respect to ψ, E, and P, considered as independent functions (the Euler-Lagrange equations), lead to the Maxwell equations and the equation that relates the polarization to the field. It should be noted that the above equations imply that F′ is independent of E and P. Of course, this assumption is not valid in the presence of an electrolyte. For two overlapping double layers, the free energy per unit area FDLVO can be written as the sum between the electrostatic energy, the chemical energy, and the entropic term of the electrolyte ions. Since the total charge density F′ is composed of the surface charges, σ, and the charge density F distributed between the surfaces, the later obeying (in the linear approximation) eq 11, the electrostatic energy per unit area becomes

Fel ) 2σψS +

2



f(P) )

)

∂P(z) + ∂z

The extremals of ψ(z) and P(z), obtained through the Euler-Lagrange equations are given by

∂ψ 0 ψ + 0E + E2 + f(P)) dV ∫V ((F′ - ∂P ∂z ) ∂z 2

The Euler-Lagrange equations of this functional, with respect to ψ and E, are (∂/∂z)(0E + P) ) F′ and E ) -∂ψ/∂z, respectively, provided that F′ and the arbitrary function f(P) do not depend on either E or ψ. Let us now obtain a functional, which represents the free energy density of a linear, homogeneous, and isotropic medium, that satisfies the constitutive equation P ) 0( - 1)E. To obtain the classical result for the free energy density, (1/2)E(0E + P), the function f(P) must acquire the form

(ψ(z))2 - ψ

P2(z) dz (A.8) 20( - 1)

∂E

Let us try to find a free energy functional of a polarizable medium, which can be extended to any constitutive relation between E and P. Since the Poisson equation in a medium is ∇(0E) ) F′ - ∇P, a natural choice for this functional is

(A.7)

Adding eqs A.5, A.6, and A.7 and writing E ) -∂ψ/∂z, one obtains

∂ψ

∂z

0

2

 1 ∂P(z) ψ+ 0 ∂z λD2

)

P(z) ∂ψ(z) + 0( - 1) ∂(z)

(A.13)

(A.14)

Since P(z) ) m(z)/v0 and R ) C1∆2/2, the system of eqs A.13 and A.14 becomes identical to the system of eqs 12a,b, while the free energy eq A.12 coincides with eq 31a. Equation 31b, for constant surface charge, can be derived in a similar manner. LA020435V

Nanodispersions

504

1114

Langmuir 2003, 19, 1114-1120

On the Chemical Free Energy of the Electrical Double Layer Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received September 26, 2002. In Final Form: December 16, 2002 The free energy of interaction between colloidal particles due to the overlap of their double layers was traditionally calculated either for systems of arbitrary geometry interacting at constant surface potential or at constant surface charge or for parallel plates interacting under arbitrary surface conditions. An expression is obtained for the change in the chemical contribution to the free energy of the double layers during a general interaction, which allows the calculation (within the Poisson-Boltzmann formalism) of the interaction free energy for systems of arbitrary geometry and surface conditions. The change in chemical free energy depends not only on the values of the surface charge and potential at the final state but also on their values at each distance between infinity and the final state. A simple approximate expression for the change in the chemical free energy contribution is also proposed, which involves only the states at infinite and final distances. Its accuracy is tested for planar and parallel surfaces, with charges generated via the dissociation of surface groups.

I. Introduction Gouy1 and Chapman2 were the first to predict the distributions of electrolyte ions in the vicinity of a charged surface, by assuming that the ions obey Boltzmann distributions and interact with a mean potential, which satisfies the Poisson equation. The distribution of ions between two planar charged surfaces (the difference in ions concentrations between middle distance and infinity) was later related to the interaction force between the two surfaces by Langmuir.3 The interaction free energy between parallel planar plates, separated by a distance l, could then be obtained by integrating the force from infinity to the distance l. The Langmuir procedure cannot be applied directly to nonplanar surfaces. However, an approximate method, the Derjaguin approximation,4 in which the interaction between curved surfaces is calculated in terms of the interactions between planar surfaces, could be employed. The method is accurate only when the radii of curvature of the surfaces are sufficiently large compared to the Debye-Hu¨ckel length and to the distance of closest approach. Major progress was achieved by Verwey and Overbeek,5 who proposed a direct method to calculate the interaction free energy between particles of arbitrary shapes, when either the surface potentials or the surface charges remain constant during the interaction. Their approach involved an imaginary charging process, in which the relation between the surface potential and the surface charge density was obtained by integrating the Poisson-Boltzmann equation for one of the above constraints. Another approach consisted of writing the free energy as the sum between the energy of the electrostatic field, an entropic term due the free ions and a chemical term * To whom correspondence may be addressed: e-mail address, [email protected]; phone, (716) 645-2911 ext 2214; fax, (716) 645-3822 (1) Gouy, G. J. Phys. Radium 1910, 9, 457. (2) Chapman, D. L. Philos. Mag. 1913, 25, 475. (3) Langmuir, I. J. Chem. Phys. 1938, 6, 893. (4) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (5) Verwey, E. J.; Overbeek, J. T. G. Theory of the stability of lyophobic colloids; Elsevier: Amsterdam, 1948.

due to the charge transfer to the surface.6 However, expressions for the chemical free energy contribution have been derived only for interactions that occur either at constant surface charge density or at constant surface potential. For these cases, the latter method was shown to be equivalent to the imaginary charging approach.6 Whereas the charging approach could be applied to any geometry but only at constant surface charge or potential, the Langmuir expression could be employed for any surface conditions but only for parallel planar plates. The addition of electrostatic, entropic, and chemical contributions would allow the calculation of the free energy of interaction for systems of any shape and any surface conditions, if one could derive a general expression for the chemical free energy contribution. There are colloidal systems for which the interactions can be well described by either constant surface charge density or potential. For example, when the surface charge is generated by the dissociation of surface groupsssuch as those of surfactantssand the dissociation constant is sufficiently large, the constraint of constant charge is a suitable one. In addition, in many cases the radii of curvature of the surfaces are large enough for the interactions to be accurately calculated by combining the Langmuir equation with the Derjaguin approximation. There are, however, colloidal systems, for which the above requirements are not fulfilled. For example, the surfaces of the proteins can have small radii, and their acidic and basic groups are not completely dissociated. In this case, the Verwey-Overbeek approach cannot be employed and the Langmuir-Derjaguin approximation is not accurate. The purpose of this article is to obtain a general expression for the chemical free energy contribution, when neither the surface potential nor the surface charge density are constant on the surfaces during the interaction. It will be shown that the change in the chemical free energy contribution depends on the trajectory ψS ) ψS(σ), where ψS is the surface potential and σ the surface charge density, during the interactions from an infinite to the final separation distance. Using this expression for the change (6) Overbeek, J. T. G. Colloids Surf. 1990, 51, 61.

10.1021/la0266132 CCC: $25.00 © 2003 American Chemical Society Published on Web 01/28/2003

Polarization Model: a unified framework for hydration and double layer interactions

Chemical Free Energy of the Electrical Double Layer

505

Langmuir, Vol. 19, No. 4, 2003 1115

in the chemical free energy, one can recover the results obtained via the imaginary charging process, for constant surface charge or potential conditions, or via the Langmuir procedure for parallel plates and arbitrary surface conditions. However, it also can be used without any of the restrictions involved in the traditional theories. Furthermore, a simple approximate expression for the chemical free energy will be suggested, which is more efficient computationally. In that approximation, the free energy of the system does not depend on the trajectory ψS ) ψS(σ), but only on the values of σ and ψS at infinite and at the final separation distance. The approximate expression is exact in the two limiting cases, namely, interactions at constant surface potential and at constant surface charge density. Finally, the approximate free energy is compared with the exact one (obtained both from the present theory and the Langmuir procedure) for two planar surfaces, when the charge is generated via the dissociation of surface groups.

Denoting by ∆µ the change in the chemical part of the electrochemical potential of an ion of charge q brought from the bulk to the surface and by n the number of ions adsorbed per unit surface area, the chemical free energy (due to the adsorption of charges) per unit area, is given by5

II. The Double Layer Free Energy

where σ is the surface charge density and ψS is the surface potential. For an interaction at constant surface charge, the charge is fixed on the surface and it is not at thermodynamic equilibrium, and the change in the chemical free energy during the interaction is zero. Hence, the chemical free energy is a constant which was taken as zero5

The total free energy of a double layer is the sum of an electrostatic free energy Fel, an entropic term due to the mobile ions Fent and a chemical free energy.6 The first one is given by Fel )

∫  (∇ψ)

1 2

0

V

2

dV

∑∫ i

V

ci ln

ci

ci0

)

- ci + ci0 dV (2)

where k is the Boltzmann constant, T the absolute temperature, ci the concentration of ions of species “i” in the double layer, and ci0 its concentration in the bulk and the subscript “i” runs over all ionic species. For an 1-1 electrolyte of concentration cE, whose ions obey Boltzmann distributions, the above expression becomes -T∆S ) 2cEkT

The thermodynamic equilibrium implies that µbulk ) µsurface + qψS

(5)

Since ∆µ ) µsurface - µbulk ) - qψS, eq 4 becomes Fch|ψS)const ) n∆µ ) -nqψS ) -σψS

Fch|σ)const ) 0

( ()

(4)

(6a)

(1)

the integral being carried out over the entire volume V of the system. In the above expression, ψ is the electric potential, 0 the vacuum permittivity, and  the relative dielectric constant. Note that in eq 1 the charges are assumed fixed (hence their entropy is not included), but the entropy of the dipoles of the medium is accounted for through . The entropic term due to the mobile ions of the electrolyte takes into account the differences between the ions distributions in the double layer and their distribution in the bulk6 Fent ) -T∆S ) kT

Fch|ψS)const ) n∆µ

eψ eψ eψ ∫ [(kT ) sinh (kT ) + 1 - cosh(kT )] dV V

(3)

where e is the protonic charge. The above two contributions to the free energy of the double layer are both positive. The third term, the chemical free energy contribution, which is responsible for the spontaneous formation of the double layer, should be negative and larger than the sum of the previous two terms. During interactions at constant surface potential ψS, the surface charge decreases when two surfaces approach each other and ion adsorption or molecular group dissociation is involved in the surface charging. The system (liquid and surfaces) is in thermodynamic equilibrium, and a transfer of charges from the liquid to the surfaces occurs to ensure the equality of the electrochemical potentials of the ions in the bulk and near the surface.

(6b)

As a result of this choice, the double layer free energy is in this case always positive. Actually this constant is negative, equal to the free energy required to charge the surfaces at infinite separation distance, and the double layer free energy is negative. Let us now derive an expression for the chemical free energy for a general interaction (when both the surface charge density and surface potential are changing during the interaction). When two particles (of arbitrary geometry) approach each other from an infinite separation, the surface charge density and the surface potential, for an infinitesimal area δA on the surface of a particle, depend on the distance and follow a trajectory ψS ) ψS(σ) (see Figure 1). This trajectory can be thought of as composed of successive infinitesimal transformations at constant surface potential (σi, ψS,i f σi+1, ψS,i) and constant surface charge density (σi+1, ψS,i f σi+1, ψS,i+1). The change of the chemical free energy during the transformations at constant surface potential is given by -ψS,i(σi+1 - σi)δA, and it is zero for the transformations at constant surface charge. Adding all the changes when two particles approach from infinity to a final separation distance l, one obtains for the change ∆Fch(l) of the chemical contribution to the free energy ∆Fch(l)δA ) (-

∫

σ(l)

σ(∞)

ψS(σ) dσ)δA

(7)

In the above integral, the Poisson-Boltzmann equation should be solved for each separation distance l and the appropriate surface boundary conditions (constant surface charge density, constant surface potential, dissociation of surface groups, or adsorption of ions). Consequently, ψS and σ are functions of l; eliminating l one obtains ψS ) ψS(σ). The function ψS(σ) describes the unique trajectory of the system during the interaction, and the integral is performed along this trajectory (the path 1 f 2 in Figure 1). The total free energy of the double layer is obtained by adding eqs 1, 3, and 7

Nanodispersions

506

1116

Langmuir, Vol. 19, No. 4, 2003

Manciu and Ruckenstein

Figure 1. The surface potential ψS as a function of the surface charge density σ when two interacting particles approach from infinite separation to the final separation distance l.

∫  (∇ψ) dV + eψ eψ eψ 2c kT ∫ [( ) sinh( ) + 1 - cosh( )] dV + kT kT kT ∫ (C - ∫ ψ (σ) dσ) dA

FDL(l) )

1 2

FDL(l)|σ)const )

2

0

V

E

V

σ(l)

A

S

σ(∞)

(8)

the last integral being taken over the entire surface of the system, and the constant C (independent of l) representing the chemical free energy at infinite separation. As shown by Overbeek, the first two terms of eq 8 can be transformed into surface integrals using Green’s theorem6 1 2

∫  (∇ψ) V

0

eψ eψ ∫ [(kT ) sinh(kT )+ eψ 1 - cosh( )] dV ) ∫ (∫ φ (ξ,l) dξ) dA kT 2

dV + 2cEkT

σ(l)

0

S

∫ (∫ A

σ(l)

0

(9)

φS(ξ,l) dξ) dA +

∫ (C - ∫ A

σ(l)

σ(∞)

σ

φS(ξ,l) dξ) dA

0

ψS(σ) dσ) dA (10)

Expression 10 is valid for a system of arbitrary shape and arbitrary surface conditions. It will be shown in what follows that one can recover the results obtained via the traditional procedures, within their domains of validity. For interactions at constant surface charge, the integral over σ in the second term vanishes and the well-known expression due to Verwey and Overbeek, which assumed C ≡ 0, is recovered5

(11a)

For interactions at constant surface potential, ψS(l) ) ψS(∞) ) ψS and using the corresponding Verwey-Overbeek choice C ≡ -σ(∞)ψS(∞), one obtains, integrating by parts the first right-hand side term of eq 10, the expression FDL(l)|ψ)const )

∫ (∫ A

σ(l)

0

φS(ξ,l) dξ) dA +

∫ (-σ(∞)ψ (∞) - ψ (∞)∫ dσ) dA ) -∫ ξ(φ ,l) dφ ) dA ∫ (φ ξ| ∫ (ψ (∞)σ(l)) dA ) - ∫ (∫ ξ(φ ,l) dφ ) dA σ(l)

A

S

A

A

To perform the integral in the right-hand side of the above expression, the Poisson-Boltzmann equation should be solved for a fixed value of l and all the “imaginary surface charges” ξ between 0 and σ(l) to obtain the “imaginary surface potentials” φS. The surface potential φS depends in this case on the values of ξ and l. From eqs 8 and 9 one obtains FDL(l) )

A

S

ξ)σ(l),φS)ψS(l) ξ)0,φS)0

S

σ(∞)

ψS(l)

S

0

S

ψS

V

A

∫ (∫

S

A

S

0

S

(11b)

which is the Verwey-Overbeek expression for the free energy for interactions at constant surface potential. The free energy of the system is always defined up to an arbitrary constant; the choice C ≡ 0 in the VerweyOverbeek approach leads to positive values for the free energy of the system at constant surface charge and negative at constant surface potential. However, the interaction free energy (the difference between the free energy of the system at the final separation distance and the free energy of the system at infinite separation) is not affected by the choice of the constant C. Let us now show that eq 10 leads to the same result as the Langmuir equation for the force between two identical planar surfaces and arbitrary surface conditions. Using eq 10, one obtains the following expression for the force per unit area, between two identical planar surfaces ΠDL(l) ) -

1 dFDL(l) d )- 2 A dl dl

∫

σ(l)

0

φS(ξ,l) dξ +

d 2( dl

∫

σ(l)

σ(∞)

ψS(σ) dσ) (12)

where the factor 2 accounts for the two identical surfaces.

Polarization Model: a unified framework for hydration and double layer interactions

507

Chemical Free Energy of the Electrical Double Layer

Langmuir, Vol. 19, No. 4, 2003 1117

The first term is the Verwey-Overbeek expression for the interactions at constant surface charge density and the second accounts for the variation of the surface charge with the separation distance. The derivative of the first term of eq 12 is given by d ( dl

∫

σ(l)

0

φS(ξ,l) dξ) )

∫

σ(l)

0

∂φS(ξ,l) dσ(l) dξ + ψS(σ) (13) ∂l dl

and the derivative of the second term by d ( dl

∫

σ(l)

σ(∞)

This expression is exact for both constant surface charge and constant surface potential interactions. One can also construct approximate expressions for the change in the chemical free energy by extending those at constant surface charge (6b) or at constant surface potential (6a) to general processes. In the first case

ψS(σ) dσ) ) ψS(σ)

dσ(l) dl

(14)

∆Fch(l) ≈ 0

which will be named in what follows “constant surface charge approximation”. In the second case, since the chemical free energy is defined in terms of σ and ψS at each distance l, a natural approximation is ∆Fch(l) ≈ -(σ(l)ψS(l) - σ(∞)ψS(∞))

Consequently, from eqs 12-14, one obtains ΠDL(l) ) -2

∫

σ(l)

0

∂φS(ξ,l) dξ ∂l

(15)

It remains only to prove that eq 15 is equivalent to the Langmuir equation

( (

ΠDL,Langmuir(l) ) 2cEkT cosh

) )

eψm(l) -1 kT

A notable consequence of eq 7 is that the values of surface charge density and surface potential at a given separation distance l are not sufficient to calculate the free energy of interaction between surfaces. To calculate that free energy, one has to solve the Poisson-Boltzmann equation at each separation distance along a trajectory from infinity to the separation distance l. Here we will propose a simple approximation for the change of the chemical free energy, which simplifies significantly the problem. As shown in Figure 1, the transformation from 1(∞) f 2(l) is bounded by two extremal processes, 1 f 3 f 2 and 1 f 4 f 2. For the first, the change in the chemical free energy, per unit area, calculated using eqs 6a and 6b, is

(20b)

∆Fch(l) ) -(σ(l)ψS(l) - σ(∞)ψS(l))

(20c)

∆Fch(l) ) -(σ(l) - σ(∞))

(17b)

Therefore, the change in the chemical free energy is bounded by ∆F132 and ∆F142. A simple approximation for the change in the chemical free energy, per unit area, is the arithmetic average of those two bounds (18)

(ψS(l) + ψS(∞)) 2

(20d)

which coincides with eq 18. This expression will be named in what follows “the constant surface potential approximation”, although it is exact when either the surface charge or the surface potential is constant. The accuracy of these approximations, when neither the surface potential nor the surface charge density are constants, will be investigated below, for two identical, parallel planar surfaces. It will be assumed that the surface charge is due to the dissociation of surface groups (for instance dissociation of surfactant molecules).8,9 In general, the electrolyte counterions are the most abundant ions in the vicinity of the surface and therefore they can control the surface charge via reassociation. Denoting by cE the concentration (in the reservoir) of an 1:1 electrolyte, by N the number of sites per unit area, and by x the fraction of dissociated sites, the dissociation equilibrium provides the expression KD )

(ψS(l) + ψS(∞))(σ(l) - σ(∞)) 2

∆Fch(l) ) -(σ(l)ψS(∞) - σ(∞)ψS(∞))

where the “constant surface potential” is assumed to be either at infinite separation distance or at the distance l. Expressions 20b and 20c are the two bounds proposed earlier (eqs 17a and 17b, respectively). A simple modality to improve the approximation is to use as “constant surface potential” in eq 6a an intermediate value between ψS(∞) and ψS(l). By employing their average, one obtains

(17a)

while for the second

∆Fch(l) ) -

which, as our calculations have shown, is a very poor approximation when the surface potential is not constant. Since eq 6a implies that ψS(l) ≡ ψS(∞), other two legitimate choices for the approximate change in the chemical free energy would be

or

III. Approximate Expressions for the Change in Chemical Free Energy

∆F142 ) ∆F14 + ∆F42 ) 0 - ψS(l)(σ(l) - σ(∞))

(20a)

(16)

where ψm(l) is the potential at the middle distance between plates. This task was performed earlier.7 For the completeness of the presentation, the details of the proof that eqs 15 and 16 are equivalent are reproduced in the Appendix. Therefore, the present treatment of the double layer interaction leads to the same results for the interaction free energy as the imaginary charging approach for systems of arbitrary shapes and constant surface potential or constant charge density and to the same results as the Langmuir equation for parallel plates and arbitrary surface conditions. It can be, however, used for systems of any shape and any surface conditions, since it does not imply any of the above restrictions.

∆F132 ) ∆F13 + ∆F32 ) -ψS(∞)(σ(l) - σ(∞)) + 0

(19)

(xN)cE+ (1 - x)N

)

xcE+ (1 - x)

)

(

)

eψS x c exp 1-x E kT

(21)

where KD is the dissociation constant, cE+ denotes the cation concentration in the liquid in the vicinity of the surface, e is the protonic charge, and the Boltzmann distribution was assumed for the counterions (note that ψS is negative). The surface charge density is therefore given by (7) Ruckenstein, E. J. Colloid Interface Sci. 1981, 82, 490. (8) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (9) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205.

Nanodispersions

508

1118

Langmuir, Vol. 19, No. 4, 2003

Manciu and Ruckenstein

Figure 2. The interaction free energy (per unit area) as a function of the separation distance l for two identical plates, planar and parallel, at T ) 300 K,  ) 80, cE ) 0.01 M and: (a) KD ) 1.0 M, N ) 6.25 × 1015 sites/m2; (b) KD )1.0 M, N ) 6.25 × 1016 sites/m2; (c) KD ) 0.001 M, N ) 3.125 × 1017 sites/m2; (d) KD ) 0.001 M, N ) 3.125 × 1018 sites/m2. The continuous thick line represents the exact result; the up triangles represent the upper bound, the down triangles represent the lower bound, the circles represent the “constant surface charge approximation”, and the crosses represent the “constant surface potential approximation”.

σ ) -exN ) -

eN cE eψS 1+ exp KD kT

(

)

(22)

When the dissociation constant KD is large, (cE/KD) exp(-eψS/kT) , 1 and σ = -eN. Then the interaction can be well approximated by that at constant surface charge density. However, for low values of KD, the interaction can no longer be approximated by assuming constant surface charge or constant surface potential. Another relation between the surface charge and surface potential is provided by the equation σ ) -0

( ) dψ(z) dz

S

(23)

where the z direction is normal to the surface. Equation 23, combined with eq 22 provides one of the boundary conditions. The other boundary condition is provided by the symmetry of the system, which implies that the derivative of the potential at the middle distance vanishes dψ(z) | )0 dz z)1/2

(24)

Consequently, one has to solve the Poisson-Boltzmann equation, which for a uni-univalent electrolyte has the form

d2ψ 2ecE eψ ) sinh 0 kT dz2

( )

(25)

for the above boundary conditions. The free energy of interaction between two identical plates with charges generated through dissociation will be calculated both exactly and approximately, and the results will be compared for various values of the dissociation constant. For a given separation distance l, the solution of the Poisson-Boltzmann equation was obtained via numerical integration. From the value of the potential at the middle distance between the plates ψm(l) ≡ ψ(l/2), the disjoining pressure was calculated using the Langmuir equation. The interaction free energy was subsequently obtained by integrating the disjoining pressure from infinite to the final separation distance l. In a second method, the free energy was calculated by adding the electric (eq 1), entropic (eq 3), and chemical contributions, and the interaction free energy was obtained by subtracting the free energy for infinite separation. The integrals were calculated using the numerical solution of the Poisson-Boltzmann equation for ψ(z,l) and σ(l). The change in the chemical free energy was calculated using several expressions, namely, eq 7 (the exact expression), eq 19 (the “constant surface charge approximation”), eq 20b (the lower bound), eq 20c (the upper bound), and eq 20d (the“constant surface potential approximation”).

Polarization Model: a unified framework for hydration and double layer interactions

Chemical Free Energy of the Electrical Double Layer

509

Langmuir, Vol. 19, No. 4, 2003 1119

Figure 3. The ratio between the approximate interaction free energy, based on the change in chemical free energy contribution provided by eq 18 and the rigorous interaction free energy. The values of the parameters are the same as those given in Figure 2.

When the exact expression was employed, the results obtained always coincided (within numerical errors), as expected, with those provided by the Langmuir procedure. For large dissociation constants (implying small cE/KD ratio), the interaction is well described, at low surface potentials, by the constant surface charge density approximation. In Figure 2a, the free energy is calculated using cE ) 0.01 M, T ) 300 K,  ) 80, KD ) 1.0 M, and N ) 6.25 × 1015 sites/m2. In this case, the calculations based on the expressions for the change in the chemical free energy (eq 19 and eqs 20b-d)) are all good approximations to the rigorous result obtained either using eq 7 for the change in the chemical energy or via the Langmuir procedure. The increase of N to 6.25 × 1016 sites/m2 (Figure 2b) displaces the σ-ψS equilibrium toward larger surface potentials. The “constant surface potential approximation” (based on eq 20d) provides in this case a much better agreement with the exact result than the “constant surface charge approximation” (based on eq 19). For the low dissociation constant KD ) 0.001 M (cE ) 0.01 M, T ) 300 K,  ) 80, N ) 3.125 × 1017 sites/m2, Figure 2c), the “constant surface charge approximation” is inaccurate even at low potentials (ψS < 0.010 V). When the site density was increased to N ) 3.125 × 1018 sites/ m2 (cE ) 0.01 M, T ) 300 K,  ) 80, KD ) 0.001 M, Figure 2d), the “constant surface charge approximation” predicted attraction between the plates. In Figure 3 the ratio between the interaction free energy calculated on the basis of the approximate (eq 20d) and the rigorous (eq 7) expressions for the change in the chemical free energy is plotted against the distance. In all the cases investigated, the approximate expression for the chemical free energy was accurate. In summary, when the surface charge was generated by the dissociation equilibrium of surface groups, the surface charge density was almost constant for large dissociation constants and low surface potentials. In this case, the neglecting of the change in the chemical free

energy (eq 19) was a good approximation. However, in the other cases (small dissociation constants or large surface potentials) it was, in general, inaccurate and sometimes even predicted double layer attraction. In all the cases investigated, the surface potential has changed, sometimes drastically, during interactions. However, a suitable choice of the surface potential (the arithmetic average of the surface potentials at infinity and at distance l) in the “constant surface potential approximation” for the change in the chemical free energy provided always a good approximation. This approximation is exact when either the surface potential or the surface charge are constant. IV. Conclusions One can calculate the free energy of a double layer by adding to the free energy of the electric field, an entropic contribution due to mobile ions and a chemical free energy contribution due to the charge transfer between bulk and interface. However, expressions for the latter contribution are provided in the literature only for interactions at constant surface charge density or constant surface potential. In this paper, an expression for the change in the chemical free energy which is valid for interactions at any surface conditions was derived; it depends not only on the values of the surface charge and potential at the final separation distance but also on their values at all distances between infinite and the final one. The present approach reduces to the traditional ones within their range of application (imaginary charging processes for double layer interactions between systems of arbitrary shape and interactions either at constant surface potential or at constant surface charge density, and the procedure based on Langmuir equation for interactions between planar, parallel plates and arbitrary surface conditions). It can be, however, employed to calculate the interaction free energy between systems of arbitrary shape and any surface conditions, for which the traditional approaches cannot be used.

Nanodispersions

510

1120

Langmuir, Vol. 19, No. 4, 2003

Manciu and Ruckenstein

Since the change in the chemical energy depends on all the values of the surface potential and surface charge density, when the particles approach from infinity to the final separation distance, the calculation of the interaction free energy involves the solution of the Poisson-Boltzmann equation at all the points along this trajectory. For systems with complicated shapes, this procedure is inconvenient. For this reason, an approximate expression for the change in the chemical free energy was also proposed, which depends only on the values of the surface potential and charge density at the infinite and final separation distances. It was shown that this approximation is accurate. Appendix Integrating eq 15 by parts, one obtains for the disjoining pressure between two identical, parallel and planar surfaces separated by the distance l ΠDL(l) ) -2

∫

-2

∂ ( ∂l

∫

σ(l)

0

∫

∫

For the integral in the last right-hand side of the equality, one can use an expression derived by Verwey and Overbeek (eq 37b in ref 5)5 -

∫

ψS

0

( ( ) ) )∫ ( ( ))

ξ(φS,l) dφS ) -cEkTl cosh

(

20cE(kT) 2

eψm(l)/kT

2 cosh(y) -

∫

0

eψm(l) kT

eψm(l)/kT

(

dy 2 cosh(y) - 2 cosh

))

1/2

(A.5)

(

eψm(l) kT

))

)1/2

( )

2 l 2cEe 2 0kT

1/2

(A.6)

By employing eq A.6, the last right-hand-side term in expression A.3 becomes

(

20cE(kT)3 2

e

(

)

( )

1/2

sinh

eψm ∂ψm kT ∂l 1

∫

eψm(l)/kT

eψS(l)/kT

(

2 cosh(y) - 2 cosh

eψm(l) kT

×

))

1/2

cEkTl sinh

dy )

( )

eψm ∂ψm (A.7) kT ∂l

(

( ( ) )

dy (A.2)

( ( ) ) ( ) ( ) ( )) | ) ( ) ∫

(

(

eψS(l) eψm(l) - 2 cosh kT kT

1/2

eψm(l) ξ(φS,l) dφS) ) -cEkT cosh -1 kT eψm ∂ψm 20cE(kT)3 1/2 cEkTl sinh + × kT ∂l e2 eψm(l) 1/2∂y (eψm(l)/kT) 2 cosh(y) - 2 cosh + kT ∂l (eψS(l)/kT) 3 1/2 20cE(kT) eψm ∂ψm eψm(l)/kT sinh × 2 kT ∂l eψS(l)/kT e 1 dy (A.3) eψm(l) 1/2 2 cosh(y) - 2 cosh kT

( (

( )

and cancels the second term of the right-hand side of eq A.3. Using expression A.5, eq A.3 becomes

the derivative of which with respect to the distance is given by ψS

))

(

)

eψm(l) 1/2 eψ(z,l) - 2 cosh kT kT (A.4)

and by direct integration, from the surface to the middle distance to the expression5

eψm(l) -1 + kT

2 cosh

∂ - ( ∂l

(

3 1/2

eψS(l)/kT

e

(

(2cE0kT)1/2 2 cosh

∂σ(l) ) ∂l

ψS(l) ∂ (φ ξ|ξ)σ(l),φS)ψS(l) ξ(φS,l) dφS) + 0 ∂l S ξ)0,φS)0 ∂ψS(l) ∂σ(l) ∂ ψS(l) 2ψS(σ) )2 ( 0 ξ(φS,l) dφS) - 2σ(ψS) ∂l ∂l ∂l (A.1)

-2

2 cosh

∂ψ(z,l) | ) ∂z z)0

σ(l) ) -0

eψS(l)/kT

φS(ξ,l) dξ) + 2ψS(σ)

1/2

which leads to the relation

∫

∂φS(ξ,l) dξ ) ∂l

σ(l)

0

( )(

2cEkT ∂ψ(z,l) )∂z 0

-

eψm(l) ∂ ψS ( ξ(φS,l) dφS) ) -cEkT cosh -1 ∂l 0 kT 3 1/2 20cE(kT) eψS(l) eψm(l) 2 cosh - 2 cosh × 2 kT kT e eψm(l) ∂ψS(l) e ∂ψS(l) ) -cEkT cosh - 1 - σ(ψS) kT ∂l kT ∂l (A.8)

(

∫

Finally, from eqs A.1 and A.8, one obtains

ΠDL(l) ) 2

))

Integrating once the Poisson-Boltzmann equation, one obtains

) ( ( ) ( )) ( ( ) )

∂ ( ∂l

∫

ψS(l)

0

∂ψS(l) ) ∂l eψm(l) 2cEkT cosh - 1 (A.9) kT

σ(φS,l) dφS) - 2σ(ψS)

( (

which is the Langmuir equation, eq 16. LA0266132

) )

Introduction to CHAPTER 7 Polarization Model and ion specificity: applications

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

M. Manciu, E. Ruckenstein: “Role of the Hydration Force in the Stability of Colloids at High Ionic Strengths,” LANGMUIR 17 (2001) 7061–7070. M. Manciu, E. Ruckenstein: “Long Range Interactions between Apoferritin Molecules,” LANGMUIR 18 (2002) 8910–8918. E. Ruckenstein, M. Manciu: “On the Stability of the Common and Newton Black Films,” LANGMUIR 18 (2002) 2727–2736. M. Manciu, E. Ruckenstein: “On possible microscopic origins of the swelling of neutral lipid bilayers induced by simple salts” JOURNAL OF COLLOID AND INTERFACE SCIENCE 309 (2007) 56–67. H. Huang, M. Manciu, E. Ruckenstein: “The effect of surface dipoles and of the field generated by a polarization gradient on the repulsive force,” JOURNAL OF COLLOID AND INTERFACE SCIENCE 263 (2003) 156–161. E. Ruckenstein, H. Huang: “Colloid Restabilization at High Electrolyte Concentrations: Effect of Ion Valency,” LANGMUIR 19 (2003) 3049–3055. H. Huang, M. Manciu, E. Ruckenstein: “On the Restabilization of Protein-Covered Latex Colloids at High Ionic Strengths,” LANGMUIR 21 (2005) 94–99. M. Manciu, E. Ruckenstein: “The polarization model for hydration/double layer interactions: the role of the electrolyte ions” ADVANCES IN COLLOID AND INTERFACE SCIENCE 112 (2004) 109–128. M. Manciu, O. Calvo, E. Ruckenstein: “Polarization model for poorly-organized interfacial water: Hydration forces between silica surfaces,” ADVANCES IN COLLOID AND INTERFACE SCIENCE 127 (2006) 29–42.

The polarization model predicts that the interaction between nanoparticles depends not only on the surface charge density, but also on the surface dipole density. As the concentration of electrolyte increases, the surface charge density decreases, due to the recombination of ions with surface groups, but the density of surface dipoles increases. At relatively low salt concentrations, the repulsion due to the dou-

ble layer is dominant and decreases with increasing ionic strength; however, at high electrolyte concentrations, the repulsive force due to surface dipoles becomes dominant and increases with increasing ionic strength. Consequently, there is a minimum in the repulsion between nanoparticles. This mechanism explains the restabilization of some colloids at high ionic strengths [7.1]. This mechanism also is employed to explain the light scattering phenomena observed for apoferritin molecules at various NaCH3COO concentrations. The second virial coefficient of apoferritin decreases to a value corresponding to hard spheres for an electrolyte concentration of 0.15 M but increases to a large value at a concentration of 0.25 M. In the traditional framework, these results can be explained either by considering that the apoferritin molecules acquire a huge charge at 0.25 M, or that the traditional hydration force increases by orders of magnitude at that ionic strength. Neither of these possibilities is, however, plausible. The polarization model can explain this behavior as being a result of the recombination of surface charges with ions and their replacement by dipoles. The increase of the surface dipole density with increasing electrolyte concentration generates a long-range repulsion at high ionic strengths [7.2]. The recombination of surface charges with ions and the corresponding increase in repulsion also has consequences regarding the stability of common and Newton black films. However, in the latter case, the repulsive thermal undulations of the films also play a role [7.3]. As revealed by experiment and intuitively expected, an increase in temperature (which in turn increases the thermal undulations) decreases the stability of the films. In contrast, the insight provided by the traditional approach suggests that the increase in thermal undulations should lead to a larger Helfrich repulsion, hence to a more stable system. However, the treatment of this repulsion by considering the membrane as composed of pieces of a suitable area implies that an increase in temperature decreases the area of the pieces, hence increases the probability of film rupture [7.3]. By combining the thermal undulations of membranes with the polarization model it is shown that an increase in

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_7, © Springer Science+Business Media, LLC 2010

512

electrolyte concentration can lead to an increase in the repulsion between neutral lipid bilayers, thus explaining their experimentally observed swelling at high electrolyte concentrations [7.4]. Whereas the coupled equations of the polarization model can be solved analytically in the linear approximation (which is valid only for small potentials), in the general case one must rely on numerical solutions [7.5]. The polarization model can explain the restabilization of proteinstabilized polymer latexes, for which the increase in the repulsive force generated by the surface dipoles more than compensates for the decrease in repulsion caused by the decrease in the surface charge and the increase in the screening of the electrostatic field by the increasing ionic strength [7.5]. The theory is extended to multivalent ions, in which case an inversion of the surface charge can occur with increasing electrolyte concentration. Depending on the sign of the surface charge, the adsorption of dipolar molecules on the surface can increase or decrease the repulsion [7.6]. When association equilibria are taken into account

Nanodispersions

for all ions, including the acidic and basic sites of a protein, our model can explain, more than qualitatively, the experimental results of Lopez-Leon et al. [Lopez-Leon, T.; Gea-Jodar, P. M.; Bastos-Gonzalez, D.; Ortega-Vinuesa, J. L. Langmuir 2005, 21, 87-93], regarding the restabilization of protein-covered latex colloids at high ionic strengths [7.7]. The polarization model is extended to account for additional interactions, not included in the “mean” field, such as the ion-hydration forces [7.8]. In order to explain the interactions between silica surfaces, the polarization model is adapted to poorly-organized surfaces. To account for the disorder induced in water by the rough surfaces of silica, the dipole correlation length λm, which is the main parameter of the polarization model, is allowed to decrease from λm=14.9Ǻ obtained for water perfectly organized in ice-like layers in the vicinity of a surface to smaller values. For λm=4Ǻ, good agreement with experiment is obtained for reasonable values of the parameters involved (such as surface dipole and charge densities) [7.9].

Polarization Model and ion specificity: applications

513

Langmuir 2001, 17, 7061-7070

7061

Role of the Hydration Force in the Stability of Colloids at High Ionic Strengths Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received May 18, 2001. In Final Form: August 28, 2001 It is shown that the repulsion between colloidal particles or emulsion droplets depends both on the surface charge density and on the surface dipole density, the latter being a result of the presence of ion pairs on the surface. For illustration purposes, one considers the case in which a surfactant, such as sodium dodecyl sulfate, is adsorbed on the surface of droplets or particles. As the concentration of electrolyte (NaCl) increases, the charge on the surface decreases, and the number of ion pairs increases, because of the association-dissociation equilibrium. At relatively low salt concentrations, the repulsion due to the double layer is dominant and decreases as the electrolyte concentration increases. At relatively high electrolyte concentrations, the hydration repulsive force due to the ion pairs present on the surface becomes dominant. Consequently, as the salt concentration increases, the total repulsion decreases and passes through a minimum, after which it increases. If the hydration repulsion is large, the emulsion or colloidal system will remain stable at any electrolyte concentration. If the hydration repulsion is small, the system will be stable only for sufficiently low electrolyte concentrations. At intermediate strength of the hydration repulsion, the stability depends on the size of the particles or droplets and the Hamaker constant. The rate of coagulation of particles of small radii and small Hamaker constants reaches a maximum and then decreases with increasing ionic strength. For particles of large radii, the increase of the maximum of the interaction energy with increasing electrolyte concentration can be so large (∼30 kT) that it can forbid the coagulation at the primary minimum; however, the particles can aggregate at the secondary minimum, which is deep.

I. Introduction The cornerstone of colloid science is the DerjaguinLandau-Verwey-Overbeek (DLVO) theory, which explains the stability of colloids in terms of a repulsive (double-layer) force and an attractive (van der Waals) one. The double-layer repulsion, due to the charge on the surface of the particles, can be indefinitely diminished by increasing the electrolyte concentration. The combination of the two interactions typically generates a potential barrier, between a low (and hence stable) primary minimum and a weak secondary minimum. The latter, with a typical depth of the order of kT, generates a weak bonding between particles (flocculation), which can be reversed by stirring or shaking the system. Coagulation of colloids occurs when two neighboring particles acquire enough energy from the random Brownian motion to overcome the potential barrier. At low electrolyte concentrations, the potential barrier is high, and this ensures a long lifetime for the colloidal dispersion; however, by adding electrolyte, the double-layer repulsion is screened and the colloidal particles eventually coagulate. While the DLVO theory has the advantage of simplicity and often offers “almost quantitative results”,1 it is wellknown to be incomplete. A striking example is offered by some colloids (e.g., silica) which do not coagulate even at high electrolyte concentrations, when the double layer is expected to collapse.1 Recently, it was also shown that the rate constant for aggregation of paraffin wax particles covered by a long-chain carboxylic acid (C22) passes through a maximum when plotted against NaCl concen* Corresponding author. E-mail: [email protected]. Phone: (716) 645-2911/2214. Fax: (716) 645-3822. (1) Hunter, R. J. Foundations of Colloid Science; Oxford Science Publications: New York, 1987.

tration.2 This means that the repulsive force passes through a minimum. The above behaviors can be explained by additional forces, which also exist between noncharged surfaces and are particularly strong at low separation distances. These forces are generically called non-DLVO forces. An important interaction, for colloids dispersed in water, was related to the organization of water in the vicinity of a surface.3 This structural force, often called hydration force, was exhaustively investigated experimentally for neutral4 or weakly charged5 phospholipid bilayers and was reasonably well described by an exponential decay, with a decay length between 1.5 and 3 Å and a preexponential factor that varied in a rather large range. While the origin of this interaction is attributed to the changes in the local organization of water, there are still debates over the theoretical models which describe the interaction at the molecular level. The interaction was associated with the orientational correlation of water molecules,6 with the mutual polarization of water layers,7 disruption of hydrogen bonding,8 protrusion interactions,9 or truncation of ion atmosphere by walls.10 A detailed examination of the above models is made in another paper dedicated to hydration forces.11 (2) Alfridsson, M.; Ninham, B.; Wall, S. Langmuir 2000, 16, 10087. (3) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Plenum: New York, 1987. (4) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (5) Cowley, A. C.; Fuller, N. L.; Rand, R. P.; Parsegian, V. A. Biochemistry 1978, 17, 3163. (6) Marcelja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (7) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (8) Attard, P.; Batchelor, M. T. Chem. Phys. Lett. 1988, 149, 206. (9) Israelachvili, J. N.; Wennerstrom, H. Langmuir 1990, 6, 873. (10) Marcelja, S. Nature 1997, 385, 689. (11) Manciu, M.; Ruckenstein, E. Langmuir, in press.

10.1021/la010741t CCC: $20.00 © 2001 American Chemical Society Published on Web 10/04/2001

Nanodispersions

514

7062

Langmuir, Vol. 17, No. 22, 2001

Manciu and Ruckenstein

The hydration force is important for distances between plates less than a few nanometers. Since the DLVO potential barrier between spherical particles or droplets is located at separation distances of the order of the Debye length, it is clear that at least at high electrolyte concentrations the hydration force becomes relevant. The qualitative effect of hydration was earlier recognized, regarding the stability of silica colloids at high electrolyte concentrations1 or the stability of amphoteric latex particles at high concentrations of some electrolytes.12 The main goal of this paper is to show that a variant of an earlier model of Schiby and Ruckenstein7 can account for the stabilization of some colloids or emulsions at high electrolyte concentrations. In the present model, the hydration force has a strong dependence on the density of ion pairs on the surface of the particles or droplets, which is enhanced by the addition of electrolyte. This is due to the reassociation on the surface, which decreases the charge but generates additional ion pairs. The dipole moments of the ion pairs polarize the nearby water molecules and thus induce a hydration force. For large surface dipole densities, the hydration repulsion is strong enough to prevent (together with the double-layer repulsion) the coagulation at any electrolyte concentration. For low surface dipole density, the system coagulates for ionic strength higher than a critical value, as predicted by the DLVO theory. At intermediate surface dipole densities, the increase in the hydration repulsion with increasing electrolyte concentration can explain the restabilization of some colloidal and emulsion systems at high ionic strength. In what follows, one considers for illustration purposes the case in which the charge is generated on the surface of colloidal particles or droplets by the adsorption of a surfactant, namely sodium dodecyl sulfate (SDS). We selected this case because information about the adsorption of SDS on an interface is available in the literature, and as it will become clearer later the number of parameters involved is smaller than in the case of silica. A more complex calculation about the silica and the amphoteric latex particles will be presented in a forthcoming paper. It involves several kinds of surface dipoles and equilibrium constants. II. Theoretical Framework In this section, the equations needed to calculate the interaction potential between two spherical particles will be presented. It will be assumed that the double-layer, hydration, and van der Waals interactions are independent of each other. II.A. Surfactant Adsorption. The surface density of surfactant has a relevant role for both the double-layer (via the surface charge density) and the hydration interaction (via the dipole moment density of the ion pairs formed). It will be computed using the Frumkin adsorption isotherm:

(

Γ Γ exp -2b2 Γ∞ Γ∞ b1CR- ) Γ 1Γ∞

)

(1)

where Γ is the surfactant surface density, Γ∞ is the surface density at saturation, b1 and b2 are empirical parameters, and CR- is the concentration of the surfactant anions in (12) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156.

the liquid in the vicinity of the interface. The surfactant in the bulk is assumed to be completely dissociated, and hence the concentration of surfactant in the vicinity of the interface is related to the bulk surfactant concentration through the expression

CR- ) Cs exp

( ) eψs kT

(2)

where Cs is the bulk surfactant concentration; ψs is the surface potential, which is negative; e is the protonic charge; k is the Boltzmann constant; and T is the absolute temperature. II.B. The Surface Charge. Denoting by R the dissociation constant, the surface charge density σ is given by

σ ) -eRΓ

(3)

For simplicity, it will be assumed that the electrolyte is 1:1 and has the same type of cations as the surfactant (e.g., NaCl and SDS). The charge is generated through the dissociation

R-X|SURFACE T R-|SURFACE + X+|LIQUID AT SURFACE where R-|SURFACE denotes the surface surfactant anion group and X+ denotes the cation in the liquid at the surface. At equilibrium, one can write

KD )

CR-,SCX+ RΓCX+ R ) ) C + CR-X (1 - R)Γ 1 - R X

(4)

where CR-,S is the surface density of the surfactant anions, CR-X is the surface density of the nondissociated surfactants, CX+ is the concentration of the cations in the liquid in the vicinity of the surface, and KD is the equilibrium constant. Assuming that the electrolyte is completely dissociated, one obtains

( )

CX+ ) (Cs + Ce) exp -

eψs kT

(5)

where Ce is the bulk electrolyte concentration. II.C. Double-Layer Interaction. The calculation of the double-layer interaction accounts for the associationdissociation at the interface. Assuming two parallel plates at a distance x apart, the potential ψm at the middle distance between plates can be very well approximated by the following equation, due to Ohshima and Kondo:13

κx κx sinh eψm 1 2 2 tanh ) γ0 - γ0 3 + 4kT κx 4 κx cosh cosh 2 2 κx 2 2 κx cosh -2 -1 2 2 γ05 κx 4 cosh5 2 κx κx 3 sinh 2 2 κx κx 1-4 tanh 2 2 6 κx 4 cosh 2

( )

{

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( )) ( )

}

(6)

where γ0 ) tanh(eψs/4kT), ψs is the surface potential, κ )

x2e2Ce/0kT

is the reciprocal Debye length for a 1:1

(13) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1988, 122, 591.

Polarization Model and ion specificity: applications

515

Role of Hydration Force in Stability of Colloids

Langmuir, Vol. 17, No. 22, 2001

7063

electrolyte, 0 is the vacuum permittivity, and  the dielectric constant of water. The surface charge density is related to the surface and midplane potentials via1

σ)-

x( ( )

κ0kT e

2 cosh

( ))

eψs eψm - cosh kT kT

(7)

Equations 1-7 can be solved simultaneously to obtain the surface charge density and the surface and midplane potentials. The double-layer force per unit area is given by the Langmuir equation,1

( ( ) )

pDL ) 2CekT cosh

eψm -1 kT

(8)

and the double-layer interaction energy per unit area, between two parallel plates at a distance H apart, is obtained by integrating the double layer force:

UDL,p(H) )

∫H pDL(x) dx ∞

(9)

For two identical spheres of radius a separated by the distance of closest approach z, the interaction energy is given, in the Derjaguin approximation, by1

UDL,s(z) ) πa

∫z UDL,p(H) dH ∞

(10)

II.D. The van der Waals Attraction. For the van der Waals attraction between identical spheres of radius a, separated by the distance of closest approach z, we will use the formula derived by Hamaker:14

UvdW ) -

{

AH 2a2 2a2 + + 6 (z + 2a)2 - (2a)2 (z + 2a)2 (z + 2a)2 - (2a)2 ln (z + 2a)2

(

)}

(11)

where AH is the Hamaker constant. The Lifshitz theory of dispersion forces, which does not imply pairwise additivity and takes into account retardation effects, shows that the Hamaker constant AH is actually a function of the separation distance. However, for the stability calculations that follow, only the values of the attraction potential at distances less than a few nanometers are relevant, and in this range one can consider that AH is constant. II.E. Hydration Interaction. A molecule in liquid water has a first-neighbor coordination number between 4 and 5, which is closer to the coordination number in ice, 4, than to that in a normal liquid, 12.15 This suggests that the tetrahedral coordination in ice, due to the hydrogen bonding, is almost preserved in the liquid state. To account for this characteristic, Nemethy and Scheraga proposed a model in which the liquid water is composed of icelike clusters of various sizes.16 It is then reasonable to assume that in the vicinity of a surface the icelike clusters are reorganized in a layered structure, similar to that of ice I, with successive layers containing out-of-plane hexagonal rings of water molecules, parallel to the external surface, as depicted in Figure 1. Each layer is composed of two planar sublayers; a water molecule from a sublayer has three hydrogen bonds with three molecules of the other (14) Hamaker, H. C. Physica IV 1937, 10, 1058. (15) Eisenberg, D.; Kauzmann, W. The Structure and Properties of Water; Oxford University Press: New York, 1969. (16) Nemethy, G.; Scheraga, H. A. J. Chem. Phys. 1962, 36, 3382.

Figure 1. The structure of ice I (reproduced from ref 16). Each site is connected by hydrogen bonding with three sites from the same layer and one site from either one or the other of the neighboring layers. Copyright 1962 by the American Institute of Physics.

sublayer of the same layer and one with a molecule belonging to a sublayer of one of the adjacent layers. While the alignment gradually decreases with the distance from the surface, it is reasonable to consider that for not too large distances this decrease can be neglected. The treatment of the hydration interaction follows the model employed earlier by Schiby and Ruckenstein,7 based on the mutual interactions of dipoles. In this model, it is considered that the dipoles of the surface polarize the water molecules of the first layer of water and the polarization propagates from layer to layer. When two surfaces approach each other, the polarized layers will increasingly overlap. As a result, the local dipole moment of the water molecules will be decreased. This increases the free energy of the system and thus generates a repulsive force. II.E.1. Mutual Polarization of the Water Molecules. Assuming that all the water molecules from the layer i have the same average polarization mi, normal to the layer, the field Eilocal, acting on a site of layer i, due to all the other polarized molecules can be formally written as

Eilocal ) (C0mi + C1(mi-1 + mi+1) + ... + Ck(mi-k + mi+k) + ...) (12) where the coefficient Ck accounts for the contribution of the dipoles of layer i ( k to the local field at a site of layer i (C0 corresponds to the field at a site of layer i generated by all the other dipoles of the same layer i). An evaluation of the coefficients C will be provided in section II.E.4. If there is no external electric field and the molecules of the first water layer have an average dipole moment m1 oriented perpendicular to the layer, caused by the surface dipoles, the average dipole moment of the molecules of the layer i situated at the distance x ) i∆, mi ) m(x)|x)i∆ is given by

mi ) m(x)|x)i∆ ) γEit ) γEilocal ) γ(C0m(x) + C1(m(x - ∆) + m(x + ∆)) + ...) (13) where ∆ is the distance between the centers of two

Nanodispersions

516

7064

Langmuir, Vol. 17, No. 22, 2001

Manciu and Ruckenstein

successive layers and γ is the molecular polarizability, as defined and calculated in the Appendix. An approximate solution can be obtained by expanding m(x(∆) in series:

m(x ( ∆) ) m(x) (

2 dm(x) 1 d m(x) 2 ∆+ ∆ ( ... dx 2 dx2

Because of symmetry the odd derivatives cancel, and when the terms of order 4 and higher are neglected, eq 13 becomes

m(x)(1 - γ(C0 + 2C1 + 2C2 + ...)) ) d2m(x) γ∆2(C1 + 4C2 + ...) dx2 which is of the form

d2m(x) dx2

-

1 m(x) ) 0 λ2

(14)

with a decay length λ given by

λ)

x

(C1 + 4C2 + 9C3 + ...)γ∆2

1 - (C0 + 2C1 + 2C2 + ...)γ

(15)

II.E.2. Polarization of the First Water Layer. In a homogeneous medium with uniform dielectric properties, an infinite, planar, uniform and continuous distribution of dipoles produces a vanishing electric field. This is due to the long range of the electrostatic interactions, which leads to the cancellation of the strong fields, generated by the nearest dipoles, by the weakest fields produced by the infinite number of remote dipoles. A discrete distribution of planar dipoles generates an oscillating field above the surface, whose average vanishes. However, the dielectric constant of the interactions of the water molecule from the first layer with the nearest dipole of the surface, denoted 1, is expected to be much smaller than the bulk dielectric constant of water, . Denoting as Ed the field generated (in a vacuum) by a surface dipole, the field generated by all the dipoles of the surface is

E)

1

 - 1 Ednearest + 1  - 1 1 1 d Ed ) Enearest ≡ Ednearest (16)  all dipoles 1 ′

d Enearest +

1

Figure 2. Sketch of three water layers between parallel plates. The dotted lines represent the two planar sublayers of the layers. Only two hydrogen bonds (thick lines) are shown for each water molecule (circles): one with a molecule from the other sublayer of the same layer and one with a molecule from an adjacent layer. The distance between two water molecules is denoted by l; ∆ is the distance between two water layers, and ∆′ is the distance between the center of the surface dipoles and the interface.

It will be assumed that the field induced by the surface dipoles becomes negligible after the first water layer. The interfaces are located at the external boundaries of the first organized water layers, while the surface dipoles are situated at a distance ∆′ below the interface (see Figure 2). There are also water molecules absorbed among the headgroups of the surface, which do not have the icelike structure but are organized differently, due to the strong interactions with the surfactant headgroups. The high local electric field generated by a dipole of the surface, as well as the small number of water molecules which are screening the water molecules of the first layer, make the local dielectric constant, 1, near the dipole small. Let us approximate the area A per surface dipole, which is polarized by the corresponding dipole, by a disk of radius R, located at a distance ∆′ from the dipole b p, which has the components p| and p⊥ parallel and normal, respectively, to the surface. The field produced by p| vanishes because of symmetry, and so does the electric field parallel to the surface. The electric field, normal to the surface, generated by the dipole, at a point whose position vector makes an angle θ with the normal, is given by

1

∑ Ed )  all others

∑

where it was assumed that the interaction with the nearest dipole is screened by a medium with the dielectric constant 1, while all the other interactions are screened by a medium with the dielectric constant . Equation 16 accounts for the fact that the average field of all the dipoles distributed on the interface, (1/) ∑all dipoles Ed, in a medium with constant  vanishes. This implies that the net average field acting on the first layer can be calculated by considering only the field generated by the nearest dipole, in a medium of effective dielectric constant ′. After some distance from the surface, the fields of all dipoles of the interface are screened by the intervening water molecules. Hence, the spatial average of the field induced by the surface dipoles vanishes, since the depolarization field of the remote dipoles is no longer negligible compared to the polarization field of the nearest dipoles.

Ed⊥ )

1 {p⊥(3 cos2(θ) - 1)} 4π′0r3

(17)

where r ) ∆′/cos(θ). For the average field produced by a surface dipole, one obtains

E h )

dF ∫0 4π′ 2πF (∆′2 + F2)3/2

1 πR2

R

0

{(

p⊥ 3

∆′2 -1 (∆′ + F2) 2

)}

)

p⊥ 1 (18) ′ 2π (R2 + ∆′2)3/2 0 II.E.3. The Hydration Interaction. The solution of eq 14 for two planar, parallel surfaces separated by the distance H (hence for the boundary condition m(-H/2) ) m1,m(H/2) ) -m1) is

x ( λ) m(x) ) -m H sinh( ) 2λ sinh

1

(19)

Polarization Model and ion specificity: applications

517

Role of Hydration Force in Stability of Colloids

Langmuir, Vol. 17, No. 22, 2001

where m1 is the dipole moment of the water molecules at the interface, which is proportional to the total field Et ) E h + E1local:

m1 ) γ(E h + E1local) ) γ(E h + m1C0 + m2C1 + ...) (20) Since from eq 19, m2 ) m[-(H/2) + ∆] ) m1 sinh[(H 2∆)/2λ]/sinh(H/2λ), eq 20 becomes

m1(H) )

( (

γE h H - 2∆ 2λ + ... H sinh 2λ

(

)

sinh

1 - γ C0 + C1

( )

)

(21)

The electrostatic energy of a water molecule with the polarizability γ, which acquires the polarization m in the field Ee, is

U)-

∫0

Ee

∫0

Ee

m(E) dE ) -

While the field produced by remote dipoles can be treated as screened by a medium with a large dielectric constant ( = 80), the screening of the neighboring dipoles is much weaker. In the present treatment, we will simply assume that Elocal is produced only by the dipoles located within a radius 2l from the given site and that the dielectric constant for them is a constant ′′. The electric field caused by a neighboring molecule is given by eq 17 (with ′′ replacing ′). It is important to emphasize that the local dielectric constant ′′ is smaller than , the bulk dielectric constant of water. The electric field Elocal i,0 generated at a site of layer i by the other dipoles from the same layer is given by

Elocal i,0 ) C0mi )

1 m2 γE dE ) - γEe2 ) 2 2γ 3

-

1

∑ 2Sall molecules

∫ 2S Volume

((

(m(x))2

1

(m(x))2 γv

γE h2

(

sinh

1 - γ C0 + C1

1

∫ (m(x))2 dx ) 2γv -H/2

)

H - 2∆

sinh

2λ

() H

2λ

H/2

)

() ()

H - λ sinh 2

4v sinh2

+ ...

H λ

H

2λ

∫z (UH,p(H) - UH,p(∞)) dH ∞

-1

1

1 x33

2

-1 3

11 3

) + 6(3(0) - 1) + 2

(x ) 8 3

)

mi

4π0′′l3

Elocal i,(1 ) C1mi (1 )

(

3 2+3

(x ) (x ) 2 3

2

8 3

-1 3

(x ) (x )

3 +6

3 11

2

11 3 2

-1

)-

(22)

3

3

3.7663 mi (24) 4π0′′l3

2 +3 53 3

()

mi(1 4π′′0l3

)

3

(x ) (x ) 2 3

2

8 3

1.8272 mi(1 4π′′0l3

-1 3

)

(25)

Only the contributions from the first 26 neighbors (12 from the same layer and 14 from the two adjacent layers) were taken into account. The field produced by the more remote dipoles is neglected, because it is screened by the intervening water molecules. III. Results III.1. General Behavior of the Interaction Energy. We will try first to obtain some general information about the interaction energy between two identical spheres of radius a, separated by the distance of closest approach z, at high electrolyte concentrations, using some simple approximations. The following expressions will be used for the interaction energies:

(23)

II.E.4. Evaluation of the Local Field. For a tetrahedral coordination of the water molecules, the distance between the two planar sublayers of the same layer of icelike structure is (1/3)l, l being the distance between the centers of two adjacent water molecules, while the distance between the centers of two adjacent layers is ∆ ) (4/3)l. The vertexes of the tetrahedron formed by the four first neighbors of a water molecule have the length lx8/3, while the planar projection of the tilted hexagonal lattice has the side lx8/9. The volume occupied in this structure by a water molecule is v ) (8/3x3)l3.

)( +

×

where V is the volume occupied by all the icelike organized water layers, v is the volume occupied by a water molecule, and S is the surface area of a plate. Here, we will assume that the electrostatic energy of the remaining water molecules, located among the headgroups of the surfactant molecules, does not depend on the separation distance H between plates and hence does not contribute to the hydration interaction. The interaction energy is obtained by subtracting the electrostatic energy at large separation distances, UH,p(∞). Using the Derjaguin approximation, the hydration interaction energy between two spheres is given by

UH,s(z) ) πa

2

The field generated by the neighbors (within a radius 2l from the given site) located in the adjacent water layers i ( 1, averaged over the two sublayer sites of layer i, is given by

)

γ

dV ) -

(

(3 3(13)

(( ) ) (x ) 3

The electrostatic interaction energy per unit area, between the two parallel plates, is given by

UH,p(H) ) -

7065

UvdW,s ) -

A Ha 12z

( λz)

UH,s ) πaλB exp -

(26a) (26b)

which are rough approximations of the interaction energies obtained with the methods described above for the van der Waals and hydration interactions (we approximate the hydration interaction energy per unit area between two parallel plates at distance x apart by UH,p ) B exp(-x/λ)). When the double-layer interaction at high ionic strengths is neglected, the interaction energy has an

Nanodispersions

518

7066

Langmuir, Vol. 17, No. 22, 2001

Manciu and Ruckenstein

absolute minimum at z ) 0 (primary minimum) and a local minimum at some finite z (secondary minimum), separated by a potential barrier. The extrema of the interaction energy are obtained from the condition of vanishing of the first derivative,

F(ξ) ) ξ2 exp(-ξ) )

( )

F1(ξ) )

which can be rewritten as

F(ξ) ≡ ξ2 exp(-ξ) )

12πBλ2

F2(ξ) )

AHa z )0 λ 12z

( )

(28)

which, together with eq 27, leads to

zmax ) λ

12πλ

U(zmax) )

(30)

(31a)

and

U(zmin) ) -

( )

AHa ξ2 - 1 12λ ξ 2 2

(34a)

AHa (η + η2) 12λ

(34b)

λξ2 2+η ξ ξ2 - 2 2 1 + η

(

)

(36a)

and

with the solutions ξ ) ξ1 ) 1 for the maximum (eq 29a) and ξ ) ξ2 ) 3.5129 for the minimum. Therefore,

zmin ) ξ2λ

λ 1+η

For the values of the parameters employed above, the factor in the front of the parenthesis is about 68 kT, and consequently even a small η produces a large increase of the energy of the maximum. For the minimum, eqs 33b and 32 lead to

(29b)

Although in this case U(zmax) - U(∞) ) 0, the secondary minimum can be sufficiently deep for the potential barrier between the latter and the maximum to hinder the coagulation at the primary minimum. By substitution of eq 29b in eq 27, the positions of the two extrema satisfy the equation

ξ2 exp(-ξ) ) exp(-1)

(33b)

ξ2 exp(1)

and

zmin )

2

ξ(2 - ξ2) + ξ22 - ξ2

zmax )

(29a)

AH exp(1)

(33a)

For the maximum, the linear approximation (eq 33a) inserted in eq 32 provides

and

B ) B0 )

ξ exp(1)

(27)

where ξ ) z/λ. F(ξ) has a maximum at ξ ) 2, F(2) ) 22 exp(-2). Hence, if B < B* ) AH/[48πλ2exp(-2)], eq 27 has no solution and there is no potential barrier to prevent coagulation. When B > B*, eq 27 has two solutions, the smaller one corresponding to the maximum and the larger one to the secondary minimum of the energy. Another useful value for the preexponential factor B of hydration is obtained from the condition of vanishing of the energy maximum,

U(z) ) πaλB exp -

(32)

Because the perturbation η is small, F(ξ) can be replaced around ξ1 and ξ2 by the linear approximations

AHa dU(z) z ) -πaB exp - + )0 dz λ 12z2

AH

exp(-1) (1 + η)

(31b)

By use of some typical values for the parameters, namely, AH )1 × 10-20 J, a ) 1000 Å, and λ ) 2.96 Å, for an absolute temperature T ) 300 K, the depth of the minimum is about -13.8 kT, and hence the transition from the secondary to the primary minimum is very slow. The depth of the minimum increases as the radius a and the Hamaker constant AH increase. We will now examine the consequences of a small increase in the hydration force, by assuming that the preexponential factor becomes B ) B0(1 + η), with η small compared to 1. Equation 27 acquires the form

U(zmin) )

(

AHa ξ2 - 1 η + 2 + O(η2) 12λ ξ2 ξ 2

2

)

(36b)

where the terms in η2 and higher were neglected. The first term represents the depth of the minimum for η ) 0. Because ξ22 ) 12.34, the increase in the energy of the secondary minimum, when the hydration increases, is about 1 order of magnitude smaller than the corresponding increase of the maximum energy. For small values of η, the increase is negligible compared with the depth of the unperturbed secondary minimum (ξ2 - 1 = 2.5129 . η). As general conclusions, one can notice the following: (1) if the preexponential factor B is smaller than the critical value B*, there is no potential barrier which can prevent aggregation; (2) if B . B0, the maximum of the interaction energy is large and no aggregation takes place at the primary minimum but can take place at the secondary minimum; and (3) small changes of B around B0 dramatically affect the value of the maximum, but much less the minimum. Because B is proportional to (p⊥/′)2, the behavior of the system is expected to be very sensitive to changes in the ratio p⊥/′. III.2. Numerical Calculations. In what follows, the equations from section II will be employed to calculate the effect of the hydration force on the stability of colloids or emulsions in water in the presence of NaCl and SDS, which represent a common electrolyte and surfactant, respectively. For the adsorption isotherm, the following values of the parameters will be used: b1 ) 881 m3/mol, b2 ) -1.53, and Γ∞ ) 5 × 10-6 mol/m2. These coefficients were obtained from experimental data at high electrolyte concentra-

Polarization Model and ion specificity: applications

519

Role of Hydration Force in Stability of Colloids

Langmuir, Vol. 17, No. 22, 2001

7067

tions.17 While the experiments have been carried out for a water-air interface, the coefficients probably provide good approximations also for the present case, because the hydrophobic bonding (the disorganization of water by the presence of surfactant) constitutes the main driving force for adsorption. The equilibrium constant for the association-dissociation at the interface is not known. The only related relevant information is the dissociation constant of the ion pair NaSO4-, which was estimated to be KD ) 10-0.7 ) 0.1995 mol/dm3.18 Because the repulsion among the headgroups increases as the dissociation increases (and this provides a positive contribution to the free energy), the dissociation constant is expected to be lower in this case. A discussion regarding the molecular polarizability is provided in the Appendix. In the present paper, we will use for the molecular polarizability expression A.7, which provides a lower bound for γ, in the absence of saturation effects. When eq A.7 is inserted into eq 15, the decay length acquires the simple form (Ck ) 0 for k >1)

λ)

x

30v( - 1)C1∆2 +2

(37)

We will use in the calculations l ) 2.76 Å (which corresponds to the distance between molecules in the structure of ice I, as compared to about 2.9 Å for molecules in water), and  ) 80. For the local dielectric constant, we will assume ′′ ) 1, which constitutes a lower bound. In a perfect tetrahedral coordination, the average distance between two successive water layers is ∆ ) (4/3)l, and the decay length of the hydration interaction calculated using eq 37 is λ ) 2.96 Å. It should be noted that the latter value is in the range determined experimentally for the hydration force between phospholipid bilayers.4 Lower values of λ can be obtained for higher ′′. For the distance between the center of the ion pair and the interface (located at the boundary of the first organized water layer), the value ∆′ ) 1.0 Å was selected. The model employed for the hydration force implies a continuum, even for low separation distances between particles. It was also assumed that the hydration, doublelayer, and van der Waals interactions are additive. The double layer affects the hydration force mainly because of the decrease of the molecular polarizability by large electric fields. However, in the present case, in the vicinity of the surface the electric field due to the double layer is smaller than E h by an order of magnitude and hence its influence on the hydration force can be neglected at small separation distances. The magnitude of the hydration force is proportional to the ratio (p⊥/′)2, which can vary in a quite large range. The dipole moments of the common salt molecules are of the order of 10 D (e.g., 8.5 D for NaCl and 10.4 D for CsCl);19 the values are expected to be larger for the ion pairs on the particle surface, because the cations are hydrated. The value of ′ is expected to depend strongly on the particular organization of the water molecules absorbed among the headgroups of the surface. To have some feelings about their order of magnitude, the ratio Γ/Γ∞, the dissociation constant R, the area per ion pair A, and the average dipole moment of the first water layer m1 are plotted in Figure 3a-d as functions of electrolyte concentration for Cs ) 0.001 mol/dm3, (p⊥/′) ) 1.0 D, large distances between particles or droplets (z f (17) Feinerman, V. B. Colloids Surf. 1991, 57, 249. (18) Davies, C. W. Ion Association; Butterworth: London, 1962. (19) Landolt-Bo¨rnstein, Springer, Heidelberg, 1985.

Figure 3. (a) The ratio Γ/Γ∞, (b) the dissociation constant R, (c) the area per ion pair A, and (d) the polarization of the water molecules near the interface, m1, as functions of the electrolyte concentration Ce for various values for the dissociation constant: (1) KD ) 0.05 mol/dm3; (2) KD ) 0.1 mol/dm3; (3) KD ) 0.1995 mol/dm3; (4) KD ) 0.5 mol/dm3.

∞), and various dissociation constants KD, ranging between 0.05 and 0.5 mol/dm3. The low dissociation of the surfactant headgroups is a consequence of the low value selected for the equilibrium dissociation constant KD; however, the results are not significantly modified even when KD varies with 1 order of magnitude. In what follows, the following values will be used for some of the parameters: KD ) 0.1 mol/dm3, b1 ) 881 m3/mol, b2 ) -1.53, Γ∞ ) 5 × 10-6 mol/ m2, λ ) 2.96 Å,  ) 80, ∆′ ) 1.0 Å, and Cs ) 0.001 mol/dm3. The other parameters, namely, a, AH, and the ratio (p⊥/′), will be varied. The values of the interaction energy at the maximum and at the secondary minimum are proportional with the radius a of the particle or droplet and depend strongly on the Hamaker constant AH and on the hydration repulsion. As shown in the previous section, small modifications in the ratio (p⊥/′) (and hence in the hydration repulsion) can lead to a large increase in the potential barrier between the primary and secondary minima, thus affecting the stability of the system. First, calculations will be presented regarding the interaction energy between two identical spherical particles of radius a ) 100 Å, Hamaker constant AH ) 2.0 × 10-20 J, and p⊥/′ ) 2.6 D. The interaction energy is plotted as a function of distance for electrolyte concentrations ranging from 0.1 to 4.0 mol/dm3 (Figure 4a). At relatively low electrolyte concentrations, the height of the barrier

Nanodispersions

520

7068

Langmuir, Vol. 17, No. 22, 2001

Manciu and Ruckenstein

Figure 4. (a) The interaction energy as a function of separation distance, for a ) 100 Å, p⊥/′ ) 2.6 D, AH ) 2.0 × 10-20 J, and various electrolyte concentrations. In the inset, the maximum value of the interaction energy is plotted versus electrolyte concentration. (b) The interaction energy as a function of separation distance, for a ) 100 Å, p⊥/′ ) 1.3 D, AH ) 2.0 × 10-20 J, and various electrolyte concentrations. In the inset, the maximum value of the interaction energy is plotted versus electrolyte concentration. (c) The interaction energy as a function of separation distance, for a ) 100 Å, p⊥/′ ) 1.9 D, AH ) 2.0 × 10-20 J, and various electrolyte concentrations. In the inset, the maximum value of the interaction energy is plotted versus electrolyte concentration. (d) The interaction energy as a function of separation distance, for a ) 200 Å, p⊥/′ ) 1.3 D, AH ) 1.0 × 10-20 J, and various electrolyte concentrations. In the inset, the maximum value of the interaction energy is plotted versus electrolyte concentration.

∆U, which is measured with respect to an infinite separation distance, ∆U ) U(zmax) - U(∞), decreases with increasing electrolyte concentration, as predicted by the DLVO theory. However, after reaching a minimum at Ce ) 0.4 M, the barrier slowly increases with increasing ionic strength (the inset presents the maximum value of the energy as a function of electrolyte concentration). Since in this case the height of the barrier is always higher than 20 kT, the system is stable at any electrolyte concentration. In Figure 4b, the interaction energy is calculated using p⊥/′ ) 1.3 D. At low electrolyte concentrations, the height of the potential barrier decreases with increasing Ce, because of the screening of the double layer. The increase in the hydration force with increasing Ce, due to the reassociation, is not sufficient to generate a high enough potential barrier to prevent the rapid coagulation. For electrolyte concentrations larger than about 0.25 mol/dm3, the system is unstable. Figure 4c provides U/kT for p⊥/′ ) 1.9 D. The maximum of the interaction energy decreases with increasing electrolyte concentration and reaches a minimum of 5.26 kT at Ce ) 0.7 mol/dm3. However, with a further increase of the ionic strength the maximum energy gradually increases and at Ce ) 4.0 mol/dm3 it becomes 7.45 kT. The

rate of aggregation at 4.0 mol/dm3 is hence decreased by a factor of the order of exp(7.45 - 5.26) ) 8.93, as compared to the maximum aggregation rate, obtained at Ce ) 0.7 mol/dm3. There is therefore a restabilization of the system with increasing electrolyte concentration. The calculations were repeated for a system with a ) 200 Å, AH ) 1.0 × 10-20 J, and p⊥/′ ) 1.5 D. As shown in Figure 4d, the minimum potential barrier is reached now at Ce ) 1.1 mol/dm3. The rate of aggregation is in this case about 3.7 times slower at Ce ) 4.0 mol/dm3 than at Ce ) 1.1 mol/dm3. The matters become more complex for large particles and large Hamaker constants. Figure 5a presents the interaction energy for a ) 1000 Å, AH ) 5.0 × 10-20 J, and p⊥/′ ) 2.45 D (while keeping the values of the other parameters as before). The maximum of the interaction energy becomes negative at Ce ∼ 0.3 mol/dm3, reaches a minimum at Ce ∼ 0.5 mol/dm3, and then strongly increases, becomes again positive at Ce ∼ 1.0 mol/dm3, and exceeds 30 kT at Ce ) 4.0 mol/dm3. However, as anticipated in the section where the general behavior was examined, the secondary minimum is extremely deep in this case. The increase of the hydration force with increasing ionic strength has only a moderate effect on the depth of the

Polarization Model and ion specificity: applications

Role of Hydration Force in Stability of Colloids

521

Langmuir, Vol. 17, No. 22, 2001

7069

paraffin particles (in the presence of a long-chain carboxylic acid, C22) increased with the concentration of NaCl at low ionic strength, reached a maximum at about 0.5 mol/dm3 NaCl, and then decreased for higher electrolyte concentrations. There are also colloidal systems (e.g., silica) whose stability at any electrolyte concentration can be explained by a large hydration repulsive force.1 It is interesting to note that there are also systems whose stability depends on the nature of the electrolyte. Healy et al.12 reported that amphoteric latex particles coagulated when the concentration of CsNO3 exceeded 0.8 mol/dm3. However, they remained stable even when concentrations up to 3.0 mol/dm3 of KNO3 or LiNO3 were employed. The effect was qualitatively interpreted in terms of a “hydration barrier”; to allow close particle-particle aggregation, the cations adsorbed on the surface should be dehydrated. In the present paper, the hydration force is considered as originating from the ability of the ion pairs formed on the surface to polarize the water layers. The cation Cs+ can be easily dehydrated because its heat of hydration is relatively low. In contrast, the cations Li+ and K+ have higher heats of hydration and it is difficult to dehydrate them. As a result, the dipole moment of Cs is smaller than those of Li and K, and hence the hydration repulsion is weaker for the former cation. As our calculations indicate, changes of the order of 10% in the ratio p⊥/′ might be sufficient to convert an unstable system into a stable one at high electrolyte concentrations. A detailed treatment of the problem will be presented in a forthcoming paper. IV. Conclusions

Figure 5. (a) The interaction energy as a function of separation distance, for a ) 1000 Å, p⊥/′ ) 2.45 D, AH ) 5.0 × 10-20 J, and various electrolyte concentrations. In the inset, the maximum value of the interaction energy is plotted versus electrolyte concentration. (b) The interaction energy as a function of separation distance, for a ) 1000 Å, p⊥/′ ) 1.95 D, and AH ) 5.0 × 10-20 J for various electrolyte concentrations, in the region where a potential barrier between the primary and the secondary minimum is formed at large electrolyte concentrations. The electrolyte concentration is varied in steps of 0.1 M. The interaction energy is plotted over larger ranges in the inset.

secondary minimum (see eq 36b and Figure 5a). Therefore, the particles will coagulate at the secondary minimum for all the electrolyte concentrations considered, with the exception of a narrow range around Ce ∼ 0.5 mol/dm3. At concentrations of the order of 0.5 mol/dm3, the particles can overcome the barrier and aggregate at the primary minimum if the frictional losses are small enough. If the frictional losses are large, an interesting situation occurs at a lower value of p⊥/′ than that used in Figure 5a, namely, p⊥/′ ) 1.95 D (see Figure 5b). When Ce ) 0.5 mol/dm3, there is no potential barrier and rapid coagulation at the primary minimum occurs. However, a relatively large potential barrier of the order of 10 kT between the primary and the secondary minima (∆U ) U(zmax) - U(zmin )) is established at very large electrolyte concentrations (Ce > 4 mol/dm3). In this case, the aggregation occurs at the secondary minimum. There is some experimental evidence that for some colloidal systems, the rate of aggregation has a maximum and then decreases with increasing electrolyte concentration. In the experiment of Alfridsson et al.,2 the rate constant of aggregation for a suspension of charged

The surface charge is responsible for the double-layer repulsion, which is attenuated by increasing the ionic strength. However, the ion pairs formed through the association-dissociation of the charges with the counterions induce a repulsive hydration force that increases with ionic strength. At the same surface density of ion pairs, the magnitude of the hydration force is related to the dipole moment, normal to the surface, of the surface ion pair, and to the local dielectric constant in the vicinity of the headgroups. A small value of p⊥/′ corresponds to a system which coagulates when the electrolyte concentration exceeds a critical value, as predicted by the DLVO theory. If p⊥/′ is large, the colloidal or emulsion system can remain stable at any ionic strength. At intermediate values of p⊥/′ and for small particles or droplets and low van der Waals forces, the increase in the hydration force with electrolyte concentration can reduce the rate of aggregation by 1 order of magnitude; for large particles or droplets, the increase in hydration force with the ionic strength can increase the maximum interaction energy by more than 30 kT. In the latter case, the coagulation at the primary minimum can occur at intermediate electrolyte concentrations but at large electrolyte concentrations can be prevented by the increase of the hydration repulsion with increasing ionic strength. The particles can however aggregate at the secondary minimum, which is deep. Appendix: Evaluation of the Molecular Polarizability We considered two models for the evaluation of the molecular polarizability in the condensed phase. One of the models follows the Lorentz-Debye theory, as summarized by Jackson20 and Frenkel.21 In a constant macroscopic field E0, the macroscopic polarization P is

Nanodispersions

522

7070

Langmuir, Vol. 17, No. 22, 2001

Manciu and Ruckenstein

given by

P ) 0( - 1)E0

(A.1)

where  is the macroscopic dielectric constant. The effective field Ee is the sum between the macroscopic field E0 and the internal field Eint, due to the other molecules of the medium. The latter can be calculated by adding the individual contributions of the other molecules close to the selected one, Elocal, and subtracting the contribution from the same molecules treated in an average continuum approximation described by the polarization P:

Eint ) Elocal - EP ) Elocal +

P 30

(A.2)

The field Elocal at a site of an icelike lattice generated by the dipoles located at the other sites of the lattice, assuming that all the water molecules have the average dipole moment m, normal to the layer, is given by

Elocal ) m(C0 + 2C1 + ... + 2Ck + ...)

(A.3)

where the coefficient Ck accounts for the contribution of the dipoles of layer i ( k to the Elocal at a site of layer i. The average dipole moment m is proportional to the effective field; hence,

(

m ) γEe ) γ E0 +

)

P + m(C0 + 2C1 + 2C2 + ...) 30 (A.4)

where γ is the molecular polarizability. Since m is related to the macroscopic polarization P via m ) Pv, where v is the volume of a water molecule, eqs A.1 and A.4 provide the following expression for the macroscopic dielectric constant:

2γ 1 - γ(C0 + 2C1 + 2C2 + ...) + 30v ) γ 1 - γ(C0 + 2C1 + 2C2 + ...) 30v

The polarizability of a polar molecule has two components: an electronic polarizability, γe, due to the displacement of electrons, and an orientational polarizability, γd, due to the rotation of the permanent dipole in an electric field. In the vapor phase, assuming that the molecule can rotate freely, the orientational polarizability is given by the Langevin function. Therefore, for low effective electric fields (pEe , kT)

γ ) γe + γd = γe +

(A.6)

where p is the permanent dipole moment of the molecule. For vapors, the local field vanishes because of the spherical symmetry and eqs A.5 and A.6 provide good agreement with experiment. However, for liquids one can no longer use eq A.6 for the polarizability in the LorentzDebye model. Indeed, for liquid water, eq A.5 diverges for values of γ about 4 times smaller than the value of γ for its vapor, which at 300 K is γ ) γe + γd ) 32.3 × 10-40 C2 m2 J-1. One can regard eq A.5 as the definition of the molecular polarizability and calculate γ in terms of the macroscopic dielectric constant . The lower value of the polarizability in liquid than in vapor can be explained in the framework of the Lorentz-Debye model by the hindered rotation of the permanent dipole moment by the neighboring molecules in the condensed state. In contrast, the Onsager-Kirkwood model provides a polarizability in polar liquids larger than that in vapors.21 This is a result of the increase of the dipole moment by the strong electric field, which is generated when a molecule with a permanent dipole moment is introduced in a polarizable medium (Onsager), and the correlation between the orientations of neighboring molecules (Kirkwood).21 In the present paper, we will use for the molecular polarizability the following expression (obtained from eq A.5):

γ) (A.5)

(20) Jackson, J. D. Classical Electrodynamics; John Wiley & Sons: New York, 1975. (21) Frenkel. J. Kinetic theory of liquids; Clarendon Press: Oxford, 1946.

p2 3kT

0v (A.7) +2 + (C0 + 2C1 + 2C2 + ...)0v 3( - 1)

which provides a lower bound for γ in the absence of saturation. In a nonuniform field, we consider that m ) γEt, where Et is the total field acting on a molecule. LA010741T

Polarization Model and ion specificity: applications

8910

523

Langmuir 2002, 18, 8910-8918

Long Range Interactions between Apoferritin Molecules Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received June 25, 2002. In Final Form: September 11, 2002 Recent light scattering experiments regarding the apoferritin molecules (the hollow shells of the ironstorage protein ferritin) indicated a surprising dependence of the repulsion between proteins on the electrolyte concentration (NaCH3COO). The second virial coefficient decreased to a value close to that corresponding to hard spheres for 0.15 M but increased to a very large value at 0.25 M. The results are difficult to be interpreted in the classical framework through the addition of double layer and hydration repulsive forces. While the double layer theory can predict the behavior of the virial coefficient at low electrolyte concentrations, only an abnormally large charge can explain the values of the virial coefficient at high ionic strengths. Alternatively, the traditional hydration force should increase with orders of magnitude between 0.15 and 0.25 M, to be consistent with experiment, and this is unlikely to happen. In this paper, it is shown that a unitary treatment of the repulsion (the double layer and the polarization-based hydration repulsions) might explain the unexpected values of the second virial coefficient and the corresponding long-ranged repulsion.

I. Introduction The interaction between charged particles immersed in an electrolyte solution is traditionally described by the Derjaguin-Landau-Verwey-Overbeek theory in terms of an attraction, due to the correlations between the instantaneous electronic dipoles of the particles (van der Waals force), and a screened Coulomb interaction, due to the charges on the particles’ surfaces (the double layer force).1 It is known that the theory is accurate only in a certain range of electrolyte concentrations (1.0 × 10-3 to 5 × 10-2 M), and a number of improvements were proposed for the calculation of the interactions, such as, for example, the accounting of the retardation2 and the field-theoretical treatment for the van der Waals interactions3 and the accounting of the dielectric saturation at high fields,4 the image forces,5 finite ion sizes,6 and the correlation between ions7 for the double layer interactions. A long time ago,Voet prepared stable sols of various metals (Pt, Pd) and salts (sulfides, halides) in highly concentrated solutions of sulfuric acid, phosphoric acid, and calcium chloride in water.8 Dilution with water induced their coagulation. More recent experiments revealed that amphoteric latex particles did not coagulate, even at high ionic strengths (above 1 M) of LiNO3.9 All the above experiments contradict the DLVO theory. The stability of some colloids at high electrolyte concentrations * Corresponding author. E-mail address: feaeliru@ acsu.buffalo.edu. Phone: (716) 645-2911/2214. Fax: (716) 645-3822. (1) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (2) Casimir, H. B.; Polder, D. Phys. Rev. 1948, 73, 360. (3) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Adv. Phys. 1961, 10, 165. (4) Henderson, D.; Lozada-Casou, M. J. Colloid Interface Sci. 1986, 114, 180. (5) Jo¨nnson, B.; Wennerstro¨m, H. J. Chem. Soc., Faraday Trans. 1983, 79, 19. (6) Levine, S.; Bell, G. M. Discuss. Faraday Soc. 1966, 42, 69. Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612. (7) Wennerstro¨m, H.; Jo¨nnson, B.; Linse, P. J. Chem. Phys. 1982, 76, 4665. Kjellander, R.; Marcelja, S. J. Chem. Phys. 1985, 82, 2122. (8) Voet, A. Thesis, Amsterdam, 1935. See also: Kruyt, H. R. Colloid Science; Elsevier: Amsterdam, 1952. (9) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156.

led to the conjecture that another repulsion (a non-DLVO one), due to the organization of the solvent around particles, should also be present. The existence of such a force was demonstrated both experimentally and theoretically for any solvent,10 being particularly strong for the polar ones. The force (called hydration force when the solvent is water) is partly responsible for the stability of multilayers of neutral lipid bilayers/water, a system that was extensively investigated because of its relevance to biology.11 One of the first quantitative models12 of the hydration force was based on the polarization of water molecules in the vicinity of a surface. Another model involving polarization, suggested by Gruen and Marcelja,13 was shown later to have some inconsistencies.14 Because Monte Carlo simulations15 indicated that the average polarization of water molecules oscillates in the vicinity of a surface, both polarization models have been contested. However, it was recently shown16 that the Schiby and Ruckenstein model12 can lead to an oscillatory profile of the polarization, if the water is assumed to be organized in icelike layers in the vicinity of the surface. This model can relate the strength of the hydration force to the surface dipole density, a dependence which was observed experimentally,17 and could also explain the restabilization of some colloids at high ionic strengths.18 The hydration repulsion is expected to be affected by the hydrogen bonding. When two surfaces approach each other, the increase of the free energy (due to the disruption of the hydrogen bonds) generates repulsion. Indeed, Monte (10) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Suface Forces; Plenum: 1987. (11) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (12) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (13) Gruen, D. W. R.; Marcelja, S. Faraday Trans. 2 1983, 211 and 225. (14) Attard, P.; Wei, D. Q.; Patey, G. N. Chem. Phys. Lett. 1990, 172, 69. (15) Kjellander, R.; Marcelja, S. Chem. Scr. 1985, 25, 73; Chem. Phys. Lett. 1985, 120, 393. (16) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582. (17) Simon, S. A.; McIntosh, T. J. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 9263. (18) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7061.

10.1021/la026126m CCC: $22.00 © 2002 American Chemical Society Published on Web 10/15/2002

Nanodispersions

524

Interactions between Apoferritin Molecules

Langmuir, Vol. 18, No. 23, 2002

Carlo simulations15 indicated that a disruption of the hydrogen bonding occurs when two surfaces approach each other, and lattice models for the calculations of such interactions have been proposed.19 We consider, however, that an important contribution to the hydrogen bonding is already contained in the dipole-dipole interactions included in the polarization model. The restabilization of colloids at high ionic strengths observed experimentally20 (the coagulation rate having a maximum with increasing electrolyte concentration) could be explained in terms of the reassociation of charges on the interface and the formation of surface ion pairs (dipoles).18 While, at low ionic strengths, the repulsion decreases with increasing electrolyte concentration because of the electrostatic screening, at high ionic strengths, the reassociation increases the density of ion pairs on the surface, and this generates a strong repulsion. A restabilization of some colloidal systems at high ionic strength could be explained if the magnitude of the hydration force increases moderately with the addition of electrolyte.18 There are, however, some recent striking experimental results regarding the interaction between apoferritin molecules,21 which are more difficult to explain. Light scattering experiments21 on solutions of apoferritin molecules (the hollow shells of ferritin, an iron-storage globular protein) in an acetate buffer at pH ) 5.0 have indicated that the dimensionless second virial coefficient (the ratio between the second virial coefficient and the volume of the molecule) first decreases with increasing electrolyte concentration, becomes close to 4 (value corresponding to hard-core interactions alone) for an electrolyte concentration in the range 0.08-0.18 M, but increases markedly at higher ionic strengths, reaching a very large value (∼13) at 0.25 M. The dimensionless second virial coefficient is defined as

∫r)0r2(1 - exp(-

˜2 ) 3 B 2a3

∞

4+

3 2a3

F(r) kT

))

dr )

∫r)2ar2(1 - exp(∞

))

F(r) kT

dr (1)

where a ) 63.5 Å represents the radius of the spherical apoferritin molecules, r the distance between the centers of two particles, k the Boltzmann constant, T the absolute temperature, and F(r) the interaction potential between particles, which was assumed to be the sum between a hard core repulsion (F(r) ) ∞ for r < 2a) and the other interactions (double layer, hydration and van der Waals). The second virial coefficient is not very sensitive to the magnitude of the repulsion, as long as the interaction energy exceeds a few kT, but is extremely sensitive to the range of the interaction (the distance between the centers of the particles at which the interaction becomes comparable to kT). At low electrolyte concentrations, the experimental results can be explained within the DLVO theory. A wellknown approximation for the double layer interaction between weakly charged spheres, at constant surface charge, is1 (19) Attard, P.; Batchelor, M. T. Chem. Phys. Lett. 1988, 149, 206. (20) Molina-Bolivar, J. A.; Galisteo-Gonzalez, F.; Hidalgo-Alvarez, R. Phys. Rev. E 1997, 55, 4522. (21) Petsev, D. N.; Vekilov, P. G. Phys. Rev. Lett. 2000, 84, 1334.

(

r - 2a λD a 2 4π0 1 + r λD

(ne)2 exp FDL(r) )

(

)

)

8911

(2)

where n is the number of charges on each protein, e the protonic charge, 0 the vacuum permittivity,  the dielectric constant, and λD the Debye-Hu¨ckel length (λD ) (0kT/ 2e2CE)1/2, CE being the concentration of the 1:1 electrolyte). Using eqs 1 and 2, the experimental virial coefficient for CE ) 0.01 M could be obtained (neglecting the van der Waals interactions) by assuming n ) -38. This value is reasonable, since the isoelectric point of apoferritin is about 4.0 and the experiments were performed at pH ) 5.0.21 This is a consequence of the long-ranged double layer repulsion at low electrolyte concentrations. However, at 0.25 M the Debye-Hu¨ckel length is about 6.2 Å and the value B ˜ 2 ) 13 can be obtained only if the charge of each molecule would be of the order of |103e|, which is most unlikely. Can the conventional hydration force be responsible for this large value of B ˜ 2? If one assumes an exponential form (a commonly used approximation)11 for the hydration interaction between two planar surfaces separated by a distance D

( Dλ )

F planar (D) ) A exp H

(3a)

where λ is the hydration decay length and A is a preexponential constant, the interaction between two spheres of radius a becomes in the Derjaguin approximation

FH(r) ) πa

r - 2a (D) dD ) πaλA exp(∫r-2aF planar H λ ) ∞

(3b) For a decay length of 14.4 Å (as selected in ref 21), a ) 63.5 Å and T ) 300 K, a value B ˜ 2 ∼ 13 could be obtained for A ) 5 × 10-3 J/m2. These values of λ and A are, however, too large, since the values for the hydration decay length reported recently are closer to 2 Å.11 Assuming that λ ) 14.4, independent of the electrolyte concentration, the value of the parameter A has to decrease by 2 orders of magnitude to provide the low value B ˜ 2 ∼ 4.5 (close to the value corresponding to the hard-core repulsion). As noted above, a small increase of A by a few percent with increasing electrolyte concentration is not unexpected (due to the charge recombination), but what mechanism could explain the 102 fold increase of A when the ionic strength changes from 0.15 to 0.25 M? If one assumes a decay length λ ) 2 Å and a preexponential factor A ) 0.05 J/m2 (which are typical values for lipid bilayers),11 one obtains B ˜ 2 ∼ 4.9. However, the preexponential factor must increase by more than 11 orders of magnitude to provide B ˜ 2 ∼ 13, which is, of course, unreasonable. To summarize the difficulties, if the hydration repulsion would be responsible for the long-ranged repulsion of apoferritin protein at high ionic strengths (0.25 M), a theory of the hydration force should be able to predict a large decay length at large separations and explain why such a repulsion was not observed in the interactions between two phospholipid bilayers. Second, it should also explain why a repulsion with a much shorter decay length (about 2 Å) has been typically determined in the latter experiments. Third (and more importantly), it should

Polarization Model and ion specificity: applications

8912

525

Langmuir, Vol. 18, No. 23, 2002

explain why there is such a strong increase of the repulsion when the electrolyte concentration changes from 0.15 to 0.25 M. As already noted, the restabilization of some colloids at high ionic strength found in previous experiments20 can be explained in the traditional framework of the additivity between double layer and hydration forces, by a slight increase of the hydration repulsion caused by the increase in surface ion pair (dipole) density with electrolyte concentration. However, the increase in repulsion due to this mechanism is much too low to explain the strong increase of the second virial coefficient. To have an idea about the range of the repulsion required to provide such a high virial coefficient, it should be noted that, if the hard-core repulsion, infinite in magnitude, is extended with 15 Å (above the 2a separation), B ˜ 2 increases from 4 to only 5.6. If the range of the hard-core repulsion is extended with 30 Å, B ˜ 2 increases to 7.55, while 60 Å leads to 12.8. From these simple estimations one can infer that the repulsion needed to explain the measured second virial coefficient for apoferritin molecules should have a much longer range than that typically observed for the traditional hydration force. A new theory for the hydration force was proposed recently by Paunov et al. and used to explain the existence of a minimum of B ˜ 2 with increasing electrolyte concentration.22 However, the predicted interactions vanish for distances that exceed two hydration diameters (about 14 Å), while the hydration force necessary to explain the high values of B ˜ 2 determined experimentally requires a much longer range. The purpose of this article is to show that a recent model,23 which provided a unitary treatment of the double layer and hydration forces, can satisfy all the above requirements and might explain the high values of the second virial coefficient of apoferritin at high ionic concentrations. The total repulsive force is described in this model by two equations for the electrical potential ψ and the average dipole moment m of a water molecule. The latter is no longer assumed to be proportional to the macroscopic electric field, as in the traditional theory, but depends also on the field generated by the neighboring dipoles. In the linear approximation, the solutions for both ψ and m are linear superpositions of functions with two distinct decay lengths. At low electrolyte concentrations, one of the decay lengths is close to the Debye-Hu¨ckel length, which is characteristic for the traditional double layer repulsion, and the other is close to a length characteristic for the traditional hydration repulsion. However, at high electrolyte concentrations, the two decay lengths characteristic of the system are markedly different from the latter two, with one of them being above 14.9 Å at any ionic strength and the other being small, below 1.67 Å. The large length is responsible for a long-ranged repulsion. At small separation distances, the terms with a shorter decay length become dominant, in agreement with the experiments on the hydration force between neutral lipid bilayers. The most important feature of this model consists of the nonadditivity of the hydration and double layer repulsions. The “double layer-like” repulsion is generated by the charge on the surface, while the “hydration-like” repulsion is generated by the surface ion pairs (dipoles), which are formed through the reassociation of the surface charges with counterions. These dipoles orient the neighboring water molecules in a direction opposite to that (22) Paunov, V. N.; Kaler, E. W.; Sandler, S. I.; Petsev, D. N. J. Colloid Interface Sci. 2001, 240, 640.

Manciu and Ruckenstein

produced by the electric field generated by the surface charge (due to the dissociated groups), because they expose an opposite charge to the water molecules. The increase of the electrolyte concentration in the acetate buffer increases the amount of Na+ ions adsorbed on the acidic sites of the surface of apoferritin, and this lowers the surface charge but enhances the surface dipole density. At a particular concentration, the electric fields generated by the surface charge and the surface dipole densities in the neighboring water molecules compensate each other to a high extent and the repulsion passes through a deep minimum. At low ionic strengths, the charges dominate the interaction, while, at high ionic strengths, the surface dipoles dominate. It will be shown below that this approach can explain the strong variation of the second virial coefficient over a relatively narrow range of electrolyte concentrations. II. Theoretical Framework II.A. Surface Association-Dissociation Equilibria. As already noted, we suggest that the behavior of the second virial coefficient of the apoferritin in acetate buffer is due to the adsorption of Na+ ions upon the negative sites of the protein surface, which depends on the concentration of the Na+ ions in the liquid in the vicinity of the surface. In what follows, the adsorption of acetate ions upon the positive sites or of neutral Na+-CH3COOpairs on the neutral sites of the protein surface will be neglected and it will be assumed that only the dipoles of the ion pairs formed through the association of Na+ to the acidic sites of the surface polarize the neighboring water molecules. The equilibrium constants for the association-dissociation of H+, Na+, and OH- to the apoferritin acidic and basic groups, respectively, are not precisely known. We will try to estimate them as follows: The apoferritin protein has NA ) 624 acidic and NB ) 576 basic amino acid residues on its surface and an isoelectric point of ∼4.0.24 The dissociation equilibria of the acidic and basic sites, at negligible electrolyte concentrations, are25

xNA a (1 - x)NA + [H+]S

(4a)

yNB a (1 - y)NB + [OH-]S

(4b)

where (1 - x) and (1 - y) represent the fractions of the dissociated acidic and basic sites and [H+]S and [OH-]S represent the concentrations of hydrogen and hydroxyl ions, respectively, in the liquid in the vicinity of the surface. At equilibrium one can write

(1 - x)[H+]S KH ) x

(5a)

and

KOH )

(1 - y)[OH-]S y

(5b)

where KH and KOH are the dissociation constants of the acidic and basic sites, respectively. The total charge q on the surface, obtained using eqs 5a and b, is given by (23) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584. (24) Petsev, D. N.; Thomas, B. R.; Yau, S.-T.; Vekilov, P. G. Biophys. J. 2000, 78, 2060. (25) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205.

Nanodispersions

526

Interactions between Apoferritin Molecules

Langmuir, Vol. 18, No. 23, 2002

(

8913

)

q ) -e((1 - x)NA - (1 - y)NB) ) NA NB e - + + (6) [H ]S [OH-]S +1 +1 KH KOH Since at the isoelectric point q ) 0, the surface potential ψS ) 0 and the concentrations of H+ and OH- in the liquid are the same in the vicinity of the surface and in the bulk. In this case, eq 6 leads to a relation between the dissociation constants:

KOH )

10-10 mol/L NB 10-4 NA - NB NA KH NA

(7)

Because the dissociation constants must be positive, the condition KOH > 0 in eq 7 provides an upper bound for KH:

KH <

NB 10-4 ) 1.2 × 10-3 mol/L NA - NB

(8)

One can determine the dissociation constants if the values of the charge are known at two distinct pH values. Unfortunately, only the value q ) 0 at pH ) 4.0 (the isoelectric point) is known. At low ionic strength, the association-dissociation equilibria of H+ and OH- are only slightly affected by the presence of the electrolyte. However, when the sodium acetate concentration increases, Na+ ions compete with the H+ ions for adsorption on the acidic sites (the adsorption of the acetate ion on the basic sites is neglected). The corresponding equilibria are

xNA a (1 - x - w)NA + [H+]S

(9a)

yNB a (1 - y)NB + [OH-]S

(9b)

wNA a (1 - x - w)NA + [Na+]S

(9c)

where wNA is the fraction of acidic sites upon which Na atoms are adsorbed and xNA is the fraction of associated acidic sites. After some algebra, one obtains25

(

q)e -

NA +

+

+

[H ]S [Na ]S + +1 KH KNa

NB

)

[OH-]S +1 KOH

(10)

where KNa is the dissociation constant of sodium. The concentrations of the ions in liquid near the surface, [Na+]S, [H+]S, and [OH-]S, are related to those in the bulk via

(

[Na+]S ) [Na+] exp -

(

[H+]S ) [H+] exp -

)

eψS kT

) ( )

[OH-]S ) [OH-] exp

(11a)

eψS kT

(11b)

eψS kT

(11c)

Let us first investigate whether the traditional double layer theory can explain the behavior of the repulsion between apoferritin molecules. For low surface potentials, the potential around a single spherical particle of radius

Figure 1. Total charge on apoferritin as a function of electrolyte concentration, for three pairs of equilibrium constants (KH - KOH) compatible with the isoelectric point of the apoferritin. The equilibrium constant KNa was varied in the range 0.04-1.0 M.

a in an electrolyte is given in the Debye-Hu¨ckel approximation by26

(

)

a r′ - a ψ(r′) ) ψS exp r′ λD

(12)

where r′ > a is the distance from the center of the particle and ψS is the surface potential. The surface potential is related to the surface charge density through the expression

(

∂ψ(r′) q 1 1 )| ) ψS + ∂r r′)a a λD 4πa20

)

(13)

Using eqs 7, 10, 11, and 13, one can determine the charge on an apoferritin molecule at any pH, provided that the values of the dissociation constants KH and KNa are known. Further, using eqs 1 and 2, one can evaluate the second virial coefficient for the interaction between two particles at constant surface charge. The dissociation constant of sodium (from the acidic sites of apoferritin) is not accurately known; a reasonable value might be 0.2 M, which is compatible with the values for the Na+ and an amino acid25 (see Figure 3 of ref 25). The charge of apoferritin molecules is plotted in Figure 1 as a function of electrolyte concentration, for three pairs of values of KH and KOH compatible with the isoelectric point (eq 7) and for KNa ) 1.0, 0.2 and 0.04 M. In some cases, the molecule, which is negatively charged at very low electrolyte concentrations, becomes positively charged at higher ionic strength, because of the adsorption of Na+ ions. The change of the charge sign might constitute an appealing explanation for the abnormal behavior of the second virial coefficient. Indeed, the double layer repulsion first decreases with the electrolyte concentration, vanishes at a concentration corresponding to neutral molecules, and then increases when the charge becomes positive. However, the charge at 0.25 M is much smaller than that required to reach B ˜ 2 ) 13. Also, a large final positive charge implies that the molecules become neutral at concentrations less than 0.01 M (see Figure 1) and not at about 0.10-0.15 M, the location of the minimum determined (26) Hunter, R. J. Foundations of Colloid Science; Oxford Science Publications: 1987.

Polarization Model and ion specificity: applications

8914

527

Langmuir, Vol. 18, No. 23, 2002

Manciu and Ruckenstein

experimentally. We performed nondenaturating electrophoretic measurements of apoferritin in an acetate buffer, and the results indicated that the molecules do not become positively charged at any acetate concentration tested (up to 0.25 M). Consequently, the large repulsion at high ionic strength cannot be due to a large positive charge acquired by the apoferritin molecules. II.B. Electrostatic Repulsion between Molecules. In the traditional double layer theory, the polarization is assumed to be proportional to the macroscopic electric field. However, this assumption is valid only when the field varies sufficiently slowly with the distance. When the variation is stronger, the interactions between neighboring dipoles become important and have to be accounted for. In addition, the dipoles present on a surface generate an electric field, which polarizes the neighboring water molecules, which in turn generate electric fields, and so on. The overlap of the polarization fields generates a repulsion when two surfaces approach each other. When both dipoles and charges are present on the surface, the traditional double layer and hydration repulsions are no longer independent, because both depend on the polarization. A unitary treatment of the interaction was presented recently23 and will only be summarized here. The average polarization of a water molecule, m(z), between two identical, charged parallel plates separated by a distance 2d is related to the macroscopic electric field, E(z), and to the local field produced by the neighboring dipoles via23

m(z) ) 0v0( - 1)E(z) + 0v0( - 1)C1∆2

∂2m(z) ∂z2

(14)

where 0 is the vacuum permittivity,  is the bulk dielectric constant, v0 is the volume occupied by one water molecule, ∆ is the distance between the centers of two adjacent icelike layers, and C1 ) 1.827/4π0′′l3 is an interaction coefficient, with ′′ denoting the dielectric constant for the interaction between neighboring water molecules and l the distance between the centers of two adjacent water molecules. Equation 14, coupled with the Poisson equation

∂2ψ(z) ∂z2

where z is measured from the middle distance between the parallel plates. The constants a j 1 and a j 2 are related to a1 and a2 through

a j 1 ) a10v0λ1 a j 2 ) a20v0λ2

( )

(15)

where the first term on the right side was obtained by assuming Boltzmann distributions for the ions, constitutes a system of differential equations for m(z) and ψ(z). For small surface potentials, hence in the linear approximation, the above system of equations becomes 2

∂ ψ(z) 2

∂z

∂2m(z) λm2 2 ∂z

)

 1 ∂m(z) ψ+ 2  ∂z λD 0v0

∂ψ(z) ) m(z) + 0v0( - 1) ∂z

λ1,2 )

(

(λ

2 D

account for the symmetry of the system, can be written for ψ and m:

() ()

() ()

z z + a2 cosh λ1 λ2

(17a)

z z m(z) ) a j 1 sinh +a j 2 sinh λ1 λ2

(17b)

ψ(z) ) a1 cosh

(18a)

(18b)

(

2 

(

+ λm2 ( λD4 + λm4 + 2λD2λm2 1 -

1/2

)) )

2

)

1/2

(19)

()

()

λD2 d d + a2λ2 sinh ) σ λ1 λ2 0

(20a)

while the average polarization of the first water layer from the surface is given by

[ (

( )

(

)(

1  - 2 (1 - 0v0( - 1)C0) × 2 λ1 λD d d-∆ sinh - 0v0( - 1)C1 sinh + λ1 λ1 0v0( - 1) d 1  sinh + a2 λ20v0 2 - 2 × λ1 λ1 λ2 λD

a1 λ10v0

( )]

(

(

))

d-∆ λ2

))

[ (

( )

)

d - 0v0( - 1)C1 × λ2 0v0( - 1) d + sinh ) λ2 λ2

(1 - 0v0( - 1)C0) sinh sinh -

where λm2 ≡ 0v0( - 1)C1∆2. The following solutions, which

1  - 2 2 λ2 λD

) )

At low ionic strengths, λ1 = λD and λ2 = λH ) λm/x, which indicates that the double layer is not affected much by the local interaction due to neighboring dipoles. The presence of a surface dipole density can either increase or decrease the repulsion, depending on the dipole’s orientation. Selecting v0 ) 30 Å3,  ) 80, and ′′ ) 1 for the dielectric constant for the interaction between neighboring water molecules, l ) 2.76 Å for the distance between the centers of two adjacent water molecules, and ∆ ) 3.68 Å,16 one obtains λH ) 1.67 Å,23 which is in agreement with the value determined experimentally for neutral lipid bilayers in water.11 However, at high ionic strength, λ1 and λ2 are markedly different from λD and λH, respectively. For CE f ∞ (λD f 0), λ1 f λm ) 14.89 Å, which indicates the presence in the system of an interaction with a much longer range than those of the traditional hydration or double layer. To obtain the solution of the system of eqs 16, two boundary conditions are needed. The overall neutrality leads to the following relation between potential and surface charge density σ:

(16a)

(16b)

1  λ12 λD2

and λ1 and λ2 are the characteristic lengths of the interaction, given by

a1λ1 sinh 2ecE eψ 1 ∂m(z) ) sinh + 0 kT 0v0 ∂z

( (

()

( )]

p v0( - 1) 1 ′ 2π ANa + ∆′2 π

(

)

3/2

(20b)

where p is the dipole moment normal to the surface of a surface dipole, ANa is the area corresponding to one surface dipole, ′ is the dielectric constant for the interaction between a dipole and the adjacent water molecules, and

Nanodispersions

528

Interactions between Apoferritin Molecules

Langmuir, Vol. 18, No. 23, 2002

8915

∆′ is the distance between the center of a dipole and the center of the first water layer. The interaction coefficients C0 and C1 are given by C0 ) -3.766/4π0′′l3 andC1 ) 1.827/ 4π0′′l3, where l is the distance between the centers of two adjacent water molecules in ice.16 We will assume, as before,23 that the free energy of the surface layer formed by the surface dipoles and the water molecules between them is independent of the distance between the two plates. Consequently, the free interaction energy, per unit area, of two planar plates separated by a distance D ) 2d is composed of a chemical term Fch, an entropic term due to the mobile counterions, and a term due to the electrostatic fields:23

Fflat ) Fch +

(

0

∫-d λ

1 2

d

(ψ)2 + 2

D

[(

0

])

)

2 ∂ψ m ∂ψ C1∆ ∂2m m 2 ∂z v0 ∂z v0 ∂z

dz (21)

For interaction at constant surface charge, Fch ≡ 0, since the (fixed) surface charge is not in thermodynamic equilibrium with the medium, while, at constant surface potential, Fch ) -2σψS, where ψS is the surface potential.1 When neither σ nor ψS is constant, the change in chemical free energy, per unit area, from infinite separation to a distance D ) 2d between plates is given by

Fch(D) - Fch(∞) ) -2

∫∞ ψS(η) dσ(η) D

(22)

The latter expression can be derived by decomposing the trajectory of ψS versus σ in a succession of small changes at constant σ followed by changes at constant ψS. At constant σ, the change in chemical energy is zero, while, at constant surface potential, the change in chemical energy is -2ψS dσ. The interaction between two identical spherical particles of radius a, separated by a distance r between their centers, is calculated using a variant of the Derjaguin approximation (see Appendix)

Fsph(r) )

∫r-2a(F(D) - F(∞))(r - D) dD

π 2

r

(23)

where the concentration of ions in liquid in the vicinity of the surface (eqs 11) is calculated using the renormalized surface potential ψS′ (see Appendix)

ψS′ )

(

ψS λD 1+ a

)

(24)

with ψS being the surface potential for planar surfaces calculated from the system (eq 16) subjected to the boundary conditions (eq 20). II.C. van der Waals Attraction. The van der Waals attraction between the hollow shells of the ferritin can be easily computed in the pairwise summation approximation. Let us consider a large sphere B made up of a small sphere b and a spherical shell S. The interaction free energy FBB between two large spheres can be written (assuming pairwise interactions) as

FBB ) F(b+S)(b+S) ) Fbb + FbS + FSb + FSS

(25)

The interaction between a large sphere B and a small one b can be separated into

FBb ) F(b+S)b ) Fbb + FbS

(26)

Figure 2. Ratio between van der Waals attraction of compact and hollow spheres of external radius a ) 63.5 Å, at various distances, as a function of the thickness of the shell.

Using eqs 25 and 26, one can calculate the interaction between two spherical shells

FSS ) FBB + Fbb - 2FBb

(27)

by taking into account that the van der Waals attraction between two spheres of radii a1 and a2, separated by a distance r between their centers, is given by26

FvdW ) -

Ha1a2 6(r - (a1 + a2))(a1 + a2) 1

1+

(

2a1a2

(

(r - (a1 + a2))(a1 + a2)

)

[

+

1+

1 + r - (a1 + a2) 2(a1 + a2)

r - (a1 + a2)

+

2(a1 + a2)

(r - (a1 + a2))(a1 + a2) (r - (a1 + a2))(a1 + a2) ln × a1a2 2a1a2

(

1+

r - (a1 + a2) 2(a1 + a2)

)]

(r - (a1 + a2))(a1 + a2) (r - (a1 + a2))2 1+ + 2a1a2 4a1a2

(28)

where H is the Hamaker constant. In Figure 2, the ratio between the van der Waals interactions of hollow and compact spheres of radius a ) 63.5 Å is plotted, at various distances of closest approach (r - 2a), as a function of the shell thickness. While the outer radius of the apoferritin molecule (a ) 63.5 Å) can be accurately determined, for example by dynamic light scattering,21 the inner radius (of the hollow shell) is more difficult to estimate. The iron cores of the ferritin have a maximum diameter of the order of 80 Å, while the six hydrophobic channels have a length of about 12 Å.27 In what follows the value 20 Å will be employed for the average thickness of the shell. The Hamaker (27) Ford, G. C.; Harrison, P. M.; Rice, D. W.; Smith, J. M. A.; Treffry, A.; White, J. L.; Yariv, J. Philos. Trans. R. Soc. London 1984, B304, 551.

Polarization Model and ion specificity: applications

8916

Langmuir, Vol. 18, No. 23, 2002

529

Manciu and Ruckenstein

constant of the apoferritin is not known; the Hamaker constants reported for proteins cover a rather large range, from 0.06kT for R-crystallin to about 10kT for R-chymotrypsin and bovine serum albumin.28 In the present calculations, the value H ) 5.0kT will be used. III. Second Virial Coefficient of Apoferritin Molecules In what follows, the model presented will be employed to calculate the dependence of the dimensionless second virial coefficient of the apoferritin molecules. Assuming that the surface dipolar density is generated only by the Na+ ions adsorbed on the surface, the area corresponding to an ion is ANa ) (4πa2/wNA). As already emphasized, one of the characteristic decay lengths of the system is larger than λm ) 14.9 Å at any electrolyte concentration; hence, a relatively long-ranged interaction is always present in the system, even at high ionic strengths. The magnitude of this interaction per unit area depends on the boundary conditions and is generally small. Apoferritin is, however, a large protein, and since in the Derjaguin approximation the interaction is proportional to the radius, the interaction is of the order of kT at large separations. Another particularity of the apoferritin is its hollow nature, which implies a relatively small van der Waals attraction. For other proteins, the attraction might overcome the weak long-ranged repulsion, leading to a small second virial coefficient at high ionic strengths. The electrostatic repulsion is calculated by combining eqs 10 and 11 for the charge (using the renormalized surface potential given by eq 24) with the system of eqs 16a and b solved with the boundary conditions (eqs 20a and b). For all separation distances D ) 2d a solution was obtained for planar surfaces by successive approximations and the corresponding free energy per unit area was calculated using eqs 21 and 22. The free energy of the repulsive interaction between two spherical apoferritin molecules was calculated using eq 23, and the van der Waals interaction was obtained from eqs 27 and 28. Figure 3a presents the interaction free energy (in kT units) as a function of the closest approach distance r 2a for various electrolyte concentrations. The following values have been employed for the parameters: KH ) 1 × 10-4 M, KOH ) 1.18 × 10-10 M, KNa ) 0.4 M, T ) 300 K,  ) 80, ′′ ) 1, (p/′) ) 5 D, ∆′ ) 1.5 Å, l ) 2.76 Å, v0 ) 30 Å3, H ) 5kT, and 20 Å for the thickness of the ferritin shell. The repulsion first decreases with increasing electrolyte concentration and then attains a minimum around 0.1 M, after which it increases. This behavior is a result of the adsorption of Na+ ions on the acidic sites. While the total charge of the protein decreases with increasing ionic strength, the strong dipole moment of the adsorbed Na+ orients the water molecules and generates a repulsion. The repulsive free energy (without the van der Waals attraction) is plotted in Figure 3b for various separations against the electrolyte concentration. The repulsion has a strong minimum at an ionic strength of about 0.08 M. The reason for this unexpected behavior is the nonadditivity of the double layer and hydration repulsions.23 When the effect of the charge is dominant, the formation of the surface dipoles decreases the free energy, because the surface dipoles and the surface charges orient the neighboring water molecules in opposite directions. Similarly, when the effect of the surface dipoles becomes dominant, the recombination of charges increases the repulsion, by (28) Broide, M. L.; Tominc, T. M.; Saxowsky, M. D. Phys. Rev. E 1996, 53, 6325.

Figure 3. (a) Interaction energy between two apoferritin molecules (in kT units) as a function of the distance of closest approach, for various electrolyte concentrations. (b) Repulsive interaction as a function of electrolyte concentration, at various distances of closest approach between particles.

increasing the number of surface dipoles and decreasing the charge. For the values of the parameters employed, about 64% of the acidic sites are occupied by Na+ ions at 0.25 M, a value which is compatible with the increase of the apoferritin mass determined by light scattering.22 For the values of the parameters employed (a relatively large Hamaker constant), the potential barrier is only a few kT or less; hence, the apoferritin should coagulate at almost all the concentrations studied. Since experiment shows that the proteins did not coagulate, another repulsion should be present, at least al low separation distances. This repulsion, while essential for the stability of the system, did not affect much, because of its short range, the behavior of the second virial coefficient. In the calculation of the second virial coefficient, it was assumed that the distance of closest approach between apoferritin proteins cannot be less than 8 Å. This value leads to a dimensionless second virial coefficient for the hard spheres repulsion of 4.8 instead of 4. The dimensionless second virial coefficient calculated from eq 1 for various values of (p/′) is compared in Figure 4 with the experimental results (circles) of ref 21. There is a minimum in the repulsion at about 0.1 M. At higher ionic strengths, the accumulation of Na+ ions via association with the acidic sites of the surface increases the surface dipole density and decreases the surface charge, leading to a higher repulsion. The value of (p/′) ) 5.5 D, which provides a good agreement between the theory and

Nanodispersions

530

Interactions between Apoferritin Molecules

Langmuir, Vol. 18, No. 23, 2002

8917

Figure 5. Geometry of two interacting spherical particles.

Figure 4. Dimensionless virial coefficient of the interaction, calculated as a function of the electrolyte concentration for various (p/′) ratios, compared with the experimental results of ref 22 (circles). The crosses represent the experimental results when the partial dissociation of the sodium acetate is taken into account.

experiment, is reasonable, since the water dipole is about 1.85 D and the dipole formed by two elementary charges separated by 1 Å is about 5 D. Good agreement with experiment can be obtained when the equilibrium constants KH and KOH are varied in a rather large range (over 1 order of magnitude), if KNa and (p/′) are suitably chosen. Small changes in the values of the Hamaker constant, shell thickness, or cutoff distance for the virial integration do not affect essentially the behavior of the second virial coefficient. While the values of the parameters chosen are reasonable, the results presented in this article should be considered as qualitative only, for the reasons outlined below. The results presented in ref 21 are, to our knowledge, the only experiments with 1:1 electrolytes in which such a strong increase of the repulsion with ionic strength was observed. If, for CE ) 0.25 M, one would disregard the last experimental point from the Debye plot, which provided the second virial coefficient (Figure 1b in ref 21), the corresponding B ˜ 2 would be about 6 instead of 13. In addition, the sodium acetate in the buffer is not completely dissociated. In water, sodium acetate forms both inner and outer sphere complexes. In the former, the ions are directly bound, while in the latter, Na+ and CH3COOions are separated by one or more water molecules. Raman spectroscopy identifies only the inner complexes as associated species, while potentiometry identifies both inner and outer complexes as associated species.29 Using the dissociation constants provided by ref 29, one can infer that, at room temperature, the inner sphere complexes are almost completely dissociated for the whole range of electrolyte concentrations (0.01-0.25 M) investigated here. However, some of the Na+ and CH3COO- remain bound in neutral pairs (outer sphere complexes) and do not participate as free charges to the electrostatic screening; hence, the concentration of truly dissociated Na+ (CE) ions differs slightly from the NaCH3COO buffer concentration. The results of our estimations for the truly dissociated Na+ are presented as crosses in Figure 4. In addition, the large dipoles of the outer sphere complexes can affect the interactions, particularly when adsorbed on the surface of apoferritin. The simplified (29) Fournier, P.; Oelkers, E. H.; Gout, R.; Pokrovski, G. Chem. Geol. 1998, 151, 69.

model presented here has neglected the influence of the dipoles of H+ and OH- ions associated with the surface and the effect of the sodium acetate neutral pairs adsorbed on the surface. It should be again emphasized that the Derjaguin approximation is not accurate at large separations, particularly at low electrolyte concentrations, when the Debye-Hu¨ckel length is not sufficiently small compared to the protein radius. At very low ionic strengths (0.01 M), the surface potential is large and the linear approximation employed here is not accurate. The main point of the article is that the outlined theory can explain the strong increase in repulsion with increasing electrolyte concentration. This is a result of the nonadditivity of the repulsions generated by the surface charge and surface dipole densities. IV. Conclusions Recent light scattering experiments on apoferritin proteins in an acetate buffer at pH 5.0 revealed an unexpected behavior of the second virial coefficient, which passed through a minimum when the electrolyte concentration was about 0.15 M but increased to a high value at a concentration of 0.25 M. The results are incompatible with the traditional theoretical framework, which assumes that the total repulsion is the sum between a “double layer repulsion” due to the charges on the interface and a “hydration repulsion” due to the structuring of the solvent near the surface. The results could have been explained by the traditional double layer theory, if the apoferritin would have acquired an unexpectedly large positive charge at 0.25 M. This possibility was, however, ruled out by nondenaturating electrophoretic experiments. The result could also have been explained by postulating that the hydration provides a very long range exponential repulsion, with a decay length about 10 times larger than what is generally accepted. Even in this case, it would still have been difficult to explain why the repulsion increased by orders of magnitude from a concentration of 0.15 M to 0.25 M. In this paper, we presented an alternate explanation for the unexpected behavior of the virial coefficient, based on our previous unitary treatment of the electrostatic repulsion between particles immersed in a polar solvent, which combined the double layer and the hydration forces into a single repulsive force.23 The main features of the model are the existence of two characteristic decay lengths for the interaction (a small one, important at small separations, and a large one, which is relevant at large separations) and the nonadditivity of the effects of surface charges and surface dipoles, which in the present case tend to orient the water molecules neighboring the surface in opposite directions. The number of Na+ ions adsorbed on the surface increases with the electrolyte concentration, decreasing the surface charge but increasing the surface dipole density. At low electrolyte concentrations, the charge effect is dominant and the interaction is well approximated by the traditional DLVO theory. At high ionic strength, the surface dipoles are mainly responsible

Polarization Model and ion specificity: applications

8918

531

Langmuir, Vol. 18, No. 23, 2002

Manciu and Ruckenstein

for the repulsion. At intermediate concentrations, the opposite fields generated by the charges and the surface dipoles in the vicinity of the surface compensate each other to some extent, and consequently the repulsion has a strong minimum. Acknowledgment. We are indebted to Prof. M. Ettinger for performing the nondenaturating electrophoretic measurements on apoferritin and to Prof. G. Frens for drawing our attention to the experimental results discussed in ref 8. Appendix The interactions between spherical particles can be calculated, in the Derjaguin approximation, in terms of the interactions between planar surfaces, by decomposing the spherical surface in small areas (which can be considered locally planar). The free energy of interaction, corresponding to each small piece, is assumed to be equal to the product between the area of the piece and the interaction free energy per unit area of some virtual infinite planar surfaces, located at the same separation distance D (see Figure 5). The total free energy of interaction between identical spherical particles of radius a is the integral over the surface of one particle:26

Fsph(r) )

∫0 (Fflat(D) - Fflat(∞))2πy dy ≈ ∞ πa∫r-2a(Fflat(D) - Fflat(∞)) dD

radii (compared to both the Debye-Hu¨ckel length and the distance of closest approach, r - 2a). Because (see Figure 5) y2 ) a2 - [(r - D)/2]2 w 2y dy ) ((r - D)/2) dD, eq A.1 becomes

Fsph(r) )

with r being the distance between the centers of the spheres. The approximation is accurate for large particle

∫r -2a(Fflat(D) - Fflat(∞))(r - D) dD r

(A2)

Another difficulty in using the Derjaguin approximation arises when the charge is related to the surface potential via various ionic equilibria. Indeed, in the linear approximation, the surface potential is related to the surface charge density, σ, of a single planar surface by

∂ψflat(z) ψS,flat σ )|z)0 ) 0 ∂z λD

(A3)

while for a spherical particle (eq 13)

(

)

σ 1 1 ) ψS,sph + 0 a λD

(A4)

To account for the effect of the radius of the particles, an approximate renormalized surface potential ψS′

ψS′ )

a

(A1)

π 2

(

ψS,flat λD 1+ a

)

(A5)

will be used to calculate the ion density in the vicinity of the surface. Both corrections (in eqs A2 and A5) become negligible at high ionic strengths, since then (λD/a) , 1. LA026126M

Nanodispersions

532

Langmuir 2002, 18, 2727-2736

2727

On the Stability of the Common and Newton Black Films Eli Ruckenstein* and Marian Manciu Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received October 17, 2001. In Final Form: January 17, 2002

The effect of electrolyte concentration on the transition from common to Newton black films and the stability of both types of films are explained using a model in which the interaction energy for films with planar interfaces is obtained by adding to the classical DLVO forces the hydration force. The theory takes into account the reassociation of the charges of the interface with the counterions as the electrolyte concentration increases and their replacements by ion pairs. This affects both the double layer repulsion, because the charge on the interface is decreased, and the hydration repulsion, because the ion pair density is increased by increasing the ionic strength. The theory also accounts for the thermal fluctuations of the two interfaces. Each of the two interfaces is considered as formed of small planar surfaces with a Boltzmannian distribution of the interdistances across the liquid film. The area of the small planar surfaces is calculated on the basis of a harmonic approximation of the interaction potential. It is shown that the fluctuations decrease the stability of both kinds of black films.

I. Introduction It is well-known that free films of water stabilized by surfactants can exist as somewhat thicker primary films, or common black films, and thinner secondary films, or Newton black films. The thickness of the former decreases sharply upon addition of electrolyte, and for this reason its stability was attributed to the balance between the electrostatic double-layer repulsion and the van der Waals attraction. A decrease in its stability leads either to film rupture or to an abrupt thinning to a Newton black film, which consists of two surfactant monolayers separated by a very thin layer of water. The thickness of the Newton black film is almost independent of the concentration of electrolyte; this suggests that another repulsive force than the double layer is involved in its stability. This repulsion is the result of the structuring of water in the vicinity of the surface. Extensive experimental measurements of the separation distance between neutral lipid bilayers in water as a function of applied pressure1 indicated that the hydration force has an exponential behavior, with a decay length between 1.5 and 3 Å, and a preexponential factor that varies in a rather large range. The interactions between the surfactant monolayers of the black film consist of a long-range van der Waals attraction, a short range repulsion due to the hydration force, and an electrostatic repulsion with a longer range than the hydration repulsion but shorter than the van der Waals attraction. Their combination can lead, in principle, to an interaction energy possessing three minima: one at a separation distance d f 0, which implies the rupture of the film, another one at a relatively large distance, which corresponds to the common black film, and finally a third one at an intermediate separation distance, which corresponds to the Newton black film. Only the minimum at d f 0 is stable; the other two, while metastable, can have relatively long lifetimes. * To whom correspondence may be addressed. E-mail address: [email protected]. Phone: (716) 645-2911/ 2214. Fax: (716) 645-3822. (1) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351.

Experiments on the stability of water/surfactant films at various pressures were performed by Exerowa et al.2,3 For a dilute aqueous solution of a nonionic surfactant,3 tetraoxyethylene decyl ether (D(EO)4 , 5 × 10-4 mol/dm3) or eicosaoxyethylene nonylphenol ether (NP(EO)20, 1 × 10-5 mol/dm3), and electrolyte (KCl), thick films (with thicknesses of the order of 100 Å) were observed at low electrolyte concentrations. With an increase of the electrolyte concentration, the film thickness first decreased, which suggests that the repulsion was caused by the double layer. This repulsive force was generated because of the different adsorptions of the two species of ions on the water/ surfactant interface. At a critical electrolyte concentration, a black film was formed, and the further addition of electrolyte did not modify its thickness, which became almost independent of the external pressure, until a critical pressure was reached, at which it ruptured. While for NP(EO)20 only one metastable equilibrium thickness was found at each electrolyte concentration, in the case of D(EO)4 a hysteresis of the film thickness with increasing and decreasing pressure (i.e., two metastable minima) was observed in the range 5 × 10-4 to 3 × 10-3 mol/dm3 KCl. The maximum pressure used in these experiments was relatively low, ∼5 × 104 N/m2, and the Newton black films did not rupture in the range of pressures employed. In the case of ionic surfactants (10-3 mol/dm3 sodium lauryl sulfate, NaLS), abrupt transitions from common black films to Newton black films were observed, for NaCl concentrations in the range 0.165-0.31 mol/dm3, by increasing the external pressure.2 The Newton black films ruptured at a pressure of about 12 × 104 N/m2. The purpose of this article is to present a model and to calculate on its basis the metastable equilibrium thicknesses of the film as a function of the applied pressure. In section II, the interaction energy of the film was calculated, assuming planar interfaces free of thermal fluctuations. The double layer interaction was calculated by accounting for the charge recombination at the surface with increasing electrolyte concentration. An approximate (2) Exerowa, D.; Kolarov, T.; Khristov, Khr. Colloids Surf. 1987, 22, 171. (3) Kolarov, T.; Cohen, R.; Exerowa, D. Colloids Surf. 1989, 42, 49.

10.1021/la011569w CCC: $22.00 © 2002 American Chemical Society Published on Web 02/20/2002

Polarization Model and ion specificity: applications

2728

533

Langmuir, Vol. 18, No. 7, 2002

Ruckenstein and Manciu

expression derived by Donners et al.4 on the basis of the Lifshitz theory was used to calculate the van der Waals interaction energy. For the hydration force, a relation derived recently5,6 was used, which accounts for the increase in the hydration force with increasing electrolyte concentration, due to a higher ion-pair (dipole) density produced by the recombination of some charges on the interfaces with counterions. This enhancement of the repulsion at relatively high ionic concentrations might be at the origin of the stability of the Newton black films, as suggested recently;7 however, in the present article, the field generated by an ion pair located at the interface is calculated more accurately and the theory is further developed to include the effect of thermal fluctuations. II. Interaction Energy of Soap Films Throughout this section, the energy of the film will be calculated by assuming planar, non-undulating, interfaces. Most of the equations of section IIA-E were taken from our previous paper.5 We include them without detailed derivation for the completeness of the presentation. II.A. The Surface Density of Surfactant. The surface density of surfactant is relevant for both the double layer (caused by the surface charge density generated via the dissociation of the surfactant adsorbed at the interface) and the hydration interaction (caused by the ion pair density of the non-dissociated surfactant molecules). The surface density of an anionic surfactant, Γ, will be related to the saturation surface density, Γ∞, via the Frumkin adsorption isotherm

(

Γ Γ exp -2a2 Γ∞ Γ∞ a1CR- ) Γ 1Γ∞

)

(1)

where a1 and a2 are empirical parameters available from experiment and CR- is the concentration of the surfactant anions in the liquid in the vicinity of the interface. The surfactant in the bulk is considered completely dissociated, and hence the concentration of surfactant in the vicinity of the interface is related to the bulk surfactant concentration through the expression

( )

eψs CR- ) Cs exp kT

R-X|SURFACE T R-|SURFACE + X+|LIQUID-AT-INTERFACE where R denotes the surfactant anion group and X the cation. At equilibrium, one can write

KD )

CR-SCX+ RΓCX+ R ) ) C + CR-X (1 - R)Γ 1 - R X

(4)

where CR-S is the surface density of the dissociated surfactant anions, CR-X is the surface density of nondissociated (ion pair) surfactant molecules, CX+ is the concentration of the cations in the liquid in the vicinity of the surface, and KD is the equilibrium constant. Assuming that the surfactant in solution and the electrolyte are completely dissociated, one can write

( )

CX+ ) (Cs + Ce) exp -

eψs kT

(5)

where Ce is the bulk electrolyte concentration. II.C. The Double Layer Interaction Energy. The calculation of the double layer interaction is based on the accurate approximation of the solution of the PoissonBoltzmann equation due to Ohshima and Kondo.9 For two parallel plates at a distance z apart, at a surface potential ψs, and with a midplane potential ψm, they obtained the equation

κz κz sinh( ) ( 2) ( ) cosh κz - cosh κz2 + (2) (2) κz κz κz κz cosh ( ) - 2( ) - 1 3( ) sinh( ) 2 2 2 2 γ × κz κz 4 cosh ( ) 4 cosh ( ) 2 2

eψm tanh ) γ0 4kT

5

0

{

1

γ03

4

2

2

5

6

(1 - 4(κz2 ) tanh(κz2 ))

(2)

where Cs is the bulk surfactant concentration, ψs is the surface potential, which is negative (hence CR- < Cs), e is the protonic charge, k is the Boltzmann constant, and T is the absolute temperature. The parameters a1, a2, and Γ∞ are provided by experiment. In what follows the data obtained by Fainerman8 at high electrolyte concentrations, namely, Γ∞ ) 5 × 10-6 mol/m2, a1 ) 881 m3/mol, and a2 ) -1.53, will be used. IIB. The Surface Charge. With R denoting the dissociation constant, the surface charge density σ is given by

σ ) -eRΓ

For simplicity, it will be assumed that the electrolyte is 1:1 and that it has the same type of cation as the surfactant (e.g., NaCl and NaLS). The charge is generated through the following dissociation equilibrium

(3)

(4) Donners, W. A. B.; Rijnbout J. B.; Vrij, A. J. Colloids Interface Sci. 1977, 60, 540. (5) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7061. (6) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582. (7) Ruckenstein, E.; Bhakta, A. Langmuir 1996, 12, 4134. (8) Fainerman, V. B. Colloids Surf. 1991, 57, 249.

}

(6)

where γ0 ) tanh (eψs/4kT), κ ) (2e2Ce/0kT)1/2 is the Debye length for a 1:1 electrolyte and 0 is the dielectric constant of water. The surface charge density is related to the surface and midplane potentials via

σ)-

[( ( )

( ))]

κ0kT eψs eψm 2 cosh - cosh e kT kT

1/2

(7)

The double layer force per unit area is given by the Langmuir equation

( (

pDL(z) ) 2CekT cosh

) )

eψm(z) -1 kT

(8a)

Equations 1-8a can be simultaneously solved to obtain the surfactant density, the degree of dissociation, and the double layer force. The double layer interaction energy (9) Ohshima, H.; Kondo, T. J. Colloids Interface Sci. 1988, 122, 591.

Nanodispersions

534

Stability of Black Films

Langmuir, Vol. 18, No. 7, 2002 2729

per unit area, UDL, is obtained by integrating the double layer force per unit area from d to ∞, where d is the thickness of the water layer

UDL(d) )

∫d

∞

pDL(z) dz

(8b)

II.D. The van der Waals Energy. The van der Waals energy per unit area for a triple layer system (hydrocarbon/ water/hydrocarbon) in air was calculated on the basis of the Lifshitz theory by Donners et al.4 and is well represented by the expression

UvdW(d) )

(

b1 + b2d 1 + b5 2 d 1 + b3d + b4d2

)

(9)

p⊥ ′

1

(Aπ + ∆′ ) 2

2π0

3/2

sinh m(z) ) m1

(

C1∆2 1 - (C0 + 2C1) γ

)

1/2

(12)

where ∆ is the distance between the centers of the adjacent layers, γ is the molecular polarizability, and Ck are interaction coefficients that account for the electric field generated by the dipoles of the layer i + k at a site of layer i

(11)

where z is the distance normal to the surface, measured from one surface (considered at the external boundary of the first water layer), d is the distance between surfaces, (10) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435.

(13)

For an icelike structure of water, the values of the coefficients are5,6

C0 ) -

3.7663 4π0′′l3

(14a)

and

C1 ) C-1 )

1.8272 4π0′′l3

(14b)

where l is the distance between the centers of two adjacent water molecules and ′′ is an effective dielectric constant for the interaction between neighboring water molecules; again ′′ < . The polarization of the first water layer is produced by the surface dipoles and the water dipoles of the first two layers, hence

m1 )

( (

γE h d - 2∆ 2λ d sinh 2λ

sinh

1 - γ C0 + C1

(10)

where p⊥ is the component, normal to the surface, of the dipole moment of an ion pair of the surface, 0 is the vacuum permittivity, 1/A is the ion pair density on the surface, ′ is an effective dielectric constant for the interaction between the water molecules of the first water layer and the nearest surface dipole, and ∆′ is the distance between the first water layer and the center of the surface dipole. It is important to emphasize that ′ is smaller than the dielectric constant of water, , because of the lower screening of the electric field by the fewer intervening water molecules. The average polarization of the water molecule is well described by

(d -2λ2z) d sinh( ) 2λ

λ)

Eilocal ) C-1mi-1 + C0mi + C1mi+1

where d is the thickness of the water layer and, for dodecane films with a thickness of 9 Å, the parameters are b1 ) -3.08 × 10-22 J, b2 ) -6.28 × 10-14 J/m, b3 ) 8.28 × 107 m-1, b4 ) 6.13 × 1015 m-2, and b5 ) -9.00 × 10-23 J. II.E. The Hydration Interaction Energy. The calculation of the hydration interaction is based on a model proposed by Schiby and Ruckenstein,10 which considers that, in the vicinity of an interface, the water is organized in layers parallel to the interface with the structure of ice. The water molecules from the first water layer are polarized by the nearest dipoles of the headgroups of the surfactant molecules; these dipoles induce an electric field into the adjacent layer, whose induced dipoles generate in turn electric fields in both adjacent layers and so on. When two surfaces approach one another, the polarization layer will increasingly overlap. This decreases the average polarization of the water molecules and hence increases the electrostatic energy, thus generating a repulsion between surfaces. We will only outline the theory here; a detailed presentation was made recently.5 The field generated by the surface dipoles, which polarize the first water layer, is given by

E h )

m1 is the polarization of the first water layer, and λ is the decay length of the polarization. For a layered structure of water, λ can be calculated using the expression5

(

)

( )

)

(15)

and the total energy, per unit area, due to the polarization of water molecules by the surface dipoles, is given by

UHtotal(d) )

( (

γE h2

1 - γ C0 + C1

d - 2∆ 2λ d sinh 2λ

sinh

(

)

( )

)

2

×

(dλ) d 4v sinh ( ) 2λ

d - λ sinh

(16)

2

where v is the volume occupied by a water molecule. The hydration interaction energy per unit area is given by

UH(d) ) UHtotal(d) - UHtotal(∞)

(17)

II.F. A Simple Model for the Interaction Energy. We will try first to obtain some information about the interaction energy by using some simple approximations.

Polarization Model and ion specificity: applications

2730

535

Langmuir, Vol. 18, No. 7, 2002

Ruckenstein and Manciu

The following approximate expressions will be used for the interaction energies per unit area:

( )

U′H ) A1 exp U′DL )

d λ1

(18a)

( )

( )

64CekT eψs d tanh2 exp(-κd) ) A2 exp κ 4kT λ2

(18b) U′vdW ) -

AH

(18c)

12πd2

which are rough approximations of the interaction energies provided by the equations presented above (eqs 17, 8b, and 9, respectively). Neglecting the double layer interaction (neutral interfaces), one obtains for the disjoining pressure (the negative of the derivative of the interaction energy) the expression

Π(d) )

( )

A1 AH d exp λ1 λ1 6πd3

(19)

Newton black films, which are the only ones that can exist in the absence of double layer interactions, can be obtained only if the disjoining pressure has a positive value (otherwise, the film will collapse). The extrema of the disjoining pressure are obtained through the derivation of eq 19 with respect to d

( )

AH d4 d exp ) ξ4 exp(-ξ) ) F1(ξ) ) (20) 4 λ1 λ1 2πA1λ12 where ξ ) d/λ1. The function F1(ξ) in eq 20 has a maximum at ξ ) 4 (see Figure 1a); therefore, eq 20 has solutions only if the Hamaker constant AH < 2πA1λ12F1(4) ) 29.46A1λ12. In such cases, eq 20 has two solutions. The smaller corresponds to a maximum disjoining pressure and the other one to a local minimum. A useful lower bound is obtained when the maximum value of the disjoining pressure vanishes, Π(d) ) 0. Using eqs 19 and 20, one obtains for the latter case

d* ) 3λ1 A1* )

(21a)

AH

(21b)

162π λ12 exp(-3)

A stable film can be obtained only if the hydration repulsion is stronger than the above critical value, hence if A1 > A1*, and the maximum disjoining pressure occurs at a distance d < d* ) 3λ1. When all three interactions are taken into account, the extrema of the disjoining pressure are given by

ξ4(exp(-ξ) + r12r2 exp(-r1ξ)) ) F2(ξ) )

AH 2πA1λ12

(22)

where r1 ) λ1/λ2 and r2 ) A2/A1. The left-hand term of eq 22 is a positive function of ξ, which vanishes at ξ ) 0 and ξ f ∞. This function is represented in Figure 1 for various values of the ratios r1 and r2. If the right-hand term is sufficiently large to exceed the maximum value of F2(ξ) (strong attraction), eq 22 has no solution and the film collapses. For r2 ) 0 (Figure 1a), F2(ξ) ≡ F1(ξ) and there are two solutions if AH < 29.46A1λ12. For r2 ) 0.05 and

Figure 1. F2(ξ) ) ξ4(exp(-ξ) + r12r2 exp(-r1ξ)) (eq 22) for (a) r2 ) 0, (b) r1 ) 0.333 and r2 ) 0.05, (c) r1 ) 0.1 and r2 ) 0.05, and (d) r1 ) 0.333 and r2 ) 0.14.

r1 ) 0.333 (Figure 1b), eq 22 still has only two solutions; hence there is only one maximum of the disjoining pressure. At metastable equilibrium, the disjoining pressure equals the external pressure. Starting with a thick film and increasing the external pressure, the thickness decreases continuously, until the film ruptures. However, if the ratio r1 is sufficiently small (r1 ) 0.1 and r2 ) 0.05, Figure 1c), there is a range of Hamaker constants (13.5A1λ12 < AH < 29.9A1λ12), in which eq 22 has four solutions. In these cases there are two maxima in the disjoining pressure, located at distances d1 and d2. Let us first consider that the maximum at d1 is higher (see Figure 2a). Starting with a thick film and increasing the external pressure, the thickness of the film decreases continuously up to the distance d2, when the external pressure reaches the value Π2. A further increase of the pressure produces a jump from the distance d2 to the distance d′, which corresponds to a stable Newton black film. On the other hand, if the maximum at d2 is the higher one (Figure 2b), then the increase of the pressure above Π2 produces the rupture of the film. However, the existence of a domain with four solutions for eq 22 does not necessarily imply that both Newton and common black films can be stable. While two maxima of the disjoining pressure exist for r1 ) 0.333 and r2 ) 0.14 ( Figure 1d), the value of the second one is negative (Figure 2c) and hence only the Newton black film is stable. II.G. Model Calculations. In what follows it will be shown that the general behavior discussed above is consistent with realistic calculations. An important parameter, which is however unknown, is the equilibrium constant of the association-dissociation equilibrium, KD. The dissociation constant of the ion pair NaSO4- was estimated to be KD ) 10-0.7 mol/dm3 ) 0.1995 mol/dm3.11 Because of the repulsion among the headgroups on the interface, the dissociation constant is expected to be lower in the present case. In what follows, we will use KD ) 0.050 mol/dm3. For this value, the pressures at which the transitions from the common to the Newton black films occur are in agreement with the experiments of Exerowa et al.2 As noted in the previous section, to obtain a transition from the common to a Newton black film, the decay lengths of the hydration and double layer repulsions must be (11) Davies, C. W. Ion Association; Butterworth: London, 1962.

536

Stability of Black Films

Nanodispersions

Langmuir, Vol. 18, No. 7, 2002 2731

Figure 3. Disjoining pressure Π(d) vs d for Γ∞ ) 5 × 10-6 mol/m2, a1 ) 881 m3/mol, a2 ) -1.53, b1 ) -3.08 × 10-22 J, b2 ) -6.28 × 10-14 J/m, b3 ) 8.28 × 107 m-1, b4 ) 6.13 × 1015 m-2, b5 ) -9.00 × 10-23 J, KD ) 0.050 mol/dm3, Cs ) 1 × 10-3 mol/dm3, λ1 ) 1.0 Å , p⊥/′ ) 3.45 D: (a) Ce ) 0.150 mol/dm3; (b) Ce ) 0.165 mol/dm3; (c) Ce ) 0.180 mol/dm3. In the inset, the peaks of the disjoining pressure are presented at a smaller scale.

Figure 2. Disjoining pressure Π(d) vs d: (a) AH ) 10-20 J, A1 ) 0.017 J/m2, λ1 ) 1.5 Å, A2 ) 0.001 J/m2, λ2 ) 15.0 Å; (b) the same values for the parameters as in (a), except A1 ) 0.016 J/m2; (c) AH ) 10-20 J, A1 ) 0.0121 J/m2, λ1 ) 1.5 Å, A2 ) 6.05 × 10-4 J/m2, λ2 ) 4.5 Å (these values correspond to the domain of four solutions of eq 22 of curve d of Figure 1; hence r1 ) 0.333, r2 ) 0.05, F2(ξ) ) 5.85).

sufficiently different. A transition from the common to the Newton black film was observed experimentally 2 to start when the electrolyte concentration exceeded 0.165 mol/dm3, and transitions were observed to take place up to the highest concentration (0.31 mol/dm3) used in experiment. For these two concentrations, the Debye lengths are λ2 ) (κ)-1 ) 7.5 and 5.5 Å, respectively. To obtain a separation between the pressure peaks, the decay length for the hydration repulsion should be much smaller. For a molecular polarizability γ ) 8.6 × 10-40 C2 m2/J, calculated using the Debye-Lorentz model, the upper bound for the hydration decay length, λ1, calculated assuming ′′ ) 1 , is 2.96 Å.5 Lower values can be obtained when the effective dielectric constant ′′ for the interaction between molecules is larger than 1. The decay length is also decreased by the structural disorder of the water layers.6 While recent experiments indicated that for lipid bilayers λ1 is close to 2 Å, 12 there are no such experimental results for soap films. In what follows, we will use ′′ ) 9, which introduced in eqs 14 and using eq 12 provides λ1 ) 1.0 Å. In Figure 3, the disjoining pressure, calculated as described above, is plotted for Cs ) 10-3 mol/dm3 and p⊥/′ ) 3.45 D. For Ce ) 0.15 mol/dm3 (Figure 3a), the first maximum of the disjoining pressure is somewhat (12) Petrache, H. I.; Gouliaev, N.; Tristram-Nagle, S.; Zhang, R.; Suter, R. M.; Nagle, J. F. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 57, 7014.

lower than the second maximum, which corresponds to the common black film. Therefore, the increase of the external pressure produces in this case the rupture of the common black film. The increase of the electrolyte concentration has two effects. First, it decreases the double layer repulsion leading to a lower height of the second peak, and secondly, the recombination of surface charges on the interfaces with counterions provides an increase in the hydration repulsion (via the increase of the ion pair density) and hence an increase of the height of the first peak. In Figure 3b (Ce ) 0.165 mol/dm3), there is a very narrow range of pressures in which the Newton black film is the only stable one (the value p⊥/′ ) 3.45 D was selected to provide about the same height for the two peaks of the disjoining pressure at Ce ) 0.165 mol/dm3). A further increase of the electrolyte concentration (Ce ) 0.18 mol/dm3, Figure 3c) lowers the transition pressure from the common to the Newton black film (because of the screening of the double layer) and also slightly increases the rupture pressure of the Newton black film (because of the increase in the hydration repulsion with increasing ionic strength). At high electrolyte concentrations, the Newton black films might be the only stable ones. The calculations presented in this section show that the behavior of the black films can be understood in terms of the interaction energy between planar films. However, they cannot explain why, for the same electrolyte concentration, the transition from the common to the Newton black film occurs at various pressures (for example, for Ce ) 10-3 mol/dm3, p ) (2.5 ÷9.8) × 104 N/m2).2 In addition, the thickness at the transition apparently does not depend on the electrolyte concentration (while the Debye length λ2 does) and is larger than the upper bound 3λ2 (which is obtained, when only the double layer and van der Waals interactions are present, using the approach employed to derive eq 21). In the next section, it will be shown that by accounting for the thermal fluctuations of the interfaces one can provide answers to these questions. III. The Role of Thermal Fluctuations of the Interfaces The first quantitative theory, which accounted for the repulsion between phospholipid bilayers due to a steric

Polarization Model and ion specificity: applications

2732

537

Langmuir, Vol. 18, No. 7, 2002

Ruckenstein and Manciu

film. The Boltzmannian distribution of the thicknesses has the form

(

F(z) ∝ exp -

)

(

SB(z - z0) SU(z) ∝ exp kT 2kT

)

2

(24)

However, as demonstrated by Helfrich,13 the distribution of thicknesses in a harmonic potential is given by

F(z) )

(

(z - z0) 1 exp 1/2 (2π) σ 2σ2

)

2

(25)

where the root mean square fluctuation of the thickness, σ, obtained from the partition function has the form13,18 Figure 4. The fluctuating interfaces are approximated by independent small, planar surfaces of area S separated by the distance z.

(rigid wall) confinement of their undulation by the neighboring bilayers, was developed by Helfrich.13 Since then a number of attempts were made to extend the Helfrich theory to arbitrary interaction potentials.14-16 The last two approaches involved the observation that the partition function of an undulating bilayer in an external potential can be calculated exactly for a harmonic potential. However, in the present case, we are interested in the equilibrium position as a function of the applied pressure in a metastable state and the interaction potential (which possesses two local minima) cannot be approximated by a harmonic one. A new approach is therefore suggested below. III.A. The Thermal Fluctuations of the Interfaces for Arbitrary Interactions. After the Helfrich initial theory,13 Helfrich and Servuss17 suggested an alternate derivation of the entropic repulsion due to the confinement of a membrane between rigid walls, by considering the lipid bilayer composed of many independent “pieces”, whose area is related to the root mean square fluctuations of the positions of the undulating bilayer. As shown below, this representation can be extended to interfaces interacting via arbitrary potentials. In what follows, the two fluctuating interfaces will be replaced by many small, independent surfaces of area S, separated by a distance z (see Figure 4). The (metastable) distribution of the distances between the surfaces, in an ensemble subjected to a constant pressure, will be assumed Boltzmannian. It will be also assumed that the fluctuating interfaces have constant total areas and an elastic bending modulus KC. Let us first consider that the interfaces interact per unit area through a harmonic potential

U(z) )

1 B(z - z0)2 2

(23)

where B is the spring constant, z is the distance between two small surfaces, and z0 is the average thickness of the (13) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (14) Evans E. A.; Parsegian V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7132. (15) Podgornik, R.; Parsegian, V. A. Langmuir 1992, 8, 557. (16) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 2455. (17) Helfrich, W.; Servuss, R.-M. Nuovo Cimento 1984, 3D, 137.

σ2 )

kT 8(KCB)1/2

(26)

Comparing eqs 24 and 25, and taking into account eq 26, one obtains

S)

( )

KC kT )8 2 B Bσ

1/2

(27)

For a constant applied pressure p, the distribution of distances between the small surfaces can be calculated using the enthalpy instead of the energy

(

)

1 S B(z - z0)2 + pz 2 F(z) ∝ exp ≡ kT

(

)

(

1 S B(z - z0′)2 + C 2 exp kT

(

)

)

(28)

where z0 is the average thickness at p ) 0, z0′ ) z0 - (p/B) is the average thickness at pressure p, and C ) z0p (p2/2B). The constant C is independent of z and hence does not affect the distribution F(z), being eliminated through normalization. As expected, the new equilibrium distance is the thickness where the elastic force equilibrates the external pressure, B(z0 - z0′) ) p. Let us first note that in this case, there is no repulsion due to the entropic confinement, because the spring constant does not depend on the applied pressure, and therefore the intersurfaces distance distribution, F(z), and the mean square fluctuation, σ2 ) 〈(z - z0′)2〉, are independent of the applied pressure and, hence, on the average distance. Because the free energy due to the entropic confinement is given by (kT)2/(128KCσ2),13,18 the undulation force is zero in this case. The above derivation will be extended to a well-behaved interaction potential, U(z), for which U(0) f ∞, U(∞) f 0 and has a minimum at a distance z0. In the vicinity of the minimum of the enthalpy (per unit area) H(z) ) U(z) + pz, the potential will be approximated by a harmonic one, with the effective spring constant B′ ) ∂2H(z)/∂z2|z)z0′ , where z0′ is the solution of ∂H(z)/∂z ) 0. It will be assumed that the areas of the small independent surfaces into which the interfaces are decomposed are still given by eq 27, but (18) Sornette, D.; Ostrowsky, N. In Micelles, Membranes, Microemulsions and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer-Verlag: Berlin, 1994.

Nanodispersions

538

Stability of Black Films

Langmuir, Vol. 18, No. 7, 2002 2733

with a new spring constant B′. In this case, the distribution of distances is given by

F(z) )

(( |) )

KC 1 exp - 8 2 N ∂ H(z) ∂z2

1/2

H(z) kT

(29)

z)z0′

where N is obtained through normalization: N ) ∫0∞F(z) dz. The average thickness, 〈z〉, is obtained from

〈z〉 )

∫0

∞

zF(z) dz

(30)

while the mean square fluctuation is given by

σ2 )

∫0

∞

(z - 〈z〉)2F(z) dz

(31)

For nonvanishing values of p, both integrals (30) and (31) converge. Using eqs 29-31, one can calculate the average thickness and the root mean square fluctuation of the intersurfaces separation as a function of the applied pressure, once the interaction potential is known. For illustration purposes, let us first apply the method to the simple interaction potential

( )

U′(z) ) A1 exp -

AH z λ1 12πz2

(32)

with A1 ) 0.07 J/m2, λ1 ) 1.0 Å, and AH ) 10-20 J, which are some typical values of the parameters. The potential is not well-behaved (in the sense described above) because U(0) f -∞ . The stable equilibrium position corresponds in this case to the trivial case z ) 0 (the film ruptures). However, we are interested in the behavior of the film at the metastable equilibrium (the secondary minimum of the enthalpy, at z0′, see Figure 5). At metastable equilibrium, the film can have a thickness between z1′, corresponding to the maximum of the enthalpy, and z ) ∞. Figure 5 presents the enthalpy and the distribution F(z) for various values of KC and (a) p ) 1.0 × 104 N/m2 and (b) p ) 1.0 × 106 N/m2. The asymmetry of the distributions depends on the pressure applied and is a consequence of the anharmonicity of the interaction potential. The lifetime of the film (until it ruptures) depends on the value of F(z) at z ) z1′. Curves 1-3 provide the ratios F(z)/F(z0′), for various values of KC. They are negligible at z ) z1′; hence the metastable state at the secondary minimum has a long lifetime. In Figure 6, the pressure (a) and the root mean square fluctuation (b) are plotted as functions of the average thickness of the film, for KC ) 10 × 10-19 J and for (1) the anharmonic (eq 32) and (2) the harmonic (eq 23) interaction potentials. The spring constant for the second case was obtained from the harmonic approximation of eq 32 around its minimum, at p ) 0. We already noted that, for a harmonic interaction, σh ) constant and 〈z〉 ) z0′ ) z0 - (p/B) varies linearly with the applied pressure. However, for an anharmonic interaction, σa is a function of the applied pressure (or, equivalently, of the average thickness of the film 〈z〉, which differs from z0′ because of the asymmetry of the distribution). The functional dependence of the pressure on the average thickness differs in the anharmonic and harmonic cases. The pressure is higher in the former case, because of the

Figure 5. The enthalpy H (thick line) and the ratio F(z)/F(z0′) vs the distance z for an anharmonic interaction potential (eq 32) with AH ) 10-20 J, A1 ) 0.07 J/m2, λ1 ) 1.0 Å: (1) KC ) 1.0 × 10-19 J; (2) KC ) 10.0 × 10-19 J; (3) KC ) 50.0 × 10-19 J. The applied pressures were (a) p ) 1.0 × 104 N/m2; (b) p ) 1.0 × 106 N/m2.

“undulation repulsion”, which increases with decreasing bending modulus. The undulation repulsion, however, depends on the interaction potential and vanishes for a harmonic interaction. The lifetime of the film depends on F(z1′); hence it decreases for interfaces with low bending moduli (less rigid interfaces are likely to fluctuate with a larger amplitude and thus can reach easier the maximum of the enthalpy). Therefore, while the undulation repulsion increases the average thickness of the film at a given external pressure (Figure 6a), it decreases its stability (Figure 5). III.B. The Role of Thermal Fluctuations on the Transition from Common Black Films to Newton Black Films. The method described in the previous section will be now applied to thin films with fluctuating interfaces, with the interaction energy calculated as in section II.G. For low values of the external pressure, the enthalpy has two metastable minima at zN and zC, and a stable one at z f 0 (the former two correspond to the Newton and to the common black films, respectively, and the latter implies the rupture of the film), separated by two maxima located at z1 and z2 (see Figure 7a). At metastable equilibrium the distances between the surfaces are distributed between z1 and z2 for the Newton black film and between z2 and z f ∞ for the common black film. The stability of the metastable states depends on the chance for a small area S of the interface to reach the

Polarization Model and ion specificity: applications

2734

539

Langmuir, Vol. 18, No. 7, 2002

Ruckenstein and Manciu

Figure 6. (a) The applied pressure p and (b) the root mean square fluctuation σ vs average thickness 〈z〉 for (1) anharmonic interaction potential (eq 32) and (2) its harmonic approximation (at p ) 0).

limits of the domain (the heights of the adjacent peaks). The lifetime of a state in the potential well is proportional to the factor exp(S∆H/kT); hence the lifetime until the common black film transforms into a Newton black film, tCfN is proportional to

(

tCfN ∝ exp

)

SC(H(z2) - H(zC)) FC(zC) ∝ kT FC(z2)

(33a)

while the lifetimes of a Newton black film until it ruptures, tNfR, or until its transition to a common black film, tNfC, are given by

(

tNfR ∝ exp

)

SN(H(z1) - H(zN)) FN(zN) ∝ (33b) kT FN(z1)

and

(

)

SN(H(z2) - H(zN)) FN(zN) tNfC ∝ exp ∝ kT FN(z2)

(33c)

respectively. Both minima of the enthalpy, while metastable, might have a relatively long lifetime, and a system prepared in one state (common black film or Newton black film) might remain in that state during the time of the experiment, if the potential barriers are sufficiently high. However, because of the thermal fluctuations, it is possible to have, for the same experimental conditions, a transition in a range of pressures, and this explains one of the experimental results of Exerowa et al.2 When both the Newton and the common black films have lifetimes exceeding the duration of the experiment in a domain of applied pressures, the common black film

Figure 7. The enthalpy H (thick line) and the ratios FN(z)/ FN(zN) and FC(z)/FC(zC) vs the distance z for the parameters Γ∞ ) 5 × 10-6 mol/m2, a1 ) 881 m3/mol, a2 ) -1.53, b1 ) -3.08 × 10-22 J, b2 ) -6.28 × 10-14 J/m, b3 ) 8.28 × 107 m-1, b4 ) 6.13 × 1015 m-2, b5 ) -9.00 × 10-23 J, KD ) 0.050 mol/dm3, Cs ) 1 × 10-3 mol/dm3, λ1 ) 1.0 Å, p⊥/′ ) 3.7 D, Ce ) 0.165 mol/dm3: (a) p ) 1.0 × 104 N/m2, KC ) (2.0, 5.0, 10.0, 20.0, and 50.0) × 10-19 J; (b) p ) 6.2 × 104 N/m2, KC ) 20.0 × 10-19 J; (c) p ) 1.6 × 105 N/m2, KC ) 20.0 × 10-19 J.

transforms into a Newton one at the upper bound of the domain, while the Newton black film can have a transition to the common one at the lower bound of the pressure domain. This results in a hysteresis in thickness with increasing and decreasing pressure. The bending modulus KC of the interfaces of the soap films is unknown; however, one expects to be comparable to those of the lipid bilayers in water, which are of the

Nanodispersions

540

Stability of Black Films

order of 10-19 J.12 In Figure 7a, the distributions of the thicknesses of the Newton and common black films, FN(z)/FN(zN) and FC(z)/FC(zC), respectively, are calculated for Ce ) 0.165 mol/dm3, p⊥/′ ) 3.7 D, p ) 1.0 × 104 N/m2, and various values of KC (2, 5,10, 20, and 50 × 10-19 J), the other parameters being as in section II.G. While the first metastable minimum (corresponding to a Newton black film) is somewhat deeper, the second one (corresponding to a common black film) is more stable to fluctuations, because FC(z2)/FC(zC) is negligible, while FN(z2)/ FN(zN) is much larger. This is a result of the stronger spatial confinement of the undulation modes in the first minimum, which leads to a higher effective spring constant. In what follows, it will be considered that the lifetime of a metastable state exceeds the duration of experiment if the potential barrier for a small surface of area S, S∆H, exceeds 3kT. For the first minimum in Figure 7a, the potential barrier is higher than 3kT for KC > 20 × 10-19 J; hence the Newton black films, once formed, remain stable in cases 4 and 5 during the experiment. In contrast, if the interfaces are more flexible (low KC, cases 1, 2, and 3), the probability for an individual small surface to reach the height which separates two minima is much higher. In this case, the metastable equilibrium of the Newton black film has a shorter lifetime; while the film can either rupture or have a transition to a common black film, the second process has a higher chance. On the other hand, the common black films are more stable, because, while the minimum of the enthalpy is higher, the interfaces have more room to fluctuate. The increase of the applied pressure decreases the stability of the common black film and the probability for a transition from a Newton to a common black film but also increases the probability of the rupture of the Newton black films. Figure 7b plots the enthalpy and the ratios FN(z)/FN(zN) and FC(z)/FC(zC), for p ) 6.2 × 104 N/m2 (KC ) 20 × 10-19 J). The common black film becomes unstable (and a transition to a Newton black films occurs), while the Newton black film has a longer lifetime. At even a higher pressure (Figure 7c, p ) 1.6 × 105 N/m2, KC ) 20 × 10-19 J) the Newton black film ruptures. Assuming that a film is stable when the potential barrier for the individual surfaces in which the interface were decomposed is higher than a critical value (selected 3kT in the present paper), one can identify the regions of stability of the films. When the ratio of the distributions at the adjacent maximum and minimum of the enthalpy, F(zmax)/F(zmin ), exceeds exp(-3) ) 0.05, a transition to the next minimum is likely. In Figure 8a exp(-S∆H/kT) is plotted for Ce ) 0.165 mol/dm3 and KC ) 20 × 10-19 J against the applied pressure for Newton black films at the first (FN(z1)/FN(zN), curve 1) and second maximum (FN(z2)/FN(zN), curve 2) and for the common black film, at the second maximum (FC(z2)/FC(zC), curve 3). Below p ) 6.0 × 104 N/m2, both the Newton and the common black films have a long lifetime. When the external pressure exceeds 6.0 × 104 N/m2 ( marked C - N in Figure 8a) the common black film is likely to have a transition to a Newton black film, and the latter has a long lifetime. However, when the pressure exceeds 1.5 × 105 N/m2 (marked N - R in Figure 8a), the Newton black film is likely to rupture. Therefore, both kinds of films have a long lifetime below 6.0 × 104 N/m2, only the Newton black films exist for pressures in the range 6.0 × 104 to 1.5 × 105 N/m2 and no black films exist at higher pressures. It is clear that the stability of the films depends on the interaction between surfaces; in addition, the domains of stability of the black films are strongly dependent on their rigidity. The calculations presented in Figure 8b differ

Langmuir, Vol. 18, No. 7, 2002 2735

Figure 8. The ratios FN(z1)/FN(zN), FN(z2)/FN(zN), and FC(z2)/FC(zN) as functions of the applied pressure for the same parameters as before (Γ∞ ) 5 × 10-6 mol/m2, a1 ) 881 m3/mol, a2 ) -1.53, b1 ) -3.08 × 10-22 J, b2 ) -6.28 × 10-14 J/m, b3 ) 8.28 × 107 m-1, b4 ) 6.13 × 1015 m-2, b5 ) -9.00 × 10-23 J, KD ) 0.050 mol/dm3, Cs ) 1 × 10-3 mol/dm3, λ1 ) 1.0 Å, p⊥/′ ) 3.7 D, Ce ) 0.165 mol/dm3): (a) KC ) 20.0 × 10-19 J; (b) KC ) 10.0 × 10-19 J.

from those of Figure 8a, only through a new value of the bending modulus, KC ) 10 × 10-19 J. However, the Newton black films are in this case unstable at any pressure; they are likely to have a transition to a common black film up to p ) 3.0 × 104 N/m2, and either to rupture or to transform into common black films for pressures between 3.0 × 104 and 4.7 × 104 N/m2. For higher pressures the Newton black films rupture immediately. The common black films are stable for pressures up to 5.5 × 104 N/m2, after which they rupture. Therefore, in this case, the black films can exist only as common black films, at low applied pressures, but rupture at higher pressures. The fluctuations increase the equilibrium thicknesses of the films (see Figures 5, 6a, and 7a), an effect which is particularly important for the common black films (see Figure 7a). This effect was found in the experiments of Exerowa et al.,2 who noted that the DLVO theory (for planar surfaces) predicts a too small repulsion and suggested that the hydration force (which has a shorter range) cannot account for the discrepancy. IV. Conclusions In the first part of the paper, the interaction energy of thin soap films was calculated, assuming planar, parallel

Polarization Model and ion specificity: applications

2736

541

Langmuir, Vol. 18, No. 7, 2002

interfaces (KC f ∞). The interaction energy (and hence the disjoining pressure) could explain qualitatively the behavior of the black films and the existence of a transition from the common black films to the Newton black films. At low electrolyte concentrations, the disjoining pressure peak due to the double layer force (corresponding to the common black film) is high and might exceed the first peak, caused by the hydration repulsion (corresponding to the Newton black film). Hence, the increase of the pressure leads to the rupture of the common black film. The increase of electrolyte concentration has two opposite consequences: the double layer repulsion decreases and, hence, the height of the second peak decreases, while the charge recombination increases the ion-pair density on the surface and, hence, increases the hydration force, raising the height of the first peak. However, the theory involving planar interfaces does not account completely for the transition from the common to the Newton black films and for the rupture of the

Ruckenstein and Manciu

common black films, which can occur in a range of pressures for the same remaining conditions. In the second part it is shown that the fluctuations of the thicknesses of the films lead to transitions which occur with different probabilities at an applied pressure. The stability of the undulating films depends not only on the difference between the corresponding local minimum of the enthalpy and the adjacent maximum but also on the shape of the enthalpy in the vicinity of the minimum. The thermal fluctuations affect especially the stability of the Newton black films, because their spatial confinement is stronger. The confinement of the fluctuating interface can drive the Newton black films either to the common black film or to rupture. The present analysis indicates that the fluctuations of the interfaces, while decreasing the stability of the film, lead to larger thicknesses, in agreement with experiment. LA011569W

Nanodispersions

542

Journal of Colloid and Interface Science 309 (2007) 56–67 www.elsevier.com/locate/jcis

On possible microscopic origins of the swelling of neutral lipid bilayers induced by simple salts Marian Manciu a , Eli Ruckenstein b,∗ a Physics Department, University of Texas at El Paso, 500 W. University Ave, El Paso, TX 79968, USA b Department of Chemical and Biological Engineering, State University of New York at Buffalo, Clifford C. Furnas Hall,

Box 604200, Buffalo, NY 14260-4200, USA Received 27 November 2006; accepted 1 February 2007 Available online 6 February 2007

Abstract It was recently suggested that the swelling of neutral multilipid bilayers upon addition of a salt can be simply explained only by the electrolyte screening of the van der Waals attractions, while assuming that the hydration force and the repulsion due to thermal undulations of membranes are unaffected by the salt. While we agree that the screening of the van der Waals interactions plays a role, we suggest that the increase in the hydration force upon addition of a salt has also to be taken into account. In a statistical model, which accounts for the membrane undulations, parameters could be found to explain the multibilayer swelling even when the van der Waals attraction is considered unaffected by the electrolyte screening. These results point out that the decrease by a factor of three of the Hamaker constant upon addition of a salt, suggested recently to be responsible for the swelling of neutral multilipid bilayers, is perhaps too large, and a smaller decrease in Hamaker constant, coupled with the above mentioned effects might explain the swelling. © 2007 Elsevier Inc. All rights reserved. Keywords: Lipid bilayers; Swelling; Effect of salt

1. Introduction The interest in lipid bilayers is due to their relevance to biological membranes [1]. They exhibit a richness of structures due to the interplay between many different inter- and intrabilayer forces. Among all the multilamellar bilayer structures, probably the most pertinent to biological membranes are the lamellar ones. Their equilibrium spacing is considered to be the result of a balance between attractive and repulsive forces. While the former forces are just the usual van der Waals interactions, the latter are composed of double layer forces (for charged bilayers) [2], hydration forces (due to the structuring of water near interfaces) [3] and repulsive forces generated by the thermal undulation of the membranes [4]. The hydration forces in neutral multilamellar lipid bilayers have been thoroughly investigated experimentally by subjecting

them to high osmotic pressures [3] and it was observed that they can be well approximated by an exponential repulsion with a decay length λH of about 1–3 Å, which is almost independent of the electrolyte concentration [3]:   z pH = AH exp − , (1) λH where z is the separation distance between bilayers in water and AH is a constant. The unbinding transition of lamellar lipid bilayers (their swelling to very large repeat distances), observed experimentally [5], cannot be explained by an exponential repulsion (which is short-ranged), but requires another long range repulsion. It was suggested that this long-range repulsion is due to the confinement of the thermal undulations of the bilayers [4]: pund = μ

* Corresponding author.

E-mail addresses: [email protected] (M. Manciu), [email protected] (E. Ruckenstein). 0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.02.001

(kT )2 , KC z3

(2)

where k is the Boltzmann constant, T the absolute temperature, KC the bending rigidity and the Helfrich proportionality con-

Polarization Model and ion specificity: applications

543

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

stant μ = 0.115. This force has a longer range than the van der Waals attraction, which at large separations becomes inverse proportional to the fifth power of the separation distance:   H 1 1 2 2b2 H pvdW (z) = + − ≈ 3 3 3 6π z (z + 2b) (z + b) πz5 (z  b), (3) where b is the bilayer thickness and H the Hamaker constant. While the increase in temperature is expected to increase the undulation repulsion (Eq. (2)), and consequently the repeat distance between the lipid bilayers, it turned out that the repeat distance increases by lowering the temperature (anomalous swelling [6]). This behavior was attributed to a strong decrease of the bending rigidity KC of the lipid bilayer as the temperature decreases [7,8] and hence to a drastic increase in the Helfrich repulsion at lower temperatures. It was also suggested that the unbinding of the lipid bilayers might be driven by a decrease in the van der Waals attraction [9]. Leibler and Lipowsky calculated a critical value for the Hamaker constant and suggested that it could be reached by modifying the polarizability of water, by the addition of a solute [9]. It was subsequently observed that the repeat distance increases upon addition of a carbohydrate solute [10] or some alkali-halide salts [11] to water, which can indeed decrease the Hamaker constant for lipid–lipid interactions through water. It is natural to consider that the swelling of neutral lipid bilayers in water upon addition of a salt can be induced by a decrease of the van der Waals attraction (effect predicted long ago [12]), and not by the increase of the hydration or undulation forces, for the following reasons. If the increase of the Helfrich repulsion Eq. (2) would be responsible for the swelling, they will dominate the van der Waals attraction at any distance and the membranes will become completely unbound, since no equilibrium position can be generated by a long range repulsion coupled with a short range attraction. On the other hand, it is very unlikely that a hydration force of the form of Eq. (1) can be responsible for such a swelling. In order to double the equilibrium distance from 20 to about 40 Å, the preexponential factor AH must increase about 1000 times, which is extremely unlikely to occur. Furthermore, the swelling of lipid bilayers upon addition of a salt remains large even for large ionic strengths; hence the charging of the membrane because of the salt cannot play a very important role. Indeed, for electrolyte concentrations larger than 1 M, the Debye–Hückel length is less than 3 Å, and therefore the double layer force has a short range effect comparable to the hydration provided by Eq. (1). While Korreman and Posselt already suggested that the decrease in Hamaker constant, because of electrolyte screening, might be responsible for the swelling of neutral lipid bilayers by salts [11], an accurate test of this hypothesis has been carried out only recently by Petrache et al. [13,14]. Upon addition of 1 M of KCl or KBr, the repeat distance in DLPC (a 12 carbonchain lipid, 1,2-dilauroyl-sn-glycero-3-phosphocholine) bilayers has increased, at 25 ◦ C, from 58 Å, to 68 and 74 Å, respectively [13]. These increases correspond to increases in the

57

water spacing from about 26 Å to about 35 and 41 Å, respectively [13]. Careful X-ray structural investigation showed that the membrane bending rigidity, KC , is not affected much by the addition of salt (see Fig. 5A of Ref. [13]), and osmotic pressure experiments have shown that, at short separations, the interactions between bilayers are only weakly affected by the salt addition. Petrache et al. also suggested that the different distances obtained for the multilamellar bilayer swellings by KCl and KBr can be explained by the preferential binding of Br− ( but not Cl− ) on the bilayers. By assuming that the “hydration force” (Eq. (1)) and the undulation repulsion (Eq. (2)) are not affected by the salt, and considering a binding constant K = 0.22 M−1 for Br− (and K = 0 for Cl− ), Petrache et al. calculated the dependence of the Hamaker constant on the electrolyte concentration required to fit their experimental data [14]. Note that the suggestion of Petrache et al. [14], that Br− binds to the surface whereas Cl− does not, might explain the differences in swelling upon addition of KBr or KCl, but cannot explain the differences upon addition of KBr and NaBr, which have been revealed by the Korreman and Posselt experiments [11]. z When plotted as functions of λDH , where z is the separation distance through water and λDH is the Debye–Hückel length, the Hamaker constants H calculated for both kinds of salts collapsed on a single curve, that can be very well represented by [14]:     z 2z H = H0 1 + 2χ exp −χ + H , (4) λDH λDH where H0 and H  represent the low-frequency and the highfrequency contributions, respectively, to the Hamaker constant and χ is an empirical correction parameter. For χ = 1, the prediction of Mahanty and Ninham [12] could be recovered; however the value χ = 0.2 was required to fit the experimental data. The work of Petrache et al. [13,14] represents, in our opinion, an important contribution to the understanding of the stability of multilamellar lipid bilayers, and we agree that the decrease in the Hamaker constant via electrolyte screening constitutes an important effect that drives the swelling of lipid bilayers upon salt addition. The “universality” of the Hamaker constant dependence on the Debye–Hückel length, e.g., its independence of ion-specificity obtained experimentally, constitutes a strong argument in favor of their hypothesis. However, whether the “universality law” holds for other salts (than KBr and KCl) is not yet clear. Until more experiments will be carried out, we like to discuss some issues that are raised by the work of Petrache et al. [13,14] and we will show that by combining the recently proposed model for the origin of hydration forces [15] and the treatment of membrane undulations [16] might clarify some of these issues. Firstly, the physical origin of the empirical parameter χ is not known. As suggested by Petrache et. al [13], it might simply represent a correction due to the interlamelar salt deficit (compared to the salt concentration in reservoir). However, since the Debye–Hückel length is inverse proportional to the square root of the ionic strength, to account for the value χ = 0.2 the

Nanodispersions

544 58

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

concentration of the interlamellar electrolyte must be about 25 times smaller than that in the reservoir. Such a salt deficit should provide a strong osmotic pressure, particularly at high ionic strengths, which would affect strongly the interactions between the lipid bilayers. Such an interaction has not been taken into account in the calculation of the Hamaker constant [14], where it was assumed that only the van der Waals interactions are affected by the salt addition. The second issue is the extent of the decrease of the van der Waals interaction. Experiment and calculation of the van der Waals interactions between polystyrene latex beads and either a bare glass plane or a polystyrene coated glass plane [17] revealed that the Hamaker constant decreases only by about 25% at complete screening, while the experiments of Petrache et al. for neutral lipid bilayers require a decrease of about 75% (from 1.2kT to 0.4kT ). Such a strong decrease of the van der Waals interaction upon addition of salt would be expected to have strong consequences in the general theory of colloid stability, and not only in the stability of lipid bilayers. The third issue is that, in determining the appropriate values of the Hamaker constant, it was supposed that both the exponential hydration (Eq. (1)) and the Helfrich repulsion (Eq. (3)) remain unchanged upon addition of a salt (and that the osmotic pressure due to the interlamellar salt deficit, discussed above, is negligible). While the exponential form for the hydration force has been determined at high osmotic pressures (corresponding to separations of around 10 Å) it is not clear that this behavior will remain the same at distances of the order of 40 Å. For a decay length of 2.2 Å [14], the magnitude of the hydration forces decreases about 106 times between 10 and 40 Å, and even a minuscule deviation from the exponential behavior will have drastic consequences in the evaluation of the Hamaker constant. The problems associated with a constant Helfrich repulsion (due to the thermal undulations of the bilayers) are even more complicated and will be addressed in detail in Section 3. The forth issue is the increase in the repulsion between bilayers at short distances. In Fig. 1, the osmotic pressure is plotted as a function of separation distance (data from Ref. [13]) for no added salt, for 1 M KCl and for 1 M KBr. They reveal an increase in repulsion at short separation distances upon addition of salt. While the relatively small difference between 1 M KCl and 1 M KBr can be attributed to the charging of the neutral lipid bilayers by the binding of Br− (but not Cl− ) [14], the relatively large difference between no salt and 1 M KCl is more difficult to explain. Even a zero value for the Hamaker constant (continuous line (2) in Fig. 1), in the 1 M KCl case, is not enough to explain the increase in repulsion, determined experimentally. The screening of the van der Waals interaction, at distances of the order of three Debye– Hückel lengths (about 10 Å) should lead, according to Petrache et al. calculations, to a decrease of only about 30% of the Hamaker constant (from 1.2kT to about 0.8kT , see Fig. 5C of Ref. [14]). Therefore, an additional mechanism to increase the hydration repulsion or the undulation force (or both) upon addition of salt should exist to explain the experiments.

Fig. 1. Experimental values for the osmotic pressure as a function of separation distance from Ref. [13] (stars: water; circles: 1 M KCl; triangles: 1 M KBr) are compared with calculations based on the simple equations ((1), (3) and (9)), with parameters reported in Ref. [13] (note that in Ref. [13] it was suggested that hydration interaction increases upon addition of salt): AH = 1.6 × 108 N/m2 , λH = 2.1 Å, H = 9.2 × 10−21 J, b = 39 Å, KC = 5.8 × 10−20 J, Afl = 1.06 Å−2 , λfl = 6.0 Å (Line 1). Even for H = 0 and the rest of the parameters as before (Line 2), the repulsion at short separations is weaker than in the experiment. This points out that either hydration and/or undulation forces must increase upon addition of electrolyte.

2. Predictions of the polarization model for hydration forces between bilayers We noted recently that the polarization model for hydration forces [18] can explain a modest increase in repulsion at short separations (mostly via the recombination of the charges of the surface with the ions of the electrolyte, as surface dipoles, which generate a polarization field in the adjacent water molecules). The polarization model predicted also an increase in repulsion at large separations, because of the correlations between neighboring dipoles and the coupling interactions between the electrolyte ions and the water dipoles [15]. The framework of the polarization model is briefly summarized in Appendix A. The coupling between the double layer and the polarization leads to the following relation between the average dipole of a water molecule m and the electric field E = − dψ dz : d2 m(z) dψ(z) = m(z) + ε0 v0 (ε − 1) , (5) dz2 dz where v0 is the volume of a water molecule, ε0 is the vacuum permittivity, ε the dielectric constant of water, and λm the correlation length of neighboring dipoles. When the water is organized in ice-like layers in the vicinity of the surface, the latter has a value of about λm ∼ 15 Å [15], which is much larger than the decay lengths corresponding to both the traditional hydration force (∼1–3 Å) and to the double layer interactions at high ionic strength (λDH ∼ 3 Å at 1 M KCl). The polarization model can be combined with ion-specific effects induced by the ions that approach the surface (with preference for the bulk water for structure making (SM) ions and λ2m

Polarization Model and ion specificity: applications

545

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

preference for the interface for structure-breaking (SB) ones), which can be well described by simple potentials of interactions between the ions and the surface [19]. The chemical potential of an ion of species “i” and of charge qi in the liquid is given by:   μi (z) = μ0i + kT ln γi ci (z) + qi ψ(z) + Wi (z), (6) where k is the Boltzmann constant, T the absolute temperature, and μ0i represents the standard chemical potential of the ions of species “i,” of charge qi and concentration ci (z). The activity coefficient γi of the ions of species “i” will be assumed unity. The interaction free energy Wi (z) should include all the other interactions of the ion with the medium, not accounted for by the mean field electrical potential ψ. Assuming that the system is in contact with an infinite reservoir in which the ion concentrations are c0i , the total charge density is provided by:    qi ψ(z) + Wi (z) ρel (z) = qi c0i exp − , (7) kT i

where Wi (z) is calculated with respect to the bulk. The Poisson equation can be combined with Eq. (7) to yield:    d2 ψ(z) qi ψ(z) + Wi (z) dm(z) ε0 =− qi c0i exp − + . dz2 kT v0 dz i (8) Equations (5) and (8) represent a system of coupled nonlinear equations, which can be solved for ψ(z) and m(z), under appropriate boundary conditions, which are discussed in Appendix A. The various contributions to the free energy are also calculated in Appendix A. In Fig. 2a, the free energies f for the hydration interaction, per unit surface, plotted as functions of the distance 2d between two identical surfaces, are calculated for various concentrations of an 1:1 electrolyte, by assuming that pε⊥ = 2 Debyes (where p⊥ is the dipole moment of the surface dipoles and ε is the dielectric constant in their neighborhood), the average area occupied by each dipole on the surface A = 50 Å2 , the surface charge density σ = 0, and neglecting the SM/SB interactions of the ions (e.g., WC ≡ WA ≡ 0, see Appendix A). While at large separations the hydration increases strongly upon addition of electrolyte, in the vicinity of the surface the repulsion actually decreases at high ionic strengths. However, this repulsion is smaller, at any distance, than the van der Waals attraction (continuous line in Fig. 2a, calculated with the values H = 9.2 × 10−21 J and b = 39 Å, employed in Refs. [13] and [14]). By assuming that for d < 4 Å, WC = 1kT and WA ≡ 0 (Fig. 2b) or that WC = 1kT and WA = 1kT (Fig. 2c), the free energy of interaction becomes larger in the vicinity of the surface, but still remains smaller than the van der Waals attraction. This means that other forces must be present to explain the equilibrium of neutral lipid bilayers. Of course, one might assume that these additional forces are due to the confinement of the membranes [4]. Their discussion is postponed for the next section. Let us also note that at large separations, the polarization model provides a very large repulsion, hence that the swelling

59

upon addition of salt might be well over-predicted for concentrations of the order of 1 M. In addition, to represents the experiments at high osmotic pressures (see Fig. 1) an exponential hydration (Eq. (1)) with a decay length of about 2.2 Å is required [14], while the polarization model generates a repulsion with a much larger decay length. There is a simple explanation for this behavior. A dipole correlation length λm = 14.9 Å has been employed for water that is strongly organized in icelike layers in the vicinity of a surface [15]. Since it is obvious that the lipid bilayers undulate thermally (otherwise, their anomalous swelling or their unbinding cannot be explained), it is impossible for the water molecules to remain organized in ice-like layers near the undulating bilayers. A model involving the assumption of a totally random orientation of water molecules, proposed recently [20], indicates that the correlation length might become in this case as low as λm = 6.27 ε Å, where ε is the dielectric constant for the screening of the interactions between neighboring atoms (much lower than the bulk dielectric constant of water, ε = 80). A value λm = 4 Å has been shown to represent well the hydration forces between silica surfaces [20]. However, a low value for λm still does not provide a viable solution for the interactions between neutral lipid bilayers; indeed, for λm = 4 Å, the free energy decreases sufficiently rapid in the vicinity of the surface (Fig. 3a, pε⊥ = 2 Debyes, A = 50 Å2 , σ = 0 and WC ≡ WA ≡ 0), but still remains much smaller that the van der Waals interaction (continuous line, Fig. 3a). This result can be easily understood. Indeed, if the coupling between dipoles is small, the energy stored in the dipole–dipole interactions is also small. To generate a hydration interaction comparable with the van der Waals attraction, huge dipoles (about 50 Debyes for each 60 Å2 of surface) must reside on the surface, which is unrealistic. To form an idea about the magnitude of the repulsion required to explain the stability of neutral lipid bilayers, in Fig. 3b the exponential free energy of hydration derived in Ref. [13] to fit the experiments z on lipid bilayers (namely, fhyd (z) = 0.0336 exp(− 2.1 ) J/m2 is compared with an exponential “hydration force” employed z recently, fhyd (z) = 0.00053 exp(− 5.4 ) J/m2 [20] to fit the experiments on the interactions between silica surfaces [21]. The later surfaces are known for their strong hydration interactions; silica colloids typically do not coagulate, regardless of the ionic strength. In the vicinity of the surface it seems that the hydration repulsion in neutral lipid bilayers is about two orders of magnitude larger than those in silica, which is again unrealistic. It is therefore likely that the thermal undulations play a considerable role for lipid bilayers, particularly at low separations. Let us therefore address in what follows in some detail the issue of the thermal undulations of the membranes. 3. Thermal undulations of lipid bilayers The role of thermal undulations of membranes in their stability [4] and particularly in the unbinding transition [5] has been early recognized. Helfrich derived the first analytical expression for the repulsion due to the entropic confinement of the membrane, by assuming hard-wall repulsions between membranes. When the membranes are far apart, they can undulate freely;

Nanodispersions

546 60

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

(a)

(b)

(c) p

Fig. 2. Hydration interaction energy as a function of distance between surfaces for various electrolyte concentrations and ε⊥ = 2 Debyes, A = 50 Å2 , σ = 0, λm = 14.9 Å, for: (a) WC ≡ WA ≡ 0; (b) WC = 1kT and WA = 0 for z < 4 Å, WC ≡ WA ≡ 0, z > 4 Å; (c) WC = 1kT and WA = −1kT for z < 4 Å, WC ≡ WA ≡ 0, z > 4 Å. The continuous line represents the van der Waals interaction (Eq. (3)), for H = 9.2 × 10−21 J and b = 39 Å.

however, when two membranes approach each other, the hard wall repulsion between them attenuates the undulations. The entropic confinement of the membranes generates a repulsion and the corresponding free energy is inversely proportional to the square of the average separation between membranes [4]. The Helfrich theory is valid for hard-wall interaction between membranes. However, at small separation additional interactions have to be taken into account. For typical membranes interacting via an exponential repulsion (hydration force) and a van der Waals interaction, it was shown that at small separations the undulation interaction has an exponential behavior [16,22,23]. However, the exponential repulsion cannot be valid at large separation, because it cannot predict the unbinding of the membranes (an exponential is always of shorter range that the van der Waals attraction). In general, the undulation repulsion is treated as an additive inter-

action, which is exponential at short separations [22]:  2   kT 1 Afl z pund = exp − , z small, 2π KC λfl λfl

(9)

where the scale factor Afl and the decay length λfl might be regarded as fitting parameters [13], while it has a power law behavior at large separations, as provided by Helfrich [4]: pund = μ

(kT )2 , KC z3

z large.

(2)

The above two different functions have been used by Petrache et al. to fit the interactions between bilayers at small (Ref. [13], Eq. (9)) and large (Ref. [14], Eq. (2)) separations. Since the ranges of validity of the above equations (Eq. (2) and Eq. (9)) are not known, it is clear that the calculation of the Hamaker constant based on either of them is not very accurate.

Polarization Model and ion specificity: applications

547

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

(a)

To account for the interdependence between thermal fluctuations and interaction potentials, we proposed recently an alternate approach [16,24,25], in which the elastic membranes were allowed to undulate thermally in a potential provided by all the interactions between membranes. Perfectly rigid membranes (i.e., KC = ∞) will align themselves parallel to each other, at a separation distance corresponding to the minimum of the interaction potential. However, when KC = ∞, the equilibrium of the membrane involves two contributions to the free energy, which generate a continuous distribution of distances between bilayers, ρund (z) [16,24] (see Appendix B). Large thermal undulations of the membranes (wide distribution ρund (z)) increase the total energy, because parts of the membranes wander into regions of higher potentials; in contrast, the suppression of the undulations (narrow distribution ρund (z), hence low values for the root mean square fluctuation σund ) increases the entropy of the system. The equilibrium between membranes is provided by the minimum of the total free energy of the system, which is the sum between an energetic and an entropic contribution [16,24,25] (see Appendix B). By using an approximate equation for the entropic contribution [16]: (−T S)und =

(b) Fig. 3. (a) Hydration interaction energy as a function of distance between surp faces for various electrolyte concentrations and ε⊥ = 2 Debyes, A = 50 Å2 ,

σ = 0, λm = 4 Å and WC ≡ WA ≡ 0. The thick line represents the van der Waals interaction (Eq. (3)), for H = 9.2 × 10−21 J and b = 39 Å. (b) The exponential approximation for the hydration interaction, required to explain the experiment for neutral lipid bilayers (Ref. [13]) is compared with the exponential approximation for the hydration interaction, required to explain the experiment for silica surfaces [21]. The thick line represents the van der Waals interaction (Eq. (3)), for neutral lipid bilayers (H = 9.2 × 10−21 J and b = 39 Å).

More importantly, the undulation force cannot be treated as an additive interaction [16]. Helfrich’ original theory assumed that the membranes interact only with a hard-wall potential, but when interactions become longer range, they affect themself the undulation of the membranes, contributing to their confinement. In this case, there is a mutual interdependence between the thermal fluctuations and the interaction potentials, which cannot be any longer assumed independent of each other, hence they cannot be simply additive.

61

(kT )2 , 2 64KC σund

(10)

where σund is the root mean square fluctuation of the interbilayer distance (see Appendix B), it was shown that the undulation repulsion can be approximated by an exponential at short separations but has a power law behavior at large separations [16]. When the system is subjected to an external pressure pext , the undulations of the membranes occur in such a manner to minimize the total Gibbs free energy of the system, which is the sum between the enthalpy of the system and the entropic contribution provided by Eq. (10) (see Appendix B):    (kT )2 G(s) = ρund (u) f (u) + pext u du + , (11) 2 64KC σund u

where f is the free energy of interaction per unit area between planar surfaces. For each value of the external pressure, an average separation between membranes can be found. This treatment was presented in details elsewhere [16,24,25], and only a brief summary is reproduced in Appendix B. It should be noted that G(s) represents the total free energy of the system, including the interaction potentials (hydration and van der Waals interactions) as well as the undulations entropy of the membranes, and s is a parameter which is used for the minimization of G. Equation (11) shows that the interactions and undulations are intermingled and that their effects are not additive. Let us evaluate the effect of this treatment of the fluctuations on the interactions between bilayers. Let us first consider the simple interaction potential between membranes, employed by Petrache et al., with the hydration interaction corresponding to Eq. (1):   z fhyd (z) = AH λH exp − (12) , λH

Nanodispersions

548 62

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

Fig. 4. The enthalpy as a function of distance between bilayers for pext = 5 × 106 N/m2 , AH = 1.6 × 108 N/m2 , λH = 2.1 Å, H = 9.2 × 10−21 J and b = 39 Å and the corresponding distribution of membrane interdistances, ρ(z), obtained from the minimization of the Gibbs free energy for KC = 1 × 10−20 J, KC = 5.8 × 10−20 J and KC = 20 × 10−20 J.

where AH = 1.6 × 108 J/m3 and λH = 2.1 Å, and a van der Waals interaction:   H 1 1 2 fvdW (z) = − + − . (13) 12π z2 (z + 2b)2 (z + b)2 For an external pressure pext , the membranes undulations minimize the Gibbs free energy, based on the enthalpy:   z h(z) = pext z + AH λH exp − λH   H 1 1 2 + + − (14) 12π z2 (z + 2b)2 (z + b)2 and the entropic term provided by Eq. (10) (see Fig. 4). Let us note that an additional, hard-wall like repulsion has to be taken into account at short separation distances, otherwise during their undulations, parts of the membranes will approach at distances less than about 1 Å, and therefore will be driven in the deep first minimum of the interaction. However, experiment shows that such a collapse does not occur, and therefore separations less than a cut-off distance dcut-off = 1 Å should be considered inaccessible to the membranes. The distances between membranes will obey a Boltzmann distribution, with the prefactor determined in such a manner to minimize the total Gibbs free energy of the system (a large prefactor corresponds to membranes confined to their minimum enthalpy, but having a large entropy because σund is small; whereas a small prefactor corresponds to a small entropy, but large enthalpy, because the membranes with higher elasticity will undulate at larger distances from the minimum of the enthalpy, see Fig. 4) For the interaction potentials provided by Eqs. (12) and (13) and various values of KC , the average separation as a function of the external pressure is compared in Fig. 5 with experiment (Ref. [13]) for neutral lipid bilayers at different salt concentrations. Note that the force due to the confinement of the undulation is not simply additive to the other interactions (hydration

Fig. 5. Experimental values for the osmotic pressure as a function of separation distance from Ref. [13] (stars: water; circles: 1 M KCl; triangles: 1 M KBr) are compared with calculations for constant parameter values for hydration (AH = 1.6 × 108 N/m2 , λH = 2.1 Å) and van der Waals (H = 9.2 × 10−21 J and b = 39 Å) interactions, but various values for KC .

and van der Waals) and that a change in the elasticity of the membrane can have a strong effect on the equilibrium distance, even if it does not affect much the interactions at short separations. One interesting questions is why the membranes never collapses, even when the undulations might drive part of them at separations less than 1 Å. One possible explanation is provided by the strong, hard-wall like, Born repulsion between the surface dipoles and their bound water molecules. Another possible explanation is that the polar headgroups of the neutral lipid bilayers, being hydrophilic, are more closely related to water than to the hydrocarbon region of the bilayers. Therefore, the van der Waals interaction between neutral bilayers might be better described by [16]:  H 1 1 + fvdW (z) = 12π (z + th )2 (z + th + 2b)2  2 − (15) (z + th + b)2 instead of Eq. (13), where th is related to the thickness of the headgroup of the lipid bilayers (it should be less than twice the physical thickness of the headgroups of each layer, but larger than zero). In this case, the van der Walls attraction does not diverge, even for z = 0, and its magnitude in the vicinity of the surface is drastically affected by the value of th . Since the membranes are typically confined closer to the minimum of the potential, the decrease of the absolute value of this minimum allows the membrane to wander in more distant regions of the potential. Consequently, the decrease in the van der Waals interaction in the vicinity of the interface has a strong effect even at large separations, because of the thermal undulations, and is felt by the system as a strong long-range repulsion. This effect is depicted in Fig. 6, where the pressure is plotted as a function of distance for KC = 5.8 × 10−20 J and various values for th .

Polarization Model and ion specificity: applications

549

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

Fig. 6. Experimental values for the osmotic pressure as a function of separation distance from Ref. [13] (stars: water; circles: 1 M KCl; triangles: 1 M KBr) are compared with calculations for constant parameter values for hydration (AH = 1.6 × 108 N/m2 , λH = 2.1 Å) and KC = 5.8 × 10−20 J, but various values for th for the van der Waals interactions (with H = 9.2 × 10−21 J and b = 39 Å).

4. Discussion and conclusions If one considers that the thermal undulations generate a constant, additive interaction, independent of salt concentration, the polarization model cannot describe the hydration interactions between neutral lipid bilayers, because for λm = 14.9 Å the interaction is too strong at large separations and large ionic strengths, and has a too large decay length at small separations and high ionic strengths. On the other hand, for smaller values of the dipole correlation lengths (e.g., λm = 4 Å) the polarization model predicts a very week hydration force between surfaces, much smaller than the van der Waals attraction. However, it was noted in a previous section that the repulsion due to thermal undulations should not be treated as an additive, constant interaction. Whenever the interactions between membranes (double layer, hydration, van der Waals) change, so does the contribution due to thermal fluctuation, even if the bending rigidity KC remains the same (which is not completely true). In addition to the traditional interactions (double layer, hydration and van der Waals) one has to account also for the Born-like repulsion between surface headgroups, as well as for the possible decrease of the van der Waals attraction between bilayers, due to the polar nature of their headgroups. While both of these effects affect the interaction potential between rigid surfaces only at very short separations (less than 10 Å), the undulations of the membranes propagate them at much larger separations. In the present treatment of the thermal undulation, it is possible to associate the swelling of the membrane with an increase in the hydration forces provided by the polarization model, even if one considers that the bending rigidity KC and the Hamaker constant are not affected at all by the addition of salt. For example, in Fig. 7 we calculated the interactions as function of the distance, using for the interaction potentials the hydration force

63

Fig. 7. Experimental values for the osmotic pressure as a function of separation distance from Ref. [13] (stars: water, circles: 1 M KCl; triangles: 1 M KBr) are compared with calculations that combine the polarization model for hydration forces with the statistical treatment of the undulation, for the followp ing values of the parameters: ε⊥ = 9 Debyes, A = 60 Å2 ,  = 1 Å, σ = 0, λm = 6 Å, dcut-off = 2.5 Å, KC = 4 × 10−20 J, th = 6 Å, H = 9.2 × 10−21 J, WK = 0.5kT for d < 4 Å, WCl = 0 and WBr = 1kT for d < 4 Å, WK , WCl and WBr = 0 for d > 4 Å.

provided by the polarization model (Eq. (A.21)) and the van der Waals interaction (Eq. (15)) using the parameters pε⊥ = 9 Debyes, A = 60 Å2 ,  = 1 Å, σ = 0, λm = 6 Å, dcut-off = 2.5 Å, th = 6 Å, H = 9.2 × 10−21 J, WK = 0.5kT for d < 4 Å, WCl = 0 and WBr = 1kT for d < 4 Å). The effect of the thermal undulations has been calculated as suggested above, by the minimization of the Gibbs free energy of the system, using KC = 4 × 10−20 J. While one can, in principle, explain the swelling of bilayers upon addition of salt even by assuming that the Hamaker constant does not depend on the electrolyte concentration, this was not the purpose of this paper. Note that the value pε⊥ = 9 Debyes employed in the previous calculations is too large for surface dipoles. We fully agree with Petrache et al. that the Hamaker constant is decreased by the addition of salt, maybe not as much as 75% but nevertheless by a measurable quantity. What we tried to emphasize here is that there are so many unknown quantities in the interactions between lipid bilayers, that it is very difficult to obtain reliable information for the dependence of the Hamaker constant on electrolyte concentrations from this type of experiments. Acknowledgments We are indebted to Horia Petrache for valuable discussions and for providing us the data published in Refs. [13] and [14]. Appendix A. Review of the polarization model The interactions of ions with all the charges contained between two planar surfaces, separated by a distance 2d, can be

Nanodispersions

550 64

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

described by a mean field potential ψ(z) which obeys the Poisson equation:  d ε0 E(z) + P (z) = ρel (z), dz

(A.1)

where ε0 is the vacuum permittivity, E ≡ − dψ dz is the electric field, P the polarization and ρel the charge density. At microscopic scale, a water molecule is subjected to a “local” electric field, which is the sum between a “macroscopic field” generated by sources from outside the medium, and a field generated by the other water molecules surrounding it [26]. For a uniformly polarized medium, the latter corresponds to the field generated in a spherical cavity in a medium, due to the polarization of the whole medium (the Lorenz field ELorenz = 3εP0 [26]). Therefore, the local field acting on a water molecule in an uniformly polarized medium is given by: Elocal = E +

P 3ε0

(A.2)

and the average dipole moment of the water molecule, m = P v0 , where v0 is the volume occupied by a water molecule, by [18]:   m m = γ Elocal = γ E + 3ε0 v0 γ = E ≡ ε0 v0 (ε − 1)E, (A.3) 1 − 3εγ0 v0 where γ is the microscopic polarizability. If the polarization is nonuniform, an additional field EP is generated by the neighboring dipoles. A simple way to calculate this field [15] is to neglect the field generated by the remote dipoles, because it is screened by the intervening water molecules, and to consider only the field generated by the neighboring dipoles, screened by an effective dielectric constant ε  , much lower than the bulk value, ε = 80. In order to calculate the additional field EP , the local structure of water must be known. A simple assumption is that the water is organized in ice-like layers of thickness in the vicinity of the surface (see Fig. A.1). The water molecules near the surface dipoles (gray area in Fig. A.1) are assumed to be bound to the surface dipoles and the distance between the center of the first ice-like water layer and the center of the surface dipoles is denoted  . The additional field in layer “i,” EP ,i is generated by the neighboring water molecules from the same layer “i,” as well as by the neighboring water molecules from the adjacent layers, “i − 1” and “i + 1.” The average dipole moment of a water molecule, mi , is constant in each layer, because the layers are assumed parallel to the surface, and m is a function of the distance from the surface z alone. Since the field generated by the dipoles is proportional to their dipole moments, the additional field can be written as: EP ,i = C1 mi−1 + C0 mi + C1 mi+1 ,

(A.4)

where [15]: C0 = −

3.766 , 4πε0 ε  l 3

C1 =

1.827 , 4πε0 ε l 3

(A.5)

Fig. A.1. A sketch of water layering in the vicinity of a dipoles-bearing surface. In the vicinity of the surface dipoles p, the bounded water (gray) has the dielectric constant ε  , the center of the first water layer is at the distance  from the center of surface dipoles.

l is the distance between the centers of two adjacent water molecules, and ε the dielectric constant for the interaction between neighboring molecules, which is expected to be much lower than 80. For a nonuniform polarization, the local field is therefore given by: Elocal,i = E +

mi + C0 mi + C1 (mi−1 + mi+1 ) 3ε0 v0

(A.6)

and the average dipole moment of a water molecule by:   m γ mi = γ Elocal = γ E + + EP = (E + EP ) 3ε0 v0 1 − 3εγ0 v0 = ε0 v0 (ε − 1)(E + EP )   = ε0 v0 (ε − 1) E + C0 mi + C1 (mi−1 + mi+1 ) .

(A.7)

By expanding in series the average dipole moment, Eq. (A.6) becomes:   d2 m(z) EP (z) = C0 m(z) + C1 2m(z) + 2 + · · · dz2 d2 m(z) ∼ , (A.8) = C1 2 dz2 where is the distance between the centers of two adjacent water layers. Equations (A.7) and (A.8) lead to: d2 m(z) dψ(z) = m(z) + ε0 v0 (ε − 1) dz2 dz with  1/2 λm = ε0 v0 (ε − 1)C1 λ2m

(A.9)

(A.10)

representing a “dipolar correlation length.” For ε = 80, = 3.68 Å, v0 = 30 Å3 , l = 2.76 Å, ε = 1, Eq. (10) provides λm = 14.9 Å. The chemical potential of an ion of species “i” and of charge qi in the liquid is given by:   μi (z) = μ0i + kT ln γi ci (z) + qi ψ(z) + Wi (z), (A.11) where k is the Boltzmann constant, T the absolute temperature, and μ0i represents the standard chemical potential of the ions of species “i,” of charge qi and concentration ci (z). The activity coefficient γi of the ions of species “i” will be assumed unity. The interaction free energy Wi (z) should include all the other interactions of the ion with the medium, not accounted

Polarization Model and ion specificity: applications

551

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

for by the mean field electrical potential ψ , such as the image forces [27], excluded volume effects [28], ion-dispersion [29] or ion-hydration forces [19]. Assuming that the system is in contact with an infinite reservoir in which the ion concentrations are ci0 , the total charge density is provided by:    qi ψ(z) + Wi (z) ρel (z) = qi c0i exp − , (A.12) kT i

where Wi (z) is calculated with respect to the bulk. Equations (A.1) and (A.12) can be combined to yield:    d2 ψ(z) qi ψ(z) + Wi (z) dm(z) ε0 = − q c exp − + . i 0i kT v0 dz dz2 i (A.13) Equations (A.9) and (A.13) represent a system of coupled nonlinear equations, which can be solved (under appropriate boundary conditions) for ψ(z) and m(z). In a homogeneous medium a surface dipole density does not generate an electric field above the surface, and consequently it is ignored by the traditional theory of the double layer, in which the double layer is generated by surface charges alone. In the present treatment, both the surface dipole density and the surface charge density generate the double layer. The first boundary condition at the surface is provided by integrating the Poisson equation (A.1) over the volume of a flat box, which includes the surface, with the large sides parallel to the surface and a vanishingly thin width. After using the Gauss theorem, one obtains:   ∂ψ(z)  m(z)  ε0 (A.14) − = −σ, ∂z  v  z=−d

0

 m(z)x=0 = 0.

65

(A.19)

Once the system of Eqs. (A.9) and (A.13) is solved for the boundary conditions ((A.14), (A.15), (A.18) and (A.19)), the total free energy of the system can be calculated by adding various contributions, due to the electric field, entropy of ions, chemical energy and interactions between ions and surfaces. The electrostatic free energy includes the usual macroscopic free energy density 12 ED (with the displacement field D = ε0 E + P ) and also the interaction of the dipoles with the additional field EP due to the nonuniform polarization [15]: d fel (2d) = −d

1 = 2

1 1 E(ε0 E + P ) − P EP dz 2 2

d  −d

 dψ(z) m(z) dψ ε0 − dz v0 dz

− C1 2

m(z) d2 m(z) dz. v0 dz2

(A.20a)

The entropic contribution of all the ion species present in the electrolyte, with respect to a homogeneous system of constant ion concentrations, is given by [30]: fent (2d) = −T S d      ci ci ln − ci + ci0 dz, ci0

= kT

(A.20b)

i −d

z=−d

where z = −d indicates one of the surfaces, the separation distance between the identical surfaces being 2d. The second boundary conditions at the surface is provided by the polarization of the water molecules from the vicinity of the surface,   which is caused by the macroscopic field E = − ∂ψ(z) ∂z z=−d , the field ES generated by the surface dipoles and the field ESP due to the neighboring water dipoles [15]:   m(0) m(0) = γ E + + ESP + ES 3ε0 v0 = ε0 v0 (ε − 1)(E + ESP + ES ) (A.15)

where “i” runs over all the ions species, and the change in the chemical energy, with respect to infinite separation, due to the charging of the surface is provided by [31]:

with [15]:

fion-surf (2d)

ESP = C0 m(0) + C1 m( ), and [15]:   p⊥ 1 1 ES = ,  2 )3/2 ε 2πε0 ( A + π

(A.16)

(A.17)

where A is the area occupied by each surface dipole. For two identical surfaces immersed in an electrolyte separated by a distance 2d, two additional boundary conditions are provided by the symmetry with respect to the middle distance, z = 0:  dψ(z)  = 0, (A.18) dz z=0

σ (l) fch (2d) = −

ψS (σ ) dσ,

(A.20c)

σ (∞)

where σ is the surface charge density and ψS the surface potential. The free energy due to the interactions between the ions and the surface is given by [19]:

  d  qA ψ − WA = WA (z)cE exp kT −d   −qC ψ − WC + WC (z)cE exp dz. kT

(A.20d)

The force per unit area is consequently obtained from the derivative: p(2d) = −

d(fel (2d) + fent (2d) + fch (2d) + fion-surf (2d)) . d(2d) (A.21)

Nanodispersions

552 66

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

Appendix B. Thermal undulation of bilayers There is only one case in which the thermal undulations of a membrane can be evaluated analytically, namely when one membrane undulates in a harmonic potential. Let us assume that the position, along the z direction, of each point of the membrane is described by the function u(x, y), with u(x, y) ≡ 0 corresponding to a nonundulating membrane (see Fig. B.1). Assuming that the total area of the membrane is constant, the Hamiltonian is given by [4,32]:   2   1 ∂ u(x, y) ∂ 2 u(x, y) 2 H = dx dy KC + 2 ∂x 2 ∂y 2  1 + Bu2 (x, y) , (B.1) 2 where KC is the bending modulus of the membrane and B is the compression modulus (the spring constant, per unit area of the bilayer of the harmonic potential of interaction). Denoting by u(q ˜ x , qy ) the Fourier transform of u(x, y), the average energy of each mode for a membrane of unit area was obtained from the equipartition principle as [4,32]:    2 1 1 1 u˜ KC q 4 + B = kT . (B.2) 2 2 2 Hence the mean square fluctuation is given by: 2 σund

∞  2   2 kT 2πq dq kT = u = u˜ = = √ 2 4 +B (2π) K q 8 KC B c qx ,qy 0

(B.3) and the distributions of the positions of the membrane is Gaussian:   1 u2 ρund (u) = √ exp − 2 . (B.4) 2σund 2πσund This distribution is formally identical with the normal (Boltzmann) distribution of small independent, planar membranes of area s in the potential 12 Bu2 [24]:

  sB Bu2 ρund (u) = exp −s , (B.5) 2πkT 2kT

if the area s of the small planar pieces of the membranes, independently moving but nonundulating, is selected as:

kT KC =8 s= . (B.6) 2 B Bσund A useful extension of this method consists in assuming that the membrane acquires a Boltzmann-like distribution of type (B.5) even for an arbitrary, nonharmonic potential f (u), which has a minimum at u = 0 [24,25]. When an external pressure pext is applied to the undulating membrane, the relevant statistical potential is the enthalpy h(u) = f (u) + pext u and the membrane distribution is provided by:   1 (f (u) + pext u) ρund (u) = exp −s , (B.7) N kT where the constant N can be obtained by normalization,  N = ρund (u) du. (B.8) u

The problem is to calculate the proper value for the area s, since Eq. (B.6) is not valid for arbitrary (inharmonic) potentials. Let us return to the elastic membrane in a harmonic potential. The partition function of the canonical ensemble in the Fourier space can be integrated [4]:       1   Z= du˜ exp −β u˜ 2 KC q 4 + B 2 qx ,qy   2πkT = . (B.9) K q4 + B c q ,q x

y

The difference between the free energies of the free membrane (which formally corresponds to B → 0) and that which interacts with the rigid wall via the harmonic potential 12 Bu2 is given by [4]:  ∞ 1 KC q 4 + B f = kT dq 2πq ln 2 (2π) KC q 4 0  kT B = . (B.10) 8 KC The entropic term of the free energy (per unit area) due to the confinement is obtained by subtracting the interaction energy per unit area from Eq. (B.10), a result which is essentially due to Helfrich [4]:  kT B 1  (kT )2 fent = − B u2 c = . (B.11) 2 8 KC 2 128KC σund

Fig. B.1. The distribution of distances between a thermal undulating membrane (2) in the harmonic potential generated by a rigid membrane (1), which is the same as that of small, independent, planar pieces of a suitable-chosen area S, Boltzmannian distributed in the same potential.

For the case when two undulating membrane interact with each other (instead of one undulating in the potential created by the other rigid membrane), the entropic contribution is provided by [16]: fent =

(kT )2 . 2 64KC σund

(B.12)

Polarization Model and ion specificity: applications

553

M. Manciu, E. Ruckenstein / Journal of Colloid and Interface Science 309 (2007) 56–67

Let us now assume that the functional form for the entropic contribution, Eq. (B.12), holds even when the membrane is subjected to an arbitrary potential, f (u). The total Gibbs free energy of the membrane at a constant external pressure pext is given by:    (kT )2 G(s) = ρund (u) f (u) + pext u du + , (B.13) 2 64KC σund u

where ρund (u) is provided by Eqs. (B.7) and (B.8), while   2 2 σund = u − u ρund (u) du. (B.14) u

Minimization of the Gibbs free energy with respect to s provides the value s0 as a function of the external pressure, which in turn provides the average distance between undulating membranes, for each value of the external pressure [25]:   (f (u)+pext u)  du u u exp −s0 kT u =  . (B.15)  (f (u)+pext u)  du u exp −s0 kT References [1] M. Bloom, E. Evans, O.G. Mouritsen, Q. Rev. Biophys. 24 (1991) 293. [2] E.J. Verwey, J.Th.G. Overbeek, Theory of Stability of Lyophobic Colloids, Amsterdam, Elsevier, 1948. [3] R. Rand, V.A. Parsegian, Biochim. Biophys. Acta 988 (1989) 351. [4] W. Helfrich, Z. Naturforsch. 33a (1978) 305. [5] M. Mutz, W. Helfrich, Phys. Rev. Lett. 62 (1989) 2881. [6] T. Honger, K. Mortensen, J.H. Ipsen, J. Lemmich, R. Bauer, O.G. Mouritsen, Phys. Rev. Lett. 72 (1994) 3911.

67

[7] N. Chu, N. Kucerka, Y. Liu, S. Tristram-Nagle, J.F. Nagle, Phys. Rev. E 71 (2005) 041904. [8] C. Pabst, H. Amenitsch, D.P. Kharakoz, P. Laggner, M. Rappolt, Phys. Rev. E 70 (2004) 021908. [9] R. Lipowsky, S. Leibler, Phys. Rev. Lett. 56 (1986) 2541; S. Leibler, R. Lipowsky, Phys. Rev. B 35 (1987) 7004. [10] B. Deme, M. Dubois, T. Zemb, Biophys. J. 82 (2002) 215. [11] S.S. Korreman, D. Posselt, Eur. Biophys. J. 30 (2001) 121. [12] J. Mahanti, B. Ninham, Dispersion Forces, Academic Press, New York, 1976. [13] H.I. Petrache, S. Tristram-Nagle, D. Harries, N. Kucerca, J.F. Nagle, V.A. Parsegian, J. Lipid Res. 47 (2006) 302. [14] H.I. Petrache, T. Zemb, L. Belloni, V.A. Parsegian, Proc. Natl. Acad. Sci. 103 (2006) 7982. [15] E. Ruckenstein, M. Manciu, Langmuir 18 (2002) 7584. [16] M. Manciu, E. Ruckenstein, Langmuir 17 (2001) 2455. [17] M.A. Bevan, D.C. Prieve, Langmuir 15 (1999) 7925. [18] M. Manciu, E. Ruckenstein, Adv. Colloid Interface Sci. 112 (2004) 109. [19] M. Manciu, E. Ruckenstein, Adv. Colloid Interface Sci. 105 (2003) 63. [20] M. Manciu, O. Calvo, E. Ruckenstein, Adv. Colloid Interface Sci. 127 (2006) 29. [21] J.J. Vale-Delgado, J.A. Molina-Bolivar, F. Galisteo-Gonzalez, M.J. Galvez-Ruiz, A. Feiler, M.W. Rutland, J. Chem. Phys. 123 (2005) 034708. [22] E. Evans, V.A. Parsegian, Proc. Natl. Acad. Sci. 83 (1986) 7132. [23] R. Podgornik, V.A. Parsegian, Langmuir 8 (1992) 557. [24] M. Manciu, E. Ruckenstein, Langmuir 18 (2002) 4179. [25] M. Manciu, E. Ruckenstein, Langmuir 20 (2004) 1775. [26] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [27] B. Jonsson, H. Wennerstrom, J. Chem. Soc. Faraday Trans. II 79 (1983) 19. [28] E. Ruckenstein, D. Schiby, Langmuir 1 (1985) 612. [29] B.W. Ninham, V. Yaminsky, Langmuir 13 (1997) 2097. [30] J.T.G. Overbeek, Colloids Surf. 51 (1990) 61. [31] M. Manciu, E. Ruckenstein, Langmuir 19 (2003) 1114. [32] D. Sornette, N. Ostrowsky, in: W.M. Gelbart, A. Ben-Shaul, D. Roux (Eds.), Micelles, Membranes, Microemulsions and Monolayers, SpringerVerlag, New York, 1994.

Nanodispersions

554

Journal of Colloid and Interface Science 263 (2003) 156–161 www.elsevier.com/locate/jcis

The effect of surface dipoles and of the field generated by a polarization gradient on the repulsive force Haohao Huang, Marian Manciu, and Eli Ruckenstein ∗ Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 9 September 2002; accepted 14 January 2003

Abstract Double-layer and hydration interactions have been coupled into a single set of equations because both are dependent on the polarization of the water molecules. The coupled equations involve the electric fields generated by the surface charge and surface dipoles, as well as the field due to the neighboring dipoles in water. The dipoles on the surface are generated through the counterions’ binding to sites of opposite charge. The equations obtained were employed to explain the restabilization observed experimentally at large ionic strengths for colloidal particles on which protein molecules were adsorbed. Polar molecules adsorbed on a charged surface of colloidal particle can generate a field either in the same direction as that generated by the charge or in the opposite direction. The effect of the sign of the dipole of the adsorbed polar molecules on the interaction between surfaces was also examined.  2003 Elsevier Science (USA). All rights reserved. Keywords: Hydration force; Double-layer interaction; Surface dipoles; Restabilization

1. Introduction The traditional double-layer theory combines the Poisson equation with the assumption that the polarization is proportional to the macroscopic electric field, and uses Boltzmann distributions for the concentrations of the ions. The potential of mean force, which should be used in the Boltzmann distribution, is approximated by the mean value of the electrical potential. The macroscopic field E and the polarization P are related via the Poisson equation dE dP =ρ− , (1) dz dz where ρ is the charge density due to the ions, ε0 is the vacuum permittivity, and z is the distance from the middle between two surfaces, which are considered planar and parallel. In the traditional treatment, E and P are assumed to be proportional: ε0

P = χE,

(2)

χ being the electrical susceptibility. Equation (2), however, can be used only when the electric field is sufficiently uni* Corresponding author.

E-mail address: [email protected] (E. Ruckenstein).

form, hence when the polarization gradient is sufficiently small. When this condition is not satisfied, the polarization gradient generates an additional electric field, because the fields acting on a water molecule due to the neighboring water dipoles do not compensate each other any longer. Only the fields induced by the neighboring dipoles are important because those caused by the more remote ones are screened by the intervening water molecules. Indeed, the dielectric constant is unity between two neighboring dipoles, but very large for the remote ones [1]. A large polarization gradient is generated in an aqueous electrolyte solution, either when the surface charge density is large or/and when there are dipoles on the surfaces. A general theory accounting for the above effects was developed in a previous article [2]. Two applications of this theory are examined in the present article. First we consider a colloidal dispersion or an emulsion stabilized by the adsorption of an ionic surfactant or a protein. In this case, the dissociation of the surfactant or protein molecules generates a charge that in the traditional theory is responsible for the stability of the system. The classic theory also predicts that, as a result of screening, the repulsive double-layer force decreases as the ionic strength increases. There are, however, experimental results [3] that show that, as the concentration of electrolyte increases, the stability ratio decreases and passes through a

0021-9797/03/$ – see front matter  2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(03)00070-5

Polarization Model and ion specificity: applications

555

H. Huang et al. / Journal of Colloid and Interface Science 263 (2003) 156–161

minimum, after which it increases. In other words, the colloidal system is restabilized at sufficiently high electrolyte concentrations. To explain the restabilization, we examine a colloidal system stabilized with adsorbed protein molecules in the presence of NaCl as electrolyte. The increase in electrolyte concentration displaces the dissociation equilibria to smaller degrees of dissociation. While the positive and negative surface charges are thus decreased, each of the neutralized charges is replaced by an ion pair (dipole). In addition, when the polarization gradient is sufficiently large, the neighboring dipoles in water generate a strong electric field. At high electrolyte concentrations the repulsion is dominated by the field generated by the surface dipoles, which propagates through water because of the local interactions between its molecular dipoles; this field is responsible for the restabilization. Second we examine the case in which polar molecules present in water are adsorbed on the surface. They can polarize the water molecules nearby in the same direction or the direction opposite of that generated by the charge. This means that the adsorption of the polar molecules is expected to affect the repulsion. The basic equations derived in previous papers [1,2] are first summarized; this is followed by examination of the above-described two cases. In contrast to the previous article in which the calculations were carried out in the linear approximation, nonlinear equations are employed in the present article.

Let us consider that near a surface the water is composed of a succession of ice-like layers, parallel to the surface, which are assumed to be planar. The electric field Ep that acts on a water molecule of layer i and is generated by the neighboring water molecules through their dipoles can be expressed as [2,4] Ep (zi ) = C1 mi−1 + C0 mi + C1 mi+1 ,

(3)

where mi is the average dipole moment in layer i, C0 and C1 are interaction coefficients given by 3.766 1.827 , C1 = , (4) 4πε0ε  l 3 4πε0 ε l 3 l is the distance between the centers of two adjacent water molecules, and ε is the dielectric constant for the interaction between neighboring molecules, taken as unity in the present calculations. Equation (3) implies that only the neighboring molecules of layers i ± 1 and layer i provide contributions. This assumption is reasonable because, as already mentioned, the more remote dipoles are screened by the intervening water molecules. Expanding in series and neglecting higher-order terms, one obtains C0 = −

Ep ≈ (C0 + 2C1 )m + C1 2

where  is the distance between the centers of two adjacent water layers, and m(z) is the dipole moment as a function of z. In traditional Lorentz–Debye theory, the local field that acts on a water dipole is given by E + P /3ε0 , where E is the macroscopic field and P is the polarization [5]. In the present case, the local field Elocal should also include the field Ep generated by the neighboring dipoles of a water molecule. Consequently, Elocal = E + Ep + P /3ε0 = E + C1 2 (∂ 2 m/∂z2 ) + P /3ε0 .

∂ 2m ∂ 2m ≈ C1 2 2 , 2 ∂z ∂z

(5)

(6)

The average dipole moment of a water molecule is given by m = γ Elocal , where γ is the polarizability. Using for γ the Clausius–Mossotti equation, γ = 3ε0 v0 (ε − 1)/(ε + 2), where v0 is the volume of a water molecule and ε is the dielectric constant of water, and because P = m/v0 , one finally obtains m(z) = ε0 v0 (ε − 1)E(z) + ε0 v0 (ε − 1)C1 2

∂ 2m . ∂z2

(7)

Assuming Boltzmann distributions for the ions, one can write ρ = −2cE e sinh(eΨ/kT ),

2. Theoretical framework

157

(8)

where cE is the bulk electrolyte concentration. By combination of Eqs. (1) and (8), the Poisson equation becomes [2]
 ∂ 2 Ψ (z) 2ecE eΨ 1 ∂m(z) = sinh + . ∂z2 ε0 kT ε0 v0 ∂z

(9)

Equations (7) and (9) constitute a system of equations for Ψ and m. To obtain particular solutions of Eqs. (7) and (9), boundary conditions must be provided. When there are dipoles on the surface, the average field that they generate and that is acting on the first water layer is given by [4]   Es = (p/ε )/ 2πε0 (A/π +  2 )3/2 ,

(10)

where p is the normal component of the dipole moment of the surface dipoles, ε is the local dielectric constant of the medium, A is the surface area per dipole of the surface (1/A is the surface dipole density), and  is the distance between a surface dipole and the center of the first layer of water molecules. The electric field that is acting on the first water layer and is induced by the nearby water molecules can be obtained from Eq. (3) in which the term containing mi−1 is absent: Ep1 = C0 m1 + C1 m2 .

(11)

Nanodispersions

556 158

H. Huang et al. / Journal of Colloid and Interface Science 263 (2003) 156–161


 (−1)i+1 eΨ + exp − 1 dz. kT

Consequently, the dipole moment in the layer of water closest to the surface is given by m|z=−d = γ Elocal = γ (E + Es + Ep1 + P /3ε0 ) = ε0 v0 (ε − 1)(E + Es + Ep )|z=−d = −m|z=d ,

(12)

where 2d is the distance between the two plates. The electroneutrality condition provides a second boundary condition:  d d
1 1 ∂ 2Ψ 1 ∂m ρ dz = ε0 2 − dz 2 2 v0 ∂z ∂z −d −d 
  1 ∂Ψ  = ε0 2 ∂z z=d    ∂Ψ  1 − − (m|z=d − m|z=−d ) ∂z z=−d v0   ∂Ψ  1 = ε0 − m|z=d . ∂z z=d v0

σ =−

(13)

Expressions for the surface charge σ and dipole densities are derived in the next section, where the dissociations of the surface groups are taken into account. The free energy of the system per unit area contains a number of contributions, F = Fchem + Fele + Fentropy ,

(14)

where Fchem is the chemical, Fele the electrostatic, and Fentropy the entropic contributions. Fchem = −2σ Ψs at constant surface potential Ψs and 0 at constant surface charge [6,7]. A general expression that is free of any constraint has the form [8] σ(2d)

Fchem = −2

Ψs dσ .

(15)

Equation (15) can be established by decomposing the pathway of Ψs versus σ in a succession of infinitesimal changes at constant charge followed by changes at constant Ψs . The electrostatic free energy due to the electric field, which includes the interactions with the neighboring dipoles, is given by

 d
1 m(z) m(z) Fele = ε0 E(z) + E(z) − Ep (z) dz. 2 v0 v0 −d (16) The excess entropy contribution to the free energy is [7]

d
  ci Fentropy = kT ci ln − ci + ci,0 dz ci,0 i −d

d   (−1)i eΨ

i=1,2 −d

The free energy of the surface layer formed by the surface dipoles and the water molecules between them is assumed independent of 2d.

3. Colloidal restabilization in protein–latex systems Molina-Bolivar and Ortega-Vinuesa have studied the stability of polystyrene colloidal particles stabilized with an adsorbed layer of protein (IgG), and observed that the stability ratio decreased with increasing ionic strength of NaCl and passed through a minimum at sufficiently large electrolyte concentrations, after which the system was restabilized [3]. This anomalous restabilization of the colloidal system at high electrolyte concentrations cannot be explained in the framework of classic DLVO theory. As the concentration of electrolyte increases, DLVO theory predicts that the colloidal system becomes increasingly unstable. In what follows it is shown that the present theory, which takes into account the effect of the fields generated by the surface dipoles and the neighboring water molecules in the liquid, can explain the restabilization. The surface charge is generated through the dissociation of the acidic and basic sites of the protein molecules. However, as the concentration of electrolyte increases, the dissociation equilibria are displaced in the direction of lower dissociations and the charges are replaced by ion pairs (dipoles). Neglecting the adsorption of Cl− anions onto the basic sites, the surface charge can be calculated by considering the equilibria NaCl → Na+ + Cl− , + SA ↔ S− A +H ,

σ (∞)

= −cE kT

(17)

kT


 (−1)i+1 eΨ exp kT

− SB ↔ S+ B + OH , + S− A + Na ↔ SA –Na,

where SA is an acidic site, S− A a dissociated acidic site, SB a basic site, S+ a dissociated basic site, and SA –Na an ion pair B formed on an acidic site. The following equilibrium equations can therefore be written: (1 − w − x)[H+]S , (18) x (1 − y)[OH− ]S KOH = , (19) y (1 − w − x)[Na+ ]S KNa = . (20) w Here x is the fraction of acidic sites occupied by hydrogen ions, w is the fraction of acidic sites occupied by sodium ions, y is the fraction of basic sites occupied by OH− ions, KH =

Polarization Model and ion specificity: applications

557

H. Huang et al. / Journal of Colloid and Interface Science 263 (2003) 156–161

and [H+ ]S , [Na+ ]S , and [OH− ]S are the concentrations of the corresponding ions at the surface. The ionic concentrations at the surface can be related to their bulk concentrations via the Boltzmann expressions
 −eΨS [H+ ]S = [H+ ] exp , (21) kT
 −eΨS [Na+ ]S = [Na+ ] exp , (22) kT
 eΨS [OH− ]S = [OH− ] exp , (23) kT where e is the protonic charge, and ΨS is the surface potential. Consequently, the surface charge density is given by   σ = e NB (1 − y) − NA (1 − x − w) , (24) where NB is the surface density of the basic sites and NA is the surface density of the acidic sites. Because we assume that surface dipoles are formed through the adsorption of Na+ only, the dipole density on the surface (1/A) is given by wNA . The isoelectric point of IgG is between 6.0 and 8.0 [3]. By selecting its value as 7.0, the surface charge becomes zero at pH 7 and Eq. (24) leads to NB (1 − y) = NA (1 − x − w).

(25)

Combining Eqs. (25), (18)–(20), one obtains KOH =

10−7 10−7 NB KH NA

−

NA −NB NA

,

(26)

which allows calculation of KOH in terms of KH and the ratio NB /NA . By use of the Derjaguin approximation, the repulsive interaction energy between two identical spherical particles of radius a is given by ∞ VR = πa

(F − F∞ ) dH ,

(27)

H0

where H0 is the minimum distance between the surfaces of the two spheres, and F∞ is the repulsive free energy per unit area between two infinitely separated plates. The van der Waals attractive interaction between two particles has the form [10] AH 2a 2 2a 2 VA = − + 6 H0 (4a + H0 ) (2a + H0 )2

H0 (4a + H0 ) + ln , (28) (2a + H0 )2 AH being the Hamaker constant. When H0  a, VA can be approximated by VA = −AH a/12H0 . The total interacting free energy VT between the two spheres is therefore given by VT = VA + VR .

(29)

159

The stability ratio W , which constitutes a measure of the stability of the colloidal system, is defined as W=

kr , ks

(30)

kr being the rate constant for rapid coagulation and ks the rate constant for slow coagulation. W , which was determined experimentally via the low-angle scattering method [3,9], is related to the interaction potential via the expression  ∞ β(u)  VT  0 (u+2)2 exp kT du W =  ∞ β(u) (31)  VA  , 0 (u+2)2 exp kT du where β is a hydrodynamic correction factor, u = H0 /a, VT is the total interaction free energy between two colloidal particles, and VA is the interaction free energy due to van der Waals attractive forces. The hydrodynamic correction factor β is taken as unity. The calculations were carried out for the conditions under which the experiments were carried out: pH 8.0 and D = 2a = 204 nm. The values of the parameters are not known. Assuming for the distance between two acidic sites or two basic sites a value of 7 Å, one obtains NA = NB = 2 × 1018 m−2 . Typical values for the pKH of amino acid residues of globular proteins are in the range 1.95–9.5 [11], and the value of 8 × 10−10 M was selected for KH (pKH = 9.1). For the KNa of human erythrocytes, values of the order of 1 M or larger fitted the experimental data reasonably well [12]. The value of 1 M was therefore selected as the KNa of IgG. The value AH = 3.2 × 10−20 J, which has the right order of magnitude, was selected for the Hamaker constant. By use of the above equations and values of the parameters, the nonlinear equations were solved numerically. The boundary value problem (Eqs. (7) and (9)), with the appropriate boundary conditions (Eqs. (12), (13) and the symmetry condition for Ψ ) was solved using the standard shooting method and the bvp4c Matlab routine. Initial guesses were provided by the exact solutions in the linear approximation. The free energy F was obtained by adding the van der Waals interaction to the double-layer interaction calculated using Eqs. (14)–(17) and (27). The stability ratio was calculated by taking 300 Å as the largest value of H0 . The results are compared with the experimental data in Fig. 1 for three values of p/ε  . There are three regions in the figure: the DLVO stabilization region, the rapid aggregation region, and the restabilization region. DLVO theory can explain the behavior in the first two regions, in which the stability of a colloidal system decreases with increasing salt concentration until the concentration reaches the so-called critical coagulation concentration (ccc). Then the colloidal system aggregates rapidly, because the repulsive electric field is suppressed by the high salt concentration. DLVO theory fails to explain the restabilization. However, by coupling the electric fields induced by the surface charge and surface dipoles and by the neighboring dipoles in water, one can obtain agreement with the experimental

Nanodispersions

558 160

H. Huang et al. / Journal of Colloid and Interface Science 263 (2003) 156–161

Fig. 2. Two different kinds of adsorption of dipoles onto a negatively charged surface.

Fig. 1. Stability ratio W versus salt concentration. The experimental data (2) are from Fig. 6 of Ref. [3]. pH 8.0, D = 2a = 204 nm, NA = NB = 2 × 1018 m−2 , KH = 8 × 10−10 M, KNa = 1 M, AH = 3.2 × 10−20 J. (a) p/ε = 1 Debye; (b) p/ε = 1.8 Debye; (c) p/ε = 3 Debye.

results and provide a possible explanation for this phenomenon. The adsorption of Na+ ion on the surface, which is stimulated by higher salt concentrations, increases the density of surface dipoles. The surface dipoles polarize the nearby water molecules, which in turn polarize the next layer and so on. The force generated by the overlap of the polarization layers of the two plates, often called hydration force, is responsible for the restabilization of the colloidal system as soon as the concentration of the electrolyte becomes higher than a critical stabilization concentration (csc).

4. Effect of the dipoles adsorbed on the surface Some polar molecules present in water can be adsorbed onto the surface of colloidal particles. They can polarize nearby water molecules and this polarization propagates from layer to layer. It is assumed that the surface has a fixed charge and that the only surface dipoles are those of the adsorbed polar molecules. Figure 2 illustrates two different kinds of adsorption. In the first one (A), the adsorbed polar molecules expose to the liquid the same charge as that of the surface. The polarization generated by the dipoles of the adsorbed molecules is in this case, in the same direction as that generated by the charge. Consequently, the polarization induced in water is larger than that generated by the surface charge alone. As a result, the repulsion between the two plates is expected to be stronger than that generated by the surface charge alone. In the second kind of adsorption (Fig. 2B), the dipole exposes to the solution a charge opposite of that of the surface. Correspondingly, the dipole generates a field opposite of that generated by the surface charge, and the adsorption is expected to decrease the repulsive force between the two plates. However, if the surface charge is small, then the dipoles of the adsorbed molecules alone generate repulsion.

Fig. 3. Effect of the kind of adsorption on the force between two plates. σ = 0.01 C/m2 , Debye length λD = 10 Å, A = 100 Å2 . (a) p/ε  = 3 Debye; (b) p/ε = 2 Debye; (c) p/ε = 1 Debye; (d) p/ε = 0; (e) p/ε = −1 Debye; (f) DLVO theory; (g) p/ε = −2 Debye.

Figure 3 presents the results of the calculations. They show that at large distances the repulsive force is always stronger than that predicted by DLVO theory, regardless the sign of the surface dipoles. This occurs because the gradient of the polarization generated by the charge induces a field in the same direction as that of the charge, and this makes the interactions to be of longer range than those predicted by the traditional Poisson–Boltzmann equation. At short distances which, however, should not be smaller than two layers of water molecules (∼8 Å) for the calculation to be physically meaningful, the repulsion is stronger than that predicted by DLVO theory when the surface dipoles act in the same direction as the charge and weaker in the opposite case. 5. Conclusions The kind of adsorption of polar molecules onto the surface of particles can affect interaction between particles. If the electric field generated by the adsorbed dipoles has the same direction as the electric field generated by the charge, the adsorbed dipoles increase the repulsion. If the electric field generated by those dipoles is in a direction opposite that generated by the charge, the adsorbed dipoles weaken the repulsion.

Polarization Model and ion specificity: applications

559

H. Huang et al. / Journal of Colloid and Interface Science 263 (2003) 156–161

The theory could successfully explain the restabilization of protein-stabilized polymer latexes; the increase in surface dipole density with increasing electrolyte concentration, generated by the binding of the counterions to sites of opposite charge, is in this case responsible for this effect. This occurs because the repulsive force generated by the surface dipoles more than compensates for the decrease in repulsion caused by screening.

[2] [3] [4] [5] [6] [7] [8] [9] [10]

References [1] D. Schiby, E. Ruckenstein, Chem. Phys. Lett. 95 (1983) 435.

[11] [12]

161

E. Ruckenstein, M. Manciu, Langmuir 18 (2002) 7584. J.A. Molina-Bolivar, J.L. Ortega-Vinuesa, Langmuir 15 (1999) 2644. M. Manciu, E. Ruckenstein, Langmuir 17 (2001) 7582. J.D. Kraus, Electromagnetics, 4th ed., McGraw–Hill, New York, 1992. E.J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. J.Th.G. Overbeek, Colloids Surf. 51 (1990) 61. M. Manciu, E. Ruckenstein, Langmuir 19 (2003) 1114. J.A. Molina-Bolivar, F. Galisteo-Gonzalez, R. Hidalgo-Alvarez, Phys. Rev. E 55 (1997) 4522. R.J. Hunter, Foundations of Colloid Science, Vol. 1, Oxford Science, Oxford, 1987. S.N. Timasheff, in: A. Veis (Ed.), Biological Polyelectrolyte, Dekker, New York, 1970. D.C. Prieve, E. Ruckenstein, J. Theor. Biol. 56 (1976) 205.

Nanodispersions

560

Langmuir 2003, 19, 3049-3055

3049

Colloid Restabilization at High Electrolyte Concentrations: Effect of Ion Valency Eli Ruckenstein* and Haohao Huang Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received October 22, 2002. In Final Form: January 3, 2003 The scope of this paper is to explain the restabilization of colloidal dispersions at sufficiently high electrolyte concentrations. This restabilization could not be explained in the framework of the traditional double layer theory. An improved theory which accounts for the field generated in water by neighboring dipoles (the polarization gradient) as well as the field induced by surface dipoles could, however, explain the experimental results. The latter dipoles are generated by the adsorption of the ions of the electrolyte upon the surface charges with opposite sign formed through the dissociation of acidic and basic sites. This adsorption changes not only the surface charge but also generates ion pairs (dipoles), which induce a field in the neighboring water molecules that propagates further in the liquid. A higher valency has an important effect upon the restabilization because it makes the field induced by the neighboring water molecules stronger and the change of the surface charge with electrolyte concentration greater. Experimental results from literature regarding the stability of a dispersion of protein-covered latexes are explained on the basis of the new theory.

1. Introduction Experiment revealed that, in most cases, the stability of a colloidal dispersion is decreased as the electrolyte concentration increases. The DLVO theory could explain, sometimes quantitatively, these experimental findings. However, many decades ago, Voet1 prepared stable sols of metals (Pt, Pd) and salts (sulfides, halides) in highly concentrated solutions of H2SO4, H3PO4, and CaCl2. Their dilution with water induced coagulation. Similar results have been reported more recently by Healy et al.,2 who observed that amphoteric latex particles did not coagulate at high ionic strength of LiNO3 (>1 M). Finally, MolinaBolivar and Ortega-Vinuesa3 have observed that the stability ratio of a dispersion of polystyrene particles stabilized with adsorbed protein molecules (Ig-G) decreased with increasing electrolyte concentration, passed through a minimum, after which it increased. Consequently, at sufficiently large electrolyte concentrations, the dispersion was restabilized. The traditional theory could not explain the above observations. In recent papers concerned with monovalent electrolytes,4,5 the restabilization was attributed to the following two effects: (i) the reassociation of charges on the surface and their replacement by ion pairs (surface dipoles), and (ii) the fields generated by the surface dipoles and in the bulk by neighboring dipoles. While the surface charge density formed through the dissociation of acidic and basic sites is decreased by the adsorption of counterions, the charges are replaced by ion pairs (dipoles) which polarize the water molecules nearby. This polar* To whom correspondence should be addressed: Telephone: (716) 645-2911 ext. 2214. Fax: (716) 645-3822. E-mail: feaeliru@ acsu.buffalo.edu (1) Voet, A. Thesis, University of Amsterdam, 1935; see also: Kruyt, H. R. Colloid Science; Elsevier: New York, 1952. (2) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156. (3) Molina-Bolivar, J. A.; Ortega-Vinuesa, J. L. Langmuir 1999, 15, 2644. (4) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584. (5) Huang, H.; Manciu, M.; Ruckenstein, E. J. Colloid Interface Sci., in press.

ization propagates, and the overlap of the polarization layers generates a repulsive force (the hydration force) which becomes dominant at high electrolyte concentrations and is responsible for the restabilization. The previously developed theory6 assumed additivity between the traditional double layer force and the hydration force generated by the surface dipoles. Because both forces depend on polarization, a new unitary theory was formulated in which they were coupled in a single repulsive force4 in which the two effects were no longer additive. It is well-known that the valency of the counterion ν has a strong effect on colloidal stability,7 since the critical coagulation concentration is proportional to ν-6. The theoretical confirmation of this Schulze-Hardy rule was one of the major successes of the DLVO theory.8 The scope of the present paper is to examine the effect of the valency of the electrolyte ions on colloidal restabilization. The valency is expected to be important for at least two reasons: (i) the strong effect which it has on the screening of the electric field; (ii) the adsorption of multivalent ions on surface charges of opposite sign which can easily cause a charge inversion on the surface. The presentation is organized as follows: First, an expression for the surface charge density will be derived by assuming that the surface charge is generated by the dissociation of acidic and basic groups and the adsorption of cations and anions of the electrolyte upon the surface charges of opposite sign. Because intuition suggests that a charge reversal should occur with increasing electrolyte concentration and this reversal is compatible with a minimum in the stability ratio, the traditional double layer theory was first employed to verify if it can predict restabilization. The conclusion was that the traditional theory cannot predict restabilization when the electrolyte concentration becomes very large. Further, it will be shown that a unitary theory, which (6) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582. (7) Overbeek, J. Th. G. In Colloid Science; Kruyt, H. R. Eds.; Elsevier: New York, 1952; Vol, 1. (8) Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.

10.1021/la026729y CCC: $25.00 © 2003 American Chemical Society Published on Web 02/27/2003

Polarization Model and ion specificity: applications

3050

561

Langmuir, Vol. 19, No. 7, 2003

Ruckenstein and Huang

involves the charge of the surface, the surface dipole density, and the fields caused by the surface dipoles and by the neighboring water dipoles in the water layer, can provide reasonable results. A comparison with experiment will conclude the paper.

ions and anions N-, and [H+]S, [Mν+]S, [OH-]S, and [N-]S are the concentrations of the corresponding ions near the surface. The surface charge density is obtained by subtracting the negative charge density from the positive one

2. Dissociation and Adsorption of Cations and Anions on Protein Stabilized Latexes

σ ) e[NB(1 - θOH - θN) + (ν - 1)NAθM NA(1 - θH - θM)] ) e[NB(1 - θOH - θN) NA(1 - θH - νθM)] (3)

Molina-Bolivar and Ortega-Vinuesa examined experimentally the stability of a dispersion of polystyrene latexes with chemisorbed protein (Ig-G) molecules upon their surface.3 A latex suspension was introduced into a buffer containing protein molecules and the latter molecules reacted with the chloromethyl groups of the latex surface. After the protein binding, the stability ratio W, the ratio of the rate constants of rapid and slow coagulations, of the latexes in various electrolyte solutions was determined using the low-angle light-scattering technique. Let us first derive an expression for the surface charge generated through the dissociation of the acidic and basic groups of the protein and adsorption of the ions of the electrolyte on the opposite charges of the surface.9 The dissociations of the protein can be represented by the equilibria

where NB and NA represent the numbers of basic and acidic sites per unit area, respectively, and e is the protonic charge. Combined with eq 2a-d and Boltzmann distributions, eq 3 becomes

(

σ)

SA T SA- + H+

(1a)

SB T SB+ + OH-

(1b)

NBKOHKN + eΨs eΨs KOHKN + [OH-] exp KN + [N-] exp KOH kT kT νeΨs ν+ NA(ν - 1)KH[M ] exp - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT (4)

where SA is an acidic site, SA- is a dissociated acidic site, SB is a basic site, and SB+ is a dissociated basic site. Since the cations and anions of the electrolyte can be adsorbed upon the dissociated acidic and, respectively, basic sites, the following adsorption equilibria should be included

where [H+], [OH-], [Mν+], and [N-] are concentrations in the bulk, Ψs is the surface potential, k is the Boltzmann constant, and T is the temperature in K If the adsorption of anions is neglected (such an approximation is reasonable for a negatively charged surface or a sufficiently large KN), eq 4 reduces to

SA- + Mν+ T (SA-Mν-1)

(1c)

SB+ + N- T SB-N

(1d)

where M is a cation of valency ν, (SA-M ) is an acidic site occupied by cation Mν+ and SB-N is a basic site occupied by anion N-. Equilibrium 1c implies that a cation of valency ν is adsorbed on a single monovalent site. Neglecting the interactions between the adsorbed species, one can write the following equilibria ν+

ν-1

KH ≡

(1 - θM - θH)[H+]S θH

(2a)

KM ≡

(1 - θM - θH)[Mν+]S θM

(2b)

(1 - θOH - θN)[OH-]S θOH

(2c)

(1 - θOH - θN)[N-]S θN

(2d)

KOH ≡ KN ≡

where KH, KOH, KM, and KN are equilibrium constants for H, OH, and the cation and anion of the electrolyte, respectively, θH and θM denote the fractions of acidic sites occupied by hydrogen ions and cations Mν+, θOH and θN denote the fractions of basic sites occupied by hydroxide (9) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205.

e

σ)e

( ) ( ( )

(

NBKOH

( ) ( ( )

eΨs KOH + [OH ] exp kT -

)

( ) (

)

)

-

)

)

νeΨs - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT (5) NA(ν - 1)KH[Mν+] exp -

(

)

3. Calculation of the Stability Ratio on the Basis of the Traditional Double Layer Theory The free energy of the system due to repulsion was calculated using the Derjaguin approximation, hence by expressing the repulsive free energy between two spherical particles in terms of that between two parallel plates. The Poisson-Boltzmann equation in one dimension has the form

d2Ψ 2

dz

)-

1

ni0νie exp(-νieΨ/kT) ∑  i

(6)

0

where Ψ is the electrical potential, z the distance from the middle distance between plates, ni0 the number of ions of species i per unit volume in the bulk solution,  the dielectric constant of the medium, and 0 the vacuum permittivity.

Nanodispersions

562

Colloid Restabilization

Langmuir, Vol. 19, No. 7, 2003

3051

The electroneutrality provides the boundary conditions

|

|

dΨ dΨ σ ))dz z)-d dz z)d 0

(7)

where 2d is the distance between the two plates. The repulsive free energy of the system is the sum of three contributions10

F ) Fele + Fentropy+ Fchem

(8)

where Fele, Fentropy, and Fchem are the electrostatic, entropic, and chemical contributions to the free energy, respectively. The quantity needed in the calculation of the stability ratio is the repulsive free energy per unit area and hence the work per unit area needed to bring the two plates from infinity to the distance 2d

∆F ) F(2d) - F(2d)∞)

(9)

The electrostatic and entropic contributions to the free energy can be calculated using the following expressions10

∫-d (dΨ dz )

1 Fele ) 0 2 Fentropy ) kT

∑i ∫-d d

d

2

( () ni ln

ni

ni

0

dz

(10)

)

- ni + ni0 dz (11)

where ni is the number ions of species i per unit volume. The chemical contribution to the free energy can be calculated by decomposing the pathway from infinity to 2d into infinitesimal steps at constant charge, for which the chemical contributions are zero,8 each followed by an infinitesimal step at constant potential, for which the chemical contributions are -2Ψs∆σ.8 The summation over the entire path leads to11

∆Fchem ) -2

σ(2d) Ψs dσ ∫σ(∞)

(12)

The free energy of the surface layer formed of the surface dipoles and the water molecules between them is considered independent of the distance 2d. If the radius of the spherical colloidal particles a is much larger than the shortest distance between the surfaces of two particles, the repulsive free energy between two identical spherical particles at a distance H0 of shortest approach is given in the Derjaguin approximation by12

VR ) πa

∫H∞ ∆F dH

(13)

0

where H ) 2d. The van der Waals attractive energy between two spheres of radius a has the form8

[

AH 2a2 2a2 VA ) + + 6 H0(4a + H0) (2a + H0)2

Figure 1. (a) Stability ratio W against electrolyte concentration C (in M). The calculations were carried out with the traditional theory. NA ) NB ) 2 × 1018 m-2, KH ) 6 × 10-10 M, AH ) 1 × 10-20J, pH ) 8.0, a ) 102 nm, and KN ) 6 × 10-5 M. Key: (1) 3:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.001 M; (3) 3:1 electrolyte with KM ) 0.01 M; (4) 3:1 electrolyte with KM ) 0.1 M; (5) 2:1 electrolyte with KM ) 0.01 M; (6) 2:1 electrolyte with KM ) 0.1 M. (b) Stability ratio W against electrolyte concentration C (in M). The calculations were carried out with the traditional theory. The adsorption of anions is neglected. NA ) NB ) 2 × 1018 m-2, KH ) 6 × 10-10 M, AH ) 1 × 10-20J, pH ) 8.0, a ) 102 nm, and KN ) ∞. Key: (1) 3:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.001 M; (3) 3:1 electrolyte with KM ) 0.01 M; (4) 3:1 electrolyte with KM ) 0.1 M; (5) 2:1 electrolyte with KM ) 0.01 M; (6) 2:1 electrolyte with KM ) 0.1 M.

The colloidal stability can be expressed through the stability ratio W, which is the ratio between the rate constants of rapid and slow coagulations13,14

∫0

∞

W)

∫0

∞

β(u) 2

(u + 2) β(u)

( ) ( )

exp

VT du kT

VA exp du 2 kT (u + 2)

(15)

where AH is the Hamaker constant.

where β is a hydrodynamic correction factor, u ) H0/a, VT is the sum of the repulsive VR and the van der Waals attractive VA free energies between two spherical particles. The hydrodynamic correction factor β was taken as unity. Parts a and b of Figure 1 examine the effect of the valency on the stability ratio in the framework of the

(10) Overbeek, J. Th. G. Colloids Surf. 1990, 51, 61. (11) Manciu, M.; Ruckenstein, E, Langmuir, in press. (12) Hunter, R. J. Foundations of Colloid Science;Clarendon Press: Oxford, England, 1986; Vol. 1.

(13) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97. (14) Derjaguin, B. V.; Muller, V. M. Dokl. Akad. Nauk SSSR (Engl. Transl.) 1967, 176, 738.

ln

H0(4a + H0) (2a + H0)2

]

(14)

Polarization Model and ion specificity: applications

3052

563

Langmuir, Vol. 19, No. 7, 2003

Ruckenstein and Huang

important for monovalent than multivalent ions. Figures 2 and 3, in which the surface potential and the surface charge density are plotted against the electrolyte concentration, confirm the above interpretation. When the cation adsorption is weak, the screening effect of the electrolyte ions is dominant at all electrolyte concentrations. The above calculations indicate that at sufficiently high ionic strengths the traditional double layer theory predicts the screening of the repulsive force. In contrast, experiment appears to indicate that at high ionic strengths a restabilization should occur. 4. The Unitary Theory and Its Application to the Calculation of the Stability Ratio

Figure 2. Reversal of the surface potential for a single plate. The calculations were carried out with the traditional theory. Key: (1) 2:1 electrolyte withKM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.01 M; (3) 2:1 electrolyte with KM ) 0.1 M. The other parameters are as in Figure 1a.

It is well-known that the traditional double layer theory is valid in a limited range of concentrations for monovalent electrolytes, but is much less valid for higher valency electrolytes.15 The traditional theory starts from the Poisson equation

0

dE dP )Fdz dz

(16)

where F is the charge density, E is the macroscopic field, and P is the polarization. The main assumption in that theory is that

P ) χE

Figure 3. Reversal of the surface charge for a single plate. The calculations were carried out with the traditional theory. Key: (1) 2:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.01 M; (3) 2:1 electrolyte with KM ) 0.1 M. The other parameters are as in Figure 1a.

classical theory. In Figure 1a, the anion adsorption equilibrium constant KN ) 6 × 10-5 M, whereas in Figure 1b, KN ) ∞. There is a small difference between the two figures as a result of a low anion adsorption in Figure 1a and zero anion adsorption in Figure 1b. For strong cation adsorption (KM ) 0.001 M), the stability ratio passes through a minimum, followed by a maximum, after which it decreases with increasing electrolyte concentration. However, for low cation adsorptions (KM ) 0.01 or 0.1 M), the stability ratio decreases monotonically with increasing electrolyte concentration. The above behavior is a result of two competing factors: one of them is the screening effect of the electrolyte ions, and the other is the change in the sign of the surface charge. At relatively low ionic strengths, the stability decreases due to ion screening. When the electrolyte concentration becomes sufficiently large, the multivalency and strong adsorption easily generate a change in the surface charge from negative to positive, and the system is restabilized. The stability decreases, however, at sufficiently large electrolyte concentrations because the screening becomes again dominant. It is important to emphasize that for monovalent ions W decays monotonically as the electrolyte concentration decreases. This occurs because the effect of charge inversion is much less

(17)

where χ is the electrical susceptibility, which is considered constant. Equation 17 is however valid only for a uniform field and hence at most when the gradient of the field is sufficiently small. Because, particularly for high valency electrolytes, the variation of the field with the distance is relatively steep, the above expression has to be modified by including the field Ep induced upon a molecule of water by the neighboring water dipoles. In the traditional Lorentz approach, on which the traditional double layer theory is based, the local field that acts upon a dipole is given by E + (P/30).16 In the present case the local field has to contain the additional field Ep induced by the neighboring dipoles

Elocal ) E + (P/30) + Ep

(18)

Ep(zi) ) C1mi-1 + C0mi + C1mi+1

(19)

where

In eq 19, the field is calculated by assuming that the water molecules are organized in icelike layers, mi being the average dipole moment of water in layer i and C0 and C1 being interaction coefficients given by6

C0 ) -

3.766 1.827 ; C1 ) 4π0′′l3 4π0′′l3

(20)

In eq 20, l is the distance between the centers of two adjacent water molecules, and ′′ is the dielectric constant for the interaction between two neighboring molecules. Only the neighboring water molecules are assumed to induce a field upon a water molecule of layer i, because the effect of the more distant molecules is screened by the intervening ones (the dielectric constant ′′ ) 1 for neighboring molecules and around 80 for the more distant ones, at room temperature). (15) Levine, S.; Bell, G. M. Discuss. Faraday Soc. 1966, 42, 69. (16) Frankl, D. R. Electromagnetic Theory; Prentice Hall: Englewood Cliffs, NJ, 1986.

Nanodispersions

564

Colloid Restabilization

Langmuir, Vol. 19, No. 7, 2003

3053

Expanding eq 19 in series, one obtains

Ep ≈ (C0 + 2C1)m + C1∆2

∂2m ∂2m ≈ C1∆2 2 (21) 2 ∂z ∂z

where ∆ is the distance between the centers of two adjacent water layers. The average dipole moment of a water molecule is given by

m ) γElocal

(22)

where γ is the polarizability for which, as in the traditional double layer theory, the Clausius-Mossotti equation

γ ) 30v0( - 1)/( + 2)

(23)

where v0 is the volume of a water molecule, will be employed. Combining eqs 18, 21, 22, and 23, yields

P(z) ) 0( - 1)E(z) + 0v0( - 1)C1∆2

∂2P ∂z2

(24)

Using Boltzmann expressions for the ion distributions, the Poisson-Boltzmann equation becomes

d2Ψ

)-

dz2

1

1 dm(z)

∑ni0νie exp(-νieΨ/kT) +  v  i 0

0 0

dz

(26)

where p is the normal component of the dipole moment of a surface dipole, ′ is the local dielectric constant near the surface, A is the surface area per surface dipole (1/A is the surface dipole density), and ∆′ is the distance between a surface dipole and the center of the first layer of water molecules. The surface area per dipole, and hence the number of dipoles per unit area, is related to the surface potential and electrolyte concentration through the following equation

A)

Equation 19a is obtained from eq 19 by taking into account that for the first layer of water near the surface the layer i - 1 is missing. Consequently, since P ) m/v0

m|z)(d ) 0 v0( - 1)(E + Es + Ep)|z)(d

(25)

Let us now derive the boundary conditions. The adsorption of the ions of the electrolyte changes the charge of the surface and generates simultaneously surface ion pairs (dipoles). Let us first assume that only the cation is adsorbed. The electric field induced by a surface dipole on the first layer of water molecules is given by6

Es ) (p/′)/2π0 (A/π + ∆′2)3/2

Figure 4. Stability ratio W against electrolyte concentration C (in M). The calculations were carried out with the new theory. The adsorption of anions is neglected. NA ) NB ) 2 × 1018 m-2, KH ) 6 × 10-10 M, pH ) 8.0, AH ) 1 × 10-20J, and a ) 102 nm. Key: (1) 3:1 electrolyte with KM ) 0.1 M; (2) 2:1 electrolyte with KM ) 0.1 M; (3) 1:1 electrolyte with KM ) 0.1 M.

KMKH + KM[H + ]s + KH[Mν+]s 1 ) NAθM KHNA[Mν+]s

The electroneutrality condition provides a second boundary condition

σ))

1 2

∫-d F dz ) 21∫-d d

d

[( |

(

0

| ) |

)

∂2Ψ 1 ∂m dz 2 v ∂z 0 ∂z

1 ∂Ψ ∂Ψ 1  - (m|z)d - m|z)-d) 2 0 ∂z z)d ∂z z)-d v0

(

|

]

∂Ψ 1 ) 0 - m ∂z z)d v0 z)d NBKOH

)e

( ) ( ( )

KOH + [OH-] exp

eΨs kT

-

)

)

νeΨs - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT (30) NA(ν - 1)KH[Mν+] exp -

(

)

In addition, due to the symmetry of the system

) {KMKH + KM[H+] exp(-eΨs/kT) + KH[Mν+] ×

m ) 0 and

exp(-νeΨs/kT)}/{KHNA[Mν+] exp(-νeΨs/kT)} (27) By inclusion of the field induced by the surface dipole into the local field that is acting upon the first layer of water molecules, the dipole moment of the first layer of water molecules near the surface is given by

m|z)(d ) γ Elocal ) γ(E + Es + Ep + P/30)|z)(d (28)

(19a)

dΨ ) 0 for z ) 0 dz

(31)

Using eqs 29-31 as boundary conditions, eqs 24 and 25 can be solved to provide the profiles of the electric potential Ψ and the dipole moment m ) Pv0. Because of the interactions with neighboring dipoles, which generate the field Ep, the electrical contribution to the free energy contains an additional term when compared to the traditional theory

where

Ep|z)(d ) -(C0m1 + C1m2)

(29)

Fele )

1 2

∫-dd

((

0E(z) +

)

)

m(z) m(z) E(z) E (z) dz (32) v0 v0 p

Polarization Model and ion specificity: applications

3054

565

Langmuir, Vol. 19, No. 7, 2003

Ruckenstein and Huang

Figure 4, in which the stability ratio is plotted against the electrolyte concentration, contains three domains: the DLVO domain, the rapid aggregation domain, and the restabilization domain. The calculations revealed that the surface dipoles and the additional contribution to the local field generated by the neighboring dipoles in the water layers can account for the restabilization. With increasing electrolyte concentration, the density of surface dipoles increases and the colloidal dispersion becomes increasingly more stable for electrolyte concentrations higher than the critical stabilization concentration. Because of the surface charge reversal as the electrolyte concentration increases, it is more realistic to also include the adsorption of anions among the adsorption equilibria, since they can have a relatively high concentration near a positively charged surface. The adsorption of anions generates surface dipoles with a sign opposite to that generated by the adsorption of cations. The total electric field induced by the surface dipoles is obtained by adding the electric fields induced by the two kinds of ion pairs (surface dipoles)

Es ) (pa/′)/2π0 (Aa/π + ∆′2)3/2 +

Figure 5. Stability ratio W against electrolyte concentration C(in M). The calculations were carried out with the new theory. The adsorption of anions is taken into account. NA ) NB ) 2 × 1018 m-2, KH ) 8 × 10-10 M, pH ) 8.0, AH ) 1 × 10-20J, a ) 102 nm, and KN ) 6 × 10-5 M. Key: (1) 1:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.001 M; (3) 3:1 electrolyte with KM ) 0.001 M.

(pb/′)/2π0 (Ab/π + ∆′2)3/2 (33) where pa, pb are the normal components of the surface dipole moments generated by the adsorptions of anions and cations, respectively, and the surface area per dipole Aa and Ab can be calculated using the following expressions

Aa )

KMKH + KM[H+]s + KH[Mν+]s 1 ) NAθM N K [Mν+] A

H

s

+

) {KMKH + KM[H ] exp(-eΨs/kT) + KH[Mν+] × exp(-νeΨs/kT)}/{NAKH[Mν+] exp(-νeΨs/kT)} (34) KNKOH + KN[OH-]s + KOH[N-]s 1 Ab ) ) NBθN N K [N-] B

)

OH

Figure 6. Relationship between the critical coagulation concentration and valency for the parameters of Figure 5.

s

KNKOH exp(-eΨs/kT) + KN[OH-] + KOH[N-] NBKOH[N-]

m|z)-d ) 0 v0( - 1)(E + Ep + (pa/′)/2π0 (Aa/π +

(35)

where [N-]s and [N-] are the concentrations of N- near the surface and in the bulk. In this case the boundary conditions acquire the forms

σ)-

(

∫-dd Fdz ) 12∫-dd

1 2

(

0

)

∂2Ψ 1 ∂m dz ) v0 ∂z ∂z2 ∂Ψ 1 0 - m ∂z z)d v0

|

|

z)d

NBKOHKN )e + eΨs eΨs KOHKN + [OH-]exp KN + [N-]exp KOH kT kT νeΨs ν+ NA(ν - 1)KH[M ] exp - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT

( ) ( ( )

)

( ) (

)

)

(36) and

∆′2)3/2 + (pb/′)/2π0 (Ab/π + ∆′2)3/2)|z)-d (37) Figure 5, in which the stability ratio is plotted against electrolyte concentration, for cations of valencies 1, 2, and 3 and monovalent anions, shows that the critical coagulation concentration increases with decreasing cation valency (Figure 6). This rule is not valid for the critical stabilization concentration. 5. Comparison with Experiment The experiments carried out by Molina-Bolivar and Ortega-Venuesa with a monovalent cation NaCl and a divalent cation CaCl2 electrolytes, described in the Introduction, are compared with calculated results in Figure 7. The adsorption equilibrium constant KOH was calculated in term of KH using the expression

KOH )

10-7 10 NB NA - NB KHNA NA -7

(38)

Nanodispersions

566

Colloid Restabilization

Langmuir, Vol. 19, No. 7, 2003

3055

which has the right order of magnitude, was selected for the Hamaker constant.12 The values of the other parameters employed in the calculations were: KCl ) 6 × 10-5 M, pa/′ ) 3 D for Ca2+, pa/′ ) 1.8 D for Na+, pb/′ ) -1 D. In these experiments, the diameter of the polystyrene particles was 204 nm, and the pH of the solution was 8.0. The calculations demonstrate that the surface dipoles and the field caused by neighboring molecules in water play important roles in colloid stability and can explain the restabilization.

Figure 7. Comparison between experimental data and calculations for the dependence of the stability ratio W on the electrolyte concentration C (in M). The calculations were carried out with the new theory. NA ) NB ) 2 × 1018 m-2, KH ) 8 × 10-10 M, pH ) 8.0, AH ) 1 × 10-20J, D ) 2a ) 204 nm, and KCl ) 6 × 10-5 M. Key: (1) NaCl with KNa ) 1 M; (2) CaCl2 with KCa ) 0.02 M. 9 and 4 are experimental data for NaCl and CaCl2 taken from Figure 6 of ref 3.

which was obtained from eq 4 at the isoelectric point and in the absence of electrolyte. The value of the pH at the isoelectric point was selected 7.0, on the basis of the experimental values, which are in the range 6.0-8.0. Selecting a distance of 7 Å between two acidic or basic sites, one obtains NA ) NB ) 2 × 1018 m-2. The pKH of amino acid residues of globular proteins are in the range of 1.95-9.5,17 and, for this reason, a value of 8 × 10-10 M was selected for KH (pKH ) 9.1). The value of 0.02 M was selected for the KCa of Ig-G. The value AH ) 1 × 10-20J, (17) Timasheff, S. N. Biological Polyelectrolyte; Veis, A., Ed.; Marcel Dekker: New York, 1970.

6. Conclusions In the framework of the traditional double layer theory, when there is a strong adsorption of a multivalent cation of the electrolyte, the stability ratio of a colloidal dispersion first decreases, passes through a minimum, followed by a maximum, after which it decreases as the concentration of the (multivalent) electrolyte increases. When the adsorption is weak, the stability ratio decreases monotonically with increasing electrolyte concentration. For a monovalent electrolyte, the stability ratio calculated in the framework of the classical double layer theory decreases monotonically with increasing electrolyte concentration. Consequently, in the traditional theory, the ion screening becomes dominant at high ionic strength, whereas experiment appears to show that the colloidal dispersion is restabilized. A new theory was proposed in which the adsorption of the ions of the electrolyte generates surface ion pairs (dipoles), which induce an electric field. In addition, the neighboring dipoles in water also generate a field in water. The restabilization of the colloidal dispersion at sufficiently high ionic strengths appears to be an effect of these fields. The charge inversion produced by the adsorption of the ions of the electrolyte plays also a role particularly for the high valency ions. LA026729Y

Polarization Model and ion specificity: applications

94

567

Langmuir 2005, 21, 94-99

On the Restabilization of Protein-Covered Latex Colloids at High Ionic Strengths Haohao Huang, Marian Manciu,† and Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received July 20, 2004. In Final Form: October 6, 2004 Recent experiments on restabilization of protein-covered latex colloids at high ionic strengths reported by Lopez-Leon et al.1 revealed strong specific anion effects. The same authors also emphasized that a recent polarization model, which involves both hydration and double layer forces, can account only for some of their experimental results but are in disagreement with other experimental results. The aim of the present paper is to show that most experimental results of ref 1 can be described, more than qualitatively, when the association equilibria for all the ions (with both the acidic and basic sites of the protein) are taken into account. As the traditional Poisson-Boltzmann approach, the polarization model neglects additional interactions between ions, and ions and surfaces, not included in the “mean field” electrical potential; therefore, a complete quantitative agreement should not be expected. While many of the discrepancies between calculations and experiment occur at low ionic strengths (10-4-10-2 M), in the range of validity of the traditional DLVO theory, the latter can neither explain them. It is suggested that the structural modifications of the protein configuration induced by the electrolyte are responsible for some of the disagreements between experiment and calculations.

1. Introduction The traditional Derjaguin-Landau-Verwey-Overbeek (DLVO) theory regards the stability of colloids as a balance between a repulsive double layer interaction due to the surface charges and an attractive van der Waals interaction.2 At high electrolyte concentrations, the electrostatic interactions are screened and the van der Waals attraction leads to the coagulation of colloids. While describing qualitatively the behavior of most colloids, this theory is in disagreement with numerous experimental results. For example, it is known that the neutral lipid bilayers3 and the strong hydrophilic colloids (e.g., silica)4 are stable regardless of the ionic strength and an additional repulsion is required to explain those experiments. This repulsion was sometimes associated with the structuring of water in the vicinity of an interface (hence called “hydration force”),4 but there is still no consensus regarding its microscopic origins. One of the models for the hydration force, the polarization model,5 assumes that the hydration force is generated by the local correlations between neighboring dipoles present on the surface and in water. The macroscopic continuum theory, in which water is assumed to be a homogeneous dielectric, predicts that there is no electric field above or below a neutral surface carrying a uniform dipolar density. However, at microscopic level the water is hardly homogeneous, and the electric interactions * Corresponding author: e-mail address, feaeliru@ acsu.buffalo.edu; phone, (716) 645 2911/2214; fax, (716) 645 3822. † Permanent address: Physics Department, University of Texas at El Paso. (1) Lo´pez-Leo´n, T.; Gea-Jo´dar, P. M.; Bastos-Gonza´lez, D.; OrtegaVinuesa, J. L. Langmuir 2005, 21, 87-93. (2) Deryaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633. Verwey, E. J.; Overbeek, J. Th. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (3) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (4) Israelachvili, J. Intermolecular and Suface Forces; Academic Press: New York, 1992. (5) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435.

between neighboring molecules are much less screened than the interactions between remote molecules; consequently, a polarization is induced in water in the vicinity of dipolar surfaces.5-7 It should be noted that Molecular Dynamics simulations reported also a polarization of water in the presence of surface dipoles,8 which is not proportional to the electric field,9 as assumed in the traditional macroscopic theory. Both results support the predictions provided by the polarization model regarding the hydration interactions. The surface dipoles are sometimes generated by the association of the cations and anions with the negative and positive sites, respectively, of the surface. These associations decrease the surface charge but increase the surface dipole densities, and the polarization induced by them has been used10,11 to explain the experimental data regarding the restabilization of protein-covered polystyrene colloids at high ionic strengths reported by MolinaBolivar and Ortega-Vinuesa.12 The present article was stimulated by the recent experimental data on protein-covered latex colloidal systems immersed in various electrolyte solutions: NaCl, NaNO3, NaSCN and Ca(NO3)2, which showed strong specific anionic effects on the restabilization curves.1 In the opinion of Lo´pez-Leo´n et al.,1 the above polarization model for double layer/hydration forces could explain only some of their experiments, but not all of them. However, they assumed that at pH ) 10 the adsorption of anions was negligible; hence “specific anion effects” could not be predicted by their association with the positive sites of the surface. Furthermore, at pH ) 4 they assumed the (6) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7061. (7) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584. (8) Perera, L.; Essmann, U.; Berkowitz, M. L. Prog. Colloid Polym. Sci. 1997, 103, 107. (9) Faraudo, J.; Bresme, F. Phys. Rev. Lett. 2004, 92, 236102-1. (10) Huang, H.; Manciu, M.; Ruckenstein, E. J. Colloid Interface Sci. 2003, 263, 156. (11) Ruckenstein, E.; Huang, H. Langmuir 2003, 19, 3049. (12) Molina-Bolivar, J. A.; Ortega-Vinuesa, J. L. Langmuir 1999, 15, 2644.

10.1021/la0481795 CCC: $30.25 © 2005 American Chemical Society Published on Web 12/08/2004

Nanodispersions

568

Colloidal Restabilization

Langmuir, Vol. 21, No. 1, 2005 95

adsorption of cations to be negligible and the “specific cation effects”, observed experimentally, could not be predicted.1 Strong specific anion effects were reported particularly at low electrolyte concentrations (10-4-10-2 M),1 a range in which the DLVO theory is considered accurate. However, as shown later, the present experimental data cannot be reproduced by the traditional theory in this range of electrolyte concentrations. In the past, no agreement could be obtained, on the basis of the traditional theory, because small changes in the values of the parameters, caused by the nonuniformity of the particles, affected strongly the stability ratio.13 The polarization model provides similar results in the above range of electrolyte concentrations, when the dipole densities are sufficiently low and cannot explain the data. It is clear that a perfect agreement with experiment cannot be provided by a theory which ignores the additional interactions between ions, and ions and surfaces, not included in the mean field potential (such as image forces,14 excluded volume effects,15 and ion-dispersion16 or ion-hydration forces17). However, it will be shown that the experimental results reported by Lo´pez-Leo´n et al.1 can be more than qualitatively reproduced for uniunivalent electrolytes by the present polarization model for hydration/double layer forces, if one accounts for the association equilibria with the surface sites for all the ions present in the electrolyte (H+, OH-, anions, and cations).11 Some additional reasons for the quantitative disagreements, involving the structural modifications of the adsorbed protein layer and the nonuniformity of the colloidal particles, will be also noted. 2. Review of the Theoretical Framework If one assumes that all the interactions between ions are mediated by a “mean field” potential ψ and that the ions are Boltzmannian distributed, the charge density F is given by

F ) - e(cE + cOH) exp

eψ (kT ) + e(c

E

(

+ cH) exp -

eψ kT

)

molecule of volume v0. Equations 1 and 2 can be combined into

0

∂2ψ(z)

( ) ( )

eψ(z) kT eψ(z) 1 ∂m(z) e(cE + cH) exp + (3) kT v0 ∂z

) e(cE + cOH) exp

In a typical “macroscopic” assumption of proportionality between polarization and applied electric field, P ) 0( - 1)E, where  is the dielectric constant, and eq 3 reduces to the traditional Poisson-Boltzmann equation (the concentrations cH and cOH being in general much smaller than cE). However, if the correlations between neighboring dipoles are taken into account, the following constitutive equation relating the polarization to the “macroscopic” electric field is obtained7

λ2m

∂2m(z) ∂z2

) m(z) + 0v0( - 1)

∂ψ(z) ∂z

(4)

where z is the direction normal to the planar surface containing the dipoles. If the “dipole correlation length” λm ) 0, the usual macroscopic constitutive relation P ) 0( - 1)E (and, consequently, the traditional PoissonBoltzmann theory) is recovered. The solutions for ψ(z) and m(z) of the system of eqs 3 and 4 depend not only on the surface charge density but also on the surface dipole densities. The boundary conditions are related to the surface charge and the surface dipoles generated by the association of the cations and H+ with some of the NA acidic surface sites per unit area of the surface and of the anions and OH- with some of the NB basic surface sites per unit area. The details are given elsewhere,11 and the results will only be briefly reviewed here. The surface charge density is given by

(1) σ)

where e is the protonic charge, cE is the bulk electrolyte concentration (assumed for the sake of simplicity uniunivalent), cOH and cH are the concentrations of OH- and H+ in the bulk, respectively, k is the Boltzmann constant, and T is the absolute temperature. The macroscopic electric field E ≡ -∇ψ obeys the Poisson equation

∇(0E + P) ) F

∂z2

(2)

where 0 is the vacuum permittivity and P ) m/v0 is the polarization, with m being the average dipole of a water (13) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1980, 73, 539. (14) Jo¨nnson, B.; Wennerstro¨m, H. J. Chem. Soc., Faraday Trans. 1983, 79, 19. (15) Levine, S.; Bell, G. M. Discuss. Faraday Soc. 1966, 42, 69; Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612; (16) Bostro¨m, M.; Williams, D. M. R.; Ninham, B. W. Phys. Rev. Lett. 2001, 87, 168103. (17) Manciu, M.; Ruckenstein, E. Adv. Colloid Interface Sci. 2003, 105, 63; Ruckenstein, E.; Manciu, M. Adv. Colloid Interface Sci. 2003, 105, 177.

eNB cOH cE eψS 1+ + exp KOH KA kT

(

) ( ) (

eNA (5) cH cE eψS 1+ + exp KH KC kT

) ( )

where KA and KC are the association-equilibria constants for anions and cations, respectively. The average areas occupied by anions, AA, and cations, AC, on the surface are provided by

1+ AA )

1+ AC )

(

)

KA cOH KA eψS + exp KOH cE cE kT NB

( )

KC cH KC eψS + exp KH cE cE kT NA

(6a)

(6b)

Polarization Model and ion specificity: applications

96

569

Langmuir, Vol. 21, No. 1, 2005

Huang et al.

The two boundary condition at the surface (z ) 0) are given by

0

∂ψ(z) m(z) | | ) -σ ∂z z)0 v0 z)0

(7a)

and

(1 - 0v0( - 1)(C0 + C1))m(z)|z)0 0v0( - 1)∆C1

[

0v0( - 1)

|

|

∂m ∂ψ + 0v0( - 1) ) ∂z z)0 ∂z z)0

( )

pA 1 1 ′ 2π0 AA + ∆′2 π

( )

(

)

3/2

+

pC 1 1 ′ 2π0 AC + ∆′2 π

(

)

]

3/2

(7b)

where pA/′ and pC/′ represent the effective dipoles generated by the association of the anions and cations with the positive and negative sites of the surface (note that the fields generated by the two types of dipoles are oriented in opposite directions), respectively, ′ is an effective dielectric constant of water in the vicinity of the surface, ∆ ) 3.68 Å is the distance between two icelike water layers in the liquid, ∆′ ) 2 Å is the distance between the surface dipoles and the first water layer, and the coupling parameters C0 and C1 are given by

Figure 1. Experimental values of the stability ratio of proteincovered latex particles as a function of electrolyte concentration, at pH ) 4.0, reported by Lopez-Leon et al.,1 compared to those calculated from the polarization-based hydration model, for the following parameter values: NA ) 1.2 × 1018 sites/m2, NB ) 2.31 × 1018 sites/m2, AH ) 0.5 × 10-20 J, KH ) 10-6 M, KOH ) 8.95 × 10-8M, KNa ) 0.021 M, (p/′)Na ) 1.8 D; (1) KSCN ) 0.65 M, (p/′)SCN ) -1.65 D; (2) KNO3 ) 0.62 M, (p/′)NO3 ) -1.8 D; (3) KCl ) 0.76 M, (p/′)Cl ) -2.3 D; circles, NaSCN; squares, NaCl; stars, NaNO3.

3. Coagulation of Protein-Covered Latex Particles

Once the system of eqs 3 and 4 is solved under the boundary conditions (7a-d), the total free energy of the system can be calculated by adding the van der Waals interactions between the surfaces to the double layer free energy composed of electrostatic, entropic, and chemical contributions,11 and the stability ratio can be calculated in the usual manner.18

Let us first analyze the stability ratio for particles immersed in an electrolyte of pH ) 4. The densities of acidic and basic sites, as well as their dissociation constants, are unknown. While one of these parameters could be determined as a function of the others from the isoelectric point (IP), even the latter is uncertain: 6.1 < IP < 8.5.19 The following values have been selected for the parameters: NA ) 1.2 × 1018 sites/m2, NB ) 2.31 × 1018 sites/m2, KH ) 10-6 M, and KOH ) 8.95 × 10-8 M. The calculated stability ratio is plotted in Figure 1, using for the cation (Na+) KNa ) 0.021 M, (p/′)Na ) 1.8 D, while for the various anions: KCl ) 0.76 M, (p/′)Cl ) -2.3 D, KNO3 ) 0.62 M, (p/′)NO3 ) -1.8 D, and KSCN ) 0.65 M, (p/′)SCN ) -1.65 D. As shown in Figure 1, a reasonable agreement between experiment and calculations was obtained for uni-univalent electrolytes by selecting a rather low value for the Hamaker constant of the van der Waals interactions between colloidal particles, AH ) 0.5 × 10-20 J. The calculations at pH ) 10.0 are compared in Figure 2 with the experimental data provided by ref 1 for NaN03 and NaCl. The only modifications in the parameter values selected above are for the number of basic sites per unit area and for the Hamaker constant, which were selected in this case as NB ) 1.62 × 1018 sites/m2 and AH ) 0.9 × 10-20 J. Again, qualitative agreement was obtained. There is no doubt that better agreement with experiment could have been obtained by other fittings of the values of the parameters to the experimental data. However, this is not very meaningful. The large number of parameters of unknown values involved in calculations raises doubt whether a good fit can be obtained only for the appropriate values of the parameters. As a matter of fact, because of the large number of parameters, a previous paper reported a good fit for the stability ratio of one of the above systems for different parameter values.10 In Figure 3, the experimentally determined stability ratio1 is plotted for low concentrations of NaSCN and pH

(18) Hunter, R. J. Foundations of Colloid Science; Oxford Science Publications: New York, 1987.

(19) Lo´pez-Leo´n, T.; Jo´dar-Reyes, A. B.; Ortega-Vinuesa, J. L.; BastosGonza´lez, D. Submitted for publication in J. Colloid Interface Sci.

C0 ) C1 )

3.766 4π0′′l3

1.827 4π0′′l3

(8a)

(8b)

l ) 2.76 Å being the shortest distance between two water molecules and ′′ ) 1 the effective dielectric constant for the electrostatic interactions between neighboring water molecules. The boundary condition (7b) takes explicitly into account the polarization generated by the surface dipoles, while eq 7a is a consequence of the Poisson equation (2). For two identical surfaces immersed in an electrolyte separated by a distance 2d, the symmetry provides the other two boundary conditions at the middle distance z )d

|

∂ψ(z) )0 ∂z z)d

(7c)

m(z)|z)d ) 0

(7d)

570

Colloidal Restabilization

Figure 2. Experimental values of the stability ratio of proteincovered latex particle as a function of electrolyte concentration, at pH ) 10.0, reported by Lopez-Leon et al.,1 compared to those calculated from the polarization-based hydration model, for the following parameter values: NA ) 1.2 × 1018 sites/m2, NB ) 1.62 × 1018 sites/m2, AH ) 0.9 × 10-20 J, KH ) 10-6 M, KOH ) 8.95 × 10-8 M, KNa ) 0.021 M, (p/′)Na ) 1.8 D; (1) KCl ) 0.76 M, (p/′)Cl ) -2.3 D; (2) KNO3 ) 0.62 M, (p/′)NO3 ) -1.8 D; stars, NaNO3; squares, NaCl.

Figure 3. Stability ratios at low concentrations of NaSCN at pH ) 10. The calculations have been carried out for NA ) NB ) 0.5 × 1018 sites/m2 and various dissociation constants KNa based on the present model (thick lines) and on the DLVO theory, with different values for KNa (dotted lines), predict almost identical results but varies much more rapidly with electrolyte concentration than the experimental values reported in ref 1 (circles). KH ) KOH ) 10-6 M, (p/′)Na ) 1.8 D, KSCN ) 0.8 M, (p/′)SCN ) -0.8 D, AH ) 0.1 × 10-20 J. (1) polarization model, KNa ) 12.5 × 10-6 M; DLVO, KNa ) 2.5 × 10-6 M. (2) polarization model, KNa ) 23.1 × 10-6 M; DLVO, KNa ) 5.83 × 10-6 M. (3) polarization model, KNa ) 54.7 × 10-6 M; DLVO, KNa ) 12.6 × 10-6 M. (4) polarization model, KNa ) 164 × 10-6 M; DLVO, KNa ) 42 × 10-6 M.

) 10. One can see that this ratio varies extremely slowly over a wide range of electrolyte concentrations. The low values of electrolyte concentrations, at which relatively rapid coagulation occurs (e.g., 1.25 × 10-4 M), suggest a low surface charge density. Therefore, in the calculations based on the polarization model (solid lines), the values NA ) NB ) 0.5 × 1018 sites/ m2 were selected, and the value of the association constant for Na, KNa, was varied to fit at least one experimental point, all the other parameter values remaining unchanged. All the calculations predict a stability ratio that varies much more rapidly with the ionic strength than the experimental one. It is

Nanodispersions

Langmuir, Vol. 21, No. 1, 2005 97

therefore unlikely to find a set of parameters that can reproduce accurately this experimental behavior. At low ionic strengths, the polarization model predicts results which are qualitatively similar to those obtained from the traditional DLVO theory. The main differences are due to the different boundary conditions, generated by the surface dipoles (ignored in the traditional theory). To obtain a quantitative agreement between the two theories, an additional surface charge should be employed in the DLVO calculations, to compensate for the field generated by the surface dipoles. The calculation of the stability ratio was repeated in the framework of the DLVO theory (dotted lines), with the KNa selected to fit each of the experimental data points. As shown in Figure 3, there is good agreement (at low ionic strengths) between the results of the two treatments, and none of them can predict the slow variation of the stability ratio observed experimentally. It should be noted that at large ionic strengths the results of the two theoretical models are drastically different and that the traditional one cannot predict restabilization. One might expect the additional interactions, not included in the electrical potential, between the large structure-breaking SCN- and interface to be responsible for this effect. However, it was shown that these additional interactions (such as ion-dispersion16 or ion-hydration17 forces) are, in general, important only at sufficiently high electrolyte concentrations (above about 0.05 M) but negligible below about 0.01 M. A modality to include in the framework of the polarization model the additional interactions between ions and surfaces, not accounted for by the mean field potential ψ, was recently proposed.20 The previous conclusion, that these additional interactions are negligible at low electrolyte concentration,17 is supported by the new model.20 We are therefore inclined to believe that the non-DLVO behavior reported in the recent experiments at low ionic strengths1 are mainly due to the nonuniformity of the colloidal particles and to the configurational changes of the adsorbed protein, triggered particularly by the added electrolyte. Let us first emphasize that the nonuniformity of the particles affects strongly the stability ratio.13 The proteins were adsorbed on latex particles at pH ) 6, then the pH of the system was changed to either pH 4 or 10 and stored up to 1 week before the light scattering experiments.19 It was observed that the amount of protein adsorbed decreased with time,19 and even if the latex particles were uniformly covered initially (which is unknown), one would expect some “voids” to be generated until the light scattering experiments. The stability ratio depends strongly on the coverage of the latex particles at the points of closest approach, and experiment provides only a statistical average over a polydisperse system. The statistical average might vary slowly with the electrolyte concentration, whereas the stability ratio calculated for identical particles (with constant numbers of acidic and basic sites), which are plotted in Figure 3, varies much more rapidly. An important issue is whether the structure of the adsorbed protein is modified by the addition of electrolyte. The electrolyte is added only at the beginning of the light scattering experiment,1 and one might expect that during the experiment (about 2 min), and at low electrolyte concentrations, not much modification in the structure of the adsorbed layer can occur. This expectation, however, is probably not accurate. The strong specific ionic effects (20) Manciu, M.; Ruckenstein, E. Adv. Colloid Interface Sci., 2004, 112, 109.

Polarization Model and ion specificity: applications

98

571

Langmuir, Vol. 21, No. 1, 2005

determined experimentally at low electrolyte concentrations, which cannot be explained by the existing theories, can be easily understood if one assumes that each type of electrolyte triggers different structural configurations of the adsorbed protein layer. Moreover, electrophoresis experiments, performed on the same system, revealed a rather low IP for the protein-covered latex particles immersed in low ionic strength electrolyte solutions (cE ) 6 × 10-3 M).19 When a protein with an IP ∼ 7 is adsorbed on a polystyrene particle with an IP ∼ 10.5, one expects an intermediate IP to result. The experimental values obtained for the IP (between 5 and 6 for the electrolytes used in the experiment)19 suggest either that many anions collapsed on the surface (perhaps on other places than on the empty basic sites) or that the structure of the protein was in such a manner modified that much more basic sites were exposed to water. A lowering by two units of the IP indicates a quite large change (by orders of magnitude) of the number of surface acidic and basic sites, and such large changes are possible only if structural modifications of the adsorbed layer occur. It should be noted that the Hamaker constant of the van der Waals interaction between particles is strongly affected by the properties of the external layer of the particles. Because the colloidal particles interact through water, the incorporation of water molecules into the adsorbed protein layer decreases strongly the van der Waals interactions between particles. It was previously reported21,22 that large decreases in the effective Hamaker constant (by orders of magnitude) could be achieved if water penetrates the external layer of the particles. Therefore, the relatively low value for the Hamaker constant in our calculations might suggest that the layer adsorbed on the latex particle contains a large amount of water. If the structure of the adsorbed layer is modified during experiment, then a more accurate model is required to take into account the complex structure of the particles. Therefore, one should not expect a perfect match between calculations and experimental results. An important issue that has to be emphasized is that the experimentally determined dependence of the stability ratio on electrolyte concentration, at low ionic strengths, exhibits (at least for NaSCN) a strongly non-DLVO behavior, in a range in which the DLVO theory is considered fairly accurate. Therefore, we are inclined to believe that the electrolyte (even at low ionic strength) induces indeed structural modifications of the adsorbed protein layer at least near the interface. Another issue which has to be clarified is the criticism in ref 1 that the hydration/double layer model fails because it assumes an icelike layered structure of water in the vicinity of an interface, while the structure-breaking ions (such as SCN-) probably destroy this layered structure.1 First, the hydration/double layer model assumes only correlations between neighboring dipoles, not necessarily an icelike structure. The correlations are perhaps stronger when the water is organized in icelike layers (and might even lead to an oscillatory behavior of the polarization)23 but do not vanish in less organized structures; they are only weaker. In fact, the value for the dipole correlation length λm was derived assuming that only a cluster of about 5 Å around the central water molecule preserves locally the structure of ice.7 The influence of disorder on dipole correlation length was discussed elsewhere.23 (21) Johnson, R. E.; Morrison, W. H. Adv. Ceram. 1987, 21, 323. (22) Ruckenstein, E. Colloids Surf. 1993, 63, 271. (23) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582.

Huang et al.

The destruction of the icelike ordering by SCN- might lower the hydration force, although such an effect was not reported for other systems (e.g., neutral lipid bilayers in water). However, the decrease in hydration due to the destruction of the icelike layers cannot explain the experiments of Lo´pez-Leo´n et al.1 The rapid coagulation at low electrolyte concentrations (10-4-10-3 M) is not due to the decrease in the hydration force: at these ionic strengths, even a small surface charge controls the coagulation. The decrease in the hydration force should affect coagulation at high ionic strength; however, at high ionic strength, the system was restabilized by adding NaSCN. Furthermore, the restabilization achieved by the addition of NaSCN is even stronger than that obtained through the addition of NaCl. However, the opposite behavior should have been observed, because the stronger structure-breaking SCN- is expected to destroy the icelike structure more than Cl- and, consequently, would lower more the hydration repulsion. Of course, one might suspect that the strong restabilization by NaSCN might be due to an unusually large negative charge generated on the surface by the accumulation of SCN-, but this hypothesis is invalidated by the electrophoretic experiments, which indicated an almost neutral surface at high ionic strengths.19 4. Conclusion It was recently suggested1 that some of the strong specific anionic effects observed regarding the coagulation of protein-covered latex particles cannot be predicted by a theory of hydration/double layer interaction,5,7 which accounts for the correlation between neighboring dipoles and the formation of dipoles on the surface.10,11 Here it is shown that the above theory can in most cases predict at least qualitatively the experimental behavior. The unusual strong specific anion effects observed for SCN- at low ionic strengths are in disagreement with both the present and the traditional DLVO theory. The present theory is an extension of the traditional theory and predicts qualitatively similar results for sufficiently charged surfaces at low ionic strengths. Notable differences occur between the two treatments either at low surface charges or at high ionic strengths, at which the double layer is screened. The strong discrepancy between experiment and the traditional DLVO theory at low ionic strengths (where the latter theory is considered to be accurate) cannot be explained by additional interactions between ions and surfaces, because they are negligible below 0.01 M. Therefore, we are inclined to believe that the structural modification of the adsorbed protein by the addition of a structure breaking ion, such as SCN- is mainly responsible for the quantitative disagreement between experiment and model calculations. The nonuniformity of the colloidal particles may be also responsible for the disagreement. It is well-known that the structure-breaking anions generate strong specific ion effects for proteins (while the type of cations does not matter much).24 In contrast, the structure-making cations (e.g., K+, Li+, Cs+) have strong specific effects for the interactions between latex25 or mica26 surfaces. An explanation for this apparent paradox might be that the structure-breaking ions are more likely to be (24) Cacace, M. G.; Landau, E. M.; Ramsden, J. J. Q. Rev. Biophys. 1997, 30, 241. (25) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156. (26) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531.

Nanodispersions

572

Colloidal Restabilization

expelled from the bulk water toward the surface, whereas the structure-making ions are in general depleted from the interface.17 The unstable surfaces, such as those of the proteins, which can be easily modified, are particularly sensitive to the ions that can approach them closely, such as SCN-, and modify the water properties near them. In contrast, the interactions between stable surfaces are particularly sensitive to the counterions depletion, because the screening of the double layer is much weaker in this case. Therefore, for negatively charged stable surfaces, one expects strong specific effects for the structure-making cations, such K+, Li+, or Cs+. The strong specific effects obtained experimentally for structure-breaking ions,1 such as SCN-, suggest that the structure of the surface is

Langmuir, Vol. 21, No. 1, 2005 99

modified by the ions; the structural modifications are also expected to occur for multivalence ions, such as Ca2+. Therefore, we suggest that the departure of the experimental data reported in ref 1 from the predictions of DLVO theory (at low ionic strengths) and from the polarization model (at all ionic strengths) are mainly due to the structural modification of the adsorbed protein layer, induced by electrolyte. This conclusion is also supported by electrophoretic experiments performed on the same systems, which showed a drastic change in the isoelectric points of the particles, dependent on the nature of the added electrolyte.19 LA0481795

Polarization Model and ion specificity: applications

573

Advances in Colloid and Interface Science 112 (2004) 109 – 128 www.elsevier.com/locate/cis

The polarization model for hydration/double layer interactions: the role of the electrolyte ions Marian Manciu1, Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, United States Available online 10 November 2004

Abstract The interactions between hydrophilic surfaces in water cannot be always explained on the basis of the traditional Derjaguin–Landau– Verwey–Overbeek (DLVO) theory, and an additional repulsion, the bhydration forceQ is required to accommodate the experimental data. While this force is in general associated with the organization of water in the vicinity of the surface, different models for the hydration were typically required to explain different experiments. In this article, it is shown that the polarization-model for the double layer/hydration proposed by the authors can explain both (i) the repulsion between neutral lipid bilayers, with a short decay length (~2 2), which is almost independent of the electrolyte concentration, and, at the same time, (ii) the repulsion between weakly charged mica surfaces, with a longer decay length (~10 2), exhibiting not only a dependence on the ionic strength, but also strong ion-specific effects. The model, which was previously employed to explain the restabilization of protein-covered latex particles at high ionic strengths and the existence of a long-range repulsion between the apoferritin molecules at moderate ionic strengths, is extended to account for the additional interactions between ions and surfaces, not included in the mean field electrical potential. The effect of the disorder in the water structure on the dipole correlation length is examined and the conditions under which the results of the polarization model are qualitatively similar to those obtained by the traditional theory via parameter fitting are emphasized. However, there are conditions under which the polarization model predicts results that cannot be recovered by the traditional theory via parameter fitting. D 2004 Elsevier B.V. All rights reserved. Keywords: Polarization model; Hydration forces; Double layer interaction; Ion-specific effects; Restabilization

Contents 1. 2.

3.

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The polarization model for hydration/double layer interactions . . . . . . . . . . . . . . . . . . 2.1. The physical basis of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The correlation between neighboring dipoles . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The effect of disorder in the water structure on the correlation length k m . . . . . . . . . 2.4. The Boltzmann distribution of ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Boundary conditions for the polarization model . . . . . . . . . . . . . . . . . . . . . . 2.6. The interaction between identical surfaces immersed in an electrolyte . . . . . . . . . . . Consequences of the polarization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Polarization skin effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Hydration forces between neutral lipid bilayers: independence of electrolyte concentration 3.3. Hydration forces between mica surfaces: ion-specific effects . . . . . . . . . . . . . . . .

* Corresponding author. Tel.: +1 716 645 2911/2214; fax: +1 716 645 3822. E-mail address: [email protected] (E. Ruckenstein). 1 Permanent address: Physics Department, University of Texas at El Paso. 0001-8686/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2004.09.001

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

110 112 112 112 114 114 115 116 117 117 117 120

Nanodispersions

574 110

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

3.4. Long-ranged interactions at high ionic strengths: restabilization 3.5. Role of ion-hydration forces in the polarization model . . . . . 3.6. The effect of the correlation length k m on the interactions . . . 4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Long ago, Gouy [1] and Chapman [2] suggested that the distribution of ions in the vicinity of a charged surface immersed in an electrolyte can be obtained by assuming that the ions interact with all the other charges in the system via a bmean fieldQ electrical potential w that obeys the Poisson equation. Assuming that the ions obey Boltzmann statistics, the bmean fieldQ w and the ion distributions can be calculated for given boundary conditions (typically, relations for the charge or potential on the surfaces). This bPoisson–BoltzmannQ formalism was employed later by Langmuir [3] to explain the force between two charged surfaces immersed in an electrolyte, and remained since then the basic equation for the behaviour of colloids. It is perhaps curious that such a simple theory is still surviving the criticism that it received for almost a century. One reason is that some parameters involved (the surface charge and surface potential) are not accurately known and can be in most cases fitted to account for the experimental data. Sometimes, the results predicted are not achievable physically. A well-known example is the prediction by the traditional Poisson–Boltzmann formalism of a counter-ion density exceeding the available volume in the vicinity of a highly charged surface [4]. Stern corrected this paradox by considering a layer of immobile counter-ions in the vicinity of the interface [4]. Since then, many such corrections have been introduced in the traditional approach, to account for the changes in the dielectric constant of water due to the presence of ions [5], or to high electric field strengths [6], to account for the finite size of the ions [7], for image forces [8], for ion-dispersion [9] and ion-hydration [10] forces, and so on. Another clear failure of the Poisson–Boltzmann approach was provided by the experiments regarding the force between neutral lipid bilayers [11]. The repulsion required to explain their stability was determined to have an almost exponential dependence, with a decay length of about 2–3 2 [11], and neither this decay length nor the magnitude of the interactions were dependent on the ionic strength. This interaction was initially attributed to the structuring of water near the surface (the bhydrationQ of the surfaces) and it is usually called bhydration forceQ [12]. The microscopic origins of this interaction are still under debate. There are good reasons for these disagreements, as it will be noted in what follows. According to the traditional Derjaguin–Landau–Verwey–Overbeek (DLVO) theory [13], most colloids coagulate when the electrolyte concentration

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

120 122 126 127 128

is sufficiently high to screen the repulsion generated by the surface charge; however, the hydration forces are strong enough to prevent the coagulation of silica particles regardless of the electrolyte concentration [12]. Therefore, one open question is why the hydration force is of paramount importance in some cases and negligible in others? Amphoteric latex particles, at pH values larger than those corresponding to the isoelectric point, coagulated when the CsNO3 concentration exceeded about 0.3 M, but remained stable up to concentrations exceeding 3 M of KNO3 [14]. The additional repulsive interaction between surfaces, which could explain the stability, was initially related to the hydration of ions [14]: to achieve contact between surfaces, the ions must lose their hydrating water molecules, a process that is energetically unfavorable. A more involved statistical model of this interaction was suggested recently [15]. Since the energetic penalty is larger for K+ than for Cs+ ion, one could understand why the system is stable with KNO3 but not with CsNO3, and also why at low pH values the system is unstable at high ionic strengths. In the latter case, the anions are attracted near the positive surfaces. Because their hydration energies are lower than those of the cations, the repulsion generated by the dehydration of anions might not be sufficient to prevent coagulation. Also, because the cations are depleted near the positively charged interface, there were no specific cation effects observed in the experiments at pH values lower than the isoelectric point of the particles [14]. The above model for the bhydration forceQ assumes that the ions (particularly, the structure making cations) play a dominant role. This conclusion is supported by many experiments with charged surfaces, which indicate that the traditional bdouble layerQ theory dominates the interactions at low and intermediate ionic strengths, while the bhydrationQ interactions occur only at large electrolyte concentrations, at which there are sufficient ions to be bdehydratedQ and also to screen the double layer. The competition of ions for hydrating water molecules (an excluded-volume effect) was shown to generate an increase in repulsion, when compared to the traditional theory, and therefore to generate a bhydration forceQ at sufficiently large ionic strengths [7,16]. However, a different picture is provided by the experiments with neutral phospholipid bilayers in water, the stability of which (presumably due to the hydration force) is almost unaffected by the electrolyte concentration. This type of bhydration forceQ certainly cannot be due to bion

Polarization Model and ion specificity: applications

575

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

dehydrationQ. Another open question is therefore whether or not the bhydration forcesQ are independent of the electrolyte concentration. A number of additional questions are further raised by experiments regarding forces between mica surfaces [17]. At short distances between surfaces, the interactions are always dominated by the divergently large van der Waals attractions. However, the potential barrier seems to increase drastically after the electrolyte concentration exceeds a particular value, which indicates the emergence of a bhydration repulsionQ at a critical electrolyte concentration. The minimum electrolyte concentration at which this bhydration forceQ occurs depends strongly on the nature of the electrolyte: 6
102 M for LiCl, 102 M for NaCl, 103 M for CsCl and 3
104 M for KCl [17]. The extremely low electrolyte concentrations required to prevent the jump of the surfaces in their primary minimum suggest that this hydration force is not necessarily related to the bdehydrationQ of cations: if a sufficiently large repulsion could be generated by 3
104 M KCl, then probably no colloidal system will coagulate at 1 M KCl, which is not the case. Furthermore, the difference of about 2 orders of magnitude between the minimum concentrations of LiCl and KCl required to generate the hydration interactions represents an enormous bspecific ion effectQ, which is unlikely to be explained by the difference in the hydration energies between the Li+ and K+ cations. Even if such a strong effect could be explained on that basis, it would be difficult to explain why this strong effect does not occur in other colloidal systems. Another interesting result of the experiments of Pashley [17] is that the decay length of the strong interaction measured at close separations is about 10 2, in contrast to the value of about 2–3 2 reported for the interaction between lipid bilayers [11]. The range of hydration interactions is also questioned by other experiments regarding the restabilization of apoferritin molecules at large ionic strengths [18]. The large values of the second virial coefficient obtained at high electrolyte concentrations cannot be explained by the sum between the traditional double layer force and a short-range hydration force [19]. Some of the experiments cited above suggest that the ions do not affect the hydration force, whereas in other experiments the ionic strength is not only important, but strong cation-specific effects are present [14,17]. However, it is well known since Hofmeister [20] that the specificity of anions influences strongly the precipitation of proteins from electrolyte solutions, while the nature of cations plays in this case a less important role. While the list of contradictory experiments about the nature of the hydration force is much longer, we will only add here some recent results about the restabilization of protein-covered latex particles [21], which indicate that the restabilization at high ionic strength (presumably, due to the hydration forces) is strongly influenced by the nature of both the anions and cations involved. It should be, however, noted that the restabiliza-

111

tion of electrosterically stabilized colloids at high ionic strength was not associated to the hydration forces, and other mechanisms were suggested instead [22]. In summary, the contradictory evidence brought by various experiments makes difficult to answer some fundamental questions about the hydration forces, such as: (i)

whether or not the hydration force depends on the electrolyte concentration? (ii) if they do depend on the ionic strength, do specific ion effects occur for anions, cations, both or none of them? (iii) what is the decay length of the hydration force and on which parameters does it depend (nature of the surface, nature of the electrolyte and electrolyte concentration)? (iv) how do the surfaces affect the magnitude of the hydration forces? It should be noted that many attempts have been made to explain the hydration force by microscopic models. For example, the disagreements between experiment and the traditional mean field theory were attributed to the ion–ion correlations (ignored by the mean field theory); one of the most successful theories is based on the anisotropic hypernetted chain method [23]. It was shown on the basis of that theory that an additional repulsion (similar to the hydration force) can indeed occur at small separations and large ionic strengths [24]. However, the results are so strongly dependent on the choice of the interaction potentials between the ion-pairs, that even a bdouble layer attractionQ could be predicted [25]. In addition, the hydration force in the absence of an electrolyte, or at moderate ionic strength, could not be explained by this model. Another model assumed the formation of a cage-like bnetworkQ of hydrogen bonds around a surface immersed in water. Because the bonds must be destroyed when two surfaces approach each other, a repulsion is generated. A model calculation was presented by Attard and Batchelor [26]. While intuitively appealing, this model cannot relate in a simple manner the hydration interactions to the nature of the surface or to the electrolyte concentration. Another model [27] attempted to explain only the hydration interactions between neutral phospholipid bilayers, because those interactions appear to be quite different from the hydration forces determined in other systems: they have a much shorter decay length and are almost independent of electrolyte concentration [11]. The theory was based on a steric repulsion between lipid molecules, which protruding from one bilayer collide with the molecules of the opposite bilayer [27]. However, it was shown that the hydration interactions are not affected much by the polymerization of bilayers, which essentially exclude molecular protrusions [28]. The present article is structured as follows. In the first section, the physical basis of the polarization model for the hydration/double layer interactions proposed by the authors

Nanodispersions

576 112

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

[29,30] will be briefly reviewed, and the model will be extended to account for the additional interactions between ions, and ions and interfaces, not included in the mean field electrical potential. In the last part of the article, it will be shown that the polarization model for hydration/double layer interactions can provide in general a reasonable qualitative agreement with the experimental results listed above.

2. The polarization model for hydration/double layer interactions

The second difficulty can be removed if one assumes that in the vicinity of an interface the water is organized in icelike layers. The electrical interactions between the water dipoles of successive layers lead indeed to an oscillatory behavior of the polarization [35]. If the surface is not perfectly flat, of if the water is not perfectly organized in water layers, the statistical average smooth out the polarization oscillations [35]. The latter results have been also supported by molecular dynamics simulation, in which the surface dipoles were allowed to move [36]. Let us now examine in detail how the correlation between neighboring dipoles occurs.

2.1. The physical basis of the model 2.2. The correlation between neighboring dipoles In some of the first models proposed to explain the microscopic origin of the hydration force, it was suggested that the hydration interactions are generated by the polarization of the water molecules located in the vicinity of a dipolar surface [29,31]. This hypothesis was supported by experiment, which revealed a direct correlation between the magnitude of the hydration force and the surface dipole density of phospholipid membranes [32], the latter being proportional to the experimentally accessible potential across the membrane (the so-called v-potential). It is intuitively appealing to assume that high surface dipolar densities correspond to strongly hydrophilic surfaces and hence that between such surfaces large hydration forces are generated. However, before establishing a relation between the v-potential and the polarization of water in vicinity of a surface, there are some difficulties that have to be addressed. First, a continuous homogeneous dipolar density on a flat surface between two continuous media does not generate any electric field above the surface. The homogeneous dipolar surface might be thought as composed of two sheets of constant and equal surface densities of opposite charges; the electric field generated by such a charge distribution vanishes everywhere but between the sheets, hence it is not clear how a polarization can be generated in the vicinity of such an interface. A second difficulty was indicated by molecular dynamics simulations, which showed that the water near an interface is indeed polarized, but the average polarization oscillated with the distance from the interface [33]. The first polarization-based models [29,31] predicted, however, an exponential decay. The first difficulty could be explained by observing that water is not homogeneous at a molecular scale [34]. The interactions with remote dipoles are screened by the intervening water molecules, because the effective dielectric constant of their interaction is comparable with the macroscopic dielectric constant (e=80). In contrast, the interaction between neighboring dipoles is much less screened, because there is no intervening medium between them. While assuming a constant q for all interactions would predict a vanishing electric field near the surface, the local value of an beffectiveQ e leads to a net electric field, which can polarize the water molecules above the surface.

The electrical field always obeys the Poisson Equation [37]: jð e0 E þ PÞ ¼ q

ð1aÞ

where e 0 is the vacuum permittivity, Eujw is the electric field, w the electrical potential, P the polarization and q the charge density. In a typical bmacroscopicQ assumption of proportionality between polarization and applied electric field, P ¼ e0 ðe  1ÞE;

ð2Þ

where e is the dielectric constant, one obtains the wellknown form of the Poisson equation Dw ¼ 

q ; ee0

ð1bÞ

which, by assuming Boltzmannian distributions for the electrolyte ions, leads to the traditional Poisson–Boltzmann equation. However, at microscopic scales, the water is not homogeneous and a water molecule is subjected to a blocalQ electric field, which is the sum between a bmacroscopic fieldQ generated by the sources from outside the medium, and a field generated by the other water molecules surrounding it [37]. For a uniform polarized medium, the latter corresponds to the field generated in a spherical cavity in a medium, due to the polarization of the whole medium (the Lorenz field E Lorenz=P/3e 0 [37]). Therefore, the local field acting on a water molecule is given by: Elocal ¼ E þ

P 3e0

ð3Þ

and the average dipole moment of the water molecule, m=P/v 0, where v 0 is the volume occupied by a water molecule, by:
 m m ¼ cElocal ¼ c E þ 3e0 v0 ¼

c Eue0 v0 ðe  1ÞE 1  3ec0 v0

ð4Þ

Polarization Model and ion specificity: applications

577

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

where c is the microscopic polarizability. The last equality was obtained using Eq. (2) because for an uniform polarized medium, the polarization is proportional to the macroscopic electric field, and corresponds to the Clausius–Mossotti relation between the microscopic molecular polarizability c and the macroscopic dielectric constant e. If the polarization is non-uniform, an additional field E P is generated by the neighboring dipoles. A simple way to calculate this field [34] is to neglect the field generated by the remote dipoles, because it is screened by the intervening water molecules, and to consider only the field generated by the neighboring dipoles, screened by an effective dielectric constant eW, much lower than the bulk value, e=80. In order to calculate the additional field E P, the local structure of water must be known. A simple assumption is that the water is organized in icelike layers in the vicinity of the surface, of thickness D (see Fig. 1). The water molecules near the surface dipoles (gray area in Fig. 1) are assumed to be bound by the dipoles and the distance between the center of the first ice-like water layer and the center of the surface dipoles is DV. The value of the additional field in layer i, E P,i , is generated by the neighboring water molecules from the same layer i, as well as by the neighboring water molecules from the adjacent layers, i1 and i+1. The average dipole moment of a water molecule, m i , is constant in each layer, because the layers are assumed parallel to the surface, and m is a function of the distance from the surface x alone. Since the field generated by the dipoles is proportional to their dipole moments, the additional field becomes: EP;i ¼ C1 mi1 þ C0 mi þ C1 miþ1

ð5aÞ

The values of the coupling constants C 0 and C 1 were calculated by assuming that only the first 26 neighbors of a water molecule in layer i (12 from the same layer i and 7 from each neighboring layer, iF1), generated a sufficiently

113

strong field, and the contributions of the more remote dipoles were neglected [34]. Under these circumstances, C0 ¼ 

3:766 4pe0 eWl 3

C1 ¼

1:827 4pe0 eWl 3

ð6Þ

with l the distance between the centers of two adjacent water molecules and eW the dielectric constant for the interaction between neighboring molecules, which is expected to be nearer to unity than to the dielectric constant of water, e=80, because there are no intervening molecules to screen the interactions. For an uniform polarized medium, (m i =m i1=m i+1), the additional field E P vanishes, because C 0+2C 1i0, and the traditional theory (Eq. (4)) is recovered. For a non-uniform polarization, the local field is given by Elocal;i ¼ E þ

mi þ C0 mi þ C1 ðmi1 þ miþ1 Þ 3e0 v0

ð7Þ

and the average dipole moment of a water molecule by:
 m mi ¼ cElocal ¼ c E þ þ EP 3e0 v0 ¼

c ð E þ EP Þ ¼ e0 v0 ðe  1ÞðE þ EP Þ 1  3ec0 v0

¼ e0 v0 ðe  1ÞðE þ C0 mi þ C1 ðmi1 þ miþ1 ÞÞ

ð8Þ

where the relation between the molecular polarizability c and macroscopic dielectric constant, Eq. (4), was employed. Eq. (8) relates the average dipole moment of the water molecules in one layer, m i , to the average dipole moments of the molecules in adjacent layers, m iF1. By expanding in series the average dipole moment, Eq. (5a) becomes:
 B2 mð xÞ EP ð xÞ ¼ C0 mð xÞ þ C1 2mð xÞ þ D2 þ . . . Bx2 B2 mð xÞ iC1 D2 ð5bÞ Bx2 where D is the distance between the centers of two adjacent water layers. Eqs. (8) and (5b) lead to: k2m

B2 mð xÞ Bwð xÞ ¼ mð xÞ þ e0 v0 ðe  1Þ Bx2 Bx

ð9Þ

with 1

km ¼ Dðe0 v0 ðe  1ÞC1 Þ 2

Fig. 1. A sketch of water layering in the vicinity of a dipoles-bearing surface. In the vicinity of the surface dipoles p, the bounded water (gray) has the dielectric constant eV, and the center of the first water layer is at the distance DV from the center of surface dipoles.

ð10Þ

representing a bdipolar correlation lengthQ. When k m =0 or B2m/Bx 2=0 the usual macroscopic constitutive relation P=e 0(e1)E is recovered. However, for e=80, D=3.68 2, v 0=30 23, l=2.76 2, eW=1, Eq. (10) provides k m =14.9 2, and, in general, the polarization and the electric field are not proportional to each other. It is of interest to note that such

Nanodispersions

578 114

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

an banomalous dielectric behaviourQ of water in the vicinity of a surface was recently observed in molecular dynamics simulations of ionic Newton Black films [38]. The above expression of k m [30] is based on the assumptions that the local cluster surrounding a water molecule has an ice-like structure and is parallel to the surface. The first approximation is reasonable, because of the small size of the local clusters (with a radius twice the distance of closest approach between water molecules and containing only the first 26 neighbors). The second assumption is less accurate; however, a simple estimation showed that the btiltingQ of the cluster with an angle up to 208 generated only a small decrease of the correlation length [35], which did not affect much the results. This insensitivity to the orientation of the cluster is due to the relatively high spherical symmetry of the first neighbors of a water molecule in an ice-like structure. It should be emphasized that the assumption of an icelike structure of water in the vicinity of the surface is only an approximation used to calculate the dipole correlation length k m (Eq. (10)). In fact, if the water would be organized in perfect ice-like layers parallel to the planar surface, the model would predict an oscillatory behaviour of the polarization in the vicinity of the surface [35].

is strong: C 1(x)=C 1. For low polarizations, m(x)Vm 0 and the coupling is vanishingly weak. Consequently, Eq. (8) becomes:

2.3. The effect of disorder in the water structure on the correlation length k m

The chemical potential of an ion of species i and of charge q i in the liquid is given by [10]:

As will be shown later, the correlation length between dipoles is responsible for the hydration force, but also for the increase of the range of the interactions at large ionic strengths. The derivation of an expression for the correlation length assumed an ordered cluster around each water molecule. The local ordering is expected to be stronger in the vicinity of the surface and at high polarizations, and the correlation length to decrease with the distance to the surface and with decreasing fields. The electrolyte ions are also expected to affect the local ordering as well as the interactions between neighboring dipoles. While a microscopic theory to account for the local disorder in the structure of water is extremely difficult to develop, simple considerations can provide an estimate of the role of the local disorder on the double layer/hydration interactions predicted by the model. The coupling between the dipoles of adjacent layers, described by the coupling constant C 1, is ultimately responsible for the propagation of the polarization through water. A simple manner to account for the decrease of the coupling with decreasing polarization is through an expression of the form:

 mð xÞ C1 ð xÞ ¼ C1 1  exp  ð11aÞ m0 where m 0 is an empirical parameter, which controls the coupling. For large polarizations, mJm 0 and the coupling

mð xÞ ¼ e0 v0 ðe  1ÞðE þ EP Þ ¼ e0 v0 ðe  1ÞðE þ C0 ð xÞmðxÞ þ C1 ð xÞmð x  DÞ þ C1 ð xÞmð x þ DÞÞ
¼ e0 v0 ðe  1Þ E þ ðC0 ð xÞ þ 2C1 ð xÞÞmð xÞ  B2 m þC1 ð xÞD2 2 þ . . . Bx Bw B2 m i  e0 v0 ðe  1Þ þ k2m ð xÞ 2 ð11bÞ Bx Bx with a dipole correlation length: 1

km ð xÞ ¼ Dðe0 v0 ðe  1ÞC1 ð xÞÞ 2

 12 m ð xÞ ¼ km 1  exp  m0

ð12Þ

which decreases with decreasing polarization. 2.4. The Boltzmann distribution of ions

li ð xÞ ¼ l0i þ kT lnðfi ci ð xÞÞ þ qi wð xÞ þ Wi ð xÞ

ð13Þ

where k is the Boltzmann constant, T the absolute temperature and l 0i represents the standard chemical potential of the ions of species i, of charge q i and concentration c i (x). The activity coefficient f i of the ions of species i will be assumed unity. The interaction free energy Wi (x) should include all the other interactions of the ion with the medium (not accounted for by the mean field electrical potential w), such as the image forces [8], excluded volume effects [7], ion-dispersion [9] or ion-hydration forces [10]. Assuming that the system is in contact with an infinite reservoir in which the ion concentrations are c i0, the total charge density is provided by:
 X qi wð xÞ þ DWi ð xÞ q ð xÞ ¼ qi c0i exp  ð14Þ kT i where DWi (x) is calculated with respect to the bulk. Eqs. (1a) and (14) can be combined to yield:
 X B 2 w ð xÞ qi wð xÞ þ DWi ð xÞ Bmð xÞ e0 ¼ qi c0i exp  þ Bx2 kT v0 Bx i ð15Þ Eqs. (9) and (15) represent a system of coupled nonlinear equations, which can be solved (under appropriate boundary conditions) for w(x) and m(x). There are various interactions between ions and surfaces. Here our focus will be on the interactions

Polarization Model and ion specificity: applications

579

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

generated by the hydration of ions, which can be accounted for in a simple manner [10]. A structure-making ion is surrounded in bulk water by a sheath of hydrating water molecules. When approaching a surface at distances of the order of molecular size, the ion has to lose some of the hydrating molecules, hence the approach to the surface is unfavorable. The hydration energy of a structure-making ion is of the order of 102kT, and even a decrease of hydration by a few percent affects strongly the ion distribution in the vicinity of the surface. In fact, the increase of the surface tension of the water/air interface with increasing electrolyte concentration led Langmuir to suggest (based on the Gibbs adsorption equation) the existence of a layer of a few 2ngstroms, completely depleted of ions [39]. Indeed, a large potential barrier with a width of a few 2ngstroms for the structure-making ions can describe fairly well the behaviour of the air/water surface tension of the simple electrolytes [10]. The ions, however, are not completely depleted near a physical surface immersed in an electrolyte, because no recombination with the surface groups could take place in this case. Instead of an infinite potential barrier near the interface, it is reasonable to expect only a potential barrier with a height of the order of kT. These repulsive ion-hydration forces acting on the structure-making ions decrease their concentration in the vicinity of the interface. In contrast, the structure-breaking ions have unfavorable interactions with the water molecules, because the bulk water molecules can better organize their hydrogen bonding network in their absence. Consequently, the structure breaking ions are expelled from the bulk toward the interface, where their concentration is increased [10]. The details of the interactions between ions and surfaces are not known, because they involve large-scale ab initio quantum mechanical calculations. A simple manner to account for the change in the free energy when the ions approach the interface is via a potential well (or a potential barrier) with a depth (height) DW(x) of the order of kT and with a width w of the order of a few 2ngstroms [10]. The results are not qualitatively affected by the shape of the interaction potentials, a square or a triangular well providing similar results as a power-law interaction [10]. 2.5. Boundary conditions for the polarization model As already noted, in a homogeneous medium, a surface dipole density does not generate an electric field above the surface and, consequently, it is ignored by the traditional theory of the double layer. In the latter theory, the double layer is generated by surface charges alone. In the present treatment, both the surface dipole density and the surface charge density generate the double layer. Assuming that the surface charge is generated by the dissociation of the acidic and basic groups of the surface, while the surface dipole are generated by the association of the electrolyte ions with the

115

corresponding sites, for a uni-univalent electrolyte the surface charge density is given by [40]: eNB     r¼ cOH S 1 þ KOH þ KcEA exp ew kT  1þ



cH KH

eNA    S þ exp  ew kT

ð16Þ

cE KC

where K A and K C are the association-equilibria constants for the anions and cations, respectively; N A and N B are the number of acidic and basic sites per unit area, respectively; c H and c OH are the bulk concentration of H+ and OH ions, respectively; and c E is the concentration of electrolyte in the bulk. The average areas, A A, occupied by anions and A C by cations on the surface are provided by:   ewS KA A cOH 1 þ KKOH þ exp  cE cE kT AA ¼ ð17aÞ NB   S 1 þ KKHC ccHE þ KcEC exp ew kT AC ¼ ð17bÞ NA The first boundary condition at the surface is provided by integrating the Poisson Eqs. (1a) and (1b) over the volume of a flat box, which includes the surface, with the large sides parallel to the surface and a vanishingly thin width. After using the Gauss theorem, one obtains:   Bwð xÞ  mð xÞ  e0 ¼ r ð18aÞ x¼0  Bx  v0 x¼0 where x=0 indicates the surface. The second boundary conditions at the surface (Eq. (18b)) is provided by the polarization of the water molecules from the vicinity of the surface, which is caused by the macroscopic field E ¼  BwBxð xÞ jx¼0 , the field generated by the surface dipoles and the field E SP due to the neighboring water dipoles: ESP ¼ C0 mð0Þ þ C1 mðDÞ;

ð19Þ

which differs from Eqs. (5a) and (5b) because there are no water dipoles behind the surface (m(D)=0). The field generated by the surface dipoles is calculated by averaging the field generated by a dipole p in a medium of effective dielectric constant eV at a distance DV above the center of the dipole over an area corresponding to the area occupied by a surface dipole (see Fig. 1). If the area occupied by the dipole on the surface is large, the field vanishes. Since the dipoles are formed through the association of the cations with the acidic surface groups, generating an effective dipolar moment p C/eV, as well as by the association of anions with the basic surface groups, of an effective dipolar moment p A/eV, the average field due to the dipoles is given by [40]: 2 3 p  1 p  1 1 1 A C 5 ES¼4 þ eV 2pe0  AA þ DV2  32 eV 2pe0  AC þ DV2  32 p

p

ð20Þ

Nanodispersions

580 116

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

Therefore, the second boundary condition at the surface is given by:


mð0Þ m ð0Þ ¼ c E þ þ ESP þ ES 3e0 v0



1 1 Eðe0 E þ PÞ  PEP dx 2 2 0  Z 2d 
1 Bwð xÞ mð xÞ Bw ¼ e0  2 0 Bx v0 Bx

Fel ð2d Þ ¼

¼ e0 v0 ðe  1ÞðE þ ESP þ ES Þ which, by expanding mðDÞ ¼ mð0Þ þ D BmBxð xÞ

with the additional field E P due to the non-uniform polarization [30]:

ð21Þ

j

Z

, becomes

2d



mð xÞ B2 mð xÞ dx v0 Bx2

C1 D2

x¼0

ð1  e0 v0 ðe  1ÞðC0 þ C1 ÞÞmð xÞjx¼0 Bm Bw  e0 v0 ðe  1ÞDC1 þ e0 v0 ðe  1Þ Bx x¼0 Bx 2 p  1 1 A ¼ e0 v0 ðe  1Þ4  eV 2pe0 AA þ DV2  32 3p p  1 1 C 5 þ eV 2pe0  AC þ DV2  32

j

ð24aÞ

The entropic contribution of all the ion species present in the electrolyte, with respect to a homogeneous system of constant ion concentrations, is given by [41]:

j

x¼0

ð18bÞ

Fent ð2d Þ ¼  T DS XZ ¼ kT

2d





ci ln

0

i

ci ci0



  ci þ ci0 dx

ð24bÞ

p

For a single surface immersed in an electrolyte, the other two boundary conditions necessary to solve the system of Eqs. (9) and (15) are provided by the vanishing of both the electric field and the polarization far away from the surface: Bwð xÞ Bx

j

¼0

¼0

rð2d Þ

ð22aÞ ð22bÞ

where r is the surface charge density and w S the surface potential. The free energy due to the interactions between the ions and the surface is given by [10]:

For two identical surfaces immersed in an electrolyte separated by a distance 2d, these boundary conditions are provided by the symmetry with respect to the middle distance, x=d:

j

Z

DFch ð2d Þ ¼ 

r ðl Þ

x¼l

m ðl Þ ¼ 0

Bwð xÞ Bx

where i runs over all the ions species (including H+ and OH), and the change in the chemical energy, with respect to infinite separation, due to the charging of the surface is provided by [42]:

ð23aÞ

wS ðrÞdr

 ew  DWA Fionsurf ð2d Þ ¼ DWA ð xÞcE exp kT 0
!  ew  DWC þDWC cE exp dx kT Z

2d




ð24cÞ




x¼d

mð xÞjx¼d ¼ 0

ð24dÞ ð23bÞ

2.6. The interaction between identical surfaces immersed in an electrolyte Once the system of Eqs. (9) and (15) is solved for the boundary conditions (18a), (18b), (23a) and (23b), the total free energy of the system can be calculated by adding various contributions, due to the electric field, entropy of ions, chemical energy and interactions between ions and surfaces. The electrostatic free energy includes the usual macroscopic free energy density (1/2)ED (with the displacement field D=e 0E+P [37]) and also the interaction of the dipoles

The force per unit area is consequently obtained from the derivative: Pð2d Þ ¼ 

BðFel ð2d ÞþFent ð2d ÞþFch ð2d ÞþFionsurf ð2d ÞÞ Bð2d Þ ð25Þ

3. Consequences of the polarization model 3.1. Polarization skin effect In the traditional double layer theory, the average dipole moment of a water molecule is proportional to the macroscopic field (the derivative of the electrical potential)

Polarization Model and ion specificity: applications

581

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

because the local field is also proportional to the latter field. In the polarization model and for a uniformly polarized medium, the field generated by the neighboring dipoles also vanishes, (C 0+2C 1)mc0, since the negative field generated by the dipoles from the same layer (C 0b0) is compensated by the positive fields generated by the dipoles from the adjacent layers (C 1N0), and the traditional theory is recovered. However, at the surface, one adjacent layer is missing, and there is no compensation anymore. The total field generated by the neighboring water molecules on the first layer is opposite in direction to the average dipole moment; hence, it induces a net depolarization, even in the absence of surface dipoles. One of the differences between the polarization model and the traditional theory is that the local field at the surface, predicted only by the former model, opposes the polarization of the medium. The solutions of the system of Eqs. (9) and (15), for low electrolyte concentrations (c E=0.001 M), with the boundary conditions (18a), (18b), (22a) and (22b), and no surface dipoles ( p C/eV=p A/eVu0) are plotted in Fig. 2a. The depolarization field at the surface strongly affects the shape of the electrical potential and average dipole moment only within a few 2ngstroms from the surface (a bskinQ effect); however, a traditional Poisson– Boltzmann treatment for the same values of the parameters predicts different results at any distance from the surface. When a suitable charge is added on the surface to compensate for this field, the results of the polarization model and the traditional theory are almost identical far away from the surface (see Fig. 2a). This result is supporting the general agreement that the traditional theory is accurate at intermediate electrolyte concentrations [12], when the surface charge is considered a fitting parameter. However, at higher ionic strength (c E=0.1 M, Fig. 2b), the polarization model and the traditional theory predict different behaviors for w(x) and m(x), which cannot be accounted for by changing the values of the surface charges. Therefore, even in the absence of surface dipoles, the polarization model for the hydration/double layer predicts qualitatively different results from those of the traditional theory at moderate and high ionic strengths. At low electrolyte concentrations, the quantitative differences between the two models, far away from the surface, can be accounted for by suitable modifications of the surface charge. The shape of the electric field and polarization within a few 2ngstroms from the surface, predicted by the two models, are however different at all electrolyte concentrations. 3.2. Hydration forces between neutral lipid bilayers: independence of electrolyte concentration In the absence of an electrolyte, Eq. (15) becomes: B 2 w ð xÞ 1 Bmð xÞ e0 ¼ Bx2 v0 Bx

ð26Þ

117

which can be integrated to lead to: m ð xÞ ¼ e 0 v 0

Bwð xÞ þC Bx

ð27Þ

The integration constant is selected in such a manner (C=0) for the polarization to vanish far away from the surface. From Eqs. (9) and (27), one obtains: k2m B2 mð xÞ ¼ mð x Þ e Bx2

ð28Þ

The solution of Eq. (28) is provided  by the sum of  two  terms, which are proportional to exp  kxH and exp kxH , kmffi with the bhydrationQ decay length kH ¼ p c1:67A˚ . For a e single surface, the boundary condition (22b) implies that the average dipole moment has the form:
 x m ¼ m1 exp  : ð29Þ kH where m 1 is the value of the average dipole moment near the surface. Since there are no surface charges, the magnitude of the polarization ( P=m/v 0) depends only on the surface dipoles. The interaction between two planar surfaces is important only when the polarization regions overlap, and increases with the magnitude and density of the surface dipoles. This result is supported by the experiments with neutral phospholipid bilayers, for which the interactions are proportional with the potential across the membrane, the latter being proportional to the surface dipole density [32]. Despite the absence of a surface charge, there is a potential in the vicinity of the surface, generated by the gradient of polarization (see Fig. 3a). In the presence of an electrolyte, the range of the polarization is increased, indicating that the hydration force predicted by the polarization model depends on electrolyte concentration. The force between neutral surfaces (with a surface dipole density) depends on the electrolyte concentrations, as shown in Fig. 3b, particularly at large separations. However, at small separations, the interaction appears to be well described by an exponential with a decay length k H. For neutral lipid bilayers, the equilibrium is reached at a distance of about 20 2, at which the attractive van der Waals interaction balances the repulsive hydration and thermal undulation interactions [43]. The experiments regarding the forces between neutral lipid bilayers [11] sample the interactions at separations smaller than 20 2, for which the dependence on ionic strength is much weaker. By adding to the total pressure a typical van der Waals disjoining pressure [12]: PvdW ð2d Þ ¼ 

AH 1 6p ð2d Þ3

ð30Þ

where A H is the Hamaker constant, one obtains the curves plotted in Fig. 3c. The value A H=0.2
1020 J was selected to provide the equilibrium distance at about 20 2 from the

Nanodispersions

582 118

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

Fig. 2. (a) The potential and the average dipole moment of water molecules in the vicinity of a charged surface, calculated using the polarization model (thick lines) for the following values of the parameters: c E=0.001 M, N A=1
1017 sites/m2, N B=1
1017 sites/m2, DW C=0, DWA=0, pH=8.0, K H=106 M, K OH=106 M, K C=0.5 M, K A=0.5 M, p C/eV=0, p A/eV=0, e=80, k m=14.9 2, m 0=0. While the calculation based on the traditional Poisson–Boltzmann theory (dotted line) provides quantitatively different results, a suitable slight adjustment of the value of the surface charge density in the Poisson–Boltzmann calculations (thin lines) led to almost identical results far away from the surfaces, for the two theories. Within a few 2ngstroms from surface, the polarization model predicts qualitatively different results (a skin effect). (b) The potential and the average dipole moment of water molecules in the vicinity of a charged surface, calculated using the polarization model (thick lines) for the same values of the parameters as in (a), except now c E=0.1 M. The results predicted by the traditional theory (dotted line) are qualitatively different at any distance from the surface. For moderate and high electrolyte concentrations, the results of the two theories cannot be made to agree by a suitable adjustment of the surface charge density.

Fig. 3. (a) The potential and the average dipole moment of water molecules in the vicinity of a charged surface, calculated using the polarization model for a neutral surface (r=0) with surface dipoles occupying an area A C=50 22, p C/eV=1 Debye, p A/eV=0, e=80, k m=14.9 2, m 0=0, DW C=0, DWA=0, for various electrolyte concentrations. (b) The force per unit area between two identical surfaces, calculated using the polarization model and the same values of the parameters as for Fig. 3a, is compared with an exponential with a decay length k=2 2. At large separations and high electrolyte concentrations, the results of the polarization model are not well described by an exponential decay with k=2 2. (c) The total force per unit area between two identical surfaces as in (b), calculated by adding to the results of the polarization model a typical van der Waals interaction between surfaces, with A H=0.2
1020 J. For most separations distances, smaller than the equilibrium separation (the only separations experimentally accessible), the repulsion is well approximated by the same exponential (with the decay length k=2 2), regardless of the electrolyte concentration. The polarization model for the hydration/double layer interactions predicts that the equilibrium separation between neutral lipid bilayers slightly increases with increasing electrolyte concentration.

Polarization Model and ion specificity: applications

583

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

Fig. 3.

119

Nanodispersions

584 120

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

surface. For most separation distances (smaller than the equilibrium separation), the pressure is apparently independent of the electrolyte concentration. However, the equilibrium separation distance predicted by the polarization model seems to increase by a few Angstroms with increasing electrolyte concentration. This result appears to be supported by recent preliminary experiments with neutral lipid bilayers, which indicate an increase of the hydration force with increasing electrolyte concentration [44]. 3.3. Hydration forces between mica surfaces: ion-specific effects In the previous section, it was shown that a hydration force with a short decay length and almost independent of the electrolyte concentration can be achieved with neutral lipid bilayers at short separation distances. At separations larger than about 20 2, the attractive van der Waals interactions dominated the interactions. A completely different situation occurs for mica surfaces, for which both charges and dipoles are present on the surface. For these surfaces, strong specific cation effects have been observed experimentally at relatively low electrolyte concentrations [17]. These specific ion effects cannot be explained by the additional interactions between ions (such as excluded volume effects) or ions and surfaces (such as ion-dispersion [9] or ion-hydration [10] forces), because these interactions are in general negligible for electrolyte concentrations smaller than about 0.05 M. The experimental data reported by Pashley [17] raise two important issues. The first one is why the critical electrolyte concentration at which the hydration force emerges, depends so strongly on the type of electrolyte (6
102 for LiCl, 3
104 M for KCl). The second issue is why the decay length of the interactions at short separations is about 10 2, about five times larger than that corresponding to the hydration force between lipid bilayers. Let us first note that, when the surface charges and surface dipoles are generated via the dissociation of the moieties of the surface, and via the association of the electrolytes ions with the dissociated sites of the surfaces, respectively, the fields generated by the surface charges and surface dipoles, in the vicinity of the surface, are oriented in opposite directions. An interesting situation occurs when the field generated by surface charges and surface dipoles are comparable to each other; in this case, even small changes in the association constants or in the effective surface dipole p/eV affect strongly the interactions. The mica surface is a complex one and many types of surface groups are present; it is therefore possible for the fields generated by the surface charges and dipoles to be oriented in the same direction and hence reinforce each other. The force between charged surfaces immersed in an electrolyte of concentration c E=0.001 M is plotted in Fig. 4a, by assuming a constant surface charge density r=0.01 C/m2, a constant area per dipole A=100 22, and various values for the effective dipole

moment, p/eV of the surface dipoles. In all the cases investigated, the interactions at large separations are well approximated by an exponential decay, with the decay length roughly provided by the Debye–Hqckel length [12]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ee kT P0 2 ; kDH ¼ ð31Þ i qi c0i while at short separations the decay length for the interaction is different. For the set of parameters employed, the later decay length is about 10 2. When a suitable van der Waals disjoining pressure (Eq. (30), A H=1
1020 J) is added to the interactions, the total interactions for systems with p/eV=0 or 3 Debyes appear to obey the traditional DLVO theory, with no additional bhydrationQ repulsion emerging at low separations. The interactions become attractive for separation smaller than about 20 2 and it appears that no hydration force exists (see Fig. 4b). However, when the field generated by the surface dipoles reinforces the field generated by the surface charge, with p/eV=5 or 7 Debyes, a bhydration forceQ occurs at low separations, with a decay length of the order of 10 2. At very low separations, the divergently large van der Waals attraction dominates; however, the maximum pressure that must be applied to fuse the mica surfaces is about two orders of magnitude larger in the latter case, than in the case when p/eV=3 Debyes. In summary, a bhydration forceQ is always predicted by the polarization model at low separations and its apparent decay length depends on the electrolyte concentration. However, if the magnitude of the repulsion is small, the total interaction (including the van der Waals attraction between surfaces) seems to be well described by the traditional DLVO theory [13]. In contrast, when the repulsion is large, a bhydration forceQ with a decay length of about 10 2 emerges in the vicinity of the interface. The transition between a bDLVO-likeQ result and a bDLVO plus a hydration forceQ result can be generated by relatively small modifications of the surface dipoles formed on the mica surface. Various cations have different association constants with the surface groups, hence can generate different surface charges and particularly different surface dipole densities. Because of the high sensitivity of the interactions on the latter density, strong ion-specific effect could be explained on this basis. 3.4. Long-ranged interactions at high ionic strengths: restabilization When the surface charges and the surface dipoles are formed via the dissociation, and via the association with electrolyte ions of the moieties of the surface, respectively, the increase of ionic strength decreases the surface charge, but at the same time increases the surface dipole density. Since the electric fields induced in the vicinity of the surface by the surface charge and surface dipoles oppose each other,

Polarization Model and ion specificity: applications

585

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

121

Fig. 4. (a) The repulsion force predicted by the polarization model between two identical surfaces, for c E=0.001 M, r=0.01 C/m2, A dipole=100 22, DW C=0, DWA=0, e=80, k m=14.9 2, m 0=0) and various values for the effective dipole moment, p/eV. In the vicinity of the surface, the force is well approximated by an exponential with a decay length k~10 2, while at large separations the force is well approximated by an exponential with the decay length k DH provided by the traditional theory. (b) The total interactions, obtained by adding to the repulsion force calculated in (a) a typical van der Waals attraction between surfaces, with a Hamaker constant A H=1
1020 J. For some values of the effective dipole moment ( p/eV=0 or 3 Debyes), the total interaction seems to obey the DLVO theory, while for the other values ( p/eV=5 or 7 Debyes), a bhydration forceQ with a decay length k~10 2 emerges at low separation distances.

the total interaction might run through a minimum at an electrolyte concentration for which these fields cancel each other. At low ionic strengths, the charges dominate the interactions, which decrease with increasing ionic strength, while at large electrolyte concentrations, the increase of ionic strength generates more dipoles on the surface, via the association of the ions with the dissociated surface groups, hence increasing the interactions. Therefore, at intermediate electrolyte concentrations, the repulsive double layer/hydration interactions predicted by the polarization model might

be lower (in absolute value) than the attractive van der Waals interactions and the system will rapidly coagulate, while at both low and high ionic strengths the repulsive interactions might be sufficiently strong to prevent coagulation [34]. This mechanism was employed to explain the restabilization of protein-covered latex particles [40,45] observed experimentally [21]. The stability of some colloids at high ionic strengths can be explained by any theory that relates the hydration force to the surface dipole density. More difficult to explain are

Nanodispersions

586 122

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

some experimental data regarding the second virial coefficient of the interactions between apoferritin molecules, which at low ionic strengths (0.01–0.075 M) decreases with increasing electrolyte concentrations, as expected from the traditional DLVO theory, but increases markedly with electrolyte concentration at large ionic strengths (~0.25 M) [18]. A short-range repulsion can prevent coagulation, if it is sufficiently strong; however, a short-range repulsion cannot lead to such a large increase in the second virial coefficient [19]. In fact, even an infinite hard-core repulsion for separations distances smaller than 50 2 between the surfaces of two apoferritin molecules is not sufficient to explain such large virial coefficients, while an exponentially decaying repulsion, with a decay length of about 2 2, must have a huge magnitude to explain the experiment. This magnitude must be by about 10 orders of magnitude larger than that determined for neutral lipid bilayers and is clearly not realistic [19]. Therefore, the interaction between apoferritin molecules cannot be explained by a phenomenological model which simply assumes that the hydration force has a fixed decay length and a magnitude proportional to the surface dipole density. A peculiar feature of the polarization model is the emergence of a long-ranged interaction at high ionic strengths. To understand qualitatively the behaviour of such an interaction, let us note than for small electrical potentials, ew kT V1, and the system of Eqs. (9) and (15) can be linearized and hence can be solved analytically. In the linear approximation, the system becomes [30]: B2 wð xÞ e 1 Bmð xÞ ¼ 2 wþ ; Bx2 e0 v0 Bx kDH k2m

B2 mð xÞ Bwð xÞ ¼ mð xÞ þ e0 v0 ðe  1Þ Bx2 Bx 

ð32aÞ

ð32bÞ 

and its solutions are proportional to exp F kxi . The values of the characteristic decay lengths k i (i=1 or 2) are determined from the characteristic equation (obtained by introducing the solutions in the system): !
 1 e k2m e1   1  2 k2 k2DH k2 k   k2 k2 ¼ 0Zk4  k2DH þ k2m k2 þ DH m ¼ 0 e

ð33Þ

and are given by: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 2 4 4 2 2 t kDH þ km F kDH þ km þ 2kDH km 1  2e k1;2 ¼ 2 ð34Þ Both solutions of Eq. (34) are real and positive, because the dielectric constant is in the range 1bebl. The two characteristic lengths as a function of the Debye–Hqckel

length k DH are plotted in Fig. 4a. For k DHJk m (k DHYl), the characteristic lengths are given by: k1 ikDH

ð35aÞ

km k2 ikH ¼ pffiffi e

ð35bÞ

which implies that two types of decay lengths coexist in the system: one roughly described by the traditional theory and the other corresponding to a short-range hydration force, independent of electrolyte concentration. However, at high ionic strengths, k DHVk m (k DHY0) the characteristic lengths are approximated by: k1 ikm

ð36aÞ

k2 Y0

ð36bÞ

which implies than an intermediate decay length (since k m =14.9 2), much larger than both k DH and k H, is present in the system at high electrolyte concentrations. The contributions of each characteristic length to the solution is determined by the boundary conditions (18a), (18b), (22a) and (22b) for one surface or boundary conditions (18a), (18b), (23a) and (23b) for two identical surfaces. While the analytical solutions (and the characteristic decay lengths) are valid only for the linear approximation of the system, the behaviour of the non-linear system is qualitatively similar, at least for sufficiently small electrical potentials. Indeed, a calculation based on the polarization model, in which the interactions between spherical molecules were calculated using the Derjaguin approximation, showed that it is possible to obtain such long-ranged interactions between apoferritin molecules, which could explain the unusually strong increase of the second virial coefficient at large electrolyte concentrations [19]. The circles in Fig. 5b represent the experimental data reported in Ref. [18], while the continuos line are the results of the calculations for various effective surface dipolar moments, reported in Ref. [19]. The details of the calculations are provided in Ref. [19]. 3.5. Role of ion-hydration forces in the polarization model There are various manners in which ion-specific effects can be accounted for by the polarization model. The first one is related to the different association constants of the ions with the dissociated surface groups. The charging of the surface depends not only on the electrolyte concentration, but also on the type of electrolyte. These ion-specific effects are similar to those predicted by the traditional theory [46]. A typical consequence of such effects is illustrated by the dependence of the isoelectric point on the nature of electrolyte [21]. A second manner of predicting ion-specific effects by the polarization model is via the magnitude and density of the

Polarization Model and ion specificity: applications

587

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

123

  kmffi Fig. 5. (a) The dependence of the characteristic decay lengths k 1, k 2 of the system on the Debye–Hqckel length kDH km ¼ 14:9 A˚ ; kH ¼ p ¼ 1:67 A˚ . (b) e The second virial coefficient for apoferritin molecules, obtained from experiment in Ref. [18] is compared with calculations based on the polarization model for hydration/double layer interactions for various values of the effective surface dipole moment. The details of the calculations are provided in Ref. [19].

dipoles formed on the surface. Even if different types of ions would have the same association constants, the magnitude of the dipoles formed on the surface should depend on the specificity of the chemical bonding between ions and surface groups, as well as on the hydration characteristics of the ions (the strongly hydrated, structure-making ions probably generating dipoles of larger magnitude, because they might retain some of the hydrating water molecules even after the association with surface groups). The surface dipoles generate a potential in the vicinity of the surface; therefore, the association equilibria of all ions are affected by the magnitude and density of the surface dipoles. The specific ion effects are particularly important when the electric fields generated by the surface charges and the

surface dipoles are comparable in the vicinity of the surface. When they are oriented in opposite directions, small changes in the magnitude of the dipoles will affect strongly the total interaction at low separations, as shown in Fig. 4b, which can be well described either by the traditional DLVO interaction at low p/eV values, or by the traditional DLVO interaction plus a short-range bhydration forceQ, at large p/eV values. There is a third manner to introduce specific ion effects via the polarization model, which will be discussed in this section. While at low electrolyte concentrations the additional interactions between ions and interfaces can be in general neglected, at moderately high electrolyte concentrations, these additional interactions have to be taken into account.

Nanodispersions

588 124

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

A simple manner to account for the interactions between ions and surfaces, DWi (x), is via a potential well (or a potential barrier) with a depth (height) of the order of kT and with a width w of the order of a few 2ngstroms [10]. Such ion potentials have been selected because of the changes in the hydration energy of ions when they approach the interface: a structure-breaking ion disorganize the structure of bulk water, and it is expelled toward the interface, because water can better organize in its absence (DW(x)b0). In contrast, a structure-making ion is strongly bound to their hydration water molecule. Since when approaching the interface, they

have to lose some of these molecules, they prefer the bulk water (DW(x)N0). While the details of these interactions are unknown, the results are not affected qualitatively by the shape selected for the potentials [10]. Most univalent anions (e.g. Br, NO3, SCN) are structure breaking, while most of the simple univalent cations (e.g. K+, Na+, Li+, Cs+) are structure making. In the model calculations that follows, it will be assumed, for the sake of simplicity, that the anions are structure-breaking (DWA(x)b0), while the univalent cations are structuremaking (DW C(x)N0). In Fig. 6a–d, the potential in the

Fig. 6. The potential in the vicinity of a surface when ion-hydration interactions affect the anions (DWA) and the cations (DW C) The other parameters are c E=0.1 M, N A=1
1018 sites/m2, N B=1
1018 sites/m2, K H=107 M, K OH=107 M, K C=104 M, K A=104 M, e=80, k m=14.9 2, m 0=0, w=5 2, and (a) pH=4, p A/ eV=0; (b) pH=4, p A/eV=3 Debyes; (c) pH=10, p C/eV=0; (d) pH=10, p C/eV=3 Debyes. The values of p C/eV do not affect the results plotted in (a) and (b), while the values of p A/eV do not affect the results plotted in (c) and (d). (e) The force between two identical surfaces for the same parameter values as in (a–d). Only the magnitude of the force is strongly dependent on the value of the surface dipoles and ion-hydration forces, but not its functional behaviour.

Polarization Model and ion specificity: applications

589

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

125

Fig. 6 (continued).

vicinity of the surface is calculated for various types of ionhydration interactions. At small pH values (Fig. 6a,b), the association of anions is important, because most acidic sites are not dissociated. Consequently, changes in the effective dipole moment of the surface dipole formed by the association of cations do not affect much the potential, while changes in the effective surface dipole formed by the association of anions affect drastically the electric potential. However, the interactions of both cations and anions with the surface affect the potential. In Fig. 6c,d, the calculations were repeated for large pH values, for which most of the basic sites of the surface are associated with OH, and consequently the association of anions is negligible. In this case, only specific cation effects can be predicted by changes in the effective dipole moment on the surface. As already noted, the interactions between

ions and surface can account for both anion and cation specificity. The interactions between two plates are plotted in Fig. 6e; the magnitude of the interaction is strongly affected by the surface charge and surface dipole densities. Therefore, specific ion effects can be accounted for through different association constants, different dipole moments of the ion pair formed on the surface, or different interactions between the ions and surfaces. However, the dependence of force on the distance appears to be much less affected by the surface conditions. In the next section, it will be shown that the forces at large separations are dominated by the largest characteristic decay length of the system. When the dipole correlation length, k m , is much smaller than the Debye– Hqckel decay length, the latter approximates well the largest characteristic decay length of the system.

Nanodispersions

590 126

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

Fig. 6 (continued).

3.6. The effect of the correlation length k m on the interactions In Section 3.3, it was shown that at low ionic strengths the interactions between surfaces, at large separations, are well described by an exponential with a decay length roughly provided by the traditional theory (c E=0.001 M, k DH=100 2). The magnitude of the interaction differs from that predicted by the DLVO theory; however, in the latter theory, the surface charge density is merely an unknown parameter, used to fit the experimental results. Therefore, the results of the polarization model at low ionic strength can in general be recovered (at large separations) by the DLVO theory when suitable values for the surface charge density are employed. The reason for this behaviour is that, in the linear approximation, the potential and the polarization are a sum of two exponentially decaying terms, and at large separations the largest term dominates. At low ionic strengths, k DHJk m c15 2, and the largest characteristic length of the system is k 1ck DH. The situation is completely different when k DH is comparable to k m and k 1 becomes considerable larger than k DH. In this case, the polarization model predicts, at large separations, an exponentially decaying interaction, but with a decay length completely different from k DH. An important issue is whether or not this increase in the decay length occurs for any colloidal system. The derivation of the value for k m =14.9 2 assumed an ice-like order for the clusters around each water molecule [30]. When this order is lowered, the value of the dipole correlation length is expected to decrease. A phenomenological model for the decrease of k m with the decrease of the average dipolar moment of water molecules is proposed in Section 2.3. In

this section, the consequences of the decrease of the dipole correlation length are estimated. The potential in the vicinity of a charged surface, calculated for m 0=0.05 Debyes (in Eq. (12)) and for c E=0.1 M, are plotted in Fig. 7a. For the sake of simplicity, the surface dipoles were neglected ( p A/eV=p C/eV=0). For large surface charge densities, a large polarization is generated in the vicinity of the surface; hence, the dipole correlation length is large and the results of the polarization model (open symbols) are very different form the results of the traditional theory (filled symbols). However, at large separations, the polarization decreases and the results of the two theories become comparable. For lower surface charge densities, the polarization in the vicinity of the interface is lower and the results of the two theories become comparable starting at smaller distances (see Fig. 7a). The same behaviour is obeyed by the forces between surfaces, plotted in Fig. 7b. Note than the change in C 1(x) (Eqs. (11a) and (11b)) affects not only the dipole correlation length k m , but also the boundary condition (18b) and the electrostatic free energy Eq. (24a). At lower surface charge densities, the force predicted by the polarization model (open symbols) roughly decays as those predicted by the traditional theory (filled symbols). However, for higher surface charge densities, the force predicted by the polarization model is much larger than the traditional one, with a decay length larger than that predicted by the Debye– Hqckel theory. In summary, when the Debye–Hqckel k DH largely exceeds the dipole correlation length k m (either because of low ionic strength, or because of disorder), the polarization model predicts, at large separations, results similar to those obtained from the DLVO theory (with suitably adjusted

Polarization Model and ion specificity: applications

591

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

127

Fig. 7. (a) The potential in the vicinity of a surface for various surface charge densities (c E=0.1 M, p C/eV=0, p A/eV=0) calculated using the polarization model (open symbols) and the traditional theory (filled symbols). The effect of disorder on the dipole correlation length is taken into account via an empirical parameter, m 0=0.05 Debyes. When the average dipole moment of the water molecule m is much larger than m 0 (at large surface charge densities and near the surface), the (disorder-free) polarization model is obtained. When mVm 0 (at low surface charge densities and far away from the surface), the DLVO theory is recovered. (b) The force between the identical surfaces of (a), immersed in an electrolyte of c E=0.1 M, for various surface charge densities, calculated using the polarization model which includes disorder (m 0=0.05 Debyes) (open symbols) and the traditional theory (filled symbols). At low surface charge densities, the traditional theory is recovered, while at large densities the polarization model predicts an interaction completely different (in both magnitude and slope) from that obtained from the DLVO theory.

boundary conditions). However, when k DH is comparable with k m , qualitatively different results are predicted by the two theories.

4. Conclusion The interactions between some colloidal surfaces immersed in an electrolyte can be well described by the

traditional DLVO theory [13], at least in a certain range of electrolyte concentrations and by employing the surface properties (surface charge, potential or dissociation constants) as fitting parameters. However, the inter-particle interactions, as well as other aspects of colloidal behaviour of many other systems cannot be understood in terms of the traditional theory, and a supplementary force (bthe hydration forceQ) is sometimes employed to account for the experimental results.

Nanodispersions

592 128

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 109–128

While the hydration force was associated with the structuring of water in the vicinity of a lyophilic surface [12], there is no consensus about its microscopic origin. This incertitude is probably due to the apparent contradictory experimental results: for phospholipid bilayers, the hydration force is apparently independent of the electrolyte concentration and has a decay length of about 2 2 [11], while for mica surfaces the hydration is strongly dependent not only on the electrolyte concentration, but also on the nature of the cation (cation-specific effects) [17] and has a decay length of about 10 2. In this article, it is shown that the polarization model proposed earlier by the authors [30] can explain both these apparently contradictory experimental results [11,17]. The physical basis of the model [29,30] is reviewed and the conditions under which the polarization model predicts results qualitatively similar to those obtained from the traditional DLVO theory are discussed. The qualitative departures from the traditional theory are also examined, and it is noted that the polarization model can explain experimental results regarding the stability of some colloids at high ionic strengths [14], and the restabilization of protein-covered latex particles at large electrolyte concentrations [21,22], via the formation of surface dipoles due to the association of the electrolyte ions with surface groups. The polarization model can also predict the large virial coefficient of apoferritin molecules at high ionic strengths [18], because the correlation between neighboring dipoles increases the range of the interactions. The polarization model is extended to account for the ion–ion and ion–surface interactions, not included in the bmean fieldQ electrical potential. The role of the disorder on the dipole correlation length k m is modeled through an empirical relation, and it is shown that the polarization model reduces to the traditional Poisson–Boltzmann formalism (modified to account for additional interactions) when k m becomes sufficiently small. References [1] [2] [3] [4] [5] [6] [7]

Gouy G. J Phys Radium 1910;9:457. Chapman DL. Phyl Mag 1913;25:475. Langmuir I. J Chem Phys 1938;6:893. Stern O. Z Elektrochem 1924;30:508. Spaarnay MJ. Recl Trav Chim Pays-Bas 1958;77:872. Gur Y, Ravina I, Babchin AJ. J Colloid Interface Sci 1978;64:333. Levine S, Bell GM. Discuss Faraday Soc 1966;42:69; Ruckenstein E, Schiby D. Langmuir 1985;1:612.

[8] Jonsson B, Wennerstrom H. J Chem Soc, Faraday Trans II 1983; 79:19. [9] Bostrfm M, Williams DMR, Ninham BW. Phys Rev Lett 2001;87: 168103. [10] Manciu M, Ruckenstein E. Adv Colloid Interface Sci 2003;105:63; Ruckenstein E, Manciu M. Adv Colloid Interface Sci 2003;105:177. [11] Rand RP, Parsegian VA. Biochim Biophys Acta 1989;988:351. [12] Derjaguin BV, Churaev NV, Muller VM. Surface forces, 1987, Plenum. [13] Deryagin BV, Landau L. Acta Physicochim URSS 1941;14:633; Verwey EJ, Overbeek JThG. Theory of Stability of Lyophobic colloids. Amsterdam7 Elsevier; 1948. [14] Healy TW, Homola A, James RO, Hunter RJ. Faraday Discuss Chem Soc 1978;65:156. [15] Paunov VN, Kaler EW, Sandler SI, Petsev DN. J Colloid Interface Sci 2001;240:640. [16] Manciu M, Ruckenstein E. Langmuir 2002;18:5178. [17] Pashley RM. J Colloid Interface Sci 1981;83:531. [18] Petsev DN, Vekilov PG. Phys Rev Lett 2000;84:1334. [19] Manciu M, Ruckenstein E. Langmuir 2002;18:8910. [20] Collinns KD, Washabaugh MW. Q Rev Biophys 1985;18:323. [21] Molina-Bolivar JA, Galisteo-Gonzalez F, Hidalgo-Alvarez R. Phys Rev E 1997;55:4522; Lo´pez-Leo´n T, Jo´dar-Reyes AB, Ortega-Vinuesa JL, Bastos-Gonza´lez D. J Colloid Interface Sci 2004 [submitted for publication]. [22] Stenkamp VS, McGuiggan PM, Berg JC. Langmuir 2001;17:637. [23] Kjellander R, Marcelja S. Chem Scr 1985;25:73 Chem Phys Lett 1985;120:393. [24] Marcelja S. Nature 1997;385:689. [25] Otto F, Patey GN. J Chem Phys 2000;113:2851. [26] Attard P, Batchelor MT. Chem Phys Lett 1988;149:206. [27] Israelachvili JN, Wennerstrom H. Langmuir 1990;6:873. [28] Binder H, Dietrich U, Schalke M, Pfeiffer H. Langmuir 1999;15:4857. [29] Schiby D, Ruckenstein E. Chem Phys Lett 1983;95:435. [30] Ruckenstein E, Manciu M. Langmuir 2002;18:7584. [31] Gruen DWR, Marcelja S. Faraday Trans. II, (1983) 211 and 225. [32] Simon SA, McIntosh TJ. Proc Natl Acad Sci U S A 1989;86:9263. [33] Berkowitz ML, Raghavan K. Langmuir 1991;7:1042. [34] Manciu M, Ruckenstein E. Langmuir 2001;17:7061. [35] Manciu M, Ruckenstein E. Langmuir 2001;17:7582. [36] Perera L, Essmann U, Berkowitz ML. Prog Colloid & Polym Sci 1997;103:107. [37] Jackson JD. Classical Electrodynamics. New York7 John Wiley & Sons; 1975. [38] Faraudo J, Bresme F. Phys Rev Lett 2004;92:236102–1. [39] Langmuir I. J Am Chem Soc 1917;39:1848. [40] Ruckenstein E, Huang H. Langmuir 2003;19:3049. [41] Overbeek JTG. Colloids Surf 1990;51:61. [42] Manciu M, Ruckenstein E. Langmuir 2003;19:1114. [43] Helfrich W. Z Naturforsch 1978;33a:305; Manciu M, Ruckenstein E. Langmuir 2002;18:4179. [44] Petrache HI, Kimchi I, Harries D, Tristram-Nagle S, Podgornik R, Parsegian VA. Biophys J 2004;86:379A. [45] Huang HH, Manciu M, Ruckenstein E. J Colloid Interface Sci 2003; 263:156. [46] Prieve DC, Ruckenstein E. J Theor Biol 1976;56:205.

Polarization Model and ion specificity: applications

593

Advances in Colloid and Interface Science 127 (2006) 29 – 42 www.elsevier.com/locate/cis

Polarization model for poorly-organized interfacial water: Hydration forces between silica surfaces Marian Manciu a,⁎, Oscar Calvo a , Eli Ruckenstein b,1 b

a Department of Physics, University of Texas at El Paso, El Paso, TX, 79968, United States Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, NY, 14260, United States

Available online 3 October 2006

Abstract The main goal of this paper is to review the theoretical models which can be used to describe the interactions between silica surfaces and to show that a model proposed earlier by the authors (the polarization model), which accounts concomitantly for double layer and hydration forces, can be adapted to explain recent experiments in this direction. When the water molecules near the interface were considered to have an ice-like structure, a strong coupling between the double layer and hydration forces (described by the correlation length between neighboring dipoles, λm) generates long range interactions, larger than the experimentally determined interactions between silica surfaces. Arguments are brought that a gel layer is likely to be formed on the surface of silica, which, by generating disorder in the interfacial water layers, can decrease strongly the value of λm. Since the prediction of λm involves a choice for the microscopic structure of water, which is often unknown, the polarization model is also presented here as a phenomenological theory, in which λm is used as a fitting parameter. Two extreme cases are considered. In one of them, the water molecules near the interface are considered to have an ice-like structure, whereas in the other they are considered randomly distributed. In the first case, the dipole correlation length λm = 14.9 Å. In the second limiting case, λm can be of the order of 1 Å. It is shown that, for λm = 4 Å, a more than qualitative agreement with the experiment could be obtained, for reasonable values of the parameters involved (e.g. surface dipole strength and density, dipole location, surface charge). © 2006 Elsevier B.V. All rights reserved. Keywords: Hydration force; Colloidal interactions; Silica

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Hydration force . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Short range interactions between silica surfaces . . . . . . . . . . . 2. Polarization model for the hydration force: a phenomenological approach . 2.1. Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Microscopic origin of the polarization model. . . . . . . . . . . . . . . . 3.1. Dipole correlation length . . . . . . . . . . . . . . . . . . . . . . 3.2. Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . 4. Interactions between silica surfaces. . . . . . . . . . . . . . . . . . . . . 5. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

⁎ Corresponding author. Tel.: +1 915 747 7531; fax: +1 915 747 5447. E-mail addresses: [email protected] (M. Manciu), [email protected] (E. Ruckenstein). 1 Tel.: +1 716 645 2911/2214; fax: +1 716 645 3822. 0001-8686/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2006.08.001

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

30 30 31 32 32 33 34 34 36 38 41 41 41

Nanodispersions

594 30

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

1. Introduction Most of the water-mediated interactions between surfaces are described in terms of the DLVO theory [1,2]. When a surface is immersed in water containing an electrolyte, a cloud of ions can be formed around it, and if two such surfaces approach each other, the overlap of the ionic clouds generates repulsive interactions. In the traditional Poisson–Boltzmann approach, the ions are assumed to obey Boltzmannian distributions in a “mean field” potential. In spite of these rather drastic approximations, the Poisson–Boltzmann theory of the double layer interaction, coupled with the van der Waals attractions (the DLVO theory), could explain in most cases, at least qualitatively, and often quantitatively, the colloidal interactions [1,2]. The limitations of this simplified model have been immediately recognized, and the first criticism [3] even preceded the full development of the DLVO theory. Since then many improvements of the theory have been proposed, to account for the finite size of the ions [4], image forces [5], dielectric corrections [6], ion correlations [7], ion-dispersion [8] and ionhydration forces [9], to name only a few. Despite the many corrections brought to the traditional DLVO theory, there are some experiments, such as those regarding the stability of neutral lipid multilayers, which could still not be explained within this framework. It is therefore commonly accepted that an additional repulsion occurs when two surfaces approach each other at a distance shorter than a few nanometers. Because this force was initially related to the structuring of water near surfaces, it is commonly named “hydration force” [10]. 1.1. Hydration force At a simple phenomenological level, the “hydration force” can be described via an exponentially decreasing force, additive to (and independent of) the DLVO double layer and van der Waals forces [10]. An alternative phenomenological description is to consider the existence of an “order parameter” and a Landau-like expansion of the free energy in that parameter. When only some of the expansion terms are retained in the latter expansion, both phenomenological descriptions lead to similar behaviors for the hydration forces [11]. The debate started when various microscopic models for the origin of the “hydration force” have been proposed. One plausible possibility is that the “hydration force” might involve in fact a plethora of different phenomena, each having a different microscopic origin. For examples, the hydration interaction measured between mica surfaces [12] exhibits an oscillatory behaviour, in contrast with the hydration force measured between neutral lipid bilayers [10]. Even more puzzling, experiment shows that the latter forces seem to be almost independent of electrolyte concentration, while in other systems they depend not only on the ionic strength, but even on the kind of ions employed, exhibiting strong specific ion effects. For example, for the interactions between mica surfaces, deviations from the DLVO theory (hence a “hydration force”) has been obtained only above some critical electrolyte concentrations, which were

about cE,critical = 10−2 M for LiCl and NaCl, cE,critical = 10− 3 M for CsCl, and cE,critical = 3 × 10− 4 M for KCl [13]. Also, amphoteric latex particles, at pH values above the isoelectric point, coagulated when the CsNO3 exceeded about 0.3 M, but remained stable up to concentrations exceeding 3 M of KNO3 [14], pointing out toward strong specific ionic effects associated with the hydration force. The earlier attempts to explain hydration forces suggested merely corrections to the Poisson–Boltzmann approach, to account for the additional effects not included in the mean field [4–9]. The more recent models proposed that the free energy of the system contain not only the traditional DLVO free energy terms, but also an additional free energy term associated with the interface. Several possible microscopic origins of the additional term have been proposed, such as (i) the structuring of water near the interface, with the corresponding disruption of hydrogen bonding [15], (ii) the dipoles present in water and on surfaces [5,16,17], (iii) the repulsion generated by ion dehydration [14,18] and (iv) the steric effect [19,20]. The disruption of hydrogen bonding constitutes an appealing possible explanation of the hydration force. Attard and Batchelor presented a two-dimensional lattice model, and concluded that both the preexponential factor and the decay length of the hydration force are determined by only one unknown parameter, the concentration w of Bjerrum defects in the vicinity of the surface [15], which are proton disorders of the ice structure, consisting of either no H atom or two H atoms between two neighboring O atoms (L and D defects, respectively). It is clear that this model cannot predict the oscillatory behavior of hydration in some systems [12] or why the ions (or even the nature of the ions) seem to govern the coagulation of colloids in other systems [13], [14]. A more serious issue is that for neutral lipid bilayers, the preexponential factor of the hydration force depends on the nature of the lipid that forms the bilayers, while the decay length (2–3 Å) seems to be almost independent of it [10]. For this reason, only one parameter (w) is perhaps not sufficient to describe simultaneously both quantities. An ion dehydration origin of hydration force has been initially proposed by Healy et al. [14] and a statistical model has been developed recently by Paunov et al. [18]. Healy et al. suggested that the approach at very short distances of two surfaces is possible only if the ions adsorbed on the surfaces have lost their hydration shell, therefore the ions with large hydration energies might generate hydration forces large enough to prevent coagulation. One problem with this model is that the range of the hydration interactions cannot exceed two diameters of the hydrated ion (about 14 Å) [18], whereas hydration forces have been currently identified at distances of up to 20–40 Å. Another difficulty is that the hydration force in this model is directly related to the ionic strength, whereas in many systems they are present regardless of the presence or absence of an electrolyte (e.g. for neutral lipid bilayers [10]). The steric repulsion, caused e.g. by coating the colloidal particles with a polymer, is well known to generate a force large enough to prevent coagulation at any electrolyte concentration [19]. It was suggested by Israelachvili and Wennerström that the statistical overlap of lipids protruded from lipid bilayers might

Polarization Model and ion specificity: applications

595

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

generate an exponentially decaying force, with a preexponential factor and decay length comparable to those experimentally measured for neutral lipid bilayers [20]. Clearly, this model cannot explain the hydration forces for other systems, such as those between mica or silica surfaces. Its validity was, however, questioned even for lipid bilayers, because the polymerization of the lipids within the bilayers, which was expected to decrease drastically the protrusion rate, did not affect the hydration force [21]. The initial microscopic models for the hydration force have been somehow related to the presence of permanent dipoles on the water molecules and on the immersed surfaces. This choice was natural, because the large dielectric constant of water (hence its solvation ability for polar molecules) is largely due to the permanent dipole moment of the water molecule. Jönsson and Wennerström suggested that the interactions between the surface dipoles and their image charges might be responsible for the hydration force [5]. However it turned out that in order to explain a typical experiment, the magnitude of the surface dipoles would have to be about two orders of magnitude larger than the normal molecular dipoles [22]. Gruen and Marcelja [17] suggested that the order parameter in the Marcelja–Radic treatment [11] might be the water polarization, and they conjectured that the Bjerrum defects might constitute the source of the polarization field, in analogy with the ions, which provide the source of the electric field [17]. The model considered the polarization and the electric fields as independent, and coupled equations for them have been obtained through a variational procedure. When a suitable value was selected for the concentration of Bjerrum defects, an exponential repulsion similar to the hydration force obtained experimentally was acquired. However, this theory did not explain how the concentration of Bjerrum defects in water is related to the physical properties of the immersed surface and it is not clear how the polarization of the surface induces the polarization of the water molecules (in a continuous model, a surface with a constant dipolar density does not generate any electric field in the fluid in which it is immersed). In addition, it was later suggested that the exponential decay of the hydration force predicted by the Gruen– Marcelja model was a result of an inconsistent choice for the nonlocal dielectric function [23]. The strongest critique of the model came, however, from Molecular Dynamics simulation, which showed that for some systems the water polarization might oscillate near an immersed surface [24,25], whereas the disruption of hydrogen bonding (the Bjerrum defects) varies monotonically [26]. Therefore, unlike the disruption of the hydrogen bonding, the polarization does not constitute a proper order parameter, in the sense of Landau's theory. The various qualitative behaviors of the hydration force in different systems (either oscillatory [12] or monotonic [10] , with various decay lengths (2–3 Å [10] or about 10 Å [13]), either independent of electrolyte concentration [10] or exhibiting strong specific ion effects [14]) appear to point out toward the existence of a number of different microscopic origins for the short range repulsions between surfaces immersed in water, in excess to those accounted by the DLVO theory. On the other hand, there are some striking similarities between the hydration forces in different systems. For example, the Molecular Dy-

31

namics simulation showed an oscillatory behaviour of the polarization of water between two layers of sodium dodecylsulfate [25] and a corresponding oscillatory force [27]. The results are comparable with the earlier Molecular Dynamics simulations of Berkowitz et al., which indicated an oscillatory polarization in the vicinity of a neutral lipid bilayer [24]. However, when the headgroups of the bilayer surface have been allowed to move, the polarization oscillations have been smoothed away and therefore a monotonic behaviour of the hydration force has arisen [28]. Therefore, the oscillatory behaviour of the hydration force, for the flat mica surfaces, and the monotonic behaviour for fluid lipid bilayers might be explained by the same molecular mechanism, namely the polarization of the water molecules in the vicinity of the surfaces. A model for this polarization was proposed by Schiby and Ruckenstein [16], who suggested that it might be a consequence of the discrete nature of the water molecules and surface dipoles [16]. A uniform, continuous surface dipolar density does not generate any electrical field in the adjacent continuous medium. In contrast, when discrete dipoles are placed on a surface lattice, an exponentially decaying field near the surface is generated [29]. Whereas this field (calculated for a continuous medium) is too weak to explain alone the hydration force, when the discrete nature of the water molecules is accounted in the dielectric response of water (which consequently becomes non-local), the correlation between the surface dipoles and the water dipoles becomes strong enough to generate a repulsion similar to the hydration force [30]. Moreover, when the water is organized in ice-like layers (presumably in the vicinity of a flat surface, like mica) the dipole correlation generates an oscillatory polarization in water, in the vicinity of the surface, and a corresponding oscillatory hydration force [31]. Within this “polarization model”, the coupling between the “hydration”, the “double layer” [32] and ion-hydration interactions [9] could explain why specific ion effects occur and why the decay length of the “hydration forces” might vary between 2 Å and about 10 Å [33]. 1.2. Short range interactions between silica surfaces The present paper has been stimulated by a recent article [22], which reported experiments on short range interactions between silica surfaces and explored possible explanations of the results by using various theoretical models for the hydration force. It is somewhat surprising that most theoretical models investigated provided a reasonable agreement with the experiment, while the best fit of the experimental data of the polarization model provided qualitatively different results (a short range attraction and a long range repulsion, instead of a short range repulsion and long range attraction). The surprise is due to the fact that the polarization model has been shown to describe the hydration forces for various systems [33] and also because it involves many unknown constants, which could be used as fitting parameters. Nevertheless, Valle-Delgado et al., concluded that the polarization model, with the values of the parameters employed for neutral lipid bilayers [32], cannot explain the hydration force measured for silica surfaces.

Nanodispersions

596 32

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

Two explanations for the discrepancy are possible. One of them is related directly and the other one indirectly to the peculiar structure of the silica surface, which increases its roughness upon immersion in water. A freshly broken quartz surface is composed of dangling silicon and oxygen bonds, which are quite unstable and can be hydroxylated within seconds if water is available, forming mostly silanol groups, which in turn interact with the neighboring water molecules, which are strongly organized by them [34]. Considering also the relatively large dissolution rate of silica surfaces [35] it is very unlikely that the surface of silica remains flat after its immersion in water. Even more drastically, it was suggested [36] that the hydroxilation of the siloxane groups (Si–O–Si) to silanol (Si– OH) groups might continue upon immersion in water, formingSi(OH)2–O–Si(OH)2–OH polymeric chains, which in turn generate a silica gel with a thickness of about 10 Å that covers the silica surface [36]. Such a gel might generate a steric repulsion between two silica surfaces, and therefore might provide a microscopic explanation for the hydration forces measured in silica [36]. The latter suggestion was, however, challenged by Yoon and Vivek [37], who observed that the hydration force disappeared when 15% methanol was added to water; they argued that the adsorption of methanol on the silica surface cannot affect too much the roughness of the surface or the structure of the polymer-like silica hairs near the surface [37], but it is likely to disrupt the hydrogen bonding between the silica surface and the neighboring water molecules, hence the organization of water near the silica surface. Consequently Yoon and Vivek suggested that the organization of the interfacial water should be responsible for the hydration force between silica surfaces [37]. The second possible explanation, that relies indirectly on the roughness of the silica surface, is based on the interaction between silica and the interfacial water. It has been suggested that the water adsorbed on silica is strongly oriented by the surface and has characteristics (entropy, mobility, dielectric constant) drastically different from those of bulk water, for at least the first three water monolayers (about 10 Å) [38]. Because the interfacial water is more disorganized than the bulk water (or an ice-like layer of water), the correlation between neighboring dipoles, which is related to their relative position, is expected to be also weakened. The consequences of a weaker dipole correlation on the polarization model for hydration forces have been investigated briefly in a previous article [33]. In the present paper, a slightly different approach is employed. In Section 2, the basics of the polarization model as a phenomenological theory will be presented, which involves only a single parameter, the dipole correlation length λm. The reason for this approach is that the evaluation of λm requires knowledge of the local structure of water, which is often missing. By employing λm as an unknown (fitting) parameter, the problem is considerably simplified. In Section 3, a microscopic justification for the dipole correlation length is presented and the effect of the structural disorder of water on λm is examined. It will be shown that the disorder can generate values for λm much smaller than λm = 14.9 Å, which was obtained for an ice-like organization of water in the vicinity of neutral lipid bilayers [32]. In Section 4,

the results provided by the polarization model for the hydration forces are compared with the experimentally determined silica interactions [22]. Using the value λm = 4 Å, it will be shown that a more than qualitative agreement with experiment can be obtained. This value is consistent with a random distribution of the nearest neighbors water molecules. The implication of the results obtained from the fit on the structure of silica surfaces will be discussed. 2. Polarization model for the hydration force: a phenomenological approach 2.1. Basic equations In a continuous and homogeneous liquid, the polarization is related to the macroscopic electric field via a local dielectric constant ε(r), Y Y PðrÞ ¼ e0 ðeðrÞ−1Þ EðrÞ;

ð1Þ

Y where ε0 is the vacuum permittivity, PðrÞ is the polarization, Y EðrÞ is the macroscopic electric field and ε(r) is the dielectric constant at point r. In the polarization model, the average polarization of a water molecule depends not only on the macroscopic electric field applied, but also on the dipoles of the neighboring molecules. If the water is considered a continuous and homogeneous dielectric, with, ε(r) ≡ ε, then the field at the location of a water molecule, created by all the dipoles of all the other molecules, should vanish. However, if the interactions between neighboring dipoles are less screened than the interactions with the more remote dipoles, there will be a net remaining field at the location of the molecule, created by all the surrounding dipoles [30]. This field polarizes the molecule and will generate a correlation in the orientation of the neighboring dipoles. Assuming that the surface immersed in water is planar and that the coordinate z is normal to the surface, the polarization m(r) = v0P(r) of a water molecule of volume v0 is provided by [32]: mðzÞ ¼ e0 m0 ðe−1ÞEðzÞ þ k2m

d 2 mðzÞ dz2

ð2Þ

Eq. (2) accounts for the correlation of neighboring dipoles through only one parameter, the dipole correlation length λm. An examination of the microscopic origin of the parameter λm (and the justification of Eq. (2)) will be presented in Section 3; until then the dipolar coupling can be thought as a phenomenological fitting parameter of the polarization model. Clearly, when λm → 0, Eq. (2) becomes the conventional relation between polarization and macroscopic electric field, Eq. (1). dwðzÞ The mean field potential ψ(z), defined by EðzÞ ¼ − , dz is related to the polarization and charge via the Poisson equation: d 2 wðzÞ qðzÞ 1 dmðzÞ ¼− þ dz2 e0 e0 m0 dz

ð3Þ

Polarization Model and ion specificity: applications

597

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

Assuming that the charge distribution is due only to the ions of a dissociated electrolyte of concentration cE, which obeys Boltzmannian distributions:
 qC wðzÞ þ DWC ðzÞ qðzÞ ¼ cE qC exp − kT
 qA wðzÞ þ DWA ðzÞ þ cE qA exp − kT

ð4Þ

In the Poisson–Boltzmann theory, the free energy per unit area of two identical planar surfaces placed at the distances z = −d and z = d, respectively, is composed of an electrostatic energy [39], Z

d −d

in which the integral is performed by considering that the surfaces approach from infinite distance to the final separation 2d, and σ is the surface charge density at each separation, and a term due to the additional interactions between ions and interfaces, not included in the mean field [41]: Z z¼d Uint ð2dÞ ¼ ðcA ðzÞDWA ðzÞ þ cC ðzÞDWC ðzÞÞdz ð5dÞ z¼−d

where cA and cC are the local concentrations of anions and cations, respectively, and ΔWA and ΔWC are defined after Eq. (4). The electrostatic free energy (Eq. (5a)) accounts for the interaction of the charges and dipoles with the macroscopic field in the conventional theory. However, in the polarization model, there is an additional contribution to the energy, because the dipoles interact with the field that generates their correlation. As it will be shown in the next section, Eq. (2) can be rewritten as [32]: mðzÞ ¼ gEtotal ¼ e0 m0 ðe−1ÞðEðzÞ þ Em ðzÞÞ

DðzÞEðzÞdz ¼

1 2

Z

d −d

e0 eE 2 ðzÞdz

ð5aÞ

Em ¼

k2m d 2 mðzÞ e0 m0 ðe−1Þ dz2

ð6bÞ

Consequently, the electrostatic free energy in the polarization model becomes: Uel ð2dÞ ¼

Z

d



−d

Z

d −d

 1 1 mðzÞ Eðe0 E þ PÞ− Em dz 2 2 m0 


 mðzÞ 1 mðzÞ e0 EðzÞ þ EðzÞ− m0 2 m0

 k2m d 2 mðzÞ dz e0 m0 ðe−1Þ dz2 #  Z "
1 d 2mðzÞ 1 mðzÞ2 ¼ e0 EðzÞ þ EðzÞ− dz 2 −d m0 2 e0 m20 ðe−1Þ 

ð5eÞ

(in which the displacement vector is considered proportional to the macroscopic electric field, D = ε0εE), an entropic term due to the ions [39], Uent ð2dÞ ¼ −T DS ¼ kT

ð6aÞ

where γ is the molecular polarizability and Em is the local field that aligns the neighboring dipoles, which is given by:

1 ¼ 2

2.2. Free energy

1 2

where the sum runs over the anions and cations present in the system, a “chemical” free energy [39,40] Z rðdÞ Uch ð2dÞ ¼ − wS ðrÞdr ð5cÞ rðlÞ

where k is the Boltzmann constant, T is the absolute temperature, qA and qC are the anion and cation charges, and ΔWA and ΔWC are free energies of interactions with the interface, not included in the mean field, of the anions and cations, respectively. Some typical examples for the latter interactions are: the image forces [5], the ion-dispersion [8] or the ion-hydration forces [9]. Eqs. (1), (3) and (4) represent a Modified Poisson–Boltzmann approach, which, when the ions interacts only via the mean field potential ψ(z) (e.g. ΔWa = ΔWc ≡ 0), reduces to the well-known Poisson–Boltzmann equation. The only change in the polarization model is the replacement of the constitutive equation Eq. (1) by Eq. (2), which accounts for the correlation in the orientation of neighboring dipoles. However, since ψ(z) and m(z) are now independent functions, four boundary conditions are needed to solve the system composed of Eqs. (2) (3) and (4). For only one surface immersed in an electrolyte, ψ(z = ∞) = 0, m(z = ∞) = 0 whereas for two identical surfaces, the symmetry of ψ and m dwðz ¼ 0Þ requires that ¼ 0 and m(z = 0) = 0 where z = 0 represents dz the midpoint between surfaces. In a phenomenological approach the surface charge (or the surface potential) as well as the water polarization near the surface can be considered fitting parameters. In Section 3, in which a microscopic model for the dipole correlation length is presented, it will be shown that knowledge of the surface properties (e.g. surface charge density, surface dipolar density, location and magnitude of the surface dipoles) provides natural boundary conditions for the equations of the polarization model.

Uel ð2dÞ ¼

33

 X Z d

ci  ci ln −ci þ cE dz cE −d i ð5bÞ

where Eq. (6a) was employed in the last substitution. In summary, the polarization model represents an extended Poisson–Boltzmann approach, in which the “hydration” and the “double layer” are not independent interactions, but are intimately coupled to each other, via an electrostatic coupling between the fields ψ(z) and m(z). These fields can be calculated by solving Eqs. (2) (3) and (4), and the total free energy of the system can be obtained by summing up the terms provided by

Nanodispersions

598 34

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

Eqs. (5b) (5c) (5d) (5e). The constitutive equation of the polarization model contains only one unknown parameter, namely λm, which expresses the correlation between neighboring dipoles. However, in order to solve the system of equations, both the surface charge density σ (or the surface potential ψ(z = −d)) as well as the polarization of water near the surface have to be known. The latter can be related, in a microscopic model that will be examined in the next section, to the surface dipoles. In the limit λm → 0, the polarization model reduces to the Poisson– Boltzmann approach, and the two boundary conditions become dependent on each other, because in this case mðzÞjz¼−d ¼ −e0 m0 ðe−1Þ

dwðzÞ e−1 ¼ m0 r dz e

ð7Þ

The polarization model has been employed previously to show that, at low electrolyte concentrations, the total interaction can be indeed calculated as the sum between two apparent independent interactions, a traditional “double layer” and an exponentially decaying “hydration force” [32]. However, at high electrolyte concentrations the coupling between the “double layer” and “hydration” can generate an unusually long range repulsion, which is perhaps responsible for the restabilization of apoferritin molecules [42]. For various combinations of the surface charge and surface dipole density, interactions with decay lengths between 2 to about 10 Å could be predicted [33]. With the supplementary hypothesis that the water is structured as icelike layers near the interface (note that in this case a discrete version of Eq. (2) had to be employed [31]), the polarization model could explain the oscillations of the polarization near the surfaces [31], [43], which were observed via Molecular Dynamics simulations [24,25] and the corresponding non-monotonic hydration force [27]. If no microscopic model is available to calculate the dipole correlation length, one might consider λm as a phenomenological parameter of the polarization model. However, before investigating the applicability of the polarization model to the interactions between silica surfaces, a more microscopic representation is used in the next section to obtain an expression for λm and to derive a relation between the surface dipoles and the polarization of water near the interface.

Fig. 1. A. A cavity in the dielectric, containing a central water molecule and its neighbors. The radius of the cavity is considered large enough, for the molecular details of the dielectric outside the cavity to be ignored. Consequently, the dielectric outside the cavity can be considered as a homogeneous and continuous medium. B. A schematic structure of the silica–water interface. Each surface dipole exerts an average field on the water molecules above him; the fields generated by the remote dipoles are neglected.

is far enough from the central molecule, to be able to ignore its molecular details. This means that the medium outside the cavity can be considered to behave as a homogeneous, continuous dielectric. The field acting on the central molecule is composed of three terms: the macroscopic field E, the Lorenz field ELorenz ¼

P 3e0

created inside the spherical cavity embedded

in a continuous dielectric with the uniform polarization P [44], and the field Em created by the neighboring molecules: P þ Em ; 3e0

3. Microscopic origin of the polarization model

Elocal ¼ E þ

3.1. Dipole correlation length

It was shown by Lorentz [44] that, for a medium of constant polarization and with a sufficient symmetric distribution of the neighboring molecules around the central one, the field Em vanishes. Then, the average dipole moment of the central molecule, m(z) = v0P(z), where v0 is the volume occupied by a water molecule, is provided by [32]:
 m g m ¼ gElocal ¼ g E þ ¼ Eue0 m0 ðe−1ÞE ð9Þ 3e0 m0 1− 3eg0 m0

Let us first review the basics of the Lorentz theory for polarization. If one assumes that a constant macroscopic field is applied to a homogeneous medium of dielectric constant ε, the polarization through the medium will be uniform. However, the polarization of a molecule is not proportional to the macroscopic electric field (created by sources external to the dielectric), but to the local electric field, which contains also the field generated by all the other molecules of the dielectric. To account for the latter, one can separate the medium in a spherical cavity (in which the central molecule and its molecular neighbors reside, see Fig. 1A) and the rest of the medium, which

ð8Þ

where γ is the molecular polarizability. The last equality is equivalent to Eq. (1), because for a uniform polarized medium the polarization is proportional to the macroscopic electric field;

Polarization Model and ion specificity: applications

599

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

it provides the Clausius–Mossotti relation between the microscopic molecular polarizability γ and the macroscopic dielectric constant ε. If the polarization is non-uniform, an additional field Em is generated by the neighboring dipoles. A simple way to calculate this field [30] is to neglect the field generated by the remote dipoles, because it is screened by the intervening water molecules, and to consider only the field generated by the first “i” neighboring dipoles, screened by an effective dielectric constant ε″, much lower than the bulk value, ε = 80. Each neighboring molecule “i”, with an average dipole moment mi, located at the position ri from the central molecule, will create at the location z0 of the central molecule the field: Y E m;i ðz0 Þ ¼

1 3Y r i ðY mid Y r i Þ−Y m i ðY r id Y ri Þ 5 4ke0 eW ri

ð10aÞ

Since we are interested only in the dipole correlation along the z directions (it is assumed that ψ(z) and m(z) do not depend on the x and y coordinates in the plane of the surface; because of this planar symmetry the x and y components of Y m ðzÞ vanish), the component of the field along the z direction has the form: Em;i ðz0 Þ ¼

mi 3cos2 hi −1 4ke0 eW ri3

ð10bÞ

where θi represents the angle between ri and the axis z. Assuming that the water polarization is a continuous function of z, the average dipole moment can be expanded around z = z0 to obtain: mi ¼ mðz0 Þ þ ðzi −z0 Þ

dmðz0 Þ 1 d 2 mðz0 Þ þ ðzi −z0 Þ2 þ …: dz 2 dz2

(12) the positions of the first 26 neighbors of a water molecule as in an ice-like layer, one can write [32]: mðz0 Þ 1:8272D2 d 2 mðz0 Þ ð−3:77 þ 2d1:83Þ þ 4ke0 eWl 3 4keWe0 l 3 dz2 2 2 1:83 16 d mðz0 Þ km d 2 mðz0 Þ c u 2 4keWe0 l 9 dz e0 m0 ðe−1Þ dz2

Em ðz0 Þ ¼

ð13Þ where Δ = (4 / 3) l is the distance between the centers of adjacent ice-like water layers. The last equality introduces a new parameter, the dipole correlation length. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1:83m0 ðe−1Þ km ¼ ð14Þ 3 keWl In Eq. (13), the term proportional to m(z0) is negligible (which shows that the field of the neighbors vanishes for a uniform polarization of the medium, as suggested long ago by Lorentz [44]), and the terms proportional to the derivatives of odd order vanish because of symmetry. Once the field Em has been determined, Eq. (9) should be replaced by:
 m g m ¼ gElocal ¼ g E þ þ Em ¼ E 3e0 m0 1− 3eg0 m0 ð15Þ 2 2 d m ue0 m0 ðe−1ÞðE þ Em Þ ¼ e0 m0 ðe−1ÞE þ km 2 dz The latter equation coincides with Eq.(2) of the previous section, employed by the polarization model. In the limit of low electrolyte concentrations cE → 0, the integration of Eq. (2) (with ρ(z) ≡ 0) leads to [32]:

ð11Þ mðzÞ ¼ e0 m0 Consequently the additional field is provided by: Em ¼

X i

Em;i

1 X 3cos2 hi −1 ¼ mi 4ke0 eW i ri3

35

dwðzÞ ; dz

ð16Þ

which was inserted in Eq. (3) provides: ð12Þ

The terms of higher order in Eq. (11) can be neglected. Indeed, the polarization of water near the surface is expected to decrease strongly and can be, in general, well approximated by an  exponential, mðzÞ ¼ m0 exp − kz ; consequently,n the term of order 1 ðzi −z0 Þ n of the expansion is proportional to . Since the field n! kn generated by the more remote dipoles is negligible (because it is screened by the intervening water molecules), for the near neighbors (zi − z0) b λ and the series is rapidly convergent. In order to evaluate the additional field Em, the local structure of water must be known. One simple assumption, which was made previously [30], was that the water is organized in ice-like layers in the vicinity of the surface, and that this structure is preserved at least for a distance 2l around a central molecule, where l is the distance of closest approach between two water molecules. Consequently, a cavity of radius 2l has been considered around the central water molecule, containing 27 water molecules. By considering in Eqs. (11) and

mðzÞ ¼

k2m d 2 mðzÞ e dz2

The solution of Eq. (17) is:
 z mðzÞ ¼ mð0Þexp − kH

ð17Þ

ð18Þ

with km kH ¼ pffiffi e

ð19Þ

The largest possible value of λm in Eq. (14) is obtained when ε″ = 1 for all the dipoles within the cavity (no screening), which provides λm = 14.9 Å [30], and Eq. (19) leads to λH = 1.67 Å. This value is comparable with the decay length of the hydration force measured in neutral lipid bilayers [10]. However, there is no reason for the dipolar correlation length to acquire its maximum value in all the systems. We estimated that the tilting of the clusters with small angles (below 20°)

Nanodispersions

600 36

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

decreases λm only by a few percents [31]. The consequences of a weaker dipole correlation for the interactions between surfaces have been also investigated earlier [33]. As shown by Valle-Delgado et al. [22], the value λ m = 14.9 Å is too large to explain the interactions between silica surfaces. Let us examine what happens in the opposite extreme case, when the water molecules near interface, instead of having the ordered structure of ice, have their neighboring water molecules at random orientations. For a random orientation, a neighboring molecule at a distance ri from the central one is situated with equal probability on a spherical shell of radius ri. Therefore, the field generated by each neighbor located inside the dielectric cavity can be obtained by averaging the field given by Eq. (10b) Z k Z 2k 1 Em;i ri2 sinhdhdu 4kri2 h¼0 u¼0 Z Z 2k 1 1 k mi 3cos2 h−1 ¼ sinhdhdu 4ke0 4k h¼0 u¼0 eW ri3 Z k 1 ¼ mi ð3cos2 h−1Þsinhdh 8ke0 eWri3 h¼0

6:27 km ¼ pffiffiffiffiffi A° eW ð20Þ

Using Eq. (11), Eq. (20) becomes: 1 8ke0 eWri3

1 ¼ 8ke0 eWri3

Z

mi ð3cos2 h−1Þsinhdh


k h¼0



Z

Z

k

ð3cos2 h−1Þsinhdh þ h¼0

k

ð3cos2 h−1Þsinhcoshdh þ h¼0



Z

ð3cos2 h−1Þsinhcos2 hdh ¼ h¼0

dmðz0 Þ 1 dz 8ke0 eWri2

d 2 mðz0 Þ 1 dz2 16ke0 eWri

k

1 d 2 mðz0 Þ 30ke0 eWri dz2

e0 ð21Þ

In Eq. (21) the terms corresponding to m(z0) and its odd order derivatives with respect to z vanish (note that the field in Eq. (10b) is proportional to the Legendre polynomial of the first order, which explains the vanishings). Assuming that on each shell of radius rj are Nj neighbors of order j, the total field is provided by: Em ¼

X

¯ m;i ¼ E

i

¼

1 30ke0

X

¯ m;j Nj E ! X Nj d 2 mðz0 Þ k2m d 2 mðz0 Þ u 2 ej Wrj dz e0 m0 ðe−1Þ dz2 j

The last equation indicates that dipolar correlations length smaller than λm = 15 Å are expected to occur in disordered systems.

As already noted, in a homogeneous medium a surface dipole density does not generate an electric field above the surface, and consequently such a field is ignored in the traditional theory of the double layer, which takes into account only the surface charges. In the polarization model, both the surface charge and the surface dipole densities generate electrostatic interactions (commonly denoted as “double layer” and “hydration “forces”). The first boundary condition at the surface is provided by integrating the Poisson Eq. (3) over the volume of a cylindrical box, which includes the surface, with the flat sides parallel to the surface and a vanishingly thin height. After using the Gauss theorem, one obtains:

dmðz0 Þ mðz0 Þ þ ri cosh dz

 1 d 2 mðz0 Þ þ ri2 cos2 h ð3cos2 h−1Þsinhdh 2 dz2 mðz0 Þ ¼ 8ke0 eWri3

ð25Þ

3.2. Boundary conditions

k h¼0

Z

By assuming that the field of the first 4 neighbors of a water molecule is screened by a medium with ε″ close to unity, while the field of the other dipoles by a much larger dielectric constant (εj″ ≫ 1), the dipole correlation length is provided by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4m0 ðe−1Þ km i ð24Þ 30kleW which, for typical values of the structural parameters of water (v0 = 32.4 Å3, l = 2.76 Å, ε = 80), becomes:

¯ m;i ¼ E

¯ m;i ¼ E

where εj″ depends on rj (with large screenings for the remote dipoles and less screening for the neighboring dipoles). Eq. (22) defines λm: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u um0 ðe−1Þ X Nj km ¼ t ð23Þ 30k ej Wrj j

j

ð22Þ

dwðzÞ dz

j

z¼−d

−

mðzÞ m0

j

z¼−d

¼ −r

ð26Þ

where z = −d indicates the surface and σ is the surface charge density. The second boundary condition at the surface is provided by the polarization of the water molecules from the vicinity of the surface, which is caused by the macroscopic field E ¼ − dwðzÞ dz jz¼−d , the average field Ēs generated by the surface dipoles and the field Em due to the neighboring water dipoles. The field generated by the surface dipoles is calculated by averaging the field generated by a dipole p in a medium of effective dielectric constant ε′, at a distance Δ′ above the center of the dipole over an area A corresponding to the area occupied by a surface dipole (see Fig. 1B). The fields generated by all the others (remote) surface dipoles are neglected, because they are screened by the intervening water molecules.

Polarization Model and ion specificity: applications

601

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

The electric field, normal to the surface, generated by a surface dipole, at a point whose position vector makes an angle θ with the normal, is given by: p Es ¼ ð3cos2 ðhÞ−1Þ 4keVe0 r3 where r ¼

DV . cosðhÞ

ð27Þ

For the average field produced by a surface

which becomes, after employing Eq. (11) with z − z0 = ricosθ: ¯ m;i ¼ E

1 4ke0 eWri3

1 ¼ 4ke0 eWri3

Z

k=2

mi ð3cos2 h−1Þsinhdh

h¼0 Z k=2
h¼0

dmðz0 Þ mðz0 Þ þ ri cosh dz

Þ

¼ ð28Þ

mðz0 Þ 4pe0 eWri3 

where A = πR02 is the average area occupied by a surface dipole. The field generated by the neighboring dipoles is provided by an expression similar to Eq. (12):

Z

Z

k=2

ð3cos2 h−1Þsinhdh þ h¼0

k=2

ð3cos2 h−1Þsinhcoshdh þ h¼0



Z

X

Em;i ¼

i

1 X 3cos2 hi −1 mi 4ke0 eW i ri3

ð29Þ

with the difference that in this case, the index “i” runs only over the neighbors which are above the interface (with z N z0, z0 being the position of the interfacial water molecule). For water that is organized (at least locally) in ice-like layers in the vicinity of the surface, with the structure preserved for at least a distance 2l around a central interfacial molecule, the average field was evaluated previously as [30,33]:
Em ¼ mðz0 Þ −

 1:9 dmðz0 Þ 1:83 þ 4ke0 eWl 3 dz 3ke0 eWl 2 d 2 mðz0 Þ 2d1:83 þ ; dz2 9ke0 eWl

ð30Þ

ð3cos2 h−1Þsinhcos2 hdh ¼

! ð31Þ

d 2 mðz0 Þ 1 dz2 30ke0 eWri

which, after using Eq. (15), becomes ¯ m;i ¼ E

mðz0 Þ dmðz0 Þ 1 þ 2 dz 16ke0 eWri2 30ke0 eWri km m0 ðe−1Þ dwðz0 Þ þ 30keWri k2m dz

ð33bÞ

and the total field Em at the location of the central interfacial molecule is given by: X ¯ m;i Em ¼ E ð34Þ where the index “i” runs only over the neighbors of the central molecule situated above the interface (z N z0). Accounting only for the first neighbors, the field becomes Em ¼

dmðz0 Þ 1:83 dwðz0 Þ 3:66m0 ðe−1Þ þ þ dz 3ke0 eWl 2 dz 9keWlk2m

mðz0 Þ dmðz0 Þ 1 m0 ðe−1Þ dwðz0 Þ þ þ : dz 8ke0 eWl 2 15keWlk2m dz 15ke0 eWlk2m ð35Þ

In the other extreme case, in which the neighboring water molecules are randomly oriented, one can assume that the water molecules neighboring an interfacial water molecule are located on a hemisphere centered on that water molecule. As in Eq. (20) with the difference that θ runs now form 0 to π / 2 instead of π, the average field generated by a neighbor “i” is provided by: Z k=2 Z 2k ¯ m;i ¼ 1 E Em;i ri2 sinhdhdu 2kri2 h¼0 u¼0 Z Z 1 1 k=2 2k mi 3cos2 h−1 ¼ sinhdhdu 4ke0 2k h¼0 u¼0 eW ri3 Z k=2 1 ¼ mi ð3cos2 h−1Þsinhdh 4ke0 eWri3 h¼0

dmðz0 Þ 1 dz 16ke0 eWri2

i

which, after employing Eq. (3), becomes: 1:9 3:66 Em ¼ mðz0 Þ − þ 4ke0 eWl 3 9ke0 eWlk2m

þ

dmðz0 Þ 1 dz 4ke0 eWri2

d 2 mðz0 Þ 1 dz2 8ke0 eWri

k=2 h¼0

Em ¼

ð33aÞ

1 d 2 mðz0 Þ þ ri2 cos2 h ð3cos2 h−1Þsinhdh 2 dz2

dipole one obtains [30]:
 Z R0 p 2kqdq D 2V E¯S ¼ 2 3 −1 3 kR0 0 4ke Ve0 ðD 2V þ q2 Þ2 D 2V þ q2 p 1 p 1 ¼ ¼   2ke0 e VðR20 þ D 2V Þ32 2ke0 eV Ak þ D 2V 32

37

ð32Þ

Finally, the second boundary condition at the surface is provided by:


 mðz0 Þ ¯ mðz0 Þ ¼ g Eðz0 Þ þ þ E S ðz0 Þ þ Em ðz0 Þ
3e0 m0  dwðz0 Þ ¯ S ðz0 Þ þ Em ðz0 Þ ¼ e0 m0 ðe−1Þ − þE ð36Þ dz _ with E s provided by Eq. (28) and Em given either by Eq. (31) (for an ice-like structuring of water near the interface), or by Eq. (35) (for the random orientation case). As noted in the previous section, for a single surface immersed in an electrolyte, the other two boundary conditions necessary to solve the system of Eqs. (2)–(4) are provided by

Nanodispersions

602 38

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

the vanishing of both the electric field and the polarization far away from the surface: dwðzÞ dz

j

¼0

ð37Þ

mðz ¼ lÞ ¼ 0

ð38Þ

z¼l

whereas for two identical surfaces immersed in an electrolyte separated by a distance 2d, the symmetry with respect to the middle distance, z = 0, provides: dwðzÞ Az

j

z¼0

¼0

mðzÞjz¼0 ¼ 0

ð39Þ ð40Þ

Once the functions ψ(z) and m(z) are obtained, one can calculate the free energy of the system by adding the free energy terms provided by Eqs. (5b) (5c) (5d) (5e). A typical application is presented in the next section, in which we will examine whether or not the polarization model can be employed to represent the interactions between silica surfaces. 4. Interactions between silica surfaces One of the reasons for the present investigation was that recent experiments on the interactions between silica surfaces seemed to be well described by most theoretical models [22], and, in contrast, the polarization model provided not only an unsuccessful fit, but also incoherent results, because it predicted a decrease of the surface dipole density with increasing electrolyte concentration [22]. We will try to explain why the polarization model provided such an unsuccessful fit. Firstly, let us note that a good fitting of the experimental data does not imply necessarily a good theoretical model. For example, the reasonable agreement between experiment [22] and the Attard and Pattey model [45] (which is an extension of the Jönnson and Wennerström model [5]) has been obtained for surface dipoles between 150 and 800 Debyes [22] (note that the permanent dipole moment of the water molecule is 1.87 Debyes). While the Paunov et al. model [18] provided good agreement [22], it predicted a very short range for the hydration force (up to about 15 Å), whereas in most cases hydration forces have been identified for up to about 20–40 Å. In the Attard and Batchelor model [15], both the preexponential factor and the decay length of the hydration force depend only on one parameter (the concentration of Bjerrum defects w); therefore, in order to fit the data, a very steep hydration force was predicted, with a decay length about one order of magnitude smaller than the experimental one. The agreement of the Marcelja and Radic model [11] with the experiment is not surprising, because the model is very similar to adding an exponential repulsion (with a preexponential factor and decay length as fitting parameters) to the DLVO interactions. Such a procedure is almost always successful [29] (a notable exception being the oscillatory hydration forces measured between mica surfaces [12]).

Before discussing the polarization model, it is useful to present the experimental procedure employed in Ref. [22]. The force measurements have been performed using the colloid probe technique, a method pioneered by Ducker et al. [46]. A silica bead with a diameter of 5 μm was glued to a standard Vshaped tipless cantilever with a spring constant of 0.12 N/m, mounted on an Atomic Force Microscope (AFM) Nanoscope III. The cantilever (with the silica sphere attached) was pressed against a silica surface, with both the cantilever and the surface immersed in an electrolyte of controlled pH and ionic strength. The AFM recorded both the displacement of the cantilever base as well as the deflection of the cantilever tip (which is proportional to the force between the silica sphere and the flat silica surface), and thus the force between silica surfaces could be recorded as a function of the relative displacement between the surfaces. Using the Derjaguin approximation [29], the force between the spherical surface of radius R and the flat surface can be related to the free energy of interactions between two flat surfaces by: Fð2dÞ ¼ 2kðU ð2dÞ−U ðlÞÞ R

ð41Þ

where U(2d ) represents the total free energy, per unit area, between two flat surfaces separated by a distance 2d, which includes the attractive van der Waals interactions, as well as the repulsive double layer and “hydration” interactions. In what follows, for the repulsive interactions we will use the polarization model, in which the two repulsive interactions are coupled. Consequently, U ð2dÞ ¼ Uent ð2dÞ þ Uch ð2dÞ þ Uint ð2dÞ þ Uel ð2dÞ þ UvdW ð2dÞ;

ð42Þ

where the first four terms are provided by Eqs. (5b) (5c) (5d) (5e) respectively, and the last one represents the van der Waals free energy. The inset of Fig. 2A shows that the experimental data from Ref. [22] (cE = 1 M, pH = 9) for distances larger than 20 Å (region that provides information particularly on the van der Waals interactions) are comparable with the resolution of the measurements with the Atomic Force Microscope, which is (in this case) 10−2 mN/m. Therefore it is difficult to obtain accurate values for the Hamaker constant. Also, for large FR values (of the order of 1 mN/m), there is an uncertainty in the distance measurements of a few Å. Because it is unlikely that those uncertainties are due to the resolution of the Atomic Force Microscope (the typical vertical distance accuracy for the AFM being higher by at least one order of magnitude), one might be tempted to assign them to a higher roughness of the silica surfaces or perhaps even to the formation of a silica gel above the surface, as suggested by Vigil et al. [36]. The latter supposition is also supported by the presence of a strong and steep, hard-wall-like interaction, at short separations, followed by a slower, exponentially-like decreasing repulsion (with a decay length of about 10 Å). A hairy gel formed near the silica interface is very likely to behave in this manner.

Polarization Model and ion specificity: applications

603

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

39

A first step in the understanding of the hydration interaction would be a fit with an exponential repulsion plus a van der Waals attraction:
 Fð2dÞ 2d AH ¼ BH exp − − ð43aÞ R kH 6ð2dÞ2 where AH is the Hamaker constant for silica–silica interactions through water while BH and λH are unknown fitting parameters. However, as the fit tentative shown in Fig. 2A, such a trial fails to provide a reasonable agreement with experimental data. The Fig. 3. The model employed for the calculations based on the polarization model in silica. The first water layer is at a distance d from the middle distance between silica surfaces. The surface dipoles of magnitude p = 4 Debyes, each occupying an area A = 50 Å2, are located at a distance Δ′ = 1 Å below the first water monolayer (ε′ = 1). The shortest distance recorded experimentally corresponds to d = δ. The surface of silica is at a distance 2t = 15 Å from the shortest experimental distance, as obtained from the best fit of the experiments with Eq. (43b), for AH = 8.3 × 10− 21 J. If δ would be equal to t, the surface dipoles would be located on the silica surface. When δ b t, the surface dipoles are located above the surface, at a distance (t − δ).

reason for the failure can be understood from Fig. 2B. As in the DLVO theory, the addition of an exponential repulsion to a power low attraction generates two minima separated by an intermediate maximum. When the cantilever of the Atomic Force Microscope exerts the maximum force possible between the silica surfaces (which is related to the force constant of the cantilever and its maximum recordable deflection), the separation between surfaces does not reach the primary minimum, but remains somewhere in between the maximum and the secondary minimum (see Fig. 2B). Therefore, the shortest separation that the microscope can reach does not provide the contact between the silica surfaces (zero separation). The separation between silica surfaces is unknown (up to an additive constant), and a more reasonable fitting function should be of the form
 Fð2dÞ 2d AH ¼ BH exp − − R kH 6ð2d þ 2tÞ2

Fig. 2. A. The force between silica surfaces at cE = 1M and pH = 9, determined experimentally in Ref. [22] (circles); the continuous line represents an unsuccessful fit with Eq. (43a). The origin of the x axis corresponds to the smallest separation distance between the two surfaces, attainable by AFM. The coordinate of the true point of contact between the surfaces cannot be obtained directly from experiment. The inset shows the region of the secondary minimum, where the magnitude of the interaction is comparable with instrumental resolution of 0.01 mN/m. B. The fit of the experimental data (cE = 1M, pH = 9) with an exponential repulsion and a van der Waals attraction (Eq. (43b)). As in Fig. 2A, the origin of the x axis corresponds to the smallest separation attained in the experiment. The true point of contact between surfaces, obtained from fit, is located at a distance which is by 2t = 15 Å larger than the smallest separation recorded by AFM.

ð43bÞ

where 2t represent the unknown distance between the point of contact of the silica surfaces and the minimum separation reached by AFM. A reasonable agreement with experiment, both at short separations (Fig. 2B, bottom inset) and near the secondary minimum (Fig. 2B, top inset) has been obtained by employing the values BH = 3.35 mN/m, λH = 5.3 Å and 2t = 15 Å in Eq. (25) For the Hamaker constant, we used the value AH = 8.3 × 10− 21 J, proposed by Valle et al. [22]. The incertitude in the position of the point of contact explains why the predicted interactions for most theoretical models investigated by ValleDelagdo et al. seem to be shifted from the experimental data by about 10 Å towards higher values for the separation distances (see Figs. 3, 5, 7 and 9 of Ref. [22]). Let us now turn our attention to the polarization model. It was shown that, in the linear approximation, the repulsive interactions can be described by the sum of two exponentially decreasing terms. At low ionic strength, their decay lengths are

Nanodispersions

604 40

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

kmffi k1 ¼ kH ¼ p and λ2 = λDH (the Debye–Hückel length), but at e high electrolyte concentrations, because of the coupling between “hydration” and “double layer”, the largest decay length λ2 becomes equal to λm [32]. By employing a value λm = 14.9 Å (corresponding to an ice-like layers organization of interfacial water), as performed in Ref. [22], it is very unlikely to obtain a good agreement of the polarization model and experimental results. However, the disorder induced in the interfacial water by the silica structure, as discussed in the previous sections, might reduce drastically the value of λm and a reasonable agreement might be obtained. Since the structure of the silica interface is unknown, we will use the following simple model, sketched in Fig. 3. It will be

Fig. 5. All experimental data exhibit a hard-wall-like repulsion at very short separations, regardless of pH or ionic strength. Note that the hydration forces increases at higher ionic strength: cE = 1 M (triangles), compared with cE = 0.01 M, pH = 3 (squares). At pH = 9 (circles) and cE = 0.01 M, the surface charges create a long range double layer repulsion.

Fig. 4. A: Results of the polarization model (continuous line) are compared with the experimental data from Ref. [22] (circles), for cE = 1 M and pH = 9. The values for the fitting parameters employed are: σ = − 0.1 C/m2, 2δ = 6 Å. The values of all the other parameters are as before (λm = 4 Å, p = 4 Debyes, ε′ = 1, Δ′ = 1 Å, A = 50 Å2, 2t = 15 Å and AH = 8.3 10− 21 J ). B: Results of the polarization model are compared with the experimental data from Ref. [22], for cE = 0.01M, for pH = 3 (squares) and pH = 9 (circles). The values for the parameters derived from fitting are: σ = − 0.0027 C/m2, δ = 8 Å (pH = 3) and σ = − 0.0065 C/m2, 2δ = 6 Å (pH = 9). The values of all the other parameters employed are as before (λm = 4 Å, p = 4 Debyes, ε′ = 1, Δ′ = 1 Å, A = 50 Å2, 2t = 15 Å and AH = 8.3 10− 21 J ).

assumed that the first water monolayer starts at z = −d. The surface dipoles of magnitude p, each occupying an area A on the surface, are located at a distance Δ′ = 1 Å behind the first water monolayer (z = − d + Δ′). Since the distance between the point of contact between silica surfaces and the smallest separation recorded by the AFM is unknown, it will be assumed that the shortest separation achieved experimentally corresponds to a distance through water of 2d = 2δ. At this point, the polarization model contains a number of unknown parameters (e.g. λm, p, A, Δ′, ε″, AH, t, δ and σ). If the microscopic structure of the surface would be known, some of these parameters would be related to each other and to the experimental details (e.g. the surface charge density and the surface dipole densities would depend on the recombination equilibria of the surface groups with the electrolyte ions, which in turn depend on cE, pH, and ψ(−d )). A fit with so many parameters would, however, not be very relevant, and in order to simplify the problem, we will employ some reasonable selected values for some of these parameters. To be consistent with the experimental data for hydration forces measured between silica interfaces, λm has to be of the order of 4 Å, a value which is compatible with Eq. (24) for a random orientation of the neighboring water molecules. Therefore, in what follows, we will employ only the value λm = 4 Å. For the van der Waals interactions, we will assume in all calculations AH = 8.3 10− 21 J and 2t = 15 Å, as obtained previously form the fit with Eq. (43b). The magnitudes of the surface dipoles will be considered 4 Debyes (about twice that of a water molecule) and it will be assumed that each dipole occupies on the surface an area of 50 Å 2 and that they are located at a distance Δ′ = 1 Å below the first water monolayer. Consequently, in what follows only two fitting parameters will be considered, the unknown surface charge densities σ and the unknown shortest separation obtained experimentally, 2δ. Figs. 4A and B shows a more than qualitative agreement with

Polarization Model and ion specificity: applications

605

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

experimental data, obtained for the following fitting parameters: for the experiment at cE = 1 M and pH = 9, the values σ = −0.1 C/m2, 2δ = 6 Å were employed (Fig. 4A); for the experiment at cE = 0.01 M and pH= 3, the values σ = −0.0027 C/m2 and 2δ = 8 Å were employed; for the experiment at cE = 0.01 M and pH = 9, the values σ = −0.0065 C/m2 and 2δ = 6 Å were employed. The values of the fitting surface charges indicate that the silica surface becomes more negatively charged at higher pH and higher ionic strengths, which is consistent with the expectation for recombination of the ionic moieties. At higher pH values, the surface becomes more negatively charged, because there are more OH − available for recombination on the surface. At higher ionic strengths, the surface potential decreases drastically (because of the increase in the Debye–Hückel screening) and therefore the Boltzmann factor for the distribution of the OH − anions near the surface is strongly increased. The most important question is, however, whether or not the polarization of water is mainly responsible for the short range interactions between silica surfaces. The fittings shown in Fig. 4 indicate that the surface dipoles are situated at a distance 2δ of about 6 Å from the shortest separation obtained experimentally. In contrast, the fitting of the van der Waals interactions (Eq. (43b), Fig. 2B) implies that the flat silica surfaces are at a distance 2t of about 15 Å from the shortest separation obtained experimentally. If a value t = δ would have been obtained, it would indicate that the surface dipoles were actually situated on the flat silica surfaces. The results suggest instead that the surface dipoles, responsible for the polarization of the first water monolayer, are situated at a distance t − δ ≈ 5 Å above each of those surfaces. This conclusion would strongly support the formation of a silica gel, with a thickness of about 5 Å, on the silica surface. The possible formation of a gel is also supported by the experimental data obtained for various pHs and electrolyte concentrations, represented for convenience in Fig. 5, which exhibit an almost hard-wall-like repulsion at short separations. On the other hand, in support of the hypothesis that the polarization of water might play an important role in the hydration force between silica surfaces, one should note that the polarization model predicts an increase in the hydration force at higher ionic strength [30], which can be indeed observed in Fig. 5, by comparing the experiments at cE = 0.01 M (pH = 3) with those at cE = 1 M. While both the gel-induced steric repulsion and the polarization model are consistent with the present experimental data, a final decision about the microscopic origin of the hydration force in the case of silica should be postponed until more accurate data or additional information regarding the nature of the silica surface will become available. 5. Conclusions This paper has been stimulated by recent experiments on the interactions between silica surfaces, which could not be represented by the polarization model proposed by the authors [32,33]. In that model, the water molecules near the surface

41

were considered to be organized in ice-like layers and consequently a large dipole correlation length, which is a measure of the range of interaction between dipoles, λm = 14.9 Å, was obtained. The main goal of this paper was to show how the dipole correlation length λm is affected by the disorder in the structure of the interfacial water, and values much smaller than 14.9 Å could be thus obtained. Since the evaluation of λm is based on a choice for the microscopic structure of water, which is often unknown, the polarization model was also presented as a phenomenological model, in which the dipole correlation length λm was used as a parameter. It was also shown that, for λm = 4 Å, a more than qualitative agreement with the experiment can be obtained, for reasonable values of the parameters involved (e.g. dipole strength, dipole density, dipole locations, surface charges), which are consistent with the surface recombination models. The value λm = 4 Å is consistent with the assumption of random orientations for the nearest neighbors of a water molecule. The results suggest that a gel might form above the surface of silica, as proposed by Vigil et al. [36], which induces disorder in the interfacial water layers. Acknowledgement We are indebted to Valle-Delgado for kindly providing us the experimental data reported in Ref. [22] for the forces between silica surfaces. References [1] Deryagin BV, Landau L. Acta Physicochim. URSS 1941;14:633. [2] Verwey EJ, Overbeek JThG. Theory of stability of lyophobic colloids. Amsterdam: Elsevier; 1948. [3] Stern O. Z Eletrochem 1924;30:508. [4] Ruckenstein E, Schiby D. Langmuir 1985;1:612. [5] Jonsson B, Wennerstrom H. J Chem Soc, Faraday Trans II 1983;79:19. [6] Spaarnay MJ. Recl Trav Chim Pays-Bas 1958;77:872. [7] Kjellander R, Marcelja S. J Chem Phys 1985;82:2122. [8] Ninham BW, Yaminsky V. Langmuir 1997;13:2097. [9] Manciu M, Ruckenstein E. Adv Colloid Interface Sci 2003;105:63. [10] Rand RP, Parsegian VA. Biochim Biophys Acta 1989;988:351. [11] Marcelja S, Radic N. Chem Phys Lett 1976;42:129. [12] Israelachvilli JN, Pashley RM. Nature 1983;306:249. [13] Pashley RM. J Colloid Interface Sci 1981;83:531. [14] Healy TW, Homola A, James RO, Hunter RJ. Faraday Discuss Chem Soc 1978;65:156. [15] Attard P, Batchelor MT. Chem Phys Lett 1988;149:206. [16] Schiby D, Ruckenstein E. Chem Phys Lett 1983;95:435. [17] Gruen DWR, Marcelja S. Faraday Trans II 1983;79:211–25. [18] Paunov VN, Kaler EW, Sandler SI, Petsev DN. J Colloid Interface Sci 2001;240:640. [19] Napper DH. Steric stabilization of colloidal dispersions. London: Academic Press; 1983. [20] Israelachvili JN, Wennerstrom H. Langmuir 1990;6:873. [21] Binder H, Dietrich U, Schalke M, Pfeiffer H. Langmuir 1999;15:4857. [22] Valle-Delgado JJ, Molina-Bolivar JA, Galisteo-Gonzalez F, Galvez-Ruiz MJ, Feiler A, Rutland MW. J Chem Phys 2005;123:034708. [23] Attard P, Wei DQ, Patey GN. Chem Phys Lett 1990;172:69. [24] Berkowitz ML, Raghavan K. Langmuir 1991;7:1042. [25] Faraudo J, Bresme F. Phys Rev Lett 2004;92:236102. [26] Kjellander R, Marcelja S. Chem Phys Lett 1985;120:393. [27] Faraudo J, Bresme F. Phys Rev Lett 2005;94:077802.

Nanodispersions

606 42

M. Manciu et al. / Advances in Colloid and Interface Science 127 (2006) 29–42

[28] Perera L, Essmann U, Berkowitz ML. Prog Colloid & Polym Sci 1997;103:107. [29] Israelachvili J. Intermolecular and surface forces. Academic Press; 1992. [30] Manciu M, Ruckenstein E. Langmuir 2001;17:7061. [31] Manciu M, Ruckenstein E. Langmuir 2001;17:7582. [32] Ruckenstein E, Manciu M. Langmuir 2002;18:7584. [33] Manciu M, Ruckenstein E. Adv Colloid Interface Sci 2004;112:109. [34] Parks GA. J Geophys Res 1984;89:3997–4008. [35] Rimstidt JD, Barnes HL. Geochim Cosmochim Acta 1980;44:1683. [36] Vigil G, Xu Z, Steinberg S, Israelachvili J. J Colloid Interface Sci 1994;165:367.

[37] [38] [39] [40] [41] [42] [43] [44]

Yoon R-H, Vivek S. J Colloid Interface Sci 1998;204:179. Sermon PA. J Chem Soc, Faraday Trans I 1980;76:885. Overbeek JTG. Colloid Surf 1990;51:61. Manciu M, Ruckenstein E. Langmuir 2003;19:1114. Ruckenstein E, Manciu M. Adv Colloid Interface Sci 2003;105:177. Manciu M, Ruckenstein E. Langmuir 2002;18:8910. Manciu M, Ruckenstein E. Langmuir 2005;21:11749. Jackson JD. Classical electrodynamics. New York: John Wiley and Sons; 1975. [45] Attard P, Patey GN. Phys Rev, A 1991;43:2953. [46] Ducker WA, Senden TJ, Pashley RM. Langmuir 1992;8:1831.

Introduction to CHAPTER 8 Polymer brushes

8.1

B. Li, E. Ruckenstein: “Statistical thermodynamics of end-attached chain monolayers,” JOURNAL OF CHEMICAL PHYSICS 106:1 (1997) 280–288. 8.2 E. Ruckenstein, B. Li: “Steric interactions between two grafted polymer brushes,” JOURNAL OF CHEMICAL PHYSICS 107: 3 (1997) 932–942. 8.3 M. Manciu, E. Ruckenstein: “Simple Model for Grafted Polymer Brushes,” LANGMUIR 20 (2004) 6490–6500. 8.4 M. Manciu, E. Ruckenstein: “On the Monomer Density of Grafted Polyelectrolyte Brushes and Their Interactions,” LANGMUIR 20 (2004) 8155–8164. 8.5 H. Huang, E. Ruckenstein: “Double-layer interaction between two plates with hairy surfaces” JOURNAL OF COLLOID AND INTERFACE SCIENCE 273 (2004) 181–190. 8.6 H. Huang, E. Ruckenstein: “Double layer interaction between two plates with polyelectrolyte brushes” JOURNAL OF COLLOID AND INTERFACE SCIENCE 275 (2004) 548–554. 8.7 H. Huang, E. Ruckenstein: “The bridging force between two plates by polyelectrolyte chains” ADVANCES IN COLLOID AND INTERFACE SCIENCE 112 (2004) 37–47. 8.8 H. Huang, E. Ruckenstein: “Steric and Bridging Interactions between Two Plates Induced by Grafted Polyelectrolytes,” LANGMUIR 22 (2006) 3174–3179. 8.9 H. Huang, E. Ruckenstein: “Effect of Steric, Double Layer, and Depletion Interactions on the Stability of Colloids in Systems Containing a Polymer and an Electrolyte,” LANGMUIR 22 (2006) 4541–4546. 8.10 M. Manciu, E. Ruckenstein: “Estimation of the available surface and the jamming coverage in the Random Sequential Adsorption of a binary mixture of disks,” COLLOIDS AND SURFACES A-PHYSICOCHEMICAL AND ENGINEERING ASPECTS 232 (2004) 1–10. A general theory, based on a simple cubic lattice model and matrix formalism is developed. The theory accounts for the chain stiffness, the nearest-neighbor bond correlations and

the intermolecular interactions. The model allows one to calculate the density profiles of the lateral, forward and backward bonds as functions of the layer number and the orientational probability for a bond to be lateral, forward or backward as functions of the bond number counted from the attached end. It is shown that the effect of the bond correlations becomes less important with decreasing chain length [8.1]. The model is employed to calculate the steric interactions between two grafted polyelectrolyte brushes, by accounting for the bond correlations and the interdigitation between brushes, and it is shown that the results are in agreement with experiment. Comparison with other theoretical models (that neglect interdigitation) indicates that the interdigitation is not negligible and can increase the repulsive force by an order of magnitude [8.2]. A simple theoretical treatment of a polymer brush is proposed, which consists of constructing its partition function by considering all possible distributions of a random walk ending at every distance z from the surface. This distance is assumed to be the position of the last monomer of the chain. A free energy minimized with respect to the local volume fraction of the monomers, calculated using the Flory-Huggins expression, is associated to each of these distributions; it is also assumed that the number of distributions with energy close to the minimum is proportional to the total number of paths that end at the distance z. The model can be employed for both good and poor solvents, and is compatible with both a parabolic-like brush profile at moderate graft densities, and with an almost step-like density for highly stretched brushes. One concludes that in good and moderately poor solvents, the interactions between grafted polymer brushes are always repulsive, while in poor solvents the interactions are repulsive at small separations but attractive at intermediate separations, results in agreement with experiment [8.3]. The model is extended to charged polyelectrolyte brushes, and the dependence of the brush thickness on electrolyte concentration is obtained. One concludes that the trapping of a fraction of counterions in the brush strongly affects the thickness of the brush. Consequently, if two brushes approach each other more rapidly than the ions diffuse in the direction parallel to the surface, the trapping of the counterions between brushes can affect the interactions by orders of magnitude [8.4].

E. Ruckenstein, M. Manciu, Nanodispersions, DOI 10.1007/978-1-4419-1415-6_8, © Springer Science+Business Media, LLC 2010

608

When hairy chains are extended into the continuous phase, they orient the water molecules and therefore lower the local dielectric constant of water. The local change in the dielectric constant has strong effects on the ion distribution (via changes in the Born hydration energy and on the Poisson equation) and therefore on the repulsion between two plates [8.5]. A model for polyelectrolyte brushes is proposed, which assumes that the polyelectrolyte chain is a rigid cylinder, on whose surface charges are generated through the dissociation of ionizable sites and adsorption of the cations of the electrolyte. In the brush region, the electric potential is described by a twodimensional Poisson-Boltzmann equation (in the plane of the surface), while in the region free of polyelectrolyte chains it is described by a one-dimensional PoissonBoltzmann equation (in the direction normal to the surface). The effects of electrolyte concentration, pH, brush thickness and chain coverage density on the repulsion between plates are examined [8.6]. The chains of the grafted polyelectrolyte brushes can be adsorbed on the other plate, thus generating bridging (hence attraction) between plates. A self-consistent field approach is employed to investigate the effect of bridging, by taking into account the van der Waals interactions between the segments of the polyelectrolyte molecules and the plates, and the electrostatic and volume exclusion interactions [8.7]. An alternate approach to steric and bridging interactions also is proposed, which consists of calculating the free

Nanodispersions

energy of the system by using a number of contributions (the Flory-Huggins mixing free energy for the grafted chains and liquid; the van der Waals interactions between segments and plates; the connectivity free energy of the segments; and the adsorption energy. For charged plates, the electrostatic free energy and the free energy of the electrolyte are also included). The free energy minimization with respect to segment concentration and electric potential provides expressions for them. Once these quantities are known, the interactions between plates with grafted polyelectrolyte molecules, possessing both steric and bridging forces, are calculated [8.8]. Experiments have shown that polystyrene latexes are restabilized at sufficiently high electrolyte concentrations in the presence of an amphiphilic block copolymer, but are again destabilized at even higher electrolyte concentrations. These phenomena occur because of steric repulsion at intermediate electrolyte concentrations and depletion attractive interactions at higher ionic strengths [8.9]. The area available to a disk (or to a sphere) to be adsorbed on a surface during a random sequential adsorption (RSA) of a binary mixture of disks (spheres) is in general estimated via Monte-Carlo simulations, which are very time consuming. A simple analytical expression is proposed, which provides good estimates of the jamming points for the RSA of a binary mixture of disks or spheres. The predictions of the simple formula have been compared with the Monte-Carlo simulations existing in the literature [8.10].

Polymer brushes

609

Statistical thermodynamics of end-attached chain monolayers Buqiang Lia) and Eli Ruckensteinb) Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York 14260

~Received 5 April 1996; accepted 25 September 1996! A simple cubic lattice model and a matrix formalism are used to calculate the segment density distribution and structural properties of monolayers consisting of end-attached chains and solvent molecules. The chain stiffness, the nearest-neighbor bond correlations, and the intermolecular interactions are taken into account. The calculations are performed for chains of various lengths, from short to polymer brushes. The main difference between the present treatment and those already available consists in the more compact formulation of the basic equations. The incorporation of the nearest-neighbor bond correlations, or higher values of the interaction parameter results in a higher average segment density and a narrower distribution profile of the chain free ends. With decreasing chain length, the effect of the bond correlations becomes less important. The model allows us to calculate the density profiles of the lateral, forward, and backward bonds as a function of the layer number, and to calculate the orientational probability for a bond to be lateral, forward, or backward as a function of the bond number counted from the attached end. © 1997 American Institute of Physics. @S0021-9606~97!50801-4# I. INTRODUCTION

The statistics of chain conformations is of importance in the investigation of surfactant monolayers, bilayer membranes, micellar aggregates as well as polymer ~adsorbed or grafted! monolayers, etc. Two cases are commonly encountered. In one of them, a compact hydrocarbon domain of uniform segment density is separated from water by polar headgroups. The hydrocarbon chains have a free end which is located in the hydrocarbon domain and another end located at the interface with water. The packing constraints result in the variation of the chain structural order within the hydrocarbon domain along a coordinate determined by the geometry of the system. In the other one, solvent molecules are present among the hydrocarbon chains and both the segment density and the chain conformation depend on the distance to the surface. Many theories1–14 for the chain conformational statistics have been developed. The early theories1–8 involved a single chain mean-field approximation. Using the rotational isomeric state scheme, Gruen1 calculated the probability distribution function of the chain conformation and the conformational free energy of the surfactant tails in an aggregate, assuming that the hydrocarbon tails form a liquidlike hydrocarbon core of uniform density. In a series of papers, Ben-Shaul et al.2–4 improved the preceding approach and provided more detailed calculations. An extension5 by the same authors considered a nonuniform segment density. Assuming a simple cubic lattice and using the generator-matrix formalism, Dill et al.6–8 developed a chain statistics for the case free of bond backtracks. All the above approaches neglected the nearestneighbor bond correlations. Some recent treatments9–14 have taken into account the effect of the nearest-neighbor bond correlations on the proba!

Permanent address: Chemical Engineering Department, Beijing, University of Chemical Technology, Beijing 100029, People’s Republic of China. b! Author to whom correspondence should be addressed. 280

J. Chem. Phys. 106 (1), 1 January 1997

ability for a site to be occupied. The self-consistent field ~SCF! theory of Scheutjens and Fleer9 could generate, on the basis of a rotational isomeric state scheme, a full set of conformations, whose weights were provided by the Boltzmann statistics. Leermakers and Scheutjens10 improved the preceding approach, generating the self-consistent anisotropic ~SCAF! theory. In the SCAF theory, the probability for a site to be occupied by a segment depends not only on the overall segment concentration in the layer in which it will be located, but also on the density of bonds in the direction of the bond to which the considered segment is connected ~nearest neighbor bond correlation!. The presence of the bond correlations results in a significant modification of the chain conformational entropy. They have applied the theory to micelles11 and to polymer brushes.12 In a paper regarding phase transitions in monolayers,13 Cantor and McIlroy incorporated the bond correlations and intermolecular interactions using an approach similar to that of Ref. 10. Finally, Cantor has incorporated the bond correlations in the generator-matrix formalism to calculate the elastic properties14 of films of athermal surfactant mixtures. In this paper, a physical picture equivalent to those of Leermakers and Scheutjens10,12 and Cantor13 is employed. However, a more simple generator-matrix methodology is used to examine the effects of the intramolecular interactions, the nearest-neighbor bond correlations, and the intermolecular interactions on the structural properties of endattached chain monolayers. While equivalent to that developed in Refs. 10 and 12, the method is more simple and compact, and hence more convenient from a computational point of view. II. THEORY A. Partition function

We consider a two-dimensional mixture which contains N end-attached chains and N 0 solvent molecules and employ the simple cubic lattice model for its representation. The lat-

0021-9606/97/106(1)/280/9/$10.00

Copyright ©2001. All Rights Reserved.

© 1997 American Institute of Physics

Nanodispersions

610 B. Li and E. Ruckenstein: End-attached chain monolayers

tice is divided into M layers parallel to the surface, numbered from the surface as i51,2,...,M , each lattice layer containing L sites. The chains consist of g11 segments which are connected through g bonds and occupy g11 sites. The first segment of each chain is constrained to be located in the first lattice layer. Each solvent molecule is considered to occupy one site. The number of lattice layers, M , is taken equal to the number of segments of the chain; consequently, one can write the relation N 0 1N ~ g 11 ! 5L ~ g 11 ! .

~1!

It is also assumed that the bulk solution contains only the pure solvent molecules. Denoting by N c the number of chains in conformation c and by N 0i the number of solvent molecules in layer i, one can write that N5 ( c N c and N 0 5 ( i N 0i . For a specified set of conformations $ N c % , the number of chains in each conformation and the number of solvent molecules in each layer are given. When only the nearest neighbor interactions are taken into account and the segments and the solvent molecules are distributed at random in each of the layers, the system has the same energy for each of the possible arrangements. The canonical partition function Q(T,L, M , $ N c % ) can be written as Q ~ T,L,M , $ N c % ! 5

S D

V DU exp 2 , V0 kT

~2!

where V is the combinatorial factor for the specified conformational profile $ N c % , V0 is the product of the combinatorial factors of the chains and solvent molecules in their reference states, and DU is the energy change of the chains and solvent molecules from their reference states to the specified conformational profile in the chain-solvent mixture. The combinatoral factor, V depends on the number of ways ~v! in which one can place successively all the chains and then the solvent molecules on the lattice. v can be factorized as follows:

v 5 v cv sv b ,

~3!

where vc is the number of ways in which one can place successively the chains and then the solvent molecules on the lattice, considering that the sites occupied by the previous segments are empty, hence, that they can be occupied again ~no excluded volume interactions are, hence, involved!; vs is the product of the vacancy probabilities for all the segments and solvent molecules to be placed one after another on the lattice; vb is the product of correction factors resulting from the effect of the nearest-neighbor bonds, already present in a given direction, on the vacancy probability for a considered segment to be a part of a bond in the same direction. Denoting by g c the number of arrangements that a chain in conformation c can have when its first segment is fixed, one can write

v c 5L N

)c

M

N

gc c

)

i51

0

L Ni .

~4!

281

In the preceding equation, g c is given by the product, g c 5P gs51 z cs11,s , where z cs11,s stands for the number of sites which segment s11 of a chain in conformation c can occupy around segment s. Indeed, the second segment has z c2,1 possible locations when the first segment is fixed, the third segment has z c3,2 possible locations, and the last segment g11 has z cg 11, g possible locations. vs arises because a lattice layer is completely empty only before the first segment is placed in the layer. When a layer is only partly empty, a vacancy probability must be included for a segment to be placed in the layer. Denoting by yi the number of segments already present in the layer, the vacancy probability for the subsequent segment to be placed in that layer is equal to (L2 y i )/L; consequently, for all the segments and all the solvent molecules to be placed on the lattice successively, one obtains M

v s5

L21

) ) i51 n 50 i

S D

L2 n i . L

~5!

vb arises because the density of bonds already present in a given direction increases the vacancy probability for the end segment of a bond to be placed in the same direction. The number of bonds in direction j which have their starting segments located in layer i is denoted by yj i . In the cubic lattice, j can be x or y for the lateral bonds, which are located in layer i in the direction x or y, and z for the vertical bonds between layers i and i11. For a bond to be placed from layer i in the direction j, a correction factor, L/(L2 y j i ) must be included in the vacancy probability because fewer sites than L are accessible. For all the chains to be placed successively on the lattice, one obtains that vb is given by M

v b5

N j i 21

)) ) i51 j n 50 ji

S

D

L , L2 n j i

~6!

where N j i is the number of bonds in direction j with the starting segments in layer i. Substituting Eqs. ~4!, ~5!, and ~6! into Eq. ~3! and accounting for the indistinguishability among the chains in the same conformation and among the solvent molecules in each layer, the combinatoral factor V is given by

S ) DS ) D F ) ) N

V5 ~ L! ! M

gc c

c

N c!

M

i51

1

N 0i !

M

i51

j

G

~ L2N j i ! ! . L!

~7! In what follows, the ordered close-packed hydrocarbon monolayer, in which each chain has an all-trans conformation and is oriented normal to the surface, is taken as the reference state of the chains. In the reference state, both the chain conformational energy and entropy are taken equal to zero. The pure solvent, with (L2N) solvent molecules in contact with the surface, is taken as the reference state of the solvent molecules. For the reference states, V051. Using Stirling’s approximation for the factorials in Eq. ~7!, the logarithm of V/V0, corresponding to the entropy of the chain-solvent mixture, is given by the expression

J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes

611

282

B. Li and E. Ruckenstein: End-attached chain monolayers

ln

S D

V 52 V0

(c

N c ln

S D

M

Nc 2 N 0i ln~ F 0i ! gcL i51

(

M

1

( (j ~ L2N j i ! ln~ 12F j i ! ,

i51

~8!

where F 0i 5N 0i /L is the fraction of sites occupied by solvent molecules in layer i and F j i 5N j i /L is the density of bonds in direction j with at least one segment in layer i. For a given conformational profile $ N c % , the mixing energy between the chains and solvent molecules, DU, involves three contributions; ~i! the change of the contact energies of the segments and solvent molecules with the surface (DU s ); ~ii! the mixing energy ~DU mix! of the chains and solvent ~intermolecular interactions!; and ~iii! the conformational energy change ~DU con! of the chains ~intramolecular interactions!. Consequently, DU5DU s 1DU mix1DU con .

The second term on the right-hand side of Eq. ~12! represents the exchange energy between the chains in their reference state and the solvent molecules covering them. The summation in the first term already includes the term N g11l1x, which accounts for the exchange energy between the last lattice layer g11 of the chain-solvent mixture and the bulk pure solvent. The chain conformational energy change is given by the following expression: DU con 5 kT

DU 52 ~ N 1 2N ! x s 1 kT

The contact energy change of the segments and solvent molecules with the surface is simply given by

~11!

where x s 5(U 0s 2U s )/kT, and U 0s 2U s represents the energy change of replacing a segment by a solvent molecule at the surface. Considering only the nearest neighbor interactions for the mixing energy of the chain-solvent mixture, one obtains M

DU mix 5 N i ^ F 0i & x 2Nl 1 x , kT i51

(

~12!

where N i stands for the number of segments in layer i, x is the Flory–Huggins segment-solvent interaction parameter, and ^F0i &, the average fraction of sites occupied by the solvent molecules around a site located in layer i, is given by

^ F 0i & 5l 1 F 0i21 1l 0 F 0i 1l 1 F 0i11 .

1

S(

D

M

i51

N i ^ F 0i & 2Nl 1 x

NcE c . kT

(c

~15!

~10!

where N 1 and N 01 are the numbers of segments and solvent molecules in the first lattice layer, respectively, and U s and U 0s are the contact energies of a segment and a solvent molecule with the surface, respectively. The first segment of a chain does not contribute to DU s because it remains connected to the surface. The last term on the right-hand side of Eq. ~10! represents the contact energy of (L2N) solvent molecules with the surface in its reference state. Since N 1 1N 01 5L, Eq. ~10! can be rewritten as DU s 52 ~ N 1 2N ! x s , kT

~14!

where E c stands for the conformational energy of a chain in conformation c. Combining Eqs. ~9!, ~11!, ~12!, and ~14!, one obtains

~9!

DU s 5 ~ N 1 2N ! U s 1N 01 U 0s 2 ~ L2N ! U 0s ,

N cE c , kT

(c

~13!

In Eqs. ~12! and ~13!, l1 and l0 are the fractions of nearestneighbor sites in an adjacent and the same layer, respectively. Since no solvent molecules are present in layers i<0 and no segments are present in layers i>g12, F0050, and F0g1251. In addition, ^ F 0i & 1 ^ F i & 512l1 for i51 and ^ F 0i & 1 ^ F i & 51 for i>2, where Fi is the fraction of sites occupied by segments in layer i.

B. Equilibrium distribution

The mixing free energy of the chains and the solvent molecules for a given chain conformational profile $ N c % is given by F52kT ln Q. The equilibrium conformational profile can be obtained by minimizing the free energy F subject to the following constraints:

(c N c g ci 1N 0i 2L50,

i51,2,..., g 11

~16!

and N2

(c N c 50.

~17!

In Eq. ~16!, gci stands for the number of segments of a chain in conformation c located in layer i. The two equations express the obvious conditions that each lattice layer must be occupied and that the total number of chains is constant. The Lagrange multiplier method is used to calculate the minimum free energy subject to the above constraints. By introducing the multipliers ai , for each of the constraints given by Eq. ~16!, and b for the constraint expressed by Eq. ~17!, one can write M

f 5ln~ Q ! 1

S

S

( a i (c N c g ci 1N 0i 2L

i51

1 b N2

(c N c

D

,

D ~18!

where ln(Q) can be obtained from Eqs. ~8! and ~15!. The equilibrium conformational profile of the chains and the equilibrium distribution of the solvent molecules can be calculated using the equations

J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Nanodispersions

612 B. Li and E. Ruckenstein: End-attached chain monolayers M

]f ] ln~ Q ! 5 1 a i g ci 2 b 50 ]Nc ]Nc i51

~19!

]f ] ln~ Q ! 5 1 a j 50, ] N 0j ] N 0j

~20!

(

and j51,2,..., g 11.

Introducing ln(Q) into the above two equations and observing that ] N i / ] N c 5 g ci and ] N j i / ] N c 5 g c j i , one obtains ln

S D

M

Nc Ec 5212r2 b 2 1 g c1 x s 1 g ci ~ a i 2 ^ F 0i & x ! Lg c kT i51

(

M

2

( ( g c j i ln~ 12F j i ! i51 j

~21!

mulation of the problem, the intramolecular and intermolecular interactions as well as the bond correlations are taken into account. e 2b can be eliminated by combining Eqs. ~17! and ~23!. One thus obtains the following equilibrium distribution: Nc Wc 5 , N J

j51,2,..., g 11.

~22!

Eliminating a j between the above two equations, yields

S D S) D F) ) G M

Nc E c g c1 5e 2 b g c exp 2 h L kT s

g

i51

p i ci

M

3

i51

j

~ p ji ! g cji ,

where

h s 5exp~ x s ! 5exp

S

U 0s 2U s

D

~23!

,

~23a!

p i 5F 0i exp@~ ^ F i & 2 ^ F 0i & ! x # ,

~23b!

kT

and p ji5

1 . 12F j i

~23c!

In the preceding equations, hs , p i , and p j i are weight factors which account for the difference between the interactions of the segments and of the solvent molecules with the surface, for the intermolecular interactions between the segments and the solvent molecules and for the nearest-neighbor bond correlations, respectively. Assuming that on average, half of the lateral bonds in layer i are in the x direction and the other half in the y direction, and denoting by FLi the total lateral bond density in layer i, one can write p xi 5 p yi 5

1 [p Li . 12F Li /2

~24!

According to Eq. ~23!, the equilibrium number of chains in conformation c depends on the number g c of arrangements of a chain in that conformation, the conformational energy E c , the surface parameter hs , as well as the weight factors due to the chain-solvent interactions (p i ) and to the nearest-neighbor bond correlations (p j i ). In the above for-

~25!

where W c represents the statistical weight of a chain in conformation c, given by

S D S ) DS ) ) D

E c g c1 W c 5g c exp 2 h kT s

M

i51

M

g p i ci

g

i51

j

p j ic j i

~26!

and J is the chain conformation partition function defined as

and ln~ F 0j ! 5211 a j 2 ^ F j & x ,

283

J5

(c W c .

~27!

C. Chain conformational statistics

The chain width of a hydrocarbon molecule ~4.6 Å! is taken as the linear dimension of a lattice site. Consequently, in the simple cubic lattice, a cube with a side equal to 4.6 Å represents the unit cell. Since the effective length per methylene group in an all-trans chain is 1.275 Å, 3.6 methylene groups are located in a cube and will be considered to represent a segment. In the lattice, a site has four nearest neighbors in the same layer and one in each of the adjacent layers. In addition, only two relative orientations of two consecutive bonds can occur, namely the collinear and the bent one at 90°. The bending energies of the collinear and bent bond pairs are taken zero and e, respectively. Denoting by s→s11 the bond which connects segment s to segment s11, there are four conformations of the bond pair (s21→s→s11) ~summarized in Table I!. ~1! The bond pair is bent, segment s21 is located in a layer adjacent to segment s and segment s11 has 4 possible locations ~z s11,s 54!; ~2! the bond pair is bent, segments s21, s, and s11 are located in the same layer and segment s11 has two possible locations ~z s11,s 52!; ~3! the bond pair is bent, segments s21 and s are located in the same layer and segment s11 has only one possible location ~z s11,s 51!; ~4! the bond pair is collinear and segment s11 has only one possible location ~z s11,s 51!. Bond zero, which connects the first segment with the surface, is considered to be forward. When the first bond is lateral, z 2,154 and an energy eh which may differ from e is involved. Denoting by n cb1 ,n cb2 ,n cb3 the numbers of bond pairs in conformation c corresponding to types 1, 2, and 3, respectively, and by n ch ~50 or 1! the number of bent bond pairs ~0→1→2! in conformation c, one can write g c 54 n cb1 2 n cb2 4 n ch , E c 5 ~ n cb1 1n cb2 1n cb3 ! e 1n ch e h . ~28! Substituting g c and E c into Eq. ~26!, the statistical weight of a chain in conformation c becomes

J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes

613

284

B. Li and E. Ruckenstein: End-attached chain monolayers TABLE I. Types of bond pairs.

qi 5p i p Li

S

0 0

pi 5p i p zi y pr

D S D S D S D a

1

0

v

0

r2 5 h s p 2 p z1 y rp ri 5p i p z,i21 y rp

05

S D 0

0

0

0

i52,3,..., g ,

,

b 11

,

1

0

v

0

1

0

v

0

i51,2,..., g ,

;

,

i52,3,..., g 21,

,

and A0 and B are the vectors A0 5 ~ a1 ,0,...,0! ; a1 5 ~ 1,0! , 05 ~ 0,0! , B5col~ c1 ,c2 ,...,cg 11 ! ; ci 5col~ p i , p i ! .

S) D M

g

n

W c 5 a h ch a n cb1 b n cb2 v n cb3 h s c1

S) ) D

g

i51

p i ci

M

3

g

j

i51

~29!

p j ic j i ,

where a h 54 v h , a54v, b52v. vh 5exp~2eh /kT! and v5exp~2e/kT! are the weight factors of a pair of bent bonds relative to a pair of collinear bonds; v characterizes the chain stiffness. A small value of v ~large bond bending energy e! corresponds to a stiff chain. Using the generator-matrix formalism of the chain conformational statistics, the chain conformation partition function J can be expressed as a product of three matrices J5

(c W c 5A0 Gg B,

S

where G is the supermatrix

G5

q1

p1

0

•••

•••

•••

0

r2

q2

p2

•••

•••

•••

0

•••







•••

•••

•••

•••

•••







•••

•••

0

•••

•••

rg 21

qg 21

pg 21

0

0

•••

•••

•••

0

qg

pg

0

•••

•••

•••

•••

•••

0

whose elements are 232 submatrices given by q1 5 h s p 1 p L1

S

0

ah

0

b 11

D

;

D

In the elements p and r, the Heaviside functions ypr and yrp are included to avoid bond backfolding since a bond cannot be backward when the previous bond was forward and vice versa. In the supermatrix G, qi characterizes a lateral bond in layer i, pi a forward bond starting from layer i, and ri a backward bond starting from layer i. qi , pi , and ri depend on three kinds of parameters. The parameters ah , a, b, and v arise from local chain stiffness and bond arrangements, p i from the intermolecular interactions and p Li , p zi from the nearest-neighbor bond correlations. For the chains whose last segment is located in layer i, the statistical weight, W g i , can be expressed as W g i 5A0 Gg Ci ,

~31!

where Ci 5col~ 0,...,0,ci ,0,...,0 ! ; ci 5col~ p i ,p i ! , 05col~ 0,0 ! . The probability P g i for a chain to end in layer i is given by P gi5

~30!

N g i W g i A0 Gg Ci 5 5 , N J A0 Gg B

~32!

where N g i stands for the number of chains whose free ends are located in layer i. For the chains with the kth bond lateral in layer i, the chains with the kth bond forward from layer i to i11 and the chains with the kth bond backward from layer i to i21, the probabilities, P kLi , and P kFi , P kBi are given, respectively, by P kLi 5A0 Gk21 Qi Gg 2k B/J,

i51,2,..., g ,

~33!

P kFi 5A0 Gk21 Pi Gg 2k B/J,

i51,2,..., g ,

~34!

P kBi 5A0 Gk21 Ri Gg 2k B/J,

i52,3,..., g 21,

~35!

and

where Qi , Pi or Ri is a supermatrix which has only the (i,i), (i,i11) or (i,i21) ~i.e., qi , pi or ri ! as a nonzero element. J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Nanodispersions

614 B. Li and E. Ruckenstein: End-attached chain monolayers

Since segment k is located in layer i if the bond k, forward, lateral or backward, starts from layer i, the probability for segment k to be in layer i is given by the sum of the probabilities for bond k to start from layer i,

N g i 12 ~ N i 2N g i ! 5N z,i21 1N zi 12N Li ,

2F i 5 P g i s 1F z,i21 1F zi 12F Li ,

1 A Gk21 ~ Qi 1Pi 1Ri ! Gg 2k B. J 0

~36!

The total number of lateral bonds in layer i (N Li ), forward bonds from layer i to i11 (N Fi ) or backward bonds from layer i to i21 (N Bi ) is given by the sum of the corresponding bonds over the bond number k, g

N Li 5N

( P kLi ,

k5i

i51,2,..., g ,

~37!

i51,2,..., g ,

~38!

g

N Fi 5N

( P kFi , k5i

N Bi 5N

( P kBi , k5i11

and g

i52,3,..., g 21.

~39!

Combining Eqs. ~33! and ~37!, one obtains for the number of lateral bonds per site in layer i ~lateral bond density, N Li /L5F Li ! the expression g

s F Li 5 A0 Gk21 Qi Gg 2k B, J k5i

(

i51,2,..., g ,

~40!

where s5N/L represents the number of chains per site at the surface ~surface chain density!. The sum of N Fi and N Bi11 , obtained using Eqs. ~34!, ~35!, ~38!, and ~39!, provides the vertical bond density between layer i and i11 [(N Fi 1N B,i11 )/L5F zi ], F zi 5

s A J 0

S( g

k5i

1

( G k5i11

k21

Ri11 G

g 2k

i51,2,..., g 11. ~43!

Since the first segment is connected to the surface and no lateral and forward bonds are present in the last layer, Fz0 5s and F z, g 11 5F L, g 11 50. Equations ~40!, ~41!, and ~43! provide a complete system of equations for the 3g11 unknown quantities. They can be numerically solved to obtain the distributions of segment and bond densities $ F i % , $ F Li % , $ F zi % as a function of layer number to the surface. The distributions of both segment and bond densities allow us to calculate any structural property of the monolayer. The bond orientational probabilities ~forward, lateral, and backward! as well as the bond order parameter of a chain as a function of bond number along the chain are given below. The sum of P kFi , the sum of P kLi ~both over all the layers from 1 to k! and the sum of P kBi over all the layers from 2 to k21 provide the probabilities for the kth bond to be forward ( P kF ), lateral ( P kL ) and backward ( P kB ), respectively, P kF 5

1 A Gk21 J 0

1 P kL 5 A0 Gk21 J

S( D S( D k

i51

k51,2,..., g ,

~44!

Qi Gg 2k B,

k51,2,..., g ,

~45!

k

i51

and P kB 5

Pi Gg 2k B,

1 A Gk21 J 0

S( D k21 i52

Ri Gg 2k B,

k53,4,..., g . ~46!

The bond order parameter as a function of bond number to the attached end is given by the following expression:

Gk21 Pi Gg 2k

g

i51,2,..., g 11 ~42!

which can be rewritten as

P ki 5 P kLi 1P kFi 1 P kBi 5

285

D

B,

i51,2,..., g .

~41!

Equations ~40! and ~41! represent a system of 2g equations which contain 3g11 unknowns, namely FLi and Fzi with i51,2,...,g, and Fi with i51,2,...,g11. In order to obtain both the segment and bond density distributions, additional equations, which are derived below, are required. For aliphatic chains, a methylene group ~–CH2 –! is connected to two covalent bonds and a terminal group ~–CH3! to a single covalent bond. Hence, a methylene group has two internal connection points and a terminal group has only one. Correspondingly, a lateral bond in layer i provides two connection points in that layer, while a vertical bond, either between layer i and i11 or between layer i and i21, provides only one connection point in layer i. A mass balance of the internal connection points provided by the segments present in layer i leads to

S k 5 32 ~ 12P kL ! 2 12 .

~47!

III. RESULTS AND DISCUSSIONS

In this section, calculations are carried out to determine the segment and bond density profiles as a function of distance to the surface, as well as the bond orientational probability and order parameter as a function of bond number to the attached end. The effects of the chain lengths, the nearest-neighbor bond correlations, the intermolecular interactions and the bond backtracks are examined. In most of the calculations, a constant chain stiffness, v5vh 50.5, corresponding to a bond bending energy of kT ln 2, a constant chain surface density, s50.1, and a constant surface parameter, hs 51, are used. The chain length is varied between 20 bonds ~21 segments! and a polymer brush with 400 bonds ~401 segments!. Figure 1 provides a comparison between the present calculations and those based on the limited bond

J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes 286

615 B. Li and E. Ruckenstein: End-attached chain monolayers

FIG. 1. A comparison between the present calculations and those of Wijmans et al. ~Ref. 12! for two polymer brushes.

FIG. 3. The probability profile of a segment in the polymer brush with 400 bonds.

flexibility model of Wijmans et al.,12 which is a cubic lattice version of the self-consistent anisotropic field ~SCAF! theory of Leermakers and Scheutjens.10 Two sets of curves are included. The first ignores the nearest-neighbor bond correlations and is for a polymer brush containing 401 segments with a chain stiffness v50.5. The second one accounts for the bond correlations and is for a polymer brush containing 400 segments with a chain stiffness v50.368 ~hence a bond bending energy e5kT!. The results of the present calculations and those of Wijmans et al.12 almost coincide. In addition, the two sets of curves are near to one another, indicating that in those cases the increase in chain stiffness is compensated by the incorporation of the nearest-neighbor bond correlations.

relations results in a higher average segment density and consequently in a smaller brush height. The effect of the bond correlations becomes more significant as the chain length increases but is negligible for the short chain with 20 bonds. Regardless of whether the bond correlations are incorporated or not, the segment density first increases and subsequently decreases as a function of the distance to the surface. Figure 3 presents the probability profiles for various segments in the polymer brush with g5400 as a function of the layer number. The profile becomes wider with increasing segment number to the attached end, indicating a higher positional disorder for larger segment numbers. One can see that for segment numbers from k550 to k5300, the profiles vary appreciably, while between k5350 and k5401 they are approximately the same.

IV. SEGMENT DENSITY PROFILE

The calculated segment volume fractions are presented in Fig. 2 as a function of layer number, for chains with 20 to 400 bonds. Two sets of theoretical curves are included. The solid curves account for the nearest-neighbor bond correlations, while the dashed ones do not involve them. One can see that the incorporation of the nearest-neighbor bond cor-

FIG. 2. The segment density profiles for chains of various length.

V. BOND DENSITY PROFILE AND BOND ORIENTATION PROBABILITY

The density profiles of lateral, forward and backward bonds as a function of layer number to the surface are pre-

FIG. 4. The density profiles of forward, lateral, and backward bonds for the polymer brush with 400 bonds.

J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Nanodispersions

616 B. Li and E. Ruckenstein: End-attached chain monolayers

287

FIG. 5. Bond orientational probability as a function of bond number accounted from the attached end.

FIG. 7. The effect of bond correlations and bond backtracks on the profile probability of chain free ends for the polymer brush with 400 bonds.

sented in Fig. 4 for a polymer brush with 400 bonds. The sum of FFi and FB,i11 provides the density of the vertical bonds between layers i and i11. Because of the low segment density in the first layer ~see Fig. 2!, both the lateral bond density in the first layer and the forward bond density from layer 1 to 2 have low values. FB1 is equal to zero because no backward bond from the first layer is possible. The backward bond density has much lower values compared to those of lateral and forward bonds, indicating that most of the bonds are forward or lateral because of the constraint imposed by the surface. In contrast to the lateral or forward bond densities which decrease rapidly with increasing distance to the surface, the backward bond density remains approximately constant in a large range of layer numbers. Bond orientational probabilities as a function of bond number to the attached end are plotted in Fig. 5 for a brush with 400 bonds. The first bond has a large forward probability. The orientational probability of a bond to be lateral increases rapidly while that of a bond to be forward decreases rapidly with increasing bond number. After a small critical layer number, the orientational probability for a bond to be lateral increases slowly while that to be forward decreases

slowly. The orientational probability for a bond to be backward always increases. The last bond has the largest lateral probability and the same probabilities to be forward or backward.

FIG. 6. The effect of bond correlations and bond backtracks on segment density profile for the polymer brush with 400 bonds.

FIG. 8. The effect of bond correlations and bond backtracks on the bond order parameter for the polymer brush with 400 bonds.

VI. EFFECT OF BOND CORRELATIONS AND BOND BACKTRACKS

For the polymer brush with 400 bonds, the profiles of the segment density and of the chain free ends are plotted in Figs. 6 and 7 as a function of the distance to the surface to examine the effects of bond correlations and bond backtracks. Figure 8 presents the bond order parameter as a function of the bond number. Each of these figures contains four curves. dc 50 in these curves indicates that no bond correlations are taken into account in the calculations and dc 51 that they are taken into account. A similar convention, db 50 or 1, is used for the bond backtracks. The bond backtracks have a significant influence on the segment density, chain free end and order parameter profiles. In Figs. 6, 7, and 8, curves 1 and 2 show that when the bond backtracks are neglected, the segment density is uniform within a large distance to the surface ~Fig. 6!, the profile of the chain free ends is narrow

J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes 288

617 B. Li and E. Ruckenstein: End-attached chain monolayers

VII. EFFECT OF INTERMOLECULAR INTERACTIONS

The effect of intermolecular interactions is characterized by the Flory–Huggins interaction parameter. For the polymer brush with 400 bonds, three values of the interaction parameter were selected in the calculations and the segment density profiles are plotted in Fig. 9. x50 corresponds to a good solvent and x50.5 to a poor one. With increasing value of x, the height of the polymer brush decreases. For the poor solvent ~x50.5!, the chain segments tend to be more uniformly distributed ~curve 3!. ~a! D. W. R. Gruen, Biochim. Biophys. Acta 595, 161 ~1980!; ~b! J. Colloid Interface Sci. 84, 281 ~1981!; ~c! J. Phys. Chem. 89, 146 ~1985!; ~d! 89, 153 ~1985!. 2 ~a! A. Ben-Shaul, I. Szleifer, and W. M. Gelbart, Proc. Natl. Acad. Sci. USA 81, 4601 ~1984!; ~b! J. Chem. Phys. 83, 3597 ~1985!. 3 I. Szleifer, A. Ben-Shaul, and W. M. Gelbart, J. Chem. Phys. 83, 3612 ~1985!. 4 I. Szleifer and A. Ben-Shaul, J. Chem. Phys. 85, 5345 ~1986!. 5 I. Szleifer, A. Ben-Shaul, and W. M. Gelbart, J. Phys. Chem. 94, 5081 ~1990!. 6 ~a! K. A. Dill, and P. J. Flory, Proc. Natl. Acad. Sci. USA 77, 3115 ~1980!; ~b! 78, 676 ~1980!. 7 K. A. Dill and R. S. Cantor, Macromolecules 17, 380 ~1984!. 8 R. S. Cantor and K. A. Dill, Macromolecules 17, 384 ~1984!. 9 ~a! J. M. H. M. Scheutjens and G. J. Fleer, J. Phys. Chem. 83, 1619 ~1979!; ~b! 84, 178 ~1980!; ~c! Macromolecules 18, 1882 ~1985!. 10 F. A. M. Leermakers and J. M. H. M. Scheutjens, J. Chem. Phys. 89, 6912 ~1988!. 11 F. A. M. Leermakers and J. Lyklema, Colloids and Surfaces 67, 239 ~1992!. 12 C. M. Wijmans, F. A. M. Leermakers, and G. J. Fleer, J. Chem. Phys. 101, 8214 ~1994!. 13 R. S. Cantor and P. M. Mcllroy, J. Chem. Phys. 91, 416 ~1989!. 14 ~a! R. S. Cantor J. Chem. Phys. 99, 7124 ~1993!; ~b! 103, 4765 ~1995!. 1

FIG. 9. The effect of intermolecular interaction parameter on the segment density profile for the polymer brush with 400 bonds.

~Fig. 7! and the bond order parameter exhibits a large horizontal transition region ~Fig. 8!. When the bond backtracks are taken into account ~curves 3 and 4 in each of the Figs. 6, 7, and 8!, the segment density ~Fig. 6! and order parameter ~Fig. 8! increase near the surface and then decrease, and the chain free ends have a wide profile. The effect of bond correlations is revealed by comparing curve 4 with 3 and curve 2 with 1. The incorporation of the bond correlations provides a higher average segment density and consequently, a narrower profile of chain free ends and higher order parameters. The order parameter decreases sharply from its maximum at the first bond to a much lower value at approximately the third bond.

J. Chem. Phys., Vol. 106, No. 1, 1 January 1997

Copyright ©2001. All Rights Reserved.

Nanodispersions

618

Steric interactions between two grafted polymer brushes Eli Ruckensteina) and Buqiang Lib) Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York 14260

~Received 10 January 1997; accepted 8 April 1997! A lattice model and a generator-matrix method are employed to calculate the interaction force profile between two grafted polymer brushes. The correlation between neighboring bonds and the interdigitation between the two brushes are taken into account. The calculations show that the effect of incorporating the bond correlations is equivalent to an increase in the value of the polymer– solvent interaction parameter when the bond correlations are ignored. The interdigitation between the two brushes decreases the free energy of the system and consequently results in a smaller steric repulsion. A complete interdigitation occurs at a separation close to half the separation between the two plates for which the interaction force is zero. The model is compared with the experimental interaction force profiles for ten systems which involve poly~2-vinylpyridine!-polyisoprene ~PVP-PI!,¬ poly~2-vinylpyridine!-polystyrene¬ ~PVP-PS! block¬ copolymers¬ as¬ well¬ as end-functionalized polystyrenes ~PS-X!. For most of the systems, the theoretical predictions are in good agreement with experiment. In addition, the present results are compared with the equations proposed by de Gennes, based on the assumption of a step distribution function for the segment density, and by Milner et al., based on the parabolic distribution of the segment density. Both equations neglected interdigitation. It is shown in this paper that the interdigitation is not negligible and that it can decrease by an order of magnitude the repulsive force. © 1997 American Institute of Physics. @S0021-9606~97!50427-2#

I. INTRODUCTION

A very effective way of stabilizing colloids is to modify the surface by grafting a polymer.1 In this manner, a high polymer density for which the chains can overlap to form a polymer brush can be obtained and this leads to a strong steric repulsion between the colloidal particles. In recent years,¬ this¬ problem¬ has¬ received¬ attention¬ both experimentally2–10 and theoretically.11–42 The theoretical investigations focused firstly on the density profile of a single brush on a flat or curved surface and secondly on the interactions between two brushes. Most of the theories were based on the self-consistent mean field approximation.12–32 The other ones include the scaling analysis,33–37 molecular dynamics simulations38,39 and Monte Carlo simulations.40,41 The self-consistent mean field theories12–32 were developed along the following three lines: ~1! on the basis of a lattice model,12–22 ~2! on the basis of a diffusion type equation,23–28 and ~3! analytical approaches.29–32 The lattice-based self-consistent mean field theories12–22 were developed mainly by the Wageningen group and can be traced back to the Scheutjens–Fleer polymer adsorption theory,12,13 which is a lattice theory that extends the Flory– Huggins approach for homogeneous polymer solutions to inhomogeneous systems. In the lattice model, all possible conformations of the grafted ~and also free! chains are generated on a lattice, each conformation being weighted by the appropriate Boltzmann factor. In the early treatments, the polymer a!

Author to whom the correspondence should be addressed. Permanent address: Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China.

b!

932

J. Chem. Phys. 107 (3), 15 July 1997

molecules were described as freely jointed chains and the chain conformations treated as step-weighted random walks in a potential field. Some improvements were made by incorporating the chain stiffness14 and the correlations between the nearest-neighboring parallel bonds.15 Wijmans et al.,17 using the rotational isomeric state scheme of Leermakers and Scheutjens15 which involves the chain stiffness and the bond correlation, provided a detailed calculation for single brushes. Fleer and Scheutjens19 extended their theory12,13 for the adsorption of homopolymers to block copolymers. The block copolymer contains an adsorbing ~or anchoring! block and a nonadsorbing block ~or tail!. The latter behaves like an end-grafted chain and at high densities, the end-grafted chains form a brush. The interaction free energy between two surfaces grafted with adsorbed di- and tri-block copolymers for completely flexible chains was also calculated. Recently, Wijmans, Leermakers, and Fleer,21 developed a selfconsistent model which enabled the calculation of the interaction potential between two flexible polymer-coated spherical particles. The second kind of self-consistent theories23–28 employed a formalism originally developed by Edwards,23 in which the polymer was described as a freely-jointed chain consisting of segments of a given length and the conformation of a chain was treated as a random walk in the presence of a potential field. A fundamental quantity in these theories is the segment probability distribution function which is governed by a self-consistent diffusion equation. Whitmore and Noolandi27 presented calculations regarding the adsorption on a single surface and on two parallel surfaces. The repulsive force acting between the two surfaces was calculated and the results were in good agreement with two sets of experiments. Recently, the diffusion self-consistent mean

0021-9606/97/107(3)/932/11/$10.00

Copyright ©2001. All Rights Reserved.

© 1997 American Institute of Physics

Polymer brushes

619 E. Ruckenstein and B. Li: Interactions between grafted brushes

field theory was used by Lin and Gast28 to calculate the steric repulsion between two spherical particles on which polymer brushes were grafted, using an approximation which improved the Derjaguin approximation, since the latter one can not be applied to small particles. Analytical self-consistent mean field theories were developed independently by Zhulina et al.29,30 and Milner et al.31,32 They are based on the assumption that for large stretchings of the grafted chains with respect to their Gaussian dimension, one can approximate the set of conformations of a stretched grafted chain by a set of most likely trajectories, and predict for such cases a parabolic density profile. In the calculations of the interactions between the two brushes, the interdigitation between the chains was ignored. The scaling analysis was suggested by Alexander33 and de Gennes.34,35 Patel and Tirrell36 compared the scaling equations with their experimental force profiles for various di- and tri-block copolymers and Pincus37 extended them to polyelectrolyte brushes. The scaling analysis assumed a stepfunction for the distribution of the polymer segment density. For a neutral brush, the interaction free energy involves two contributions; an osmotic term which depends on the segment density and an elastic one which arises because of chain stretching. For a polyelectrolyte brush, the Poisson– Boltzmann equation plays also a part. While there is good agreement between theory and experiment regarding the dependence of the steric repulsive force on distance, the equations contain two parameters which have to be provided by experiment. In this paper, we calculate on the basis of a previously developed formalism for a brush42 the interaction force profile between two identical polymer brushes grafted to two crossed cylindrical surfaces. As noted42 previously, the method is equivalent but different formally from that developed in Refs. 15 and 17, being more simple and compact. A detailed comparison with recent experiments,9,10 which involve ten systems, is carried out. The Derjaguin approximation is combined with the flat geometry results to calculate this force profile. Particular emphasis is on the effects of the interdigitation between the chains of the two brushes and of the nearest-neighboring bond correlation. II. THEORY A. Chain conformational statistics

We consider a system consisting of two identical polymer brushes ~grafted to two parallel plates! and solvent molecules. The cubic lattice model is employed to represent the mixture of polymer chains and solvent ~schematically shown in Fig. 1!. Each polymer segment and each solvent molecule occupies one lattice site. The lattice is divided into 2m layers parallel to the plates, numbered from any of the two plates as 1, 2,..., i,...,m, m11,...,2m112i,...,2m. Each lattice layer contains L sites. Since the two brushes are considered identical, there exists a plane with respect to which the segment density and bond conformations are symmetrical. Thus, layers i and 2m112i with i51,2,...,m contain the same numbers of segments, of solvent molecules and have the same

933

FIG. 1. Schematic representation of the lattice model.

bond conformations. We consider only monodisperse brushes. Each brush contains N chains and each chain g segments which are connected through g b bonds and occupy g sites ( g 5 g b 11). The first segment of each chain is constrained to be located in layer 1 or 2m. In the cubic lattice, only two relative orientations of two consecutive bonds can occur, namely the collinear and the bent one at 90°. The bending energies of the collinear and the bent bond pairs are taken to be zero and e, respectively. The chain stiffness is characterized by the parameter v 5exp(2e/kT). A small value of v ~large bond bending energy e! corresponds to a stiff chain. Taking into account four types of bond pairs in the cubic lattice, depending on whether a bond pair is collinear or bent and on how many ways the bond pair can arrange on the lattice, the statistical weight of a chain in conformation c can be expressed42 as

S ) DS ) ) D 2m

n

W c 5 a h ch a n cb1 b n cb2 v n cb3

2m

g

i51

p i ci

g

i51

j

p j ic j i

~1!

which differs from the corresponding equation for a single brush only because 2m replaces m. In Eq. ~1!, n ch 50 or 1 when the first bond is forward or lateral in conformation c, n cb1 , n cb2 , and n cb3 denote the numbers of bond pairs in conformation c corresponding to three different bent bond pairs. The quantities a h , a, and b are given by the expressions a h 54 v h , a54v, and b52v with v h 5exp(2eh /kT) and v 5exp(2e/kT), where e h stands for the bending energy of the first bond near the surface which may differ from e. g ci stands for the number of segments in layer i provided by one chain in conformation c and g c j i for the number of bonds in direction j with at least one segment in layer i provided by one chain in conformation c. In the cubic lattice, j can be x or y for the lateral bonds and z for the vertical one. p i is the weight factor due to the segment–solvent intermolecular interactions, given by p i 5F 0i exp@~ ^ F i & 2 ^ F 0i & ! x 1 ~ d 1i 1 d 2mi ! x s # , i51,2,...,2m,

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

~2!

Nanodispersions

620 934

E. Ruckenstein and B. Li: Interactions between grafted brushes

where F 0i and F i are the fractions of sites occupied by the solvent molecules and segments, respectively, the brackets ^ & stand for the nearest neighbor average, x is the polymer– solvent interaction parameter, x s 5(U 0s 2U s )/kT ~with U 0s and U s denoting the contact energies of the solvent molecules and segments with the surface, respectively!, d is the Delta function. p j i is the weight factor due to the correlation between the nearest-neighboring parallel bonds, expressed as p ji5

1 ,¬ i51,2,...,2m, 12F j i

~3!

where F j i is the number of bonds per lattice site ~bond density! in direction j with at least one segment in layer i. Assuming that on average, half of the lateral bonds in layer i are in the x direction and the other half in the y direction, and denoting by F Li the total lateral bond density in layer i, one can write for the lateral bonds, p xi 5 p yi 5

1 [p Li ,¬ i51,2,...,2m. 12F Li /2

~4!

It is worth noting that the surface parameter h s 5exp(xs) which appears in the corresponding equation of the preceding paper42 does not appear in Eq. ~1! because in the present derivation, the effect of x s was included in the expression for the weight factor of a segment located in either layer 1 or 2m. Using the generator-matrix formalism of the chain conformational statistics, the chain conformation partition function, J, can be expressed using a product of three matrices,42,43,44 J5

B5col@ c1 ,c2 ,...,cn # , ci 5col@ p i , p i # . In the above equations, n stands for the highest layer which can be reached by a chain. For the present case, n 52m. In the supermatrix G, ri represents a backward bond starting from layer i, qi a lateral bond in layer i, and pi a forward bond starting from layer i. In the elements ri and pi , the Heaviside functions y rp and y pr are included to avoid bondfolding, since a bond can not be backward when the previous bond was forward and vice versa. The elements ri ,qi , and pi depend on three kinds of parameters. The first kind of parameters a, b, and v arise from the local chain stiffness and bond arrangements, the second p i from the segment– solvent interactions, and the third kind p j i from the correlations between nearest-neighboring parallel bonds. The supermatrix G differs from that proposed by DiMarzio and Rubin,43 and employed by Scheutjens and Fleer12 and Cosgrove et al.16 since it accounts for the bond correlation and chain stiffness and excludes via the Heaviside functions the bondfolding. Compared to the supermatrix of Dill,44 it accounts, in addition, for the bond backtrack and bond correlation. The statistical weight (W g i ) and the probability ( P g i ) for a chain to end in layer i are given by W g i 5AGg b Ci and P gi5

~5!

b

where G is a n3n supermatrix, q1

p1

•••

•••

•••

0

r2

q2

p2

•••

•••

0

G5 ••• ••• 0¬ •••

•••

•••

•••

•••

•••

rn 21

qn 21

pn 21

0¬ •••

•••

•••

rn

qn

F G F G F G F G

qi 5 p i p Li

0

0

0

0

0

0

0

a

0

b 11

pi 5 p i p zi y pr

05

1

v

1

0

v

0

N gi W gi 5 , N J

G

and N g i stands for the number of chains whose free ends are located in layer i. For the chains with the kth bond lateral in layer i, the chains with the kth bond forward from layer i to i11 and the chains with the kth bond backward from layer i to i 21, the probabilities P kLi , P kFi , and P kBi are given, respectively, by

,

i52,3,..., n ,

P kLi 5

1 AGk21 Qi Gg b 2k B,¬ i51,2,..., n , J

~8!

P kFi 5

1 AGk21 Pi Gg b 2k B,¬ i51,2,..., n 21, J

~9!

P kBi 5

1 AGk21 Ri Gg b 2k B,¬ i52,3..., n , J

and

,¬ i51,2,..., n ,

,

~7!

Ci 5col@ 0,...,ci ,...,0# , ci 5col@ p i , p i # , 05col@ 0,0# ,

whose elements are the 232 submatrices ri 5 p i p zi21 y rp

~6!

where

(c W c 5AGg B,

F

A5 @ a,0,...,0# , a5 @ 1,0# , 05 @ 0,0# ,

i51,2,..., ~ n 21 ! ,

.

A and B are vectors containing n elements

~10!

where Qi , Pi or Ri is a supermatrix which has only the (i,i), (i,i11) or (i,i21) ~i.e., qi , pi or ri ! as a nonzero element. Since no bond can be forward from layer n and no bond can be backward from layer 1, P kF n and P kB1 are zero. Because the kth segment of a chain is located in layer i if the kth bond of the chain is forward, lateral, or backward

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes

621 E. Ruckenstein and B. Li: Interactions between grafted brushes

from layer i, the probability for the kth segment to be in layer i is equal to the sum of the probabilities for the kth bond to start from layer i, P ki [

N ki 1 5 AGk21 ~ Qi 1Pi 1Ri ! Gg b 2k B,¬ i51,2,..., n , N J ~11!

where N ki represents the number of chains whose kth segment is located in layer i. Eq. ~11! can be applied to the segments from 1 to g b . For the end segment ~chain free end!, Eq. ~7! should be used. According to Fig. 1, the number of segments provided by the lower brush in layer i can be obtained by summing N ki from segment i to the end segment g, while that provided by the upper brush is given by summing N k2m112i from segment 2m112i to the end segment g. Consequently, the total number of segments in layer i, N i , is given by g

N i5

g

( N ki 1 k52m112i ( N k2m112i . k5i

~12!

Using Eqs. ~7! and ~11!, Eq. ~12! becomes F i 5 s ~ P g i 1 P g 2m112i !

S( gb

1s

D

gb

k5i

P ki 1

(

k52m112i

P k2m112i .

~13!

Similarly, one can obtain, for the lateral bond density (F Li ) in layer i and for the vertical bond density between layers i and i11 (F zi ), the following equations:

S( S( gb

F Li 5 s

k5i

gb

F zi 5 s

k5i

(

k52m112i

P kL2m112i ,

gb

P kFi 1

gb

1

D

gb

P kLi 1

~14!

gb

( P kB2m112i k52m122i

D

.

layer. Consequently, a mass balance of the connection points provided by the segments present in layer i provides the equation 2N i 2N g i 2N g 2m112i 5N zi21 1N zi 12N Li .

~15!

Since the two brushes are symmetrical, only the distributions of the segment density and bond densities along the distance from layer 1 to m, namely F i , F Li , and F zi with i51,2,...,m, are required. They can be obtained by numerically solving Eqs. ~13!, ~14!, and ~15!, each containing m equations. Instead of Eq. ~13!, a more simple set of equivalent equations is derived below. One can demonstrate that the new set of equations is a linear combination of Eqs. ~13!, ~14!, and ~15!. In a linear chain, a middle segment is connected to two bonds and a terminal segment to a single bond; in other words a middle segment has two connection points and a terminal one only one. A lateral bond in layer i provides two connection points in that layer, while a vertical, either between layer i and i11 or between layer i and i21, provides only one connection point in layer i. In addition, a terminal segment located in layer i, whether connected to a lateral or to a vertical bond, provides only one connection point in that

~16!

Dividing by L, the above equation becomes 2F i 5 s ~ P g i 1 P g 2m112i ! 1F zi21 1F zi 12F Li .

~17!

Equations ~14!, ~15!, and ~17! will be used to calculate the quantities F i , F Li , and F zi . The obtained distributions of the segment density and bond densities allow to calculate any structural property of the two interacting polymer brushes as well as the mixing free energy. The latter quantity is used to obtain the interaction force profile. When the effect of bond correlations is neglected, only Eq. ~13! is required to calculate the distribution of segment density from layer 1 to m; the p j i ’s, which are present in the elements ri , qi , and pi to represent the weight factors due to the bond correlations, become in that case equal to unity.

B. Interaction free energy of two brushes

Denoting by N c the number of chains in a brush in conformation c, one can write N5 ( N c . Since the two brushes are symmetrical, the whole system has 2N c chains in conformation c. For a specified set of conformations $ N c % of the N chains of any brush, the canonical partition function Q(T,L,2m, $ 2N c % ) can be written as Q ~ T,L,2m, $ 2N c % ! 5

( P kBi11 1 k52m2i ( P kF2m2i k5i12

935

S D

V DU exp 2 , V0 kT

~18!

where V is the combinatorial factor for the specified set of conformations $ N c % , V 0 is the product of the combinatorial factors of the chains and solvent molecules in their reference states, DU is the energy change of the chains and the solvent molecules from their reference states to the specified set of conformations. The ordered close-packed brush, in which each chain has an all-trans conformation ~hence both the conformational energy and entropy are taken zero!, is taken as the reference state of the chains, while the pure solvent as the reference state of the solvent molecules. For the reference states, V 0 51. Extending the corresponding equations of the preceding paper42 for a single brush to the present case involving two interacting brushes, one can write

S) D S) D F) ) G N

V5 ~ L! !

gc c

2m

c

2m

3

i51

j

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Nc!

2

2m

i51

~ L2N j i ! ! , L!

1

N 0i !

~19!

Nanodispersions

622 936

E. Ruckenstein and B. Li: Interactions between grafted brushes

F

m

DU ~ L2N ! U s 52 N 01 x s 1 1 N i ^ F 0i & x kT kT i51 1

(c

(

G

N cE c , kT

~20!

where g c is the number of arrangements that a chain in conformation c can have when its first segment is fixed, N 0i is the number of solvent molecules in layer i, N j i is the number of bonds in direction j with at least one segment in layer i, ^ F 0i & is the average fraction of sites occupied by the solvent molecules around a site located in layer i and E c is the conformational energy of a chain in conformation c. At equilibrium, N c 5NW c /J. Introducing the later expression in Eqs. ~19! and ~20!, one finally obtains for the segment-solvent mixing free energy A52kT ln(Q), the expression,

S D

2m

2m

A N 52N ln 1L ln~ F 0i ! 1 N i ^ F i & x 2L kT LJ i51 i51

(

(

2m

3

(( i51 j

~21!

ln~ 12F j i ! .

Taking into account the symmetry of the two brushes, the mixing free energy per surface site ( f 5A/2L) becomes

SD

m

m

f s 5 s ln 1 ln~ F 0i ! 1 F i^ F i& x kT J i51 i51

(

m

2

(( i51 j

ln~ 12F j i ! 1

(

1 ln~ 12F zm ! , 2

~22!

where s 5N/L stands for the number of chains per surface site ~surface density!. The mixing free energy f becomes independent of the separation D between the two surfaces for distances larger than 2L 0 ~twice the equilibrium thickness of an uncompressed brush L 0 ! for which the steric repulsion is zero. Denoting by f 0 the mixing free energy at D>2L 0 , the interaction free energy per unit area (E) between two grafted brushes is given by E ~ D ! 52 ~ f 2 f 0 ! /l 2 ,

Derjaguin approximation. For the present comparisons with experiment, the Derjaguin approximation is considered to be applicable, since the curvature radius of the surface ~cylindrical surface! is much larger than the separation between the two surfaces ~105 times!. III. RESULTS AND DISCUSSIONS

In this section, the theory is compared with the experimental data available regarding the interaction force profile between two crossed cylinders bearing two identical grafted polymer brushes. The results will be also compared with the equation proposed by de Gennes,35 which involves a step function for the segment density distribution, and that proposed by Milner et al.31 which involves a parabolic distribution. The interdigitation between the two brushes has been neglected in both preceding equations. In the present paper, the distributions of segment density, bond densities and chain free ends as well as the steric repulsion force are calculated taking into account the interdigitation between the two brushes and the correlations between the nearestneighboring parallel bonds. The comparison between theory and experiment is made for the following systems: end-functionalized polystyrenes @polystyrene-(CH2!3N1~CH3!2~CH2!3SO2 3 denoted as PS-X# of various molecular weights and block copolymers @poly~2vinylpyridine!-polyisoprene denoted as PVP-PI and poly~2vinylpyridine!–polystyrene denoted as PVP-PS# over a range of molecular weights of each of the blocks. The end group X or the PVP block serves as an ‘‘anchor,’’ which attaches the PS or PI chains to the surface forming grafted polymer brushes in a solvent ~here toluene!. A. Chain characteristic properties and lattice parameters

The application of the lattice model to various systems requires a procedure to determine the lattice size and the chain bending energy, hence the number of segments and the chain stiffness in the lattice. Based on the equivalence between the contour length, volume, and gyration radius of a real polymer chain and that in the lattice model, the following equations were suggested:22

~23! 2

where l stands for the length of a site and l is the area per site. The interaction free energy can be converted to the interaction force (F) between two crossed cylinders of radius R, which is measured experimentally, using the Derjaguin approximation, F ~ D ! /R52 p E ~ D ! .

~24!

Recently,21,28 two kinds of self-consistent mean field models were developed to calculate the interaction potential between layers of flexible polymer chains grafted to spherical surfaces whose radii are comparable to the separation between the two surfaces. The calculations showed that for small particles the repulsion is less steep than that provided by the Derjaguin approximation, while with increasing radii, the interaction potential becomes close to that given by the

g l5l b M /M s ,

~25!

M g l 35 ¯ , rN

~26!

pl 2 g /65a 2 M .

~27!

and

In the above equations, M is the molecular weight of the polymer, M s is the molecular weight of the monomer, l is the lattice spacing, l b the bond length between two monomers ~estimated from the structure of the monomer!, r the polymer density, ¯ N Avogadro’s number, a 2 5R g /M 1/2 with R g the gyration radius, and p is the ratio between the Kuhn length and the bond length l in the lattice model ~lattice spacing!. The ratio p is related to the bond bending energy e through the following expression.22

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes

623 E. Ruckenstein and B. Li: Interactions between grafted brushes

937

TABLE I. Chain characteristics and interaction parameters. Ms lb ra ~g/mol! ~Å! (g/cm3)

Polymer

v

l x aFH ~Å! ~F51.0–0.0!

x8

x9

Polystyrene¬ 104¬ 2.53¬ 1.130¬ 0.27 7.77¬ 0.67–0.04¬ 0.20 0.03 Polyisoprene¬ 68¬ 5.07¬ 0.913¬ 0.27 4.94¬ 0.36–0.32¬ 0.35 0.18 a

Reference 45.

p511

S D

1 e 1 exp 511 , 2 kT 2v

~28!

where v 5exp(2e/kT) is the chain stiffness parameter. Using Eqs. ~25!–~28!, one obtains for the lattice spacing l, the chain stiffness parameter v and the number of segments g the expressions

S D S

Ms l5 rl ¯ N

1/2

~29!

,

b

v5

1 6a 2 M s 21 2 ll b

g5

lbM . lM s

D

21

~30!

,

and ~31!

Equations ~29! and ~30! show that l and v are independent of the chain length; this is as expected. For the PS chains, a50.3 Å, 22 M s 5104 g/mol, l b 52.53 Å, and r 51.13 g/cm3; 45 hence l and v are 7.77 Å and 0.27, respectively. For the PI chains, no value for a is available and R g had to be used to calculate v. The following expression for the R g of a PI ~also PS! chain is available,10 R g 5b ~ M /M s ! 0.595

~32!

with b51.79 Å for the PI chain and 1.86 Å for the PS chain. Combined with Eq. ~32!, Eq. ~30! yields

v5

F S D G

1 6b 2 M 2 ll b M s

FIG. 2. The effect of bond correlations in the PS-X ~140 K!-toluene system. Free of bond correlations ~solid curves!: 1, x850.3; 2, x850.2; 3, x850.1 and with the bond correlations ~dashed curves!: 1; x950.2; 2; x950.03.

0.19

21

21

~33!

.

According to Eq. ~33!, the chain stiffness parameter v decreases weakly with the chain length. Consequently, the polymer chain becomes slightly stiffer with increasing chain length, which is unreasonable. For this reason, an average of the values provided by Eq. ~33! for the four PI chains employed was used. For the PI chains, M s 568 g/mol, l b 5 5.07 Å, r 50.913 g/cm3; 45 consequently l54.94 Å and v50.29 ~calculated as the arithmetic average!. All the parameters for PS and PI chains are listed in Table I, which also includes three polymer–solvent interaction parameters for reasons explained below. B. Interaction parameters and bond correlations

Two sets of theoretical curves are plotted in Fig. 2 to examine the effect of the interaction parameter on the theoretical interaction force profile between two identical PS-X ~140 K! brushes. The first set was calculated by ignoring the

bond correlations and the second by incorporating them. x8 and x9 denote the interaction parameters in the two cases. In each of the two cases, the theoretical force profile curve is moved up with decreasing interaction parameter, indicating, as expected, that a good solvent results in a higher steric repulsion between the two brushes. By comparing the two sets of curves, one can note that for the same value of the interaction parameter, the incorporation of the bond correlations reduces the repulsion between the two brushes. This happens because the correlations between the nearestneighboring parallel bonds lead to a more efficient packing of the segments, and as a result the segment density increases. Consequently, the bond correlations have the same effect as that caused by an increased value of the interaction parameter. Indeed, almost the same profiles of the interaction force, the segment density and the chain free ends were obtained by neglecting the bond correlations but by using a larger value for the interaction parameter( x 9 5 x 8 10.17). Such a suggestion was made previously by Wijmans et al.,17 who demonstrated that, for sufficiently low polymer concentrations, x95x810.17. They applied this observation to single brushes. For the PS-X ~140 K! brushes and the two cases ~without and with bond correlations!, one can see from Fig. 2 that the theoretical force profile ~solid curve 2! for the case free of bond correlations and x850.2. and that ~dashed curve 28! which incorporates the bond correlations and uses for x 9 the value 0.03 almost overlap. In addition, the theoretical curves 2 and 28 fit well the experimental data of Taunton et al.9 In the present calculations, the interaction parameter ~x8 or x9! is the only undetermined parameter. For the PI chains–toluene interactions, x8 is selected 0.35, which is close to the experimental values of the Flory–Huggins interaction parameter45 ~x FH50.32– 0.36 in the concentration range F50–1 and in the temperature range 25–55 °C!. For the PS chains–toluene interactions, x8 is taken 0.2, which is in the lower range of the experimental Flory–Huggins interaction parameters45 ~x FH50.04– 0.67 in the concentration

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Nanodispersions

624 938

E. Ruckenstein and B. Li: Interactions between grafted brushes

TABLE II. Molecular weights, number of segments, and surface densities.

Polymers

Molecular weight ~M PI or M PS! ~31023 g/mol)

The number of segments ~g!

Surface density s 3103

PVP-PI 63-39¬ PVP-PI 26-50¬ PVP-PI 38-69¬ PVP-PI 30-217¬ PVP-PS 60-60¬ PVP-PS 60-90¬ PS-X ~26 K! PS-X ~58 K! PS-X ~140 K! PS-X ~375 K!

38.8¬ 50.4¬ 69.3¬ 217.0¬ 60.0¬ 95.0¬ 26.3¬ 57.8¬ 139.8¬ 374.8¬

588¬ 754¬ 1041¬ 3274¬ 185¬ 292¬ 81¬ 178¬ 430¬ 1153¬

2.12 3.05 2.44 0.83 4.95 5.31 24.43a 13.74a 7.79 2.72a

a

Value calculated as noted in the text.

range F50–1 and in the temperature range 162–229 °C!. For either of the two kinds of chains ~PI or PS!, the difference x82x950.17. The values of the interaction parameters are listed in Table I. C. Comparison with experimental data

In this section, the calculated results are compared with experimental data for the PS-X polymers and for the PVP-PI and PVP-PS block copolymers. The two-dimensional surface density ~number of chains per surface site! in the lattice model s is proportional to the surface grafting density s e ~number of chains per unit area!, namely s 5 s e l 2 . For the systems involved in the calculations, the surface density s and the number of segments g are listed in Table II. For the experiments carried out by Watanabe and Tirrel,10 which involved the PVP-PI and PVP-PS block copolymers, we use the surface grafting densities of the PI and PS chains calculated by them. For the experiments performed by Taunton et al.9 involving the PS-X polymers, only the surface grafting density of PS-X ~140 K! (3 60.5 mg/m2! was given. The average value 3 mg/m2 was therefore used in our calculations. For the other three PS-X polymers, the surface grafting densities were obtained by us on the basis of the theoretical analysis provided by the authors and we do not know how accurate the values are. The comparison between theory and experiment is presented in Figs. 3–6. Since when the bond correlations are ignored and the interaction parameter x8 is used, or when the bond correlations are incorporated and the interaction parameter x95x820.17 is employed, the theoretical curves almost overlap, we plot in the figures only the theoretical curves for the case with bond correlations. Since for the block copolymer PVP-PI 30-217, which has a long chain ~g53274!, the incorporation of bond correlations will require a timeconsuming computation, only the calculations for the case free of bond correlations were performed, using, however, a larger interaction parameter. For comparison, Figs. 3–6 include the theoretical curves based on de Gennes35 and Milner et al.31 equations. de Gennes equation is an extension to two interacting brushes of Alexander’s ideas33 regarding the single grafted brushes. In

FIG. 3. A comparison among the present calculations, and the equations of de Gennes and Milner et al. The curve based on the de Gennes expression was obtained in Ref. 4 by fitting experimental results, while that based on the Milner et al. expression, which does not involve any fitting parameter, was provided in Ref. 32.

Alexander’s treatment, the grafted brush is assumed to have a uniform segment density ~step function distribution!, and each chain to consist of connected semidilute ‘‘blobs.’’ The osmotic repulsion between blobs tends to stretch the chains, while the elastic free energy of the chains has the opposite effect. For a single brush, the minimization of the overall free energy with respect to the brush thickness yields the equilibrium brush thickness L 0 , given by L 0 5s ~ R F /s ! 5/3,

~34!

where s and R F are the mean spacing between grafting points and the Flory radius, respectively. In the above equation, a numerical prefactor of order unity was ignored. The quantity s was calculated from the surface grafting density s e (s5 s 21/2 ) and R F (R F 56 1/2R g ) from the radius of gye ration R g , given by Eq. ~32!. Considering two contributions, a repulsive osmotic one due to the increase in the polymer concentration, and an

FIG. 4. Interaction force profiles for PVP-PI 69-39 and PVP-PI 38-69 in toluene. The parameters à and v in the Milner et al. expression and the prefactors k L and k F in the de Gennes expression are obtained by fitting the experiments for PVP-PI 38-69.

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes

625 E. Ruckenstein and B. Li: Interactions between grafted brushes

FIG. 5. Interaction force profiles for PVP-PI 26-50 and PVP-PI 30-217 in toluene. The parameters à and v in the Milner et al. expression and the prefactors k L and k F in the de Gennes expression are obtained by fitting the experiments for PVP-PI 38-69.

elastic one due to the decrease of the elastic free energy, de Gennes35 suggested for the interaction force per unit area the following equation; f G~ D ! 5

kT s3

FS D S D G 2L 0 D

9/4

2

D 2L 0

3/4

.

~35!

A prefactor in the above equation was ignored. In order to test the validity of de Gennes8 scaling model, one can examine if the two prefactors in Eqs. ~34! and ~35! remain constant when the polymer system is changed. We introduce two prefactors k L and k F in Eqs. ~34! and ~35!, respectively, and calculate the interaction free energy per unit area using the expression, E~ D !5

E

2L 0

D

f G ~ D 8 ! dD 8 .

~36!

FIG. 6. Interaction force profiles for PVP-PS 60-60, PVP-PS 60-90, and PS-X ~140 K! in toluene. The parameters à and v in the Milner et al. expression and the prefactors k L and k F in the de Gennes expression are obtained by fitting the experiments for PS-X~140 K!.

939

The two prefactors, k L and k F , can be obtained by fitting the experimental force-distance data for a given system. The fitted results show that for the four PI chains considered in the calculations, k L remains approximately constant ~0.91– 1.15!, while for the six PS chains k L has the values 0.74– 0.81. However, k F varies within a wider range. For the PI chains, k F varies from 0.68 to 0.90 while for the PS chains, k F varies between 0.98 and 2.82. These calculations show that k L depends to some extent on the nature of the chain, and that k F depends on both the nature of the chain and the chain length. In contrast to de Gennes’ model which involves a step function for the segment density distribution, the model of Milner et al., based on a self-consistent mean field approach, predicts a parabolic distribution of the segment density distribution in either a free brush or in a brush interacting with another one. Neglecting the interdigitation, the free energy per unit area of a brush as a function of separation is given by f M~ u !5

S

D

5 0 1 u5 f M 1u 2 2 , 9 u 5

~37!

where u5D/2L 0 and f 0M is the free energy per unit area of the free brush of thickness L 0 The quantities L 0 and f 0M can be calculated using the expressions L 05 f 0M kT

S D S DS D S DS D

5

12 p2

1/3

9 p2 10 12

M Ms

s eà v

1/3

,

M ~ s 5 Ã 2 v ! 1/3, Ms

~38! ~39!

where à ~with dimensions of length3! is an excluded volume parameter and v ~with dimensions of length22 ! is a coefficient present in the chain elastic contribution. Milner32 used the model to interpret the experimental force profile obtained by Taunton et al.4 for the PS-X ~141 K!. à and v were calculated from the osmotic pressure p~F! and the gyration radius R g using the expressions p (F)5ÃF 2 /2 and 2R 2g 5M /(M s v ). Using a surface grafting density of 3.5 mg/m2 ~the upper end of the experimental range 3 60.5 mg/m2!, good agreement with experiment was obtained. For the PS-X ~140 K! chain, the present calculation and the equations of Milner et al. and de Gennes are plotted in Fig. 3. In the figure, two sets of experimental data obtained by the same authors4,9 in two different papers are included. One can see that the model of Milner et al. and that of de Gennes fit well the early experiments, while the present calculation is in good agreement with the new ones. Regardless of the large difference between the two sets of experimental data, the models of de Gennes and Milner et al. fit well either the old or the new experiments by adjusting the parameters. Indeed, using Ã545.2 Å 3 and v 50.069 Å 22 instead of those calculated ~Ã533.7 Å 3 and v 50.052 Å 22 !, the model of Milner et al. fits well the new experiments. When the de Gennes model is fitted to the new experiments, a smaller prefactor k F 50.8 replaces k F 51.5.

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Nanodispersions

626 940

E. Ruckenstein and B. Li: Interactions between grafted brushes

FIG. 7. Interaction force profiles for PS-X~26 K!, PS-X~58 K!, and PSX~375 K! in toluene. The parameters à and v in the Milner et al. expression and the prefactors k L and k F in the de Gennes expression are obtained by fitting the experiments for PS-X~140 K!.

A comparison between the three models and experiments involving ten systems is made in Figs. 4–7. In the comparison, the parameters à and v in the model of Milner et al. and the prefactors k L and k F in de Gennes’ model are taken the same for the same kind of chains ~PI or PS!. For the PS chains, they are obtained by fitting the experiments for the PS-X ~140 K!, while for the PI chains, by fitting those for PVP-PI 38-69. The values of the parameters are listed in Table III. Figures 4 and 5 present comparisons for four PI chains of¬ various¬ molecular¬ weights¬ ~from¬ 3.883104 to 2.173105 !. For the PVP-PI 38-69-toluene system which was selected to determine the optimum parameters in the Milner et al. and de Gennes’ models, the present calculations, which do not involve any fitted parameters, are in very good agreement with experiment over the entire range of the force profile. For both the PVP-PI 69-39 with a short PI chain and the PVP-PI 30-217 with a long PI chain, the present calculations fit well the experimental force profiles, while the other two models provide lower values for the former system over the entire range of separation and higher values for the latter system in a wide separation range. For the PVP-PI 26-50, the three models provide higher values than experiment. For the PVP-PI systems, the models of Milner et al. and de Gennes provide similar results. Figures 6 and 7 present a comparison for six PS chains of various molecular weights ~from 2.633104 to 3.75 3105 !. For the PS-X ~140 K!-toluene system which was selected to determine the optimum parameters in the models of Milner et al. and de Gennes, the present calculations and TABLE III. Fitted parameters in the equations of Milner et al. and de Gennes. Polymer¬

à ~Å!

v (Å 22 )¬

Polystyrene¬ Polyisoprene¬

45.2¬ 48.3¬

0.069¬ 0.029¬

kL 1.15¬ 0.76¬

kF 0.42 0.80

FIG. 8. The effect of the interdigitation between two brushes.

the other two models provide approximately the same results within a wide region, in good agreement with experiment; at larger separations, the other two models deviate from experiment. For the PVP-PS 60-60 and PVP-PS 60-90 block copolymers ~see Fig. 6! and PS-X ~58 K! ~see Fig. 7!, the theoretical force profiles provided by the present calculations are lower than those provided by experiment, but can be shifted along the ~logarithmic! vertical axis to fit the experimental data. For the PS-X ~26 K! containing 185 segments, the present calculations fit well the experimental results, while for the PS-X ~375 K! containing 1153 segments, there is a slight deviation ~Fig. 7!. For all the PS chains, the equations of Milner et al. and de Gennes lead to larger deviations from the data than the present model. Particularly, at large separations, the interaction force given by the models of Milner et al. and de Gennes decrease rapidly with increasing separation D. As a result, the two models predict a smaller separation 2L 0 at which the interaction force between the two brushes becomes zero than the experimental one. The comparisons plotted in Figs. 4–7 show that for most systems, the present model is in good agreement with experiment; for some systems, such as PVP-PI 26-50, PVP-PS 60-90, and PS-X ~58 K!, a good fitting with experiment can be obtained by parallel shifting up the theoretical force profile curves. This happens probably because of inaccurate data regarding the surface density.

D. The interdigitation between two brushes

The effect of the interdigitation between the polymer chains of two brushes on the interaction force profile is plotted in Fig. 8 for three polymers PVP-PI 69-39, PVP-PI 3869, and PS-X ~140 K!. The solid curves present the results in which the interdigitation was taken into account, while the dashed ones those in which the interdigitation was neglected. For the former, the chain conformational partition function @Eq. ~5!# involves a 2m32m supermatrix G for which the chain free ends can be located in any lattice layer between the two surfaces. For the latter, the conformational partition function involves a m3m supermatrix G with chains con-

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes

627 E. Ruckenstein and B. Li: Interactions between grafted brushes

941

FIG. 9. The distribution of chain free ends in the PS-X~140 K!-toluene system.

FIG. 10. The distribution of segment density in the PS-X~140 K!-toluene system.

fined between the middle plane and the surface on which the chains are grafted; the interaction energy between the two brushes is in this case the sum of the contributions of the two separate brushes. One can see that the theoretical curves free of interdigitation are always located above those which account for the interdigitation, indicating that by neglecting interdigitation a much larger repulsion is obtained. The absolute deviation between the two theoretical results increases with decreasing separation. The interdigitation can not be neglected over the entire range of separations. It can decrease the repulsion by more than an order of magnitude. The equations of Milner et al. and de Gennes ignore the interdigitation, which is accounted only indirectly through the fitting parameters.

surface increases starting from zero. For the small separation ~155.4 Å!, the theoretical distribution curve becomes symmetrical with respect to the middle plane. For the same system and the same separation distances as in Fig. 9, the distributions of segment densities are given in Fig. 10. The filled point represents again the position of the middle plane. The segment density at the middle plane is close to zero at the large separation of 1554 Å ~curve 5! and increases with decreasing separation. The separation at which the interactions start, which is twice the thickness of the single brush, is equal to 1554 Å. The above theoretical value is slightly larger than the experimental one 13006100 Å. With decreasing separation, the segment density becomes more uniformly distributed. At a separation smaller than 388.5 Å ~curve 2!, one can achieve an almost uniform segment density distribution in the middle region. For all the separations, the segment density decreases close the surfaces, indicating a depletion of polymer chains at the surface.

E. The distributions of chain free ends and segment density

For the PS-X ~140 K! and five different separations, the distribution probabilities of the chain free ends are plotted in Fig. 9 as a function of the distance to the surface on which the chains are grafted. In the figure, the filled points denote the position of the middle plane. Therefore, the part of the theoretical curve at the right of the filled point provides the distribution of chain free ends of one brush in the other one. The thickness of the interdigitation region is equal to twice the distance from the middle plane to the point at which the distribution probability of chain free ends becomes zero. As expected, as the separation decreases from a large value ~1554 Å! to a smaller one ~155.4 Å!, the filled point moves up and reaches a point located at the maximum of the distribution probability. There is a critical separation at which the chain free ends of one brush can reach the grafting surface of another brush and a full interdigitation occurs. For the PS-X ~140 K! shown in Fig. 9, a full interdigitation occurs at separations smaller than 777 Å ~curve 3!, which is approximately half of the separation 2L 0 . With decreasing separation from 777 Å to 155.4 Å ~see curves 3, 2, and 1!, the distribution probability of the chain free ends located on the opposite

IV. CONCLUSIONS

On the basis of a cubic lattice model and generatormatrix formalism, the interaction force profile between two identical polymer brushes grafted on two curved surfaces is calculated. This model allows to examine in detail the effect of the interdigitation between the two brushes. As expected, the presence of interdigitation results in a smaller interaction free energy and hence decreases the steric repulsion between the two brushes. The calculations show that complete interdigitation, at which the free end of a chain grafted on a surface reaches the opposite surface, is reached at a separation L 0 , which represents the thickness of a single brush in thermodynamic equilibrium. The effect of the correlations between the nearest neighboring parallel bonds is important. However, it is shown that the calculations can be simplified by ignoring these correlations and by replacing the interaction parameter employed when the correlations are included with a value larger by 0.17.

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Nanodispersions

628 942

E. Ruckenstein and B. Li: Interactions between grafted brushes

The calculations for four PI chains of various molecular weights, from 3.883104 to 2.173105 , and six PS chains of various molecular weights, from 2.633104 to 3.753105 , are in good agreement with experiment for most systems. D. Napper, Polymeric Stabilization of Colloidal Dispersions ~Academic, London, 1983!. 2 A. Halperin, M. Tirrell, and T. P. Lodge, Adv. Polym. Sci. 100, 31 ~1992!. 3 G. Hadziioannou, S. Patel, S. Granick, and M. Tirrell, J. Am. Chem. Soc. 108, 2869 ~1986!. 4 H. J. Taunton, C. Toprakcioglu, L. J. Fellers, and J. Klein, Nature 332, 712 ~1988!. 5 H. J. Taunton, C. Toprakcioglu, L. J. Fetters, and J. Klein, Colloid Surf. 31, 151 ~1988!. 6 H. J. Taunton, C. Toprakcioglu, L. J. Fetters, and J. Klein, Polym. Prepr. 30, 368 ~1989!. 7 M. A. Ansarifar and P. F. Luckham, Polymer 29, 328 ~1988!. 8 J. Marra and M. L. Hair, Colloid Surf. 34, 215 ~1988!. 9 H. J. Taunton, C. Toprakcioglu, L. J. Fetters, and J. Klein, Macromolecules 23, 571 ~1990!. 10 H. Watanabe and M. Tirrell, Macromolecules 26, 6455 ~1993!. 11 S. T. Milner, Science 251, 905 ~1991!. 12 J. M. H. M. Scheutjens and G. J. Fleer, J. Phys. Chem. 83, 1619 ~1979!. 13 J. M. H. M. Scheutjens and G. J. Fleer, J. Phys. Chem. 84, 178 ~1980!. 14 F. A. M. Leermakers and J. M. H. M. Scheutjens, J. Chem. Phys. 89, 3264 ~1988!. 15 F. A. M. Leermakers and J. M. H. M. Scheutjens, J. Chem. Phys. 89, 6912 ~1988!. 16 T. Cosgrove, T. Heath, B. van Lent, F. A. M. Leermakers, and J. M. H. M. Scheujtens, Macromolecules 20, 1692 ~1987!. 17 C. M. Wijmans, F. A. M. Leermakers, and G. J. Fleer, J. Chem. Phys. 101, 8214 ~1994!. 18 C. M. Wijmans, J. M. H. M. Scheutjens, and E. B. Zhulina, Macromolecules 25, 2657 ~1992!. 1

G. J. Fleer and J. M. H. M. Scheutjens, Colloid Surf. 51, 281 ~1990!. C. M. Wijmans and E. B. Zhulina, Macromolecules 26, 7214 ~1993!. 21 C. M. Wijmans, F. A. M. Leermakers, and G. J. Fleer, Langmuir 10, 4514 ~1994!. 22 C. M. Wijmans and B. J. Factor, Macromolecules 29, 4406 ~1996!. 23 S. F. Edwards, Proc. Phys. Soc. 85, 613 ~1965!. 24 A. K. Dolan and S. F. Edwards, Proc. R. Soc. London, Ser. A 337, 509 ~1974!. 25 A. K. Dolan and S. F. Edwards, Proc. R. Soc. London, Ser. A 343, 427 ~1975!. 26 M. Muthukumar, and J.-S. Ho, Macromolecules 22, 965 ~1989!. 27 M. D. Whitmore and J. Noolandi, Macromolecules 23, 3321 ~1990!. 28 E. K. Lin and A. P. Gast, Macromolecules 29, 390 ~1996!. 29 E. B. Zhulina, O. V. Borisov, and V. A. Priamitsyn, J. Colloid Interface Sci. 137, 495 ~1990!. 30 E. B. Zhulina, O. V. Borisov, and L. Brombacher, Macromolecules 24, 4679 ~1991!. 31 S. T. Milner, T. A. Witten, and M. E. Cates, Macromolecules 21, 2610 ~1988!. 32 S. T. Milner, Europhys. Lett. 7, 695 ~1988!. 33 S. Alexander, J. Phys. ~Paris! 38, 983 ~1977!. 34 P.-G. de Gennes, Macromolecules 13, 1069 ~1980!. 35 P.-G. de Gennes, C. R. Acad. Sci. Paris 300, 839 ~1985!. 36 S. Patel, M. Tirrell, and G. Hadziioannou, Colloids Surf. 31, 157 ~1988!. 37 P. Pincus, Macromolecules 24, 2912 ~1991!. 38 M. Murat and G. S. Grest, Phys. Rev. Lett. 63, 1074 ~1989!. 39 M. Murat and G. S. Grest, Macromolecules 22, 4054 ~1989!. 40 A. Chakrabarti and R. Toral, Macromolecules 23, 2016 ~1990!. 41 P. Y. Lai and K. Binder, J. Chem. Phys. 95, 9288 ~1991!. 42 B. Li and E. Ruckenstein, J. Chem. Phys. 106, 280 ~1997!. 43 E. A. DiMarzio and R. J. Rubin, J. Chem. Phys. 55, 4318 ~1971!. 44 K. A. Dill and R. S. Cantor, Macromolecules 17, 380 ~1984!. 45 J. Bandrup and E. J. Immerght, Polymer Handbook, 3rd ed. ~Wiley, New York, 1989!. 19 20

J. Chem. Phys., Vol. 107, No. 3, 15 July 1997

Copyright ©2001. All Rights Reserved.

Polymer brushes

629

6490

Langmuir 2004, 20, 6490-6500

Simple Model for Grafted Polymer Brushes Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received January 23, 2004. In Final Form: April 29, 2004 The first theories of grafted polymer brushes assumed a step profile for the monomer density. Later, the real density profile was obtained from Monte Carlo or molecular dynamics simulations and calculated numerically using a self-consistent field theory. The analytical approximations of the solutions of the self-consistent field equations provided a parabolic dependence of the self-consistent field, which in turn led to a parabolic distribution for the monomer density in neutral brushes. As shown by numerical simulations, this model is not accurate for dense polymer brushes, with highly stretched polymers. In addition, the scaling laws obtained from the analytical approximations of the self-consistent field theory are identical to those derived from the earlier step-profile-approximation and predict a vanishing thickness of the brush at low graft densities, and a thickness exceeding the length of the polymer chains at high graft densities. Here a simple model is suggested to calculate the monomer density and the interaction between surfaces with grafted polymer brushes, based on an approximate calculation of the partition function of the polymer chains. The present model can be employed for both good and poor solvents, is compatible with a parabolic-like profile at moderate graft densities, and leads to an almost steplike density for highly stretched brushes. While the thickness of the brush depends strongly on solvent quality, it is a continuous function in the vicinity of the Θ temperature. In good and moderately poor solvents, the interactions between surfaces with grafted polymer brushes are always repulsive, whereas in poor solvents the interactions are repulsive at small separations and become attractive at intermediate separation distances, in agreement with experiment. At large separations, a very weak repulsion is predicted.

1. Introduction One of the first important applications of polymer brushes was to the stabilization of colloidal dispersions.1 The brushes were generated through the grafting of polymers on the surface of colloidal particles. When two particles approach each other, the overlap of their polymer brushes generates a steric repulsion, which can prevent their coagulation. A simple theory for polymer brushes was proposed by Alexander,2 who assumed that the grafted polymer is composed of a sequence of blobs. The sectional area of each blob was provided by the area occupied by each grafted polymer on the surface, and it was assumed that, inside the blobs, the relation between the chain length and the occupied volume is the same as for a free polymer in a solvent. The repulsion due to the overlap of two polymer brushes was subsequently evaluated by de Gennes3 up to an arbitrary proportionality constant, from scaling considerations. The simple Alexander-de Gennes theory, which assumed a steplike monomer density in the brush, captured the dependence of the interaction on the physical parameters (length of the polymer, density of grafting, quality of the solvent) and provided a satisfactory approximation for the calculation of the steric repulsion. However, new applications of grafted polymers on surfaces, such as the control of the catalytic selectivities of some chemical reactions by varying the thickness of a brush,4 the prevention of the adsorption of proteins on surfaces (a condition required for biocompatibility),5 or the control of * Corresponding author. E-mail: [email protected]. Telephone: (716) 645 2911/2214. Fax: (716) 645 3822 (1) Napper, D. H. Polymeric stabilization of colloidal dispersion; Academic Press: San Diego, CA, 1983. (2) Alexander, S. J. Phys. (Paris) 1977, 38, 983. (3) de Gennes, P. G. Adv. Colloid Interface Sci. 1987, 27, 189. (4) Ruckenstein, E.; Hong, L. J. Catal. 1992, 136, 178. (5) Carignano, M. A.; Szleifer, I. Colloids Surf. B 2000, 18, 169.

a membrane selectivity by changing the pH of the solution6 required a more accurate theory of polymer brushes. The properties of grafted polymer brushes could be determined accurately via the traditional Monte Carlo7 or molecular dynamics8 simulations. Since one should account not only for the monomer-monomer interactions but also for the interactions with the solvent molecules, in general only calculations for systems of relatively small size could be performed. A modality to increase the calculation efficiency is to employ simplifying assumptions; one of the most successful was proposed by Dolan and Edwards,9 who considered that the monomers of the grafted polymer are distributed according to a random walk in an external potential. This potential is generated by monomer-monomer and monomer-solvent interactions and depends on the local monomer concentration. Since the monomer concentration in the brush depends on the potential, which in turn depends on the monomer concentration, the potential can be determined only selfconsistently via successive approximations. A convenient modality to implement self-consistent calculations10 is based on a lattice model due to Scheutjens and Fleer.11 The lattice-based self-consistent field theory was later improved to account for anisotropy, bond correlations, chain stiffness, and intermolecular interactions.12-14 (6) Idol, W. K.; Anderson, J. L. J. Membr. Sci. 1986, 28, 269. (7) Muthukumar, M.; Ho, J.-S. Macromolecules 1989, 22, 965. (8) Seidel, C. Macromolecules 2003, 36, 2536. (9) Dolan, A. K.; Edwards, F. R. S. Proc. Royal Soc. London A 1974, 337, 509. (10) Hirz, S. Modeling of Interactions Between Adsorbed Block Copolymers. M.S. Thesis, University of Minnessota, Minneapolis, MN, 1988. Cosgrove, T.; Heath, T.; van Lent, B.; Leermakers, F.; Scheutjens, J. Macromolecules 1987, 20, 1692. (11) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619 and 1980, 84, 178. (12) Leermakers, F. A. M.; Scheutjens, J. M. H. M. J. Chem. Phys. 1988, 89, 6912. (13) Cantor, R. S.; Mcllroy, P. M. J. Chem. Phys. 1989, 91, 416.

10.1021/la049781y CCC: $27.50 © 2004 American Chemical Society Published on Web 06/16/2004

Nanodispersions

630 Simple Model for Grafted Polymer Brushes

Both the Monte Carlo and molecular dynamics simulations, as well as the numerical solutions of the selfconsistent field (SCF) equations, are time-expensive and do not offer a clear insight regarding the relations between the parameters of the grafted polymers and the properties of the brushes. A major progress in the analytical approximation of the SCF solution was suggested by Milner et al. (the MWC theory),15 based on the observation that, at high stretching, the partition function of the brush is dominated by the “classical path” (the most probable distribution). This approach is similar to the semiclassical approach of quantum mechanics, the Wentzel-KramersBrillouin (WKB) approximation. Under these assumptions, it turned out that, for equal-length polymer chains, the self-consistent field is parabolic, which leads to a parabolic distribution of the monomer density for neutral brushes. The simplicity of this result became extremely appealing, and most subsequent theories of the polyelectrolyte brushes (such as those of Miklavic and Marcelja,16 Misra et al.,17 and Zhulina and Borisov,18 to name only a few of them) were based on the “parabolic field” approximation. A more intuitive model to describe the parabolic distribution of monomers in grafted polymer brushes was proposed by Pincus,19 who considered that the profile of the brush is a result of the competition between an excluded-volume free energy and the decrease of the entropy of the polymer in the stretched configuration. Assuming that the excluded volume interactions are proportional to the square of the local monomer concentration and also a Gaussian elasticity of the polymer, this model led also to a parabolic distribution of the monomers in the brush.19 It should be noted that the “parabolic field” was derived by assuming a high stretching of the polymers; it provides, however, a better description of polymer brushes in the intermediate-stretching regime (good solvents and moderate graft densities). In contrast, the strongly stretched brushes, as obtained from molecular dynamics simulations,8 seem to be better described by the Alexander step profile. As a matter of fact, the analytical approximations of the self-consistent field theory provided the same scaling laws for the brush thickness as those derived earlier by Alexander in the framework of the step-profile approximation.19 These laws clearly fail at both very low graft densities (by predicting a vanishing small thickness of the brush) as well as at very high graft densities (by predicting a brush thickness larger than the length of the grafted polymer).20 The limitation of the parabolic profile description only to brushes with moderate graft densities immersed in good solvents was also recognized by Shim and Cates, who proposed an ad hoc functional form for the free energy of stretching of the chain and consequently obtained a density profile much flatter than a parabola.21 Because the neutral polymer brushes are unlikely to be very strongly stretched, the MWC approach usually provides in this case an accurate approximation. The situation is however different for the highly charged grafted polyelectrolytes, which are almost completely (14) Li, B. Q.; Ruckenstein, E. J. Chem. Phys. 1997, 106, 280. Ruckenstein, E.; Li, B. Q. J. Chem. Phys. 1997, 107, 3. (15) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules, 1988, 21, 2610. (16) Miklavic, S. J.; Marcelja, S. J. Phys. Chem. 1988, 92, 6718. (17) Misra, S.; Varanasi, S.; Varanasi, P. P. Macromolecules 1989, 22, 4173. (18) Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107, 5952. (19) Pincus, P. Macromolecules 1991, 24, 2912. (20) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (21) Shim, D. F. K.; Cates, M. E. J. Phys. (Paris) 1989, 50, 3535.

Langmuir, Vol. 20, No. 15, 2004

6491

stretched.8,22 The purpose of this article is to present a simple method to calculate the thickness of the strongly stretched grafted polymer brushes and the interactions between surfaces covered by strongly stretched brushes. The calculations are based on an approximate Monte Carlo method, involving the assumptions that the configurations of minimum free energy are dominating the partition function and that the probability that the farthest monomer from the surface located at the distance z is approximated by the probability that the last monomer of the chain reaches the same distance. Under this hypothesis, in good solvents, the model predicts a constant density of monomers for each configuration that ends up at distance z, which is similar to the approximation successfully employed by Flory to explain the scaling laws of the free polymers in good solvents.23 2. The Profile of Neutral Polymer Brushes The traditional Monte Carlo method implies the generation of all possible configurations of grafted polymers and the calculation of the following partition function for a polymer

Z)

( ) Ui

∑i exp -kT

(1)

where k is the Boltzmann constant, T the absolute temperature, “i” runs over all possible configurations and Ui represents the total energy of a configuration, which includes the interactions between the monomers of the same and neighboring polymer chains and the interactions between the monomers and the grafting surface as well as their interactions with the solvent molecules. A usual modality to generate the configurations of a polymer composed of N monomers of length “a” is to assume that the polymer is composed of Nl independent pieces of persistence length “l” (where lNl ≡ aN), that are connected but can assume any relative orientations. This model constitutes the statistical equivalent of a random walk. Since the length of the grafted polymers perpendicular to the surface is much larger than their size parallel to the surface, one can assume that the in-plane monomer concentration is constant. In the absence of any monomer interactions, the probability ψ′(z) for the last monomer of the chain to reach the position z is well approximated (for large N) by the analytical solution of a noninteracting random walk with a perfect-reflecting wall at the origin (see Appendix)24

ψ′(z) =

x

( )

2 z2 exp - 2 2 πσ 2σ

0 < z < Na

(2a)

where σ ) lxNl is the root-mean square distance of the random walk of Nl steps of length l. When additional interactions are present, the probability for a configuration to end at a distance z is multiplied by its corresponding Boltzmann factor

( ) ( )

ψ(z) ∝ exp -

z2

2σ

Ui(z)

∑i exp - kT 2

(2b)

where the sum is performed over all possible configurations (22) Huang, H.; Ruckenstein, E. J. Colloid Interface Sci. 2004, in press. (23) Flory, P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1971. (24) Chandrasekhar, S. Rev. Mod. Phys. 1943, 15, 3.

Polymer brushes

631

6492 Langmuir, Vol. 20, No. 15, 2004

Manciu and Ruckenstein

“i” whose last monomer of the chain is located at z and have the total free energies Ui(z). The random walk analogy was employed in the derivation of eq 2b only to provide the number of all possible configurations that end up at the distance z. The main difficulty is to calculate the free energies for each of the possible configurations. A first approximation, the mean field approach, consist in assuming that the total free energy density of a configuration (which includes all the interactions between monomers and solvent molecules as well as the chain entropy of mixing) can be described in term of the local monomer volume fraction φ by the FloryHuggins mixing free energy density20,23

kT φ ln(φ) + (1 - φ) ln(1 - φ) + χφ(1 a3 N kT 1 1 1 φ) = 3 (χ - 1)φ + - χ φ2 + φ3 + φ4... (3a) 2 6 12 a

FFH(z) )

)

( (

(

)

)

where a is the length of a monomer, χ is the Flory interaction parameter,20,23 and the term proportional to 1/N was neglected because N is large. The term linear in φ, while nonnegligible for the mixing free energy density eq 3a, is not important for the total Flory-Huggins mixing free energy of the configuration, since its integral over the volume of the brush

s

∫0

2

Na

(

)

Na kT kTs2 (χ 1)φ(ζ) dζ ) (χ - 1) 0 φ(ζ) dζ ) 3 3 a a kT (χ - 1)Nv ) const (3b) a3

∫

provides (see eq 4b) the same constant for each possible configuration, and the same additive constant in the free energy of each configuration does not modify the result. For this reason, a convenient approximation of the FloryHuggins free energy density is given by20,25

FFH(φ) =

kT 1 2 1 3 - τφ + wφ 6 a3 2

(

)

(3c)

where τ is a dimensionless excluded volume parameter (τ ) (2χ - 1) with τ < 0 for a good solvent and τ > 0 for a poor solvent) and w is the third virial coefficient, which is positive and typically on the order of unity.25 The latter coefficient should roughly account for the higher order terms in eq 3a (all of them being positive), as well as for the additional interactions between monomers, not included in the traditional Flory-Huggins mixing free energy density eq 3a. The second approximation employed here is to consider that the configurations of the minimum energy provide most of the contributions to the average Boltzmann factor in eq 2b, the well-known saddle point approximation of statistical mechanics. For a good solvent, the Flory-Huggins free energy density is minimized by the lowest possible volume fraction of monomers, which is in turn obtained for a completely stretched chain. In the stretched configuration that ends up at the distance z < Na a random walker moves away from the surface until reaches a distance z′ > z and then returns to z (such as the length of the path is z′ + (z′ z) ) Na; hence, z′ ) (Na + z)/2. However, there is only one such configuration among the 2CNk ) 2N!/k!(N - k)! configurations that ends up at z ) ka, where the factor 2 multiplying the binomial coefficient accounts for the (25) Ross, R. S.; Pincus, P. Europhys. Lett. 1992, 19, 79.

reflecting wall at the origin. While this particular configuration has the highest Boltzmann factor, its contribution to the sum in eq 2b is in most cases not significant, because of its low statistical weight. In what follows, a third approximation will be employed, namely that we will take into account only the configurations for which the last monomer of the chain is also the farthest from the surface. As will be shown below, under these assumptions, the minimum Flory-Huggins free energy in good solvents is provided by the configurations with a constant distribution of monomers. Such a distribution was successfully employed by Flory to derive a relation between the rootmean-square distance between the ends and the number of monomers of a polymer in good solvents,23 and it provided the correct value for the scaling coefficient in one dimension and values within a percent of the most accurate numerical results in 2 and 3 dimensions.20 The Flory-Huggins free energy of a configuration for which the monomers are confined between 0 and z is given by

UFH(z) ) s2

2

z τ - φ2(ζ) + ∫0z FFH(ζ) dζ ) s akT 3 ∫0 ( 2

w 3 φ (ζ) dζ (4a) 6

)

where the monomer volume fraction φ obeys the normalization condition

∫0 φ(ζ) dζ ) Nv s2 z

(4b)

where N is the number of monomers of volume v in a chain, s2 is the area corresponding to a grafted chain on the surface, and Na is the maximum length of a polymer. The function φ(z) which minimizes the Flory-Huggins free energy (eq 4a) with the constraint (eq 4b) is provided by the extremum of the functional 2

τ w - φ2(ζ) + φ3(ζ) + λφ(ζ)) dζ ∫0 J (φ) dζ ) s akT 3 ∫0 ( 2 6 z

z

(5a)

where λ is a Lagrange multiplier. The corresponding Euler-Lagrange equation

∂ ∂J ∂J )0 ∂ζ ∂φ ∂φ ∂ ∂ζ

( )

(5b)

leads to

- τφ +

w 2 φ +λ)0 2

(5c)

The Euler-Lagrange equation (eq 5c) does not depend on ζ, and the continuous solution is a constant obtained from the constraint (4b):

φ(ζ,z) )

Nv s2z

(5d)

While (5d) is the only continuous solution that is an extremal of the functional (5a), a discontinuous solution, which corresponds to the separation into two homogeneous phases, might lead to a lower value of the free energy. Therefore, we will consider configurations with volume fraction φa on a domain (not necessarily connect) of length za < z/2 and with volume fraction φb on the remaining

Nanodispersions

632 Simple Model for Grafted Polymer Brushes

Langmuir, Vol. 20, No. 15, 2004

domain of length zb ) z - za. The restriction that za < z/2 does not reduce the generality, since the other case is obtained by interchanging the subscripts a and b. The Flory-Huggins free energy of such a configuration is given by

(( ) (( ) (

s2kT τ w U(z,φa,za) ) 3 za - φa2 + φa3 + (z - za) × 2 6 a 2 3 Nv Nv - φaza - φa z a 2 2 τ s w s + 2 z - za 6 z - za

))

Nv , za ) z, which is equivalent to eq 5d s2z (7a)

2τ 2τ Nv Nv 2τ z - za - 2 - za w w w s s2 (ii) φa ) , φb ) z - 2za z - 2za

(7b)

Introducing the pair of solutions (eq 7b) in eq 6, the free energy becomes a function of za, with its derivative with respect to za given by

(

)

Nv w - τz ∂U(z,za) 2 s2 ) ∂za 3w2 z - 2za

3

(8)

For τ < 0 (good solvent), the derivative is always positive; hence, the minimum corresponds to za ) 0, which leads to zb ) z and φb ) Nv/s2z, which is the same as eq 5d. For τ > 0 (poor solvents), the derivative is positive for z smaller than

1 Na - z 2 τ s2a -1 w v

φa ) φmax )

2τ v (on a domain of length za,max) w s2a (10c)

φb ) φmin )

v (on a domain of length z - za,max) s2a (10d)

Parts c and d of eq 10 simply state that the system separates into two phases, if the solvent is poor (τ > 0) and the confinement is weak (z > zmin ) (w/τ) Nv/s2), but not when the confinement is strong (z < zmin). While the diluted polymer in the bulk solvent undergoes a phase separation as soon as τ becomes positive (at the Θ temperature), only the configurations of the grafted polymer that are sufficiently long undergo a phase separation. This occurs because at high volume fractions, the excluded volume interactions (due to the positive third virial coefficient w) dominate the Flory-Huggins free energy and prevent the polymer collapse. If the grafting density is sufficiently high (small s) in eq 9, the minimum length zmin required for phase transition might exceed the maximum length of the polymer Na regardless of the value τ, therefore none of the physically allowed chain configurations undergoes phase separation. In summary, the configurations of minimum FloryHuggins free energy for monomers distributed between 0 and z correspond to the following. (i) Good solvents (τ < 0) and 0 < z < Na:

φ(ζ,z) )

(9)

and negative for z > zmin. In the first case (z < zmin, poor solvent), the minimum free energy is obtained again for za ) 0 (a constant distribution of monomers between 0 and z) while in the latter case (z > zmin, poor solvent), the Flory-Huggins free energy is a monotonically decreasing function of za. However, the volume fraction φb in eq 7b, which also decreases with increasing za, cannot become smaller than that corresponding to a completely stretched chain:

φmin )

Nv v ) 2 2 s Na s a

The maximum value physically allowed for za, obtained from the condition φb ) φmin,

Nv 2τ - za w s2 v ) 2 z - 2za sa is given by

(10a)

Nv s2z

(5d)

(ii) Poor solvents (τ>0) and 0 < z
φ(ζ,z) ) w Nv zmin ) τ s2

(10b)

which employed in eq 7b leads to the volume fractions:

(6)

where the normalization eq 4b (zaφa + (z - za)φb ) Nv/s2) was taken into account. The extremal of U(z,φa, za) with respect to φa leads to two pair of solutions:

(i) φa )

za,max )

6493

Nv s2z

(5d)

(iii) Poor solvents (τ>0) and zmin ) (Nv/s2)(w/τ) < z < Na:

φ(ζ,z) ) φmax )

2τ v on an interval of length w s2a 1 Na - z za,max ) (10c) 2 τ s2 a -1 w v

φ(ζ,z) ) φmin )

v on an interval of length z-za,max s2a (10d)

Therefore, the minimum total Flory-Huggins energy of a chain configuration confined between 0 and z is given by the following. (i) Good solvent in the range 0 < z < Na or poor solvent in the range 0 < z < zmin:

UFH(z) ) s2

(

2 2

τN v ∫0z FFH(ζ) dζ ) kT a3 2s2z

+

)

wN3v3 6s4z2 (11a)

(ii) Poor solvent in the range zmin< z< Na:

Polymer brushes

633

6494 Langmuir, Vol. 20, No. 15, 2004

UFH(z) )

Manciu and Ruckenstein

kTs2 τ w za,max - φmax2 + φmax3 + 3 2 6 a τ w (z - zamax) - φmin2 + φmin3 2 6

(

(

)

(

)) (11b)

The probability for the last monomer of the chain to reach point z is consequently obtained from eq 2b:

ψ(z) )

( ) (

)

UFH(z) 1 z2 exp - 2 exp , 0 < z < Na C1 kT 2σ (12a)

where σ ) lxNl is the root-mean-square length of the random walk of persistence length l and C1 is a normalization constant which obeys

∫0

Na

ψ(z) dz ) 1

(12b)

The monomer volume fraction in the brush Φ(z) is the statistical average of the contributions of configurations ending up at different positions. In poor solvents, the configurations with z > zmin that minimize the FloryHuggins energy do not have a constant monomer density. However, the monomers are equally like to accumulate at a volume fraction φmax in any region between 0 and z; therefore, an average over all the possible configurations (of the same minimum energy) would lead to a constant monomer density between 0 and z. For z < zmin (in poor solvents) or for good solvents, the configuration of minimum energy corresponds to a constant distribution of monomers between 0 and z, φ(ζ,z) ) Nv/s2z for 0 < ζ z. Consequently, the chains that end up at distances smaller than z do not contribute to the volume fraction of the monomers at z, whereas the chains which end up at distances larger than z contribute φ(ζ,z) ) Nv/s2z with the probability ψ(z) to the total monomer fraction Φ(z). Therefore, the monomer volume fraction is given by

Φ(z) )

∫zNa ψ(ζ)φ(ζ,ξ) dξ )

1 C1

1 C1

( ) ( 2

ξ exp - 2 ∫zNa Nv 2 sξ 2σ

exp -

)

UFH(ξ) dξ (13) kT

The results will be illustrated through this paper with model calculations for a grafted brush with the following values of the parameters: s ) 10 Å, a ) 1 Å, N ) 1000, l ) 10, v ) 1 Å3. In parts a and b of Figure 1, the probability for the last monomer of the chain to be at z (Figure 1a) and the local volume fraction of the monomers in the brush (Figure 1b) are plotted as a function of the distance from the surface for good solvents (τ < 0). In the absence of interactions between monomers τ ) w ≡ 0, ψ(z) is Gaussian and the profile of the brush can be roughly approximated by a parabola. The brush is swollen because of the repulsive Flory-Huggins interactions, which increase with increasing volume fraction φ. At low graft densities (large values of s, hence low values of φ) or for moderately good solvents (small absolute value of τ and w) the brush profile can be still approximated by a parabola. However, by increasing the graft density (small s) or solvent quality (large absolute value of τ) a strongly stretched brush is generated, with an almost constant density profile over most of the brush, but with an exponential-like tail, in agreement with the numerical simulations.8 The most likely chain configuration in the brush is obtained from the condition of vanishing of the derivative of ψ(z) with respect to z in eq

Figure 1. (a) Probability ψ(z) for the last monomer of a grafted chain to be located at the distance z from the surface and (b) the local volume fraction of monomers in a good solvent, for s ) 10 Å, a ) 1 Å, N ) 1000, l ) 10 Å, v ) 1 Å3, and various solvent qualities.

12a. Neglecting the second term in the Flory-Huggins interaction (w ) 0) leads to

(

zj ) -

τl2NlN2v2 2s2a3

)

1/3

(14)

If the length of the most probable chain configuration zj is assumed to be equal to the thickness of the brush L, eq 14 provides the same dependence between the graft density and thickness as the Alexander theory. Whereas eq 14 accounts explicitly for the persistence length of a polymer (its degree of flexibility), with the simplified assumptions that l ∼ a, Nl ∼ N, v ∼ a3 the Alexander scaling law is completely recovered:

( )

L∝N

τva2 2s2

1/3

(15)

Here τv represents an “excluded volume” of the interaction.19 However, as shown in Figure 1a, the approximation of the thickness of the brush by the length of the most probable configuration is in many cases arbitrary, since configurations that end up at different distances can contribute notably to the brush thickness. In addition, the derivation of eq 14 implies that the random walk can be described by a Gaussian extended from z ) 0 to z ) ∞,

Nanodispersions

634 Simple Model for Grafted Polymer Brushes

while the random walk should not exceed the maximum chain length Na. Equations 12 and 13 do account for this inconsistency (by a cutoff in the integral limits) and do not predict a brush thickness that exceeds Na. Another feature of the present model is that whereas at vanishing Flory-Huggins interactions (τ ) w ) 0) the most probable configuration ends up at z ) 0, the brush does not collapse on the surface, as suggested by the scaling laws (eqs 14 or 15), because the configurations extended to z > 0 have nonvanishing probabilities. The present model is based on several assumptions: (i) the possible configurations of the grafted chain are described by a random walk; (ii) their free energy densities are expressed as functions of the local monomer volume fraction alone; (iii) the configurations of minimum energy dominate the partition function of the system; (iv) only the configurations with monomers distributed between the surface and the position of the last monomer of the chain, assumed to be the farthest one, are taken into account. The latter assumption basically implies that the probability that the most distant monomer from the surface reaches the distance z is equal to the probability that the last monomer of the chain reaches this distance; this approximation clearly fails when z is in the vicinity of the surface. However, in swollen brushes the behavior of the monomers in the vicinity of the surface is less important than the behavior of the distant monomers, which are primarily responsible for the brush thickness and for the interactions between brushes. As a matter of fact, in good solvents, the above assumptions predict that only configurations with constant monomer density between 0 and z contribute essentially to the partition function. This is equivalent to the Flory model for polymer swelling in a good solvent, which was remarkably successful, especially for one dimension.20,23 It is therefore to be expected that the model is reasonably accurate in good solvents and fails in extremely poor solvents, because the configurations in the vicinity of the surface dominate the collapsed brush. It should be however noted that the sharp distinction between good and poor solvent at the Θ temperature occurs only at high dilutions, whereas for high volume fractions the Flory-Huggins free energy is dominated by the volume-exclusion interaction between monomers and increases with φ even for poor solvents. Therefore, at high grafting densities, the brush might be swollen even at temperatures below Θ; hence, the present model is applicable. The probability ψ(z) that the last monomer of the chain is located at the distance z, and the volume fraction Φ(z) of the monomers in the grafted brush immersed in a moderately poor solvent are plotted in Figure 2, parts a and b, using for the parameters the values selected above (s ) 10 Å, a ) 1 Å, N ) 1000, l ) 10, v ) 1 Å3), a fixed value for the third virial coefficient w ) 1 and various values for the interacting parameter τ. It is of interest to note that the brush thickness is modified continuously when the quality of the solvent changes from good (τ < 0) to poor (τ > 0). This occurs because the Flory-Huggins free energy is minimized by a constant monomer distribution in both good solvents and poor solvent for configurations with z < zmin ) Nv/s2 w/τ. For small values of τ, zmin (for which phase separation occurs) exceeds the maximum length of the chain Na. Therefore, in these cases, the polymer behaves as immersed in a good solvent, even below the Θ temperature. This uniform collapse of the brushes induced by changing the solvent quality is in agreement with the results reported by Ross and Pincus.25 The most probable configurations in Θ solvents (χ ) 1/2, τ ) 0), obtained from the extremum of eq 12, occurs at a

Langmuir, Vol. 20, No. 15, 2004

6495

Figure 2. (a) Probability ψ(z) for the last monomer of a grafted chain to be located at the distance z from the surface and (b) the local volume fraction of monomers in a poor solvent, for s ) 10 Å, a ) 1 Å, N ) 1000, l ) 10 Å, v ) 1 Å3, and various solvent qualities.

distance given by

zj )

(

wN3Nlv3l2 3a3s4

)

1/4

(16)

which differs from the Alexander scaling law, eq 15. In Figure 3, the length of the most probable chain configuration (which roughly accounts for the thickness of the brush), is plotted as a function of the quality of the solvent τ for various values of the third virial coefficient w. For small values of the third virial coefficient, there is a large variation in the thickness of the brush (albeit continuous) in the vicinity of the Θ temperature, as observed experimentally.26 3. The Interaction between Grafted Neutral Brushes A. Good Solvents. When two surfaces with grafted polymer brushes approach each other, the overlap of the neutral brushes generates an interaction between surfaces. In good solvents, the Flory-Huggins mixing free energy density (eq 3c) increases with the monomer concentration; therefore, one expects that the overlap of the brushes would lead always to repulsion. In the present (26) Kilbey, S. M.; Watanabe, H.; Tirrell, M. Macromolecules 2001, 34, 5249.

Polymer brushes

635

6496 Langmuir, Vol. 20, No. 15, 2004

Manciu and Ruckenstein

ψ(z) )

( ) (

)

UFH(z) 1 z2 exp - 2 exp C2 kT σ

(19a)

The Flory-Huggins free energy for the nonoverlapping configurations (0 < z < d) is given by

UFH(z) ) 2z

( ( ) ( ))

kTs2 τ Nv 3 2 a s2z

2

+

w Nv 6 s2z

3

0
which is the sum of the energies of the independent configurations that end up at z1 ) z2 ) z, while for z > d

( ( ) ( ))

kTs2 τ Nv UFH(z) ) 2d 3 2 a s2 d Figure 3. Length L ≡ zj of the most probable chain configuration as a function of solvent quality for various values of the third virial coefficient w.

model, the profile of each brush depends not only on the physical properties of the individual brushes (number of monomers N, monomer length a, monomer volume v, persistence length l, excluded volume parameter τ, third virial coefficient w, grafting area s2), but also on the distance 2d between surfaces. The probability for a chain grafted on surface “1” to end up at the distance z1 from the first surface, whereas a chain which starts on surface “2” to end up at the position z2 from the surface “2” is given by (see Appendix):

ψ(z1,z2) )

1

∑

ψ1′(z1)ψ2′(z2) C2 i,j

(

)

UFHi,j(z1,z2) exp kT

(17a)

where the sum is performed over all possible configurations which end up at z1 and z2, respectively, ψk′(zk) (k ) 1, 2) represents the probability that a random walk, constrained by two reflecting walls at 0 and 2d, which starts on the surface “k” and ends up at the distance zk from that surface (see Appendix), and the normalization constant C2 is obtained by taking into account that both chains should end up somewhere between 0 and 2d:

∫0 ∫0 2d

2d

ψ(z1,z2) dz1 dz2 ) 1

(17b)

Assuming, as previously, that the Boltzmann factor of a configuration is dominated by the configuration corresponding to a minimum energy UFH(z1,z2) with the constraints that all the monomers of the first chain are distributed between the first surface and z1 while all the monomers of the second chain are distributed between the second surface and z2, the total free energy of the overlapping brushes, per unit area, is the statistical average over all possible configurations (see Appendix)

F(2d) )

1 s2

∫02d ∫02d ψ(z1,z2)UFH(z1,z2) dz1 dz2

(18)

and the force per unit area is provided by its derivative with respect to the distance 2d. The calculations are considerable simplified if only the symmetric configurations (z1 ) z2) would be taken into account and the paths that reach z after reflections on the opposite walls would be ignored. In this case, eq 17a becomes

2

+

w Nv 6 s2d

3

d < z < 2d (19c)

which implies that, for two overlapping configurations, the minimum Flory-Huggins free energy in good solvents is obtained for a constant distribution of monomers between 0 and 2d. The free energy per unit area of the system is given by

F(2d) )

1 s2

∫0

2d

ψ(z)UFH(z) dz

(19d)

and the force per unit area

Π(2d) ) -

∂F(2d) ∂(2d)

(19e)

In Figure 4a, the force per unit area between two identical surfaces with grafted polymer brushes immersed in good solvents, with the parameters selected as above (s ) 10 Å, a ) 1 Å, N ) 1000, l ) 10 Å, v ) 1 Å3) is calculated for various solvent qualities using all possible configurations (continuous line) as well as the simplified version which accounts only for the symmetric configurations. The ratio between the forces calculated in the two different manners, represented in Figure 4b, shows that the simpler calculation is fairly accurate for high overlaps (strong interactions). B. Poor Solvents. When the exponential tails of two brushes immersed in a poor solvent overlap, one might expect the local increase in the monomer volume fraction to lead to a more negative Flory-Huggins free energy and therefore to a lower total free energy of the system, which corresponds to an attraction between surfaces. However, as will be shown below, this is not always true. Earlier experiments on polystyrene adsorbed on mica immersed in cyclohexane showed that below the Θ temperature the force was attractive for large separations between the mica plates, but became repulsive for small separations.27 However, it was found, for the same system, that the attraction persisted even for temperatures higher than Θ. For this reason, its origin was attributed to the formation of polymer bridges between the surfaces.28 Later, another experiment, in which the bridging was essentially eliminated, has shown that at large separations the force was attractive for T < Θ and repulsive for T > Θ.29 First, let us note that the attractive interaction could not be predicted by theories which did not consider the interpenetration of the chains from opposing surfaces but (27) Klein, J. Nature 1980, 288, 248. (28) Israelachvili, J. N.; Tirell, M.; Klein, J.; Almog, Y. Macromolecules 1984, 17, 204. (29) Hadziioannou, G.; Patel, S.; Granick, S.; Tirell, M. J. Am. Chem. Soc. 1986, 108, 2869.

Nanodispersions

636 Simple Model for Grafted Polymer Brushes

Figure 4. (a) Force per unit area between two identical surfaces with grafted polymers, separated by a distance 2d for s ) 10 Å, a ) 1 Å, N ) 1000, l ) 10 Å, v ) 1 Å3 in a good solvent, calculated using all possible chain configurations, or using only the symmetric configurations. (b) Ratio between the two forces, showing that the restriction to symmetric configurations is a good approximation, particularly at large interactions.

assumed instead that each brush was distributed between the surface and the middle distance.3 Indeed, in the latter model, the total free energy of the compressed brush should be higher than that of the free brush; otherwise, the brush would have been adopted spontaneously a compressed configuration, even in the absence of the other surface. Therefore, the interactions between surfaces with noninterpenetrating brushes were always repulsive; in those cases, the attraction could be explained only if additional interactions (such as those due to polymer bridging) were taken into account. Let us now analyze how the interactions between brushes occur in the present model. Our system is described by the statistical average of all possible configurations of chains that start from opposite surfaces and end up at the distances z1 and z2, respectively. For each possible path that is not reflected by the opposite wall, it is assumed that the monomers of the chains grafted on surfaces “1” and “2” are distributed between 0 and z1 and between z2 and 2d, respectively. The paths that are reflected by the opposite walls have the monomers distributed everywhere between 0 and 2d. Only the configurations that minimize the Flory-Huggins free energy are taken into account. Since a complete evaluation of all possible combinations is quite tedious, we will focus

Langmuir, Vol. 20, No. 15, 2004

6497

here only on the symmetric configurations, neglecting also the paths reflected on the opposite walls. As in the previous section, the limitation to symmetric configurations does not affect the results qualitatively. For the highly confined configurations that end up at distances smaller than zmin ) (w/τ)(Nv/s2) in a poor solvent, the minimum Flory-Huggins energy is obtained for an uniform (constant) monomer distribution. These configurations behave as immersed in a good solvent even below the Θ temperature (the monomer does not separate from solvent). For the confined configurations, with monomer volume fractions exceeding φmax ) 2τ/w, corresponding to a separation 2dmin ) (w/τ)(Nv/s2), the increase of the local volume fraction due to the overlap of the brushes leads to an increase of the Flory-Huggins mixing free energy. These unfavorable interactions (dominated by the excluded-volume repulsion) are responsible for the strong, hard-wall-like repulsion, which occurs always at high overlaps (2d < 2dmin), regardless of the quality of the solvent. It is of interest to note that below (but in the vicinity of) the Θ temperature, the minimum confinement distance for which the overlap reduces the free energy might exceed the maximum length of the chains, 2dmin > 2Na. Because for 2d > 2Na there are no interactions between brushes, since the chains cannot overlap, the interactions occur in this case for 2d < 2Na < 2dmin and hence are repulsive. Therefore, the brushes repel each other at all separations, as if they would have been immersed in a good solvent. This might explain the repulsion (and no attraction) between brushes observed recently in near-Θ solvents.26 When the separation between surfaces exceeds 2dmin, the Flory-Huggins mixing free energy decreases for all overlapping (z > d) configurations. Using the same procedure as in section 2, one can show that the FloryHuggins free energy of a configuration whose chains end at z (with z > d) is minimized by a constant monomer distribution for 2d < 2zmin ) 2(w/τ)(Nv/s2) and for 2d > 2zmin by two constant monomer distributions

2τ 2τ Nv 2d - za - 2 2 w w s φa ) z - 2za (over a domain of length za) (20a) Nv 2τ - za w s2 φb ) z - 2za (over a domain of length zb ) z - za) (20b) 2

and the condition φb ) φmin ) v/s2a leads to the solution

z2d )

1 2Na - 2d 1 2Na - 2z >2za,max ) (20c) 2 τ s2 a 2 τ s2a -1 -1 w v w v

The values of φmax and φmin,obtained by introducing eq 20c in eq 20, parts a and b, are the same as those in section 2

φa ) φmax )

2τ v (over a domain of length z2d) w s2a (20d)

φb ) φmin )

v (over a domain of length z-z2d) s2 a (20e)

Polymer brushes 6498 Langmuir, Vol. 20, No. 15, 2004

637 Manciu and Ruckenstein

Figure 6. (a) Force per unit area between two identical surfaces with grafted polymers, separated by a distance 2d for s ) 10 Å, a ) 1 Å, N ) 1000, l ) 10 Å, and v ) 1 Å3 in a poor solvent, calculated using only the symmetric configurations. At small separations, there is a strong repulsion due to excluded-volume interactions (carried by the third virial coefficient). At intermediate separations there is an attractive region due to the decrease of the free energy of configurations due to the chain overlap. At large separations, the swelling of the brush due to chain overlap (the increase of ψ(z) at large z) favors configurations with high Flory-Huggins mixing free energies, thus increasing the total energy of the system and therefore generating a repulsion. The absolute value of this repulsion is, however, smaller by orders of magnitude than the attraction that occurs at intermediate separations and smaller by many orders of magnitude than the strong repulsion that occurs at small separations.

Figure 5. (a) Probability ψ(z) for a chain to end up at distance z in a poor solvent differs in noninteracting brushes (2d ) 2Na ) 2000 Å) and overlapping brushes (2d ) 60 Å). The swelling of brushes in poor solvents due to overlapping increases ψ(z) at large z on the expense of decreasing ψ(z) at moderate z. Because the former configurations have higher Flory-Huggins mixing free energies, they might increase the total free energy of the system. (b) Function ψ(z)UFH(z), the integral of which provides the total free energy. The balance between its increase (in absolute value) for large values of z and its decrease for small values of z provides the sign of the interactions between surfaces.

In this case, the energy of the overlapping configurations decreases because the length of the interval over which the monomers accumulate to the energetically favorable φmax value increases. The separation of the monomers from the solvent is favored by the presence of the other brush. Since the overlapping of the stretched configurations lowers their free energy, one expects the overlapping of the brushes to always decrease the total energy of the system at large separations. Surprisingly, this does not happen. In Figure 5a, the probability ψ(z) for a configuration to end at the distance z is compared for large separation distances with the same probability at small separations, when the configurations overlap. The decrease of the energy of the configurations for large values of z leads to an increase in the probability for the chain to reach this distance, at the expense of decreasing ψ(z) for small values of z. However, the “stretched” configurations have higher Flory-Huggins energies that the “collapsed” ones, and the increase of the number of stretched configurations might increase the total free energy of the system. The product ψ(z)UFH(z), whose

integral with respect to z provides the total free energy,is plotted in Figure 5b. In Figure 6, the force between brushes is calculated for various solvent qualities. At small overlaps, the decrease of ψ(z) for small z dominates, and the interaction is repulsive. However, at intermediate separations an attraction between brushes occurs, followed by a much stronger repulsion at small separations, in agreement with experiment.27 It should be noted that the very weak repulsion that occurs at large separations, which was not reported experimentally, is by a few orders of magnitude smaller than the attraction and by many orders of magnitude smaller than the strong repulsion at small separations. This weak repulsion is a consequence of the statistical average over possible configurations employed in this article. 4. Conclusions The traditional analytical treatments of the distribution of monomers in a grafted neutral polymer brush and the interaction between two surfaces with grafted polymer brushes are based either on the assumption of a step profile brush or on a parabolic profile brush. The latter assumption is based on an approximate solution of the selfconsistent field equation (MWC).15 Both approaches predict the same scaling relations between the thickness of the polymer brush and the graft density, which fail at both very low and very high graft densities. In this article an alternate approach was suggested, based on an approximate Monte Carlo procedure. It is assumed that the most contributions to the Boltzmann factor of the configurations that end up at the distance z are provided by the configurations of minimum Flory-Huggins mixing free energy, with a monomer distribution restricted between the surface and the distance z. In a good solvent, these configurations correspond to a constant volume

Nanodispersions

638 Simple Model for Grafted Polymer Brushes

Langmuir, Vol. 20, No. 15, 2004

fraction of monomers, an approach that was successfully employed by Flory to predict the scaling laws of a polymer in a good solvent. The method is particularly valuable for strong-stretched polymers, for which the parabolic approximation is not accurate. Because at high volume fractions of monomers (typical for a high-density polymer graft), the Flory-Huggins free energy increases with the monomer volume fraction and the brushes are swollen even for poor solvent qualities, the method can be applied to high-density brushes immersed in both good and poor quality solvents. When the graft density is not very high, or when the interaction parameters (τ and w) are small, the profile of the brush can be well approximated by a parabola. At higher interactions (either higher graft densities or larger excluded-volumes), the monomer density is almost constant in the vicinity of the surface, but decays exponentially at large distances, in agreement with molecular dynamics simulations and the numerical solutions of the self-consistent field equations. By assuming that the thickness of the brush is equal to the length of the most probable chain configuration (a rather poor approximation), one can recover for good solvents the scaling laws derived by previous analytical treatments; however, the brush thickness obeys a different scaling law at the Θ temperature. The interactions between two surfaces with grafting polymer brushes immersed in good solvents are always repulsive, with a magnitude dependent on the quality of the solvent. For marginally poor solvents, the model predicts also repulsive interactions at any separation. However, for poor solvents, the present theory predicts a very weak repulsion at large separations, followed by a (much stronger) attractive minimum of the free energy at intermediate separation distances, and a strong repulsion at small separations. Appendix The probability for an one-dimensional, free and unconstrained random walker, starting from the origin, to reach the distance z after Nl steps of length l is well approximated (at large Nl) by

ψU(z) )

( )

z2 exp - 2 , 0 < z < Nll 2σ x2πσ2 1

(A1)

where σ ) lxNl is the root-mean square distance of the random walk.24 The probability for a free random walker that starts at 0 to arrive at the distance z in the presence of a reflecting wall at the distance d1 is equal to the sum of the probabilities for an unconstrained random walker to arrive at the point z and at its “image” point 2d1 - z, in the absence of the reflecting wall.24 When the reflecting wall is located at the origin, the probability for the “reflected” walk to reach a distance z > 0 is simply twice the probability to reach the same distance in the absence of the reflecting wall (eq 2a). The general solution of a random walk on a lattice between perfect reflecting walls was provided by Feller.30 In what follows, a simpler approximate solution will be constructed using the method of images.24 When the walk is confined between two reflecting walls, one at the origin and the other at the distance 2d, the probability for the constrained walk to reach the distance z, ψ′(z), can be written as a sum of probabilities ψU for the unconstrained walk to reach various “image” points, as follows. (30) Feller, W. An Introduction to Probability Theory and its Application; John Wiley & Sons: New York, 1968; Vol. 1, Chapter XVI.

6499

ψU(z): The possible paths either do not cross the points 0 and 2d or cross them an even number of times. The presence of reflecting walls at 0 or 2d does not modify the probability to reach z, because after two successive reflections on the same wall the unconstrained path is always recovered. ψU(-z): The possible paths cross an odd number of times the origin. Because of the reflecting wall at 0, the constrained walker arrives at z instead (some paths cross an even number of times 2d, but inserting a reflecting wall there does not affect the probability, as noted above). ψU(4d - z): The possible paths cross the point 2d an odd number of times; because of reflections on that wall, the walker arrives at the point z instead. ψU(-4d + z): The possible paths cross the origin an odd number of times, followed by crossing the point at -2d an odd number of times. Because of the reflection on the walls at origin and at 2d, the constrained walker reaches the point z instead, and so on. Since the length of the walk cannot exceed Nll ≡ Na, only the image points at shorter distances from origin than Na should be taken into account. For the monomer distribution, a distinction has to be made between paths of type a, which are not reflected by opposite walls, and paths of type b, which return to z after reflecting at least once on the opposite wall. In the first case

ψa(z) ) ψU(z) + ψU(-z)

(A2a)

and in the calculation of the configuration of minimum energy it will be assumed that the monomers can be distributed between the starting point 0 and the end point z. For the remaining configurations

ψb(z) ) ψU(4d - z) + ψU(4d + z) + ψU(8d - z) + ψU(8d + z) + ... (A2b) Since the chain reaches both walls at least once, it will be assumed that its monomers are distributed between the walls. The energy of the system of overlapping brushes, per unit area, becomes

ψ(z1,z2) ) =

[

1 C2

∑ i,j

ψ1′(z1)ψ2′(z2)

(

(

)

UFHi,j(z1,z2) exp kT

)

(14a)

UFHaa(z1,z2) 1 a ψ (z1)ψa(z2) exp + C2 kT

( )

bb

)

UFH (z1,z2) + ψa(z1)ψb(z2) × kT UFHab(z1,z2) UFHba(z1,z2) exp + ψb(z1)ψa(z2) exp kT kT (A3) ψb(z1)ψb(z2) exp -

(

(

)]

where in eq 14a the sum is performed over all possible configurations i, j with chains that start on the surface “1” and “2” and end up at z1 and z2, respectively, while in eq A3 only the configurations of minimum Flory-Huggins free energy are taken into account. In the latter case, the minimum depends on whether the chain reaches the opposite wall (type b) or not (type a). For the configurations which do not overlap, the minimum Flory-Huggins energy is obtained for constant monomer distributions between 0 and z1, and between the location z2 and 2d, respectively. Figure 7 presents the

Polymer brushes

639

6500 Langmuir, Vol. 20, No. 15, 2004

Manciu and Ruckenstein

should be located between 0 and z1 or z2 and 2d, respectively, it will be assumed that in the overlapping region, the number of monomers belonging to each grafted brush is proportional to the volume fraction of those monomers in the nonoverlapping regions. The result is a constant volume fraction φ1 between 0 and z1 and a constant volume fraction φ2 between z1 and 2d, where φ1 > φ2 (see Figure A.1b). The volume fractions in each region obey therefore the conservation equations:

(2d - z2)φ1 + R(z1 + z2 - 2d)φ1 )

Nv s2

(A4a)

(2d - z1)φ2 + R(z1 + z2 - 2d)φ2 )

Nv s2

(A4b)

Rφ1 + Rφ2 ) φ1

(A4c)

Equation A4, parts a-c, can be solved for R, φ1, and φ2. For the case bb, when both chains undergo reflections on the opposite walls, the Flory-Huggins free energy is minimized by a constant distribution of monomers between walls. In the case ab, when the second chain is reflected by the opposite wall, but not the first chain, the proportional redistribution of the excess monomers leads to a configuration with volume density φ1 between 0 and z1 and φ2 between z1 and 2d, which obey the equations

Rz1φ1 )

Nv s2

(2d - z1)φ2 + Rz1φ2 ) Figure 7. Local volume fractions of the monomers for two configurations of chains which end up at z1 and z2, respectively: (a) simple addition of the volume fractions of the brushes; (b) configuration that minimizes the Flory-Huggins mixing free energy for the conditions mentioned in the text.

monomer distribution obtained from the sum of the volume fractions of two overlapping configurations. This result clearly does not correspond to a minimum Flory-Huggins free energy, because of the high value of φ in the overlapping region. A smaller free energy is obtained for the configurations in which the excess volume fraction is redistributed to the nonoverlapping regions. In addition to the requirements that all the monomers of each brush

(A5a) Nv s2

Rφ1 + Rφ2 ) φ1

(A5b) (A5c)

and are similar for the ba case. The average free energy density is consequently calculated from

F(2d) =

1

∫ ∫0 C 0 2

2d

2d

∑

(

UFHi,j(z1,z2)ψi(z1)ψj(z2) exp

i)a,b;j)a,b

-

UFHi,j(z1,z2) kT

)

dz1 dz2 (A.6)

where the indices i and j run now over the two types of paths, a and b. LA049781Y

Nanodispersions

640

Langmuir 2004, 20, 8155-8164

8155

On the Monomer Density of Grafted Polyelectrolyte Brushes and Their Interactions Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received March 31, 2004. In Final Form: June 30, 2004 Most of the modern theories of grafted polyelectrolyte brushes are valid only for moderate stretching of the polyelectrolyte. However, particularly at low ionic strength and high grafting densities, even a moderate charge of the polyelectrolyte can generate a strong stretching. A simple mean field model for strongly stretched grafted polyelectrolyte brushes is suggested, based on an approximate calculation of the partition function of a polyelectrolyte chain. It is shown that the average Boltzmann factor of a possible chain configuration can be approximated by the Boltzmann factor of a configuration with a constant monomer distribution, for which the free energy can be readily obtained. The monomer density in the brush and the interaction between two surfaces with grafted polyelectrolyte brushes could be calculated as a statistical average over all possible configurations. Some simple analytical results are derived, and their accuracy is examined. The dependence of the brush thickness on the electrolyte concentration is investigated, and it is shown that the trapping of a fraction of counterions in the brush influences strongly the thickness of the brush. When two surfaces with grafted polyelectrolyte brushes approach each other more rapidly than the ion diffusion parallel to the surface, the trapping of the counterions between the brushes can affect the interactions by orders of magnitude.

1. Introduction The grafting of neutral polymers on colloidal surfaces is a well-known method to enhance the stability of colloids, since the steric repulsion between brushes is not highly sensitive to changes in ionic strength or pH, unlike the double layer repulsion induced by surface charges. The predictions of the monomer density distribution and of the thickness of neutral brushes have been challenging problems, the earlier scaling theories assuming a constant density of the monomers within the brush,1 whereas the most recent ones, originating from a model due to Dolan and Edwards2 in which the neutral polymer follows a random walk from the surface in a field generated by the polymer itself, suggested a parabolic density distribution of monomers.3 During the same period, the charged grafted polyelectrolyte chains received considerable less attention than the neutral ones. One of the reasons is that the sufficiently strong charged grafted polyelectrolyte chains are almost completely stretched because of the electrostatic repulsion between their charges, which implies that a step-profile density of the monomers is an accurate approximation.4 However, the grafted polyelectrolytes received recently renewed attention because of a plethora of potential applications, such as the possibilities (i) to control the catalytic selectivity of some chemical reactions by varying the thickness of the brush,5 (ii) to prevent the adsorption * Corresponding author: e-mail address, feaeliru@ acsu.buffalo.edu; phone, (716) 645-2911(extention 2214); fax, (716) 645-3822. (1) Alexander, S. J. Phys. (Paris) 1977, 38, 983. (2) Dolan, A. K.; Edwards, F. R. S. Proc. R. Soc. London, Ser. A 1974, 337, 509. (3) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (4) Huang, H.; Ruckenstein, E. J. Colloid Interface Sci. 2004, 275, 548. (5) Ruckenstein E.; Hong, L. J. Catal. 1992, 136, 178.

of proteins on surfaces,6 and (iii) to design porous filters for pH-controlled gating.7 The recent theories of grafted polyelectrolytes8-10 are based on an extension to charged brushes of the approximate solution of the self-consistent field equations obtained for neutral brushes,3 which predicted a parabolic shape for the self-consistent field. However, as pointed out by Shim and Cates,11 the parabolic shape of the self-consistent field is accurate only for a moderate stretching but is not valid for very high and very small stretchings. It should be noted that the MilnerWitten-Cates theory3 leads to the same scaling laws for the thickness of the brush as the step profile (the Alexander model)12 and that these scaling laws predict vanishing thicknesses for small grafting densities and thicknesses that exceed the length of the grafted polymers at strong stretching. However, the grafted polyelectrolytes are often strongly stretched, because such stretchings can be sometimes generated by only a moderate charging of the polyelectrolyte. This fact can be understood qualitatively using the following simple analysis. Let us assume that to each grafted chain corresponds an area s2 on the surface and a parallelepipedic atmosphere with the same bottom area and a length equal to the thickness of the brush. The flux of the electric field through the sides of the parallelepiped is proportional to the inside charge ne. This charge is generated through the dissociation of the polyelectrolyte but also includes the fraction of the polyelectrolyte counterions trapped in the brush and the electrolyte ions. (6) Carignano, M. A.; Szleifer, I. Colloids Surf., B 2000, 18, 169. (7) Ito, Y.; Ochiai, Y.; Park, Y. S.; Imanishi, Y. J. Am. Chem. Soc. 1997, 119, 1619. (8) Miklavic, S. J.; Marcelja, S. J. Phys. Chem. 1988, 92, 6718. (9) Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107, 5952. (10) Misra, S.; Varanasi, S.; Varanasi, P. P. Macromolecules 1989, 22, 4173. (11) Shim, D. F. K.; Cates, M. E. J. Phys. (Paris) 1989, 50, 3535. (12) Pincus P. Macromolecules 1991, 24, 2912.

10.1021/la049168e CCC: $27.50 © 2004 American Chemical Society Published on Web 08/14/2004

Polymer brushes

641

8156 Langmuir, Vol. 20, No. 19, 2004

The electric field vanishes on the lateral sides of the atmosphere because of the periodicity of the grafting and also on its bottom surface, because of the overall electroneutrality of the system. Therefore, a large electric field is generated at the top of the brush, of the order of E ) ne/0s2, where  is the dielectric constant within the brush and 0 is the vacuum permittivity. The force required to stretch a polymer is of the order of kT/l, where l is the persistence length.13 The force acting on an elementary charge e located at the end of a chain is given by eE = ne2/0s2 ≈ kT/l, which for typical values of the parameters ( ) 80, s2 ) 1000 Å2, l ) 10 Å) leads to a value of the order of unity for n. This result implies that a few charges within the atmosphere are in general sufficient to exert a sensible stretching of the grafted polyelectrolyte. It should be emphasized that the length of the parrallelepipedic atmosphere (the thickness of the brush) is in general much larger than its lateral length s, therefore the strong stretching is a result of the confinement of the flux of the electric field to a relatively small area. This strong stretching is in contrast to those of a free polyelectrolyte in a solvent or to a single grafted polyelectrolyte, cases in which much larger charges are required to stretch the polyelectrolyte. In this paper, a simple modality to calculate the density distribution of the monomers in a grafted polyelectrolyte brush as well as the interaction between surfaces with grafted polyelectrolyte brushes is suggested. The model is an extension to charged polymer brushes of a treatment proposed recently for neutral polymer brushes.14 It is based on a simplifying assumption applied to a standard Monte Carlo procedure, namely, that the average Boltzmann factor of a possible configuration is dominated by the configuration with the lowest free energy. For a neutral brush immersed in a good solvent, this configuration corresponds to a uniform density of monomers.14 This constant monomer distribution approximation for each possible configuration was successfully employed by Flory to explain the scaling relation between the length and the gyration radius of a polymer in a good solvent.15 In polyelectrolytes, the configuration of minimum free energy would be provided by a constant monomer density only if the electrical potential is constant through the brush. Because, as it will be shown in what follows, this condition is almost satisfied, the approximation of a constant monomer distribution is accurate and simplifies considerably the calculations. 2. Monomer Density and Interaction between Polyelectrolyte Brushes General Framework. The traditional approach to calculate the density distribution of a polyelectrolyte brush consists of generating all the possible configurations of the polyelectrolyte and calculating the total energy corresponding to each configuration. The probability that such a configuration occurs is proportional to its corresponding Boltzmann factor. A simple modality to generate the configurations of a polymer composed of N monomers of length a grafted on a surface is to assume that the polymer is composed of Nl independent pieces of persistence length l, which are connected but can assume any relative orientations. In the absence of any other interactions between monomers (apart from their mutual bonding), the probability that the last monomer of the chain (13) Marko J. F.; Siggia E. D. Macromolecules 1995, 28, 8759. (14) Manciu, M.; Ruckenstein, E. Langmuir 2004, 20, 6490. (15) Flory, P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1971.

Manciu and Ruckenstein

reaches the distance z is provided by the solution of a random walk16

Ξ0(z) ) 0

(1a)

z<0

( ) ( ) 2 πσ2

Ξ0(z) )

1/2

exp -

z2 2σ2

(1b)

z >0 where σ ) lNl1/2 is the root-mean-square distance of a one-dimensional random walk of Nl steps of length l, if one assumes that the surface represents a perfect reflecting wall for the random walk.16 In the presence of additional interactions, the probability Ξ0(z) for a configuration to end up at a distance z must be multiplied by the corresponding Boltzmann factor

Ξ(z) )

( )〈 (

U(z) 1 z2 exp - 2 exp C1 kT 2σ

)〉

(2a)

where U(z) is the total free energy of a configuration and the average 〈 〉 is performed over all possible chain configurations whose last monomer is located at z. The constant C1 is provided by the normalization

∫0

Na

Ξ(z) dz ) 1

(2b)

where Na ) Nll represents the maximum length of the polyelectrolyte. The main difficulty in eqs 2 is to calculate the Boltzmann factor averaged over all possible configurations. As previously suggested,14 it will be assumed that the configuration of lowest free energy dominates the partition function (the saddle-point approximation of statistical mechanics) and therefore the average Boltzmann factor of all the configurations which end up at a distance z can be approximated by a Boltzmann factor corresponding to the configuration of lowest free energy. A further approximation is to consider that the configuration of lowest free energy that ends up at the distance z corresponds to a constant density of monomers φ(ζ,z) distributed between the surface and the distance z

φ(ζ,z) )

N s2z

(3a)

0<ζ
(3b)

ζ>z where N is the number of monomers in a chain and s2 is the area corresponding to a grafted polyelectrolyte on the surface. The above expressions provide a minimum free energy for neutral chains immersed in good solvents and for charged chains, if the electrical potential would be constant through the brush. Since the calculations show that the potential is almost constant through the brush (see eq 14a and Figure 1), the last condition is almost satisfied and eqs 3 are accurate approximations of the monomer distributions corresponding to the minimum of the free energy. (16) Chandrasekhar, S. Rev. Mod. Phys. 1943, 15, 3.

Nanodispersions

642

Monomer Density of Polyelectrolyte Brushes

Langmuir, Vol. 20, No. 19, 2004

Fel(z) )

s2 2

∫0

∞

0

(∂ψ∂ζ )

2

8157

dζ

(6a)

and the entropy of electrolyte ions of concentration ci(ζ), Fent17

Fent(z) ) kTs2

∑i ∫0

∞

( () ci ln

ci

cE

)

- ci + cE dζ (6b)

while the Flory-Huggins free energy is given by12,18

FFH(z) ) s2

Figure 1. The electric potential as a function of the distance from the surface for various values of z, and N ) 1000, a ) 1 Å, η ) 0.005, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80: thick continuous line, exact result; dotted line, linear approximation.

Assuming that each monomer carries an average charge ηe, where η is the degree of dissociation and e is the elementary charge, and that the brush is immersed in an electrolyte solution in contact with an infinite reservoir of concentration cE, the potential inside the brush obeys the Poisson equation

∂2ψ 2ecE eψ ηe N ) sinh 0 kT 0 s2z ∂ζ2

( )

(4a)

∂2ψ 2ecE eψ ) sinh 0 kT ∂ζ2

( )

(4b)

ζ>z where ψ(ζ) is the electrical potential, ζ the coordinate measured from the surface,  the dielectric constant, 0 the vacuum permittivitty, k the Boltzmann constant, and T the absolute temperature and Boltzmannian distributions were assumed for the electrolyte ions. The solution of eqs 4 is obtained by assuming the continuity of the electric potential and its derivative at ζ ) z, and the boundary conditions

σS ∂ψ )∂ζ ζ)0 0

|

(5a)

ψ|ζf∞ ) 0

(5b)

and

where σS represents the surface charge density. Once the potential has been determined, the total free energy corresponding to a polyelectrolyte configuration that occupies an area s2 on the surface and whose last monomer is located at the distance z from the surface can be calculated as the sum between a Flory-Huggins free energy, due to the monomer-monomer and monomersolvent interactions, which does not depend on the charging of the polyelectrolyte and a double layer free energy, due to the charging of the polyelectrolyte and of the grafted surface. The latter one is composed of the contribution of the electric field17

∫0

z

(

-

)

1 2 2 1 τv φ + wv3φ3 dζ ) 2 6

(

2 2

)

kT τN v wN3v3 + (6c) a3 2s2z 6s4z2 where a is the length of a monomer, v its volume, τ is the dimensionless excluded volume parameter (τ < 0 for a good solvent and τ > 0 for a poor solvent), and w is the third virial coefficient (typically positive and of the order of unity). In this paper it will be assumed that the charges of the surface and polyelectrolyte do not depend on the separation between the two grafted polyelectrolyte brushes and, hence, that the chemical free energy of the double layer is a constant.17 The probability for a configuration to end up at the distance z is therefore given by

Ξ(z) )

0<ζ
kT a3

( ) (

Fel + Fent + FFH 1 z2 exp - 2 exp C1 kT 2σ

)

(7)

The average monomer density Φ(z) of the brush is the statistical average of the contributions of all configurations that end up at all possible positions between z and Na. The contribution at the location ζ of a configuration which ends up at a distance z equals φ(ζ,z) ) N/s2z when ζ < z and φ(ζ,z) ) 0 when ζ > z; therefore the statistical average of configurations is given by14

Φ(z) )

∫z

Na

Ξ(σ)

N dσ s2σ

(8)

When two surfaces with grafted polyelectrolyte approach each other, the probability for a chain grafted on surface “1” to end up at the distance z1 from the first surface, whereas a chain which starts on surface “2” to end up at position z2 from the surface “2” is given by

Ξ(z1,z2) )

( ) ( )〈 (

z1 2 z2 2 1 exp - 2 exp - 2 C2 2σ 2σ

exp -

)〉

U(z1,z2) kT

(9a) z1,z2

where the average is performed over all possible configurations which end up at z1 and z2, respectively. The normalization constant C2 is obtained by taking into account that neither chain can be longer than Na and (17) Overbeek, J. T. G. Colloids Surf. 1990, 51, 61. Manciu, M.; Ruckenstein, E. Langmuir 2003, 19, 1114. (18) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.

Polymer brushes

643

8158 Langmuir, Vol. 20, No. 19, 2004

Manciu and Ruckenstein

both of them should end up somewhere between 0 and 2d

∫0 ∫0 ξ

ξ

Ξ(z1,z2) dz1 dz2 ) 1

(9b)

the electric potential and its derivative at ζ ) z, one obtains

ψ(ζ) )

where ξ is the shortest of the distances Na and 2d. Assuming, as previously, that the partition function is dominated by the configurations with constant monomer density between each of the surfaces and the position z1 and z2, respectively, eq 9a becomes

Ξ(z1,z2) )

( ) ( ) (

)

z12 z22 Fel + Fent + FFH 1 exp - 2 exp - 2 exp C2 kT 2σ 2σ (10)

where the integrals of the electric (6a), entropic (6b), and excluded volume contributions (6c) are this time integrated between 0 and 2d. The total free energy of the interacting brushes, per unit area, is the statistical average over all possible configurations, given by

Fbrushes )

1 s2

∫0ξ ∫0ξ (Fel + Fent + FFH)Ξ(z1,z2) dz1 dz2 (11a)

The force per unit area between surfaces can be obtained from the derivative of the free energy per unit area, eq 11a, with respect to the separation distance 2d

Π(2d) ) -

(11b)

∂(2d)

A. Linear Approximation. The linear approximation of the Poisson-Boltzmann equation (sinh(eψ/kT) = eψ/ kT) is in general accurate only for small potentials (ψ < kT/e). However, it offers insight into the qualitative behavior, even at larger potentials. In the linear approximation, the Poisson-Boltzmann equations (4) become

∂2ψ ψ ηe N ) ∂ζ2 λD2 0 s2z

(12a)

0<ζ
(12b)

ζ>z where λD ) (0kT/2e2cE)1/2 is the Debye-Hu¨ckel length, with the solutions

ψ(ζ) )

()

( )

ηe N ζ ζ λ 2 + A1 exp + A2 exp 0 s2z D λD λD

(13a)

0<ζ
()

ψ(ζ) ) A3 exp

ψ(ζ) )

( )

ζ ζ + A4 exp λD λD

(13b)

ζ>z Assuming a neutral grafted surface (σS ) 0), with the boundary conditions eqs 5a,b, and using the continuity of

(14a)

() ( )

(14b)

ηe N z ζ λ 2 sinh exp 0 s2z D λD λD ζ>z

The electrical potential through the brush (eq 14a) is almost constant, since for (z - ζ)/λD . 1, cosh(ζ/λD) exp(-z/λD) ≈ (1/2) exp(-(z - ζ)/λD) , 1 and varies notably only when (z - ζ)/λD is of the order of unity. A typical solution of eqs 14 is compared in Figure 1 to the results obtained from the numerical integration of the PoissonBoltzmann equations (4a,b) for various values of z (the maximum length of a configuration) and the following values of the parameters: N ) 1000, a ) 1 Å, η ) 0.005, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80. The linear approximation is in general not accurate for small values of z, which correspond to large potentials; however, these configurations have also large double layer energies and, therefore, small contributions to the final statistical average. In the linear approximation, the double layer energy of a configuration of distance z, obtained from the integration of eqs 6a and 6b, is given by

∫0z (Fel + Fent) dζ = s2

3. Monomer Distribution in a Grafted Polyelectrolyte Brush Immersed in an Infinite Reservoir

( ) ( )]

ζ
FDL(z) )

∂Fbrushes

[

ηe N ζ z λ 2 1 - cosh exp 0 s2z D λD λD

λD2η2e2N2 2 2

40s z

0 2

(

∫0

z

(( ) ( ) dψ(ζ) dζ

2

4z + 4λD exp -

+

ψ2(ζ) λD2

)

dζ )

() ( ))

2z z cosh2 - 3λD λD λD λD exp -

4z λD

(15)

For a charged polyelectrolyte brush, the double layer energy, eq 15 is in general much larger than the excluded volume interaction, eq 6c. The probability for a brush to end up at a distance z (eq 7) is represented in Figure 2a, for the above values of the parameters and l ) 10 Å, σ ) lNl1/2 ) 100 Å, τ ) -1.0, w ) 0.0, v ) 1.0 Å3. Since the selected area corresponding to a grafted chain on a surface is relatively small (s2 ) 1000 Å2), low dissociation fractions (of the order of a few percent) are sufficient to generate large electrical potentials and consequently strong stretching of the brush. A qualitative analysis starting from eq 14a shows that the potential at the grafting surface, for a strong stretching (z = Na), is of the order of

ψ(0) =

2 2 ηe NλD ηe λD ≈ 0 s2z 0 s2a

(16)

This implies that even at moderate electrolyte concentrations (cE ) 0.01 M, λD ) 30 Å) and small dissociation fractions η, ψ(0) > kT/e and the linear approximation of the Poisson-Boltzmann equations fails. An approximation, which is more accurate at large potentials, is suggested below. B. Nonlinear Approximation. The numerical solution of the Poisson-Boltzmann system of eqs 4a and b, as well as the analytical solutions (eqs 14) of its linear approximation (eqs 12) indicate that the potential is almost constant through each configuration, until a distance of

Nanodispersions

644

Monomer Density of Polyelectrolyte Brushes

sinh

Langmuir, Vol. 20, No. 19, 2004

8159

2λD eψ h ηN sinh ) z 2kT 2cEs2z

(17c)

eψ h (kT )+

( )

from which a numerical solution for ψ h can be readily obtained. When the potential is sufficiently large, sinh(eψ/ 2kT) , sinh(eψ/kT) (in many cases λD is also much smaller than z), and eq 17c reduces to

eψ h (kT ) ) ηN sz

2cE sinh

(17d)

2

which simply states that the constant potential ψ h is such that the electrolyte ions compensate exactly the charge of the polyelectrolyte. The solution of eq 17d is provided by

ψ h =

((( ) )

kT ln e

2

ηN 2cEs2z

+1

1/2

-

ηN 2cEs2z

)

(18)

As a matter of fact, the approximate surface potential predicted by the linear analysis can be derived using the same assumption of a vanishing charge distribution near the grafting surface. Indeed, ηN/s2z ) 2cE sinh(eψ(0)/kT) = 2ecEψ(0)/kT ) 0ψ(0)/eλD2, which coincides with eq 16. In the region of constant potential, the electrostatic energy vanishes (because dψ/dζ ≡ 0) while the entropic contribution is given by

FDL(z) ) kTs2

∑i ∫0

cEkTs2z Figure 2. (a) The probability for a chain whose last monomer is located at a distance z for N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, v ) 1 Å3, and various degrees of dissociation η: thick continuous line, exact result; dotted line, linear approximation; continuous line, nonlinear approximation. (b) The monomer density in brushes with various degrees of dissociation for N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, and v ) 1 Å3: thick continuous line, exact result; dotted line, linear approximation; continuous line, nonlinear approximation.

the order of λD from the maximum length of the configuration (see Figure 1). A simple approximation consists of assuming a constant potential ψ h between 0 and z. Outside the brush (for z < ζ < ∞), the traditional PoissonBoltzmann equation is valid and can be integrated to provide19

-

( )

|

dψ(ζ) eψ(z) 2kT ) sinh dζ z λDe 2kT

(17a)

where the electric field at z is related to the charge distribution between 0 and z (due to the polyelectrolyte and electrolyte ions), which is given by

∫0z

(

( ))

2ecE ηeN eψ h sinh dζ ) 2 0 kT 0s z

()

ηeN 2ecEz eψ h sinh (17b) 0 kT 0s2

By combining eqs 17a and b, one obtains

z

( () ci ln

ci

( () eψ h

kT

exp

eψ h

-

kT

eψ h kT

( )

exp

eψ h

kT

)

- ci + cE dζ )

cE

( ) ( )

exp -

- exp -

eψ h

kT

eψ h

kT

-

)

+ 2 (19)

The contribution to the double layer free energy from the region between z and infinity can be calculated by solving the corresponding traditional Poisson-Boltzmann equation. However, the value of ψ h given by eq 18 is an overestimate of the potential inside the brush (since it implies a potential large enough to cancel the charge density in the brush) and consequently eq 19 is also an overestimate of the free energy of the double layer between 0 and z. Therefore, eq 19 can be used as an approximation of the total double layer free energy for the configuration that ends up at the distance z. In Figure 2a, the probability for a brush to end up at a distance z is calculated using the simple nonlinear approximation based on eqs 18 and 19; this approximation provides in general a much better agreement with the numerical calculations based on the nonlinear equations than the linear approximation at high potentials. The corresponding monomer density distributions, statistically averaged over all possible configurations, are plotted in Figure 2b. The thickness of the grafted polyelectrolyte brush can be described roughly by the length of the most probable configuration, which is provided by the solution of

( ( ) (

))

FDL(z) + FFH(z) d z2 exp - 2 exp dz kT 2σ

) 0 (20)

(19) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987.

Polymer brushes

645

8160 Langmuir, Vol. 20, No. 19, 2004

Manciu and Ruckenstein

Figure 4. A configuration of a polyelectrolyte molecule, grafted on a surface, whose last monomer is located at the distance z, immersed in a finite reservoir of length L.

of the electrolyte ions through the expressions

∫0

L

∫0

L

∂2ψ ) ∂ζ2

eψ eψ (kT ) - c exp(- kT )) -

(

e c1 exp

2

0

0

( )

eψ(ζ) dζ ) ηL kT

| |

∂2ψ ∂ζ2

ζ)0

∂2ψ ∂ζ2

ζ)L

(

( )

(

))

( )

z<ζ
(22b)

(22c)

)0

(23a)

)0

(23b)

An analytical solution of the above system can be obtained in the linear approximation as follows. The Poisson-Boltzmann equations become

∂2ψ ) a1ψ + a2 - a3 ∂ζ2

(24a)

0<ζ
(24b)

z<ζ
(

e2 N c + c2 + R 2 0kT 1 sL

a2 )

(

e N c - c2 + R 2 0 1 sL

0<ζ
(22a)

Assuming as above that the grafting surface is not charged (σS ) 0), the electroneutrality of the system leads to the boundary conditions

a1 )

ηe N ReN eψ + exp (21a) 0 s2z  s2L kT

)

eψ(ζ) dζ ) cEL kT

( )

R exp

4. Monomer Distribution in a Grafted Polyelectrolyte Brush Immersed in a Finite Reservoir A. Linear Approximation. As noted in previous sections, even dissociation of the order of a few percent of the grafted polyelectrolyte brushes might be enough to stretch the chains, particularly at low electrolyte concentrations. In an infinite reservoir, the density of the counterions generated by the dissociation of the polyelectrolyte groups vanishes, and only the electrolyte ions are left to compensate the brush charge. An interesting situation occurs when the brush is immersed in a finite reservoir, because a fraction of the counterions of the polyelectrolyte remains trapped inside the brush, neutralizing a part of its charge. In this case, assuming a finite reservoir of length L > Na (see Figure 4), the Poisson-Boltzmann equations become

eψ(ζ) dζ ) cEL kT

∫0L c2 exp(-

Figure 3. The thickness of the brush (the length of the most probable configuration) vs the Debye-Hu¨ckel length for various degrees of dissociation and N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, and v ) 1Å3.

The dependence of the length of the most probable configuration on the electrolyte concentration is plotted in Figure 3. The brush is strongly stretched for most ionic strengths. As already noted, this occurs because at lower electrolyte concentrations there are not many electrolyte counterions to screen the polyelectrolyte charges; therefore the potential in the brush is high, even for small degrees of dissociation.

( )

c1 exp

a3 ) η

eN 0s2z

)

)

(25a)

(25b)

(25c)

The solution of the linear system (24), which obeys the boundary conditions (23a,b) and has ψ and dψ/dζ continuous at z is determined up to an arbitrary constant potential, which is irrelevant for the problem. It should be, however, noted that eqs 22 imply that the ion concentrations reach the values c1, c2, and RN/s2L at the point where ψ ) 0. Selecting this point to be at ζ ) L

Nanodispersions

646

Monomer Density of Polyelectrolyte Brushes

Langmuir, Vol. 20, No. 19, 2004

8161

(the far side of the reservoir), one obtains the solution

ψ(ζ) ) a3 sinh(za11/2) sinh((L - z)a11/2) 1cosh(ζa11/2) a1 sinh(La 1/2) sinh(La 1/2)

(

1

1

)

(26a) 0<ζ
ψ(ζ) )

a3 sinh(za11/2) (cosh((L - ζ)a11/2) - 1) a1 sinh(La 1/2)

(26b)

1

z<ζ
∫0

L

( )

exp

eψ(ζ) e dζ ) L + kT kT L+

∫0

L

ψ(ζ) dζ )

(

)

sinh(za11/2) e a3 z-L (27) kT a1 sinh(La 1/2) 1

Equations 22 become

c1 )

c2 )

R)

(

cE

(

cE

(

η

)

(28a)

)

(28b)

)

(28c)

sinh(za11/2) e a3 z 1+ kT a1 L sinh(La 1/2) 1

sinh(za11/2) e a3 z 1kT a1 L sinh(La11/2) sinh(za11/2) e a3 z 1+ kT a1 L sinh(La 1/2) 1

Figure 5. The dependence of the brush thickness (the length of the most probable configuration) on the finite reservoir size L for various degrees of dissociation and N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, and v ) 1.0 Å3.

B. Nonlinear Approximation. The linear approximation in general is not accurate for potentials larger than kT/e. A simple approximation, which is more accurate than the linear one at larger potentials, can be obtained by assuming that the electrical potential is constant along the configuration that ends at the distance z and vanishes in the rest of the reservoir. This average potential can be obtained, as in section 3B, from the condition of vanishing of the total charge density inside the brush

η

( ) (

The conservation of the number of ions in the reservoir (eqs 22) leads to

c1 )

It should be again emphasized that the linear approximation is accurate only for ψ , kT/e. Whereas the solutions for the potential (eqs 26) are qualitatively (but not quantitatively!) valid even for ψ > kT/e, eqs 28 fail in this case and might even predict negative concentrations of co-ions. Equations 28a-c can be solved for the constants c1, c2, and R as functions of η, cE, z, and L. Then the double layer free energy of a configuration that ends up at the distance z in a reservoir of finite length L is given by

( ) ( ) ( )

s20 L dψ(ζ) 2 dζ + 0 2 dζ c1 ec2 c2 L ec1 eψ eψ kTs2 0 exp ln exp ln kT kT cE kT kT cE eψ eψ c1 exp - c2 exp + 2cE + kT kT e RN eψ R RN eψ ηN exp ln - 2 exp + 2 (29) kT s2L kT η kT sL sL

FDL(z) )

∫

[

∫

( ) ( ( ) ()

(

)

( )

)

()

]

where it was taken into account that not only the electrolyte ions but also the counterions of the polyelectrolyte chains contribute to the entropy of the double layer, because, as a result of the double layer formation, their distribution differs from the average density ηN/s2L.

) ( )

N eψ h N eψ h + c2 exp - c1 + R 2 exp )0 2 kT kT sz sL (30a)

c2 )

cEL eψ h z exp +L-z kT

( )

cEL eψ h z exp +L-z kT

R)

(

)

ηL eψ h z exp +L-z kT

( )

(30b)

(30c)

(30d)

The solution of ψ h from eqs 30a-d can be used to obtain the following approximate expression for the double layer free energy

[

()

ec1 c1 eψ exp ln kT kT cE ec2 c2 eψ eψ eψ exp ln - c1 exp - c2 exp + kT kT cE kT kT

FDL(z) ) kTs2z

(

2cE

()

)

( )

( ) ( ) eψ R eψ e RN RN ηN + exp( ) ln( ) exp( ) + kT s L kT η kT sL s L] 2

2

2

(31)

In Figure 5, the length of the most probable configuration is plotted as a function of the reservoir length L for

Polymer brushes

647

8162 Langmuir, Vol. 20, No. 19, 2004

Manciu and Ruckenstein

N ) 1000, a ) 1 Å, l ) 10 Å, σ ) lNl1/2 ) 100 Å, η ) 0.01, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, τ ) -1.0, w ) 0.0, and v ) 1.0 Å3. For small reservoirs, most counterions are not able to leave the brush; hence the overall charging is small. This situation is not unlike some molecular dynamics simulations,20 where the counterions were explicitly taken into account, but the size of the system was limited (because of computational requirements). However, when the brush is embedded in an infinitely large reservoir and chemical equilibrium is reached, there are no counterions left in the brush, and a stronger stretching is achieved.

conditions at ζ ) z is given by

ψ(ζ) )

[

ηe N ζ λ 2 1 - cosh 0 s2z D λD

| |

∂ψ )0 ∂ζ ζ)0

(32a)

∂ψ )0 ∂ζ ζ)d

(32b)

and the requirements of continuity of ψ(ζ) and dψ(ζ)/dζ at ζ ) z. The free energy of each configuration that ends up at the distance z is obtained from eqs 6a-c, while the free energy of the system is obtained from eqs 9-11, by considering only the symmetric configurations

Fbrushes )

∫0

2d

(Fel(z) + Fent(z) + FFH(z))Ξ(z) dz

(33)

For small charges of the brush or large ionic strengths, one can employ the linear approximation of the PoissonBoltzmann equations (eqs 12) with the boundary conditions (eqs 32). The solution that obeys the continuity (20) Seidel, C. Macromolecules 2003, 36, 2536.

d-z λD d sinh λD

0<ζ
]

() ( ) ()

ηe N d-ζ ψ(ζ) ) λ 2 cosh 0 s2z D λD

5. The Interactions between Surfaces with Grafted Polyelectrolyte Brushes: The Role of the Rate of Approach of the Surfaces A. Thermodynamic Equilibrium (Slow Approach of the Surfaces). The general formalism for the interaction between two surfaces with grafted polyelectrolyte brushes, presented in this paper in section 2, implies that, for each separation distance 2d, the Poisson-Boltzmann equations have to be solved for every possible configuration of the chains, ending up at z1 and z2, where 0 < z1 < ξ, 0 < z2 < ξ, ξ being the smallest between the maximum length of a chain, Na, and the distance 2d between surfaces. Since such calculations are tedious, here a much simpler analysis is used for the interaction between surfaces. It was shown previously14 that for neutral brushes and good solvents, the symmetric configurations (for which z1 ) z2) are the most important contributions to the free energy, and (at least at strong repulsion) the contributions of the other configurations can be neglected. Such an approximation is reasonable because for a single brush only the configurations around the most probable one have significant probabilities Ξ(z) (see Figure 2a). For two brushes, Ξ(z1,z2) ≈ Ξ(z1)Ξ(z2) and for the asymmetric configurations (z1 * z2), at least one of the probabilities is small. For the symmetric configurations with z < d, a constant monomer density between surface and z will be considered for each brush, while for z > d (physical overlap of the brushes) it will be assumed that all the monomers are homogeneously distributed between the two surfaces. In the absence of a surface charge density on the plates (σS ) 0), the solution of the Poisson-Boltzmann equations (4a and b) are obtained for the boundary conditions

( ) () () sinh

z λD d sinh λD

(34a)

sinh

(34b)

z<ζ
FDL(z) )

η2 e 2 40

N2λD2

( )[

( )

2d (2z - λD) sinh + d λD s z sinh λD 2d - 2z 2z λD sinh + sinh - 2z (35) λD λD 2 2

( (

2

)

( )) ]

In Figure 6a, the force per unit area between surfaces with grafted polyelectrolyte brushes, plotted as a function of their separation distance 2d, calculated in the linear approximation, is compared with the numerical solution of the nonlinear Poisson-Boltzmann equations, for a system with N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, l ) 10 Å, σ ) 100 Å, τ ) -1.0, w ) 0.0, v ) 1.0 Å3 and various degrees of dissociation. As expected, even moderate degrees of dissociation of the polyelectrolyte generate large potentials, for which the linear approximation fails. B. Quasi-Thermodynamical Equilibrium (Rapid Approach of the Surfaces). The motion of the counterions along the distance perpendicular to the surfaces is in general sufficiently rapid to reach the thermal equilibrium along this direction during the interaction. If the approach of the surfaces is slow, the counterions of the polyelectrolyte chains can also reach thermal equilibrium with the infinite reservoir. However, if the surfaces are sufficiently large, and the approaching of the plates sufficiently rapid, the diffusion of the ions in the planes parallel to the surface toward the infinite reservoir might be much slower than the rate of approach of the surfaces. In this case, the ions reach thermal equilibrium along the direction perpendicular to the surfaces but can remain practically trapped between the surfaces. A simple modality to describe such a behavior is to consider a system with infinitely large surfaces, with all the ions (the counterions of the polyelectrolyte chains and the electrolyte ions) located between the surfaces. Each half of the system obeys the same equations as the brush embedded in a finite reservoir treated in the previous paragraph, if the reservoir thickness L is replaced by the (variable) half distance d between surfaces. In the linear approximation, the solution for the potential (up to an arbitrary constant) is provided by eqs 26a,b, while the corresponding double layer free energy is given by eq 29. In Figure 6b, the force per unit area between surfaces with grafted polyelectrolyte brushes, plotted as a function of their separation distance 2d, provided by the linear approximation is compared with the numerical result

648

Monomer Density of Polyelectrolyte Brushes

Figure 6. (a) The force per unit area between two identical plates with grafted polyelectrolyte brushes, as a function of the separation distance, for N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, and v ) 1.0 Å3 calculated in the linear approximation is compared to the numerical result obtained from the nonlinear equations, assuming that the counterions diffuse into an infinite reservoir (slow approach). (b) The force per unit area between two identical plates with grafted polyelectrolyte brushes, as a function of the separation distance, for N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, and v ) 1.0 Å3 calculated in the linear approximation is compared to the numerical result obtained from the nonlinear equations, assuming that the counterions remained trapped between the plates (rapid approach).

obtained from the integration of a nonlinear system for a system with the same parameter values as above (N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, l ) 10 Å, σ ) 100 Å, τ ) -1.0, w ) 0.0, v ) 1 Å3) and various degrees of dissociation. Because of the counterions trapping, the potentials are in general smaller than in the previous case; hence the linear approximation is a better one. The most interesting results are the changes in the interactions between plates as a function of the rate of approach. The extreme cases, when all the counterions are trapped between plates (rapid approach of the surfaces) and when the counterions diffuse into the infinite reservoir (slow approach of the surfaces) are compared in Figure 7a for a system with the same parameter values as above (N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, l ) 10 Å, σ ) 100 Å, τ ) -1.0, w ) 0.0, v ) 1 Å3) and various degrees of dissociation, the force being calculated numeri-

Nanodispersions

Langmuir, Vol. 20, No. 19, 2004

8163

Figure 7. (a) The force per unit area between two identical plates with grafted polyelectrolyte brushes, as a function of the separation distance, for N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, v ) 1.0 Å3, and various degrees of dissociation, calculated by assuming that the counterions either diffuse into an infinite reservoir (slow approach) or remain trapped between the plates (rapid approach) (b) The ratio between the interactions calculated assuming that the ions reach thermal equilibrium with an infinite reservoir (slow approach of the surfaces) and those calculated assuming that all the ions remain trapped between the surfaces (rapid approach of large surfaces), for N ) 1000, a ) 1 Å, cE ) 0.01 M, s2 ) 1000 Å2,  ) 80, σ ) 100 Å, τ ) -1.0, w ) 0.0, v ) 1.0 Å3, and various degrees of dissociation. At small separation distances and even moderate charging, the interactions can differ by orders of magnitude. At large separations, the differences become negligible.

cally in each case, using the nonlinear equations and only the symmetric configurations. Even for moderate degrees of dissociation, there is a difference of orders of magnitude between the two cases. The large difference is due mainly to the strong stretching of the brushes in the absence of the polyelectrolyte counterions. The double layer interaction roughly decays exponentially with a decay length λD = 30 Å (for cE ) 0.01 M), and variations in brush thicknesses of the order of 100 Å lead to changes of orders of magnitude in the interactions. In addition, when the counterions are trapped between plates and compensate a part of the polyelectrolyte charge, a smaller potential is generated. At very large separations between plates, as expected, the difference between the two cases vanishes, because the counterions have sufficient room to leave the brush (see Figure 7b).

Polymer brushes

649

8164 Langmuir, Vol. 20, No. 19, 2004

6. Conclusion A simple mean field theory for polyelectrolyte brushes is suggested, based on an approximate Monte Carlo-type calculation. The method can be applied at high polyelectrolyte stretchings, for which the traditional treatments based on the self-consistent-field fail. It is assumed that the Boltzmann factor for the configurations that end up at a distance z from the surface can be approximated by a Boltzmann factor of the configuration that ends at the same distance and has a constant density distribution of monomers. The free energy of the latter can be calculated by solving the appropriate Poisson-Boltzmann equations. Both the density distribution of the monomer in the brush, calculated as the statistical average of all possible configurations, as well as the interactions between surfaces can be provided by this method. Simple expressions for the free energy of a configuration have been obtained, both in the linear approximation (valid at small

Manciu and Ruckenstein

potentials) and in a nonliner approximation, which is more accurate at high potentials. The dependence of the brush thickness (defined as the length of the most probable configuration) on electrolyte concentration was investigated. A simple modality to calculate the interaction between surfaces with grafted polyelectrolyte brushes using only symmetric configurations has been suggested. The possibility for a part of the counterions to remain trapped in the brush was also examined, and it was shown that even at moderate degrees of dissociation of the polyelectrolyte chains, it might reduce the force between brushes by orders of magnitude at small separations. Since for plates approaching rapidly (or having large surfaces) the counterions might not have sufficient time to diffuse into the infinite reservoir, the trapping of counterions can play an important role in colloid coagulation. LA049168E

Nanodispersions

650

Journal of Colloid and Interface Science 273 (2004) 181–190 www.elsevier.com/locate/jcis

Double-layer interaction between two plates with hairy surfaces Haohao Huang and Eli Ruckenstein ∗ Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 8 July 2003; accepted 29 August 2003

Abstract In most theoretical treatments of colloidal particles with hairy surfaces, only the steric effect is taken into account. The steric force is a shortrange interaction and acts only when the chains on different particles begin to interpenetrate each other. However, since the hairy chains are extended into the continuous phase, they constrain the orientation of the water molecules near the surface and, as a result, the dielectric constant in that region can become very different from that in the bulk. The low dielectric constant affects the distributions of ion concentrations and the gradient of the electric field. Therefore, the double-layer interactions between two plates with hairy surfaces cannot be calculated on the basis of the classical Gouy–Chapman theory, which involves a uniform dielectric constant in the Poisson–Boltzmann equation. A model which accounts for the difference in dielectric constants in the hairy region and outside that region is therefore proposed. The ion specificity is also taken into account by using Born’s expression for the free energy of hydration of ions. The repulsive forces calculated via the Gouy– Chapman theory and via the new model are compared. The hairy region can have a long range effect on the repulsive double-layer interactions.  2003 Elsevier Inc. All rights reserved. Keywords: Hairy surface; Double-layer interaction; Hydration energy; Specific ion effect

1. Introduction Polymer molecules are often employed to stabilize colloids [1]. In most theoretical treatments of the effect of polymer adsorption [2–5], only the steric force is taken into account, and the steric force and the traditional double-layer force for particles devoid of “hairs” are assumed to be additive. The steric force is a short-range interaction which acts only when the chains on the surfaces of the two particles interpenetrate [6–8]. However, in addition to this short-range interaction, a hairy surface can also generate another effect, because it can change the dielectric constant in the vicinity of the surface. Approaches have been developed which modify the traditional double-layer theory by assuming that the dielectric constant depends upon position. This dependence is considered to occur because the dielectric constant is affected by the electric field [9–12]. However, the dielectric constant can also be changed by the presence of hairs on the surface. Indeed, hairy chains on the surface constrain the orientation of the water molecules nearby, and, as a result, in the region near the surface the dielectric constant becomes lower * Corresponding author.

E-mail address: [email protected] (E. Ruckenstein). 0021-9797/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.08.060

than in the bulk. In fact, the water molecules are ordered very near the particle surface even in the absence of a hairy surface [13] and this ordered structure leads to a lower local dielectric constant near the surface than in the bulk. The thickness of the latter region is, however, very small. In what follows it will be shown that, by changing the dielectric constant near the surface, the hairy chains can have a long-range effect, much longer than the range of steric repulsion. In this paper, a model is proposed that involves two different dielectric constants in the calculation of the electrical potential between two plates, one for the region near the surface and the other outside this region. The ion specificity will be also taken into account via Born’s expression for the hydration free energy of ions. The repulsive force was calculated numerically at constant surface charge density at constant surface potential and for a surface charge generated through dissociation equilibrium. The effects of electrolyte concentration, of the dielectric constant in the hairy region, of the thickness of the hairy region, and of the nature of the electrolyte were investigated. The repulsive forces calculated via the Gouy–Chapman theory and those based on the new model are compared. An analytical expression for the repulsive force in the framework of the linear approximation is provided.

Polymer brushes

651

182

H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

2. Theoretical framework The region near the surface (region I), of thickness δ, is considered to have a dielectric constant εI different from that in the remaining region between the plates (region II) εII , due to the constraint on the water molecules induced by the hairy surface. For simplicity, the dielectric constants in both regions are assumed to be constant, with that in region II equal to the bulk dielectric constant. The electrical potential distribution between the two plates is given by the Poisson equations ∂ 2Ψ ρ =− I 2 ∂x ε ε0

(for region I)

(1)

(for region II),

(2)

and ∂ 2Ψ ρ = − II 2 ε ε0 ∂x

where Ψ is the electrical potential, x is the distance from the middle between the two plates, ρ is the charge density, εI and εII are the dielectric constants in regions I and II, respectively, and ε0 is the vacuum permittivity. According to the Born theory, the hydration free energy of ion i can be expressed as Ai , (3) ε where Ai is a hydration constant, which is specific to each ion and could be evaluated from experimental hydration free energies [14,15], and ε is the dielectric constant of the medium in which ion i is located. The values of Ai for various ions used in the calculations are listed in Table 1. The equalities between the chemical potentials in region I and the bulk and region II and the bulk lead to Uih =

kT

ln nIi

Ai Ai + ezi Ψ + I = kT ln n0i + II ε ε

(4)

(for region I)

(6)

and

  ezi Ψ 0 nII = n exp − i i kT

Ai Ai = kT ln n0i + II , εII ε

(5)

where n0i is the bulk concentration of ion i, k is the Boltzmann constant, T is the absolute temperature, e is the protonic charge, zi is the ion valency (positive for cations and negative for anions), and nIi and nII i are the concentrations of Table 1 The value of Ai for different ions [11] Ion

Ai (kcal/mol)

Li+ Na+ K+ F− Cl− Br−

127.1 103.0 85.1 84.4 70.8 63.8

(for region II).

(7)

Substituting Eqs. (6) and (7) into Eqs. (1) and (2), respectively, the Poisson–Boltzmann equations become    Ai  1 e i n0i zi exp − ezkTi Ψ − kT − ε1II ∂ 2Ψ εI =− εI ε0 ∂x 2 (for region I), (8) e ∂ 2Ψ =− ∂x 2

 i

  n0i zi exp − ezkTi Ψ εII ε0

(for region II).

(9)

For surfaces that have a constant potential Ψ0 , the boundary condition at the surface is Ψ |x=−d = Ψ0 ,

(10)

where 2d is the distance between the plates. For surfaces at constant charge density σ , the boundary condition at the surface becomes

∂Ψ

σ =− I . (11)

∂x εε x=−d

0

The symmetry condition provides an additional boundary condition,

∂Ψ

= 0. (12) ∂x x=0 At the boundary between the two regions, the electric potentials should be the same, Ψ |Ix=−d+δ = Ψ |II x=−d+δ ,

and kT ln nII i + ezi Ψ +

ion i in regions I and II, respectively. Consequently,
  ezi Ψ Ai 1 1 I 0 ni = ni exp − − − kT kT εI εII

(13)

and if there is no charge accumulation, the Gauss law provides the equation

II

∂Ψ

I II ∂Ψ

εI = ε , (14)

∂x x=−d+δ ∂x x=−d+δ where the superscripts I and II indicate regions I and II, respectively. An equation of the form (8) was previously employed [11], in a different context. Equations (8) and (9) and the boundary conditions (10) or (11) and (12)–(14) allow the profile of the electric potential to be calculated numerically. The Langmuir equation cannot be used to calculate the repulsive force, because it implies ideal behavior of ions and the presence of the term Ai (1/ε I − 1/ε II ) in Eq. (6) implies nonideality. Consequently, to obtain the repulsive force between two plates, it is necessary to calculate the free energy

Nanodispersions

652 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

183

of the system as a function of the distance 2d between them, its derivative with respect to 2d providing the force. The free energy of the system F is composed of four contributions, F = Fele + Fentropy + Fche + Fhyd ,

(15)

where Fele is the electrostatic, Fentropy the entropic, Fchem the chemical, and Fhyd the hydration contribution. The electrostatic contribution comes from the electrical potential energy of the charge in the field [16]: d Fele = −d

  εε0 ∂Ψ 2 dx 2 ∂x

−d+δ 

=2

 ε I ε0 ∂Ψ ∂x

2

−d

2

d−δ 

dx +

 εII ε0 ∂Ψ

−d+δ

∂x

2

2 dx.

(16) The entropic contribution to the free energy is given by [16] 

d

Fentropy = kT

 ni ln

i −d

ni n0i



 − ni + n0i

3. Results and discussion dx.

(17)

The chemical contribution Fchem = −2Ψs σ at constant surface potential Ψs and = 0 at constant surface charge. A universal expression free of any constraints has the form [17] σ(2d)

Fchem = −2

Ψs dσ .

(18)

σ (∞)

Equation (18) was derived by decomposing the pathway of Ψs vs σ in a succession of infinitesimal changes at constant charge followed by changes at constant Ψs [17]. The hydration contribution is provided by the difference between the hydration free energies of ions in the region with low dielectric constant and in the bulk, −d+δ    1 1 Fhyd = 2 ni Ai I − II dx. ε ε i

(19)

−d

For εI = εII , the hydration contribution Fhyd becomes zero. Since the dielectric constant and the ion distributions are provided by different expressions in the two regions, the electrostatic contribution and the entropic contribution should be calculated separately for the two regions, the total contributions being their sum. The repulsive force f between the two parallel plates is given by the derivative of the free energy with respect to the distance 2d between them, f =−

∂F . ∂(2d)

Fig. 1. The repulsive force between two plates at constant surface charge and various bulk NaCl concentrations. The following parameters were employed in the calculations: σ = −0.01 C/m2 , εI = 10, εII = 80, δ = 10 Å. (1) 0.001; (2) 0.01; (3) 0.1 M NaCl. The solid lines are for the force predicted by the new model and the dashed lines for the force predicted by the Gouy–Chapman theory.

(20)

In the numerical calculations, the force provided by the classical theory was obtained by assuming εI = εII = 80.

3.1. Constant surface charge density 3.1.1. The effect of electrolyte concentration on double-layer interaction The double-layer repulsive force calculated using Eqs. (7), (8), and (10)–(20) is plotted in Fig. 1 for various NaCl concentrations. In the calculations, the dielectric constant for region I was taken to be 10 and for region II 80; the thickness of region I was taken to be δ = 10 Å and the surface charge density σ = −0.01 C/m2 . As expected, the force decreases with increasing electrolyte concentration. The difference between the forces calculated on the basis of the new model and predicted by the traditional theory increases with increasing electrolyte concentration. At concentration 0.1 M and a distance of 40 Å, the force is 10 times larger than predicted by the Gouy–Chapman theory. However, at 0.001 M and a distance of 40 Å, it is only about twice as large. In Fig. 2, the ion distributions are plotted for various electrolyte concentrations. Compared with those predicted by the Gouy–Chapman theory, the ion concentrations in region I are extremely low. This occurs because the hydration free energies of the ions are inversely proportional to the dielectric constant (see Eq. (3)). As a result, the ions have higher affinity for region II than region I. Even though the surface is negatively charged, the concentration of cations is still lower near the surface than in the bulk. Fig. 2 also shows that at high salt concentrations the anion concentration is higher than that of the cation in a fraction of region I, even though the electrical potential is negative. This behavior is a result of the larger hydration constant of the cations than of the anions. For a dielectric constant of 10 in region I, the difference between the hydration free energies of Na+ and Cl− is 4.8kT (Table 1). Only for electrical potentials smaller than −0.062 V (= (ACl /εI − ANa /ε I )/2e) can the difference between the electrostatic energies of the anions

Polymer brushes 184

653 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

and cations compensate for the difference in hydration free energies, and the cation concentration becomes larger than that of the anion. However, for electrical potentials larger than −0.062 V, the total potential energy of the cation becomes larger than that of the anion. Consequently, in spite of the negative electrical potential, the cation concentration becomes lower than that of the anion for electrical potentials larger than −0.062 V. Since the electrical potential decreases with decreasing electrolyte concentration, the thickness of the fraction of region I with negative charge density decreases and becomes zero at a concentration of 0.001 M. The electrical potential profile for various electrolyte concentrations is plotted in Fig. 3, which shows that its magnitude decreases with increasing electrolyte concentration. The surface potential is more negative than for a uniform

Fig. 2. The ion distributions between two plates for various bulk NaCl concentrations. The distance between the two plates is 100 Å. The other parameters employed are: σ = −0.01 C/m2 , εI = 10, εII = 80, δ = 10 Å. (1) 0.001; (2) 0.01; (3) 0.1 M NaCl. In the figure, A and C indicate Cl− and Na+ , respectively. The solid lines are for the ion distributions predicted by the new model and the dashed lines for those predicted by the Gouy–Chapman theory.

Fig. 3. The electrical potential distributions between two plates for various bulk NaCl concentrations. The distance between the two plates is 100 Å. The other parameters employed are σ = −0.01 C/m2 , ε I = 10, εII = 80, δ = 10 Å. (1) 0.001; (2) 0.01; (3) 0.1 M NaCl. The solid lines are for the electrical potentials predicted by the new model and the dashed lines for the electrical potentials predicted by the Gouy–Chapman theory.

dielectric constant between plates. The electrical field in region I is almost constant, and its magnitude is much larger than in region II, due to the lower charge density in the former region caused by the lower dielectric constant. The electric field in region I did not change as the electrolyte concentration increased because the surface charge density was considered constant and the electrical field depended little on position in region I. 3.1.2. The effect of the dielectric constant on double-layer interaction Fig. 4 presents the effect of the dielectric constant of region I on the double-layer force. The force increases with decreasing dielectric constant. As the dielectric constant increases, the hydration free energies of the ions decrease, the ion concentrations in region I increase (Fig. 5), and the magnitudes of the surface potential and of the electric field in

Fig. 4. The repulsive force between two plates for various values of the dielectric constant εI . The parameters employed in the calculations are: σ = −0.01 C/m2 , εII = 80, δ = 10 Å, NaCl concentration = 0.01 M. (1) εI = 10; (2) ε I = 40; (3) εI = 60; (4) εI = 80.

Fig. 5. The ion distributions between two plates for various values of the dielectric constant ε I . The distance between the two plates is 100 Å. The parameters employed in the calculations are: σ = −0.01 C/m2 , εII = 80, δ = 10 Å, NaCl concentration = 0.01 M. (1) ε I = 10; (2) ε I = 20; (3) εI = 40. In the figure, A and C indicate Cl− and Na+ , respectively. The solid lines are for the ion distributions predicted by the new model and the dashed lines for those predicted by the Gouy–Chapman theory.

Nanodispersions

654 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

Fig. 6. The electrical potential distributions between two plates for various dielectric constants. The distance between the two plates is 100 Å. The other parameters employed are: σ = −0.01 C/m2 , εII = 80, δ = 10 Å, NaCl concentration = 0.01 M. (1) εI = 10; (2) εI = 20; (3) ε I = 40; (4) εI = 80 (the Gouy–Chapman theory).

185

Fig. 8. The effect of the nature of the anion on the force for low and high dielectric constants. The parameters employed in the calculations are: σ = −0.01 C/m2 , εII = 80, δ = 10 Å, electrolyte concentration = 0.01 M. The curves for NaF, NaBr, and NaCl are almost the same for both εI = 20 and 40.

region I decrease (Fig. 6). The calculations have shown that the electrostatic and the entropic contributions to the free energy decrease with increasing dielectric constant. However, the hydration contribution first increases, because of the drastic increase of the number of ions in region I, and then decreases, because of the decrease of the hydration free energies with increasing dielectric constant. Because the magnitude of the surface potential decreases, the total free energy and the force decrease with increasing dielectric constant in region I.

undetectable. However, for a dielectric constant of 60, the nature of the cation provides a specific effect on the force. Although the hydration free energy of each ion is larger at low dielectric constants, the total hydration free energy of the colloidal system is in such cases too small to affect the force between plates, because of the low ion concentrations in region I. This low total hydration free energy makes the system insensitive to specific ion effects. As the dielectric constant increases, the hydration free energy of each ion moderately decreases, whereas the ion concentrations in region I drastically increase. For this reason, the total hydration free energy of the system is not negligible and differences in the forces for various cations become noticeable, but small. Li+ has a higher hydration free energy than Na+ or K+ ; furthermore, calculations have shown that the dielectric constant affected much more than the hydration constant the concentrations of ions in region I. For these reasons, the total hydration free energy is larger for Li+ than for Na+ or K+ cations and the force for LiCl is larger than for NaCl or KCl. Fig. 8 presents the effect of hydration free energy of the anion (NaF, NaCl, and NaBr) on the force between the plates. For both high and low dielectric constants, the differences between forces for various anions are negligible. This happens because the concentration of anions in region I is very small and the differences in the hydration constants of the anions are also small. Therefore, the free energy and the force are almost the same for these anions.

3.1.3. The effect of the natures of ions on the force Table 1 shows that different ions have different hydration free energies in media with the same dielectric constant. Fig. 7 presents the dependence of the force on the distance between plates for various cations (LiCl, NaCl, and KCl) and for high and low dielectric constants in region I. When the dielectric constant is low (10 or 20), the differences between the forces for various cations are small, even

3.1.4. The effect of the thickness of region I on the force Since region I has a different dielectric constant, its thickness will significantly affect the repulsive force between the two plates. Fig. 9 shows that the force increases as the thickness of region I increases. This occurs because the electrical potential becomes more negative with increasing δ and because of the larger contribution of region I to the total free energy.

Fig. 7. The effect of the nature of cations on the force for low and high dielectric constants. The parameters employed in the calculations are: σ = −0.01 C/m2 , εII = 80, δ = 10 Å, electrolyte concentration = 0.01 M. For εI = 20, the three curves for LiCl, KCl, and NaCl are almost the same.

Polymer brushes 186

655 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

Fig. 9. The effect of the thickness of region I on the force. The parameters employed in the calculations are: σ = −0.01 C/m2 , ε I = 10, εII = 80, NaCl concentration = 0.01 M. (1) δ = 10 Å; (2) δ = 20 Å; (3) δ = 30 Å.

Fig. 10. The repulsive force between two plates at constant surface potential and various NaCl concentrations. The following parameters are employed in the calculations: Ψ0 = −0.05 V, εI = 10, ε II = 80, δ = 10 Å. NaCl concentration: (1) 0.001 M; (2) 0.01 M. The solid lines are for the force predicted by the new model and the dashed lines are for the force predicted by the Gouy–Chapman theory.

3.2. Constant surface potential 3.2.1. The effect of the dielectric constant on the force In the case of constant surface charge density, the force was larger than predicted by the classical Gouy–Chapman theory. In contrast, for constant surface potential, as shown by Fig. 10, the force predicted by the new model is smaller than that provided by the classical theory. Because of the low dielectric constant in region I, the charge density near the surface is much smaller in the new model than in the classical one.1 As a result, the electrical potential varies more rapidly with the distance to the surface (and its slope is larger near the surface) for the former model. However, because the ratio between the dielectric constant in region I and in the classical model is small, the surface charge is smaller in the new model, and the repulsion force lower. Indeed, Fig. 11 1 This can be demonstrated starting from the Poisson equation and tak-

ing into account that ρnew model  ρclassical model .

Fig. 11. The electrical potential distributions between two plates. The distance between the two plates is 100 Å. The other parameters employed are: Ψ0 = −0.05 V, εI = 10, εII = 80, δ = 10 Å, NaCl concentration = 0.01 M. (1) The new model; (2) the Gouy–Chapman theory.

Fig. 12. The repulsive force between two plates for various dielectric constants εI . The parameters employed in the calculations are: Ψ0 = −0.05 V, εII = 80, δ = 10 Å, NaCl concentration = 0.01 M. (1) ε I = 10; (2) ε I = 20; (3) εI = 40; (4) ε I = 60.

shows that the ratio of slopes in the new and classical models is 2.5, whereas the ratio of the dielectric constants is 1/8. Fig. 12 provides the dependence of the force on the dielectric constant in region I. In contrast to the case of constant surface charge density, at constant surface potential, the higher the dielectric constant of region I, the larger is the repulsion between plates. 3.2.2. The effect of ion nature on the force At high or low dielectric constants, there are small differences among the forces for various cations or anions (Figs. 13 and 14). For a dielectric constant in region I of 40, the force between the plates at a distance of 45 Å is 4.83 × 104 for LiCl, 4.78 × 104 for NaCl, and 4.77 × 104 N/m2 for NaF, hence almost the same. 3.2.3. The effect of region I thickness on the force Fig. 15 shows that the force between the two plates decreases as the thickness of region I increases. This behavior occurs because the electrical potential between the plates,

Nanodispersions

656 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

Fig. 13. The effect of the nature of cation on the force at low and high dielectric constants. The parameters employed in the calculations are: Ψ0 = −0.05 V, εII = 80, δ = 10 Å, electrolyte concentration = 0.01 M. The curves for LiCl, KCl, and NaCl are almost the same.

187

Fig. 15. The effect of the thickness of region I on the force. The parameters employed in the calculations are: Ψ0 = −0.05 V, ε I = 10, ε II = 80, NaCl concentration = 0.01 M. (1) δ = 10 Å; (2) δ = 20 Å; (3) δ = 30 Å.

where Keq is the adsorption equilibrium constant, θ is the fraction of surface sites unoccupied by cation M+ , and [M+ ]s is the concentration of M+ near the surface, which can be related to the surface potential and the bulk concentration via the equality of the chemical potentials. The surface charge density is given by σ = θ σ0 ,

(23)

where σ0 is the surface charge density when there is no adsorption of M+ onto the (negative) surface sites. Substituting Eq. (23) into Eq. (22), a boundary condition for the surface charge density generated through the dissociation equilibrium is obtained. Equations (8) and (9) can be solved numerically for the boundary conditions (11)–(14), (23), to calculate the force between plates. Fig. 14. The effect of the nature of the anion on the force at low and high dielectric constants. The parameters employed in the calculations are: Ψ0 = −0.05 V, εII = 80, δ = 10 Å, electrolyte concentration = 0.01 M. The curves for NaF, NaBr, and NaCl are almost the same.

and hence the surface charge density, increases as δ increases. Therefore, the thickness of region I has a different effect on the force at constant surface charge density and constant surface potential. 3.3. The surface charge generated through the dissociation equilibrium Let us assume that there is a dissociation equilibrium at the surface of the plate, S− + M+ ↔ S–M,

(21)

where S− is a negatively charged site on the surface, M+ is a cation which can be adsorbed on a negatively charged site, and S–M is the surface site occupied by cation M+ . At equilibrium, Keq =

1−θ , [M+ ]s θ

3.3.1. The effect of electrolyte concentration on the force As the electrolyte concentration increases, the magnitude of the surface charge density decreases, whereas the screening effect increases. However, because of the lower dielectric constant in region I, the magnitude of the surface charge density does not decrease as much as in the Gouy–Chapman theory and the screening effect is not so strong. Therefore, at high electrolyte concentrations, the force predicted by the new model is much larger than that provided by the Gouy– Chapman theory. In Fig. 16, which is for a concentration of 0.1 M NaCl, the force obtained is two orders of magnitude larger than that predicted by the Gouy–Chapman theory. It should be emphasized that the comparison was made for the same adsorption equilibrium constants, even though, in reality, they depend on the dielectric constant.

(22)

3.3.2. The effect of the dielectric constant on the force As the dielectric constant of region I increases, the concentrations of ions near the surface increase, the magnitude of the surface charge density decreases, and the force between the two plates decreases (Fig. 17). Compared to the constant surface charge case, for which only the ion concentrations in the double layer increased with increasing dielec-

Polymer brushes 188

657 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

Fig. 16. The repulsive force between two plates with adsorption equilibrium at various NaCl concentrations. The following parameters are employed in the calculations: σ0 = −0.01 C/m2 , ε I = 10, ε II = 80, δ = 10 Å, Keq = 1000. NaCl concentration: (1) 0.001; (2) 0.01 M. The solid lines are for the force predicted by the new model and the dashed lines are for the force predicted by the Gouy–Chapman theory.

Fig. 18. The repulsive force between two plates for various equilibrium constants. The parameters employed in the calculations are: σ0 = −0.01 C/m2 , εI = 10, ε II = 80, δ = 10 Å, NaCl concentration = 0.01 M. (1) Keq = 20; (2) Keq = 100; (3) Keq = 1 × 105 . The solid lines are for the forces predicted by the new model and the dashed ones for those predicted by the Gouy–Chapman theory.

4. Comparison with the Langmuir expression

Fig. 17. The repulsive force between two plates for various dielectric constants εI . The parameters employed in the calculations are: σ0 = −0.01 C/m2 , εII = 80, δ = 10 Å, NaCl concentration = 0.01 M. (1) εI = 20; (2) εI = 40; (3) εI = 60.

tric constant, in the present case the magnitude of the surface charge density has also increased. 3.3.3. The effect of adsorption constant on the force It is obvious that a larger adsorption constant will lead to a lower force. However, due to the low ion concentrations in region I, the adsorption constant has different effects on the equilibrium in the new model and in the Gouy–Chapman theory. In the new model, the ion concentrations near the surface are very low. Therefore, only a large value of the adsorption constant can affect the surface charge density. For sufficiently small adsorption constants, the magnitude of the surface charge density acquires its maximum value σ0 . In the Gouy–Chapman theory, the cation concentration near the surface is comparatively large. If the adsorption constant is also large, the magnitude of the surface charge density will be very small (Fig. 18).

As already noted, the Langmuir equation is not rigorously valid in the present case because of the nonideal term (Ai /ε I ), which appears in the chemical potential. Nevertheless, we carried out calculations using that equation and obtained in all cases differences with respect to the new model of at most 5%. This can be explained by observing that the mentioned term appears only in region I and not in region II, where the ideal approximation is considered valid. In what follows, a detailed comparison will be carried out in the Debye–Hückel approximation, for which simple analytical expressions can be derived. Expanding the right-hand side terms of Eqs. (8) and (9) in series and neglecting the higher order terms, one obtains

    ∂ 2 y κI2 1 1 1 1 = − +y + ∂x 2 2 w− w+ w− w+ (for region I)

(24)

and ∂ 2y = κII2 y ∂x 2

(for region II),

(25)

where y = eΨ /(kT ), 1/κI and 1/κII are the Debye lengths in region I and II, respectively, and

  A− 1 1 w− = exp − , (26) kT εI εII

  A+ 1 1 w+ = exp − . kT εI εII

(27)

Equations (24) and (25) can be solved analytically for the boundary condition (10) or (11) and (12)–(14). The electrical

Nanodispersions

658 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190

 potential distribution in region I is given by √  w− − w+ w− + w+ y= + C1 exp √ κI x w− + w+ 2w− w+  √  w− + w+ + C2 exp − √ κI x 2w− w+

 (w+ + w− ) (λ3 − λ2 )(e2κII(d−δ)+λ1δ − 1)

 + (λ2 + λ3 )(e 2κII(d−δ) − eλ1 δ ) ,

(28)

and in region II by y = 2g cosh(κII x),

189

(29)

(A.1)



  √ √ λ1 (d − δ) C1 = 2 w− w+ exp − 2
  λ1 δ × (λ2 + λ3 )(w− + w+ )µ exp 2

  εII   + (w+ − w− )λ2 κII exp 2κII (d − δ) − 1 εI   √ √ λ1 (d − δ) + 2 w− w+ exp − (λ2 − λ3 ) 2   λ1 δ × (w− + w+ )µ exp 2κII (d − δ) + 2 

where 2g is the electrical potential at the middle distance divided by kT /e. Expressions for the constants g, C1 , and C2 are given in Appendix A. An equation for the force is obtained by substituting Eqs. (A.1) or (A.4) into the Langmuir equation [18]:   f = 2n0 kT cosh(2g) − 1 . (30) For the case in which the surface charge is a result of dissociation equilibrium, one can use the same equation as for constant surface charge combined with the equilibrium equations (22) and (23) and the equality of the chemical potential at the surface and in bulk (kT ln[M+ ]s + ezi Ψ0 + Ai /εI = kT ln[M+ ] + Ai /ε II ). These calculations have shown that the differences between the exact numerical results and Eq. (30) are less than 5%.

κI (w− + w+ )3/2     × (λ2 − λ3 ) 1 − exp 2κII (d − δ) + λ1 δ      + (λ2 + λ3 ) exp 2κII (d − δ) + exp(λ1 δ) , (A.2) 

5. Conclusions In this paper, a model which takes into account the effect of a lower dielectric constant near a surface due to the presence of hairs on the surface is developed. At constant surface charge, the force between the two plates is larger than predicted by the Gouy–Chapman theory. As the dielectric constant increases, the force decreases. The nature of cations has a specific effect on the force between plates. At constant surface potential, the force provided by the new model is smaller than predicted by the Gouy–Chapman theory. In contrast, at constant surface charge density, the dielectric constant and the thickness of region I have effects opposite to those at constant surface potential. When the surface charge is generated through dissociation equilibrium, the difference between the forces calculated using the new model and the Gouy–Chapman theory is much larger than for constant charge density or constant surface potential.

Appendix A For constant surface charge density  g = eκII (d−δ) λ2 (eδλ1 − 1)(w+ − w− ) √ + 4εI(w+ + w− )µ w− w+ eδλ1 /2

C2 =

  √ √ λ1 d λ2 κII εII 2 w− w+ exp (w− − w+ ) 2 εI
    λ1 δ λ1 δ × exp − exp 2κII (d − δ) + 2 2   √ √ λ1 d + 2 w− w+ exp (w− + w+ )µ 2      × (λ2 + λ3 ) exp 2κII (d − δ) + (λ2 − λ3 )

 κI (w− + w+ )3/2     × (λ2 − λ3 ) 1 − exp 2κII (d − δ) + λ1 δ      + (λ2 + λ3 ) exp 2κII (d − δ) + exp(λ1 δ) , (A.3) and for constant surface potential  g = λ2 κII (d − δ) (eδλ1 + 1)(w− − w+ )   + 2eδλ1/2 (y0 − 1)w− + (y0 + 1)w+   (w+ + w− ) (λ2 − λ3 )(e2κII(d−δ)+λ1δ + 1)  + (λ2 + λ3 )(e2κII (d−δ) + eλ1 δ ) , (A.4)

Polymer brushes 190

659 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 273 (2004) 181–190



  λ1 (−d + 3δ)  C1 = exp (λ2 − λ3 )y0 (w− + w+ ) 2

where  √ λ1 = − 2(w− + w+ )κI / w− w+ ,

+ (λ3 − λ2 )(w− − w+ )   λ1 (−d + 2δ) + exp − 2κII (d − δ) 2  × (λ2 + λ3 ) y0 (w− + w+ ) − w− + w+   + exp −2κII (d − δ) λ3 (w− − w+ )   + λ3 (w+ − w− )

λ2 =



    × (λ2 + λ3 ) exp −2κII (d − δ) + δλ1 + 1      + (λ2 − λ3 ) exp 2κII (−d + δ) + exp(λ1 δ) , (A.5)

  λ1 d exp − 2κII (d − δ) (λ2 − λ3 ) 2    λ1 d + exp (λ2 + λ3 ) 2  × y0 (w− + w+ ) − w− + w+  + exp −2κII (d − δ) + λ1 δ λ3 (w+ − w− )  + exp(λ1 δ)λ3 (w− − w+ )

 (w− + w+ )      × (λ2 + λ3 ) exp −2κII (d − δ) + δλ1 + 1      + (λ2 − λ3 ) exp 2κII (−d + δ) + exp(λ1 δ) , (A.6)

(A.8) (A.9)

µ=

eσ , kT ε I ε0

(A.10)

y0 =

eΨ0 . kT

(A.11)

(w− + w+ )

C2 =

2(w− + w+ )κI εI ,

√ λ3 = 2κII εII w− w+ ,








(A.7)

References [1] D.H. Napper, Polymeric Stabilization of Colloidal Dispersions, Academic Press, New York, 1983. [2] R.J. Hunter, Foundations of Colloid Science, Vol. 1, Oxford Science Publications, Oxford, 1987. [3] H. Casanova, J. Chen, E. Dickinson, B.S. Murray, P.V. Nelson, M. Whittle, Phys. Chem. Chem. Phys. 2 (2000) 3861. [4] N.S. Bell, J. Sindel, F. Aldinger, W.M. Sigmund, J. Colloid Interface Sci. 254 (2002) 296. [5] Y.L. Ong, A. Razatos, G. Georgiou, M.M. Sharma, Langmuir 15 (1999) 2719. [6] S. Alexander, J. Phys. 38 (1977) 983. [7] P.G. de Gennes, Adv. Colloid Interface Sci. 27 (1987) 189. [8] E. Ruckenstein, B. Li, J. Chem. Phys. 107 (1997) 932. [9] B.E. Conway, J.O’M. Bockris, I.A. Ammar, Trans. Faraday Soc. 47 (1951) 756. [10] M.J. Spaarnay, Rec. Trav. Chim. 77 (1958) 382. [11] Y. Gur, I. Ravina, A.J. Babchin, J. Colloid Interface Sci. 64 (1978) 333. [12] S. Basu, M. Sharma, J. Colloid Interface Sci. 165 (1994) 355. [13] G. Nemethy, H.A. Scheraga, J. Chem. Phys. 36 (1962) 3382. [14] R.M. Noyes, J. Am. Chem. Soc. 84 (1962) 513. [15] R.H. Stokes, J. Am. Chem. Soc. 86 (1964) 979. [16] J.Th.G. Overbeek, Colloids Surf. 51 (1990) 61. [17] M. Manciu, E. Ruckenstein, Langmuir 19 (2003) 1114. [18] I. Langmuir, J. Chem. Phys. 6 (1938) 893.

Nanodispersions

660

Journal of Colloid and Interface Science 275 (2004) 548–554 www.elsevier.com/locate/jcis

Double layer interaction between two plates with polyelectrolyte brushes Haohao Huang and Eli Ruckenstein ∗ Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 19 December 2003; accepted 24 February 2004 Available online 1 April 2004

Abstract When polyelectrolyte chains are grafted to colloidal particles, the electric field between particles is affected by the charges of the chains. In some previous theoretical attempts, the charge density of the polyelectrolyte chains per unit length was considered constant, and its effect was accounted for by introducing an additional constant charge density into the unidimensional Poisson–Boltzmann equation, which was evaluated assuming that it is uniformly distributed in the polyelectrolyte volume of the brush. In this paper, a more detailed model is employed for the calculation of the electrical potential between two plates on which polyelectrolyte brushes are present. In this model, the polyelectrolyte chain is viewed as a rigid cylinder, on the surface of which charges are generated through the dissociation of ionizable sites and adsorption of the cations of the electrolyte. To each of the chains an atmosphere is attached which for simplicity is assumed cylindrical. In the brush region, the electrical potential is described by a two-dimensional Poisson–Boltzmann equation, while in the region free of polyelectrolyte chains by a unidimensional Poisson–Boltzmann equation. Such a model is physically suitable when the charges of the chains are sufficiently large for the repulsion they generate to ensure that the chains are fully extended. Such cases are quite frequent, because relatively low charges lead to an almost complete extension of the chains. In this paper, both the plate surface and the surface of the cylinders are considered charged. The effects of electrolyte concentration, pH, brush thickness and chain coverage density on the repulsion between plates are examined.  2004 Elsevier Inc. All rights reserved. Keywords: Polyelectrolyte; Repulsion force; Cylindrical model

1. Introduction Colloidal dispersions can be stabilized by attaching polymer chains to their surface [1–9]. When neutral polymer chains grafted on two parallel plates interpenetrate, a steric repulsion is generated. If the polymer chains grafted to the plates are charged, the double layer interaction between the two plates is also affected by the presence of the chains. Using a self-consistent field theory, Misra et al. [10] examined the effect of the brush charge, electrolyte concentration, and surface charge density on the brush thickness. They extended the self-consistent field polymer brush theory suggested by Milner et al. [11] to the case of a polyelectrolyte brush. The theory involves a parabolic monomer concentration profile rather than the step-function suggested by Alexander [12] and de Gennes [13,14]. The repulsion force * Corresponding author. Fax: +1-716-645-3822.

E-mail address: [email protected] (E. Ruckenstein). 0021-9797/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.02.058

between two brushes in poor solvents was also calculated using a similar approach [15]. In some previous calculations [16–18] of the interactions involving polyelectrolyte chains grafted to two surfaces, the charge of the polyelectrolyte chains was assumed to be constant and this fixed charge density was introduced into a one-dimensional Poisson–Boltzmann equation to calculate the electrical potential profile. All the above treatments involved a unidimensional model. In this paper, a two-dimensional model is proposed, in which the polyelectrolyte chains are treated as rigid cylinders normally grafted to each of the plates. On the surface of the cylinders, ionizable sites are uniformly distributed and the surface charge of the cylinder is generated through their dissociation. Using this two-dimensional model, the repulsive force between two plates is calculated as a function of electrolyte concentration, surface density of polyelectrolyte chains on the plate, thickness of the polyelectrolyte brushes, and bulk pH (denoted in what follows as pH0 ).

Polymer brushes

661 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 275 (2004) 548–554

549

2. The model Polyelectrolyte chains possessing two kinds of ionizable sites, such as the protein molecules, are considered to be grafted to a charged plate surface. The positive and negative charges on the chain surface arise from the dissociation of two kinds of ionizable sites. For sufficiently high charge densities, the polymer chains become fully extended and hence the chains can be viewed as cylinders normal to the plate surface. One end of the cylinder is located on the plate and its height is equal to the length of the polyelectrolyte chain. To each of the cylinders an atmosphere is associated by dividing the space between cylinders in equal parallelepipeds. To simplify the treatment, the external boundary of the atmosphere will be replaced by a cylinder of equal volume of radius R2 (Fig. 1). The height h of this cylindrical atmosphere is equal to the completely extended length of the polyelectrolyte molecules. The space between the plates is divided into two regions. One of them, region I, contains the polyelectrolyte chains, whereas the other one, region II, is free of chains. Because in region I, the boundary of the atmosphere around each polymer chain was approximated with a cylinder, the electrical potential is related to the electrolyte concentration via a Poisson–Boltzmann equation in cylindrical coordinates,
 1 ∂ ∂Ψ ∂ 2Ψ 2(c0 + ch )eN0 sinh(eΨ/kT ) r + = , r ∂r ∂r ∂z2 εε0

(1)

where Ψ is the electrical potential, r and z are the radial and normal coordinates, c0 is the bulk electrolyte (NaCl) concentration, ch is the bulk acid concentration which is employed to change the pH, N0 is the Avogadro number, e is the protonic charge, k is the Boltzmann constant, T is the absolute temperature, ε is the dielectric constant of water, and ε0 is the vacuum permittivity. Equation (1) will be solved for the cell between the two cylinders of radii R1 and R2 and height h using suitable boundary conditions which are provided below. The boundary condition for the Poisson–Boltzmann equation on the chain surface is  ∂Ψ  σ (z) =− , ∂r r=R1 εε0

(2)

where R1 is the radius of the cylindrical chain, and σ is the surface charge density on the surface of the cylinder of radius R1 . At the plate surface, the boundary condition is  σp ∂Ψ  =− ,  ∂z z=0 εε0

(3)

where σp is the free plate surface charge density which is assumed constant in the calculation.

Fig. 1. Schematics of the stretched polyelectrolyte chains on the plate surface. The stretched polyelectrolyte chain is represented as a cylinder. The chains are assumed to have the same length.

For reason of symmetry, at the boundary of the cylindrical atmosphere,  ∂Ψ  = 0. (4) ∂r r=R2 The dissociation of the two kinds of ionizable sites provides the chain surface with positive and negative charges through the dissociation equilibria [19] + Ga H ↔ G− a +H , +

+

(5)

Gb H ↔ Gb + H ,

(6)

Ga Na ↔ G− a

(7)

+

+ Na ,

where Ga H and Gb are the ionizable sites which generate the + negative G− a and positive Gb H charges, respectively, and + + H and Na are the hydrogen and sodium ions, respectively. By neglecting the interactions between species, one can write the equilibria (1 − θa − θNa )[H+ ]S , θa (1 − θb )[H+ ]S Kb ≡ , θb (1 − θa − θNa )[Na+ ]S KNa ≡ , θNa Ka ≡

(8) (9) (10)

where Ka , Kb , and KNa are equilibrium constants (the values pKa = 3.4, pKb = 12, and KNa = 1 × 108 M, obtained

Nanodispersions

662 550

H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 275 (2004) 548–554

in Ref. [19] by fitting the experimental data for proteins, are employed in the calculation), θa and θNa denote the fractions of sites Ga occupied by H+ and Na+ , respectively, θb is the fraction of sites Gb occupied by H+ , and [H+ ]S is the concentration of hydrogen ions near the surface of the chains. The net charge density σ on the chain cylindrical surface is the difference between the positive and negative surface charge densities   σ = e Nb θb − Na (1 − θa − θNa ) , (11) where Na and Nb are the numbers of ionizable sites Ga H and Gb per unit area, respectively. Combined with Eqs. (8)–(10) and assuming Boltzmann distributions, Eq. (11) becomes   s Nb [H+ ]0 exp − eΨ σ kT =   s e Kb + [H+ ]0 exp − eΨ kT −

Na Ka KNa

(KNa [H+ ]0

[H+ ]

,   s + Ka [Na+ ]0 ) exp − eΨ kT + Ka KNa (12)

[Na+ ]0

where are the bulk concentrations of the 0 and corresponding ions and Ψs is the surface electrical potential. In the second region, region II, which is outside the polyelectrolyte brush, the electrical potential will be assumed to be a one-dimensional function of z, independent of r. Two boundary conditions at the boundary between the two regions can be written. In one of them, the electrical potential at the boundary in the outside region is assumed to be equal to the average of the electrical potential at the boundary of the brush region. In the other one, the electric field at the boundary of the outside region is assumed to be equal to the average of the field at the boundary of the brush region, II Ψ

2 = 2 z=h R2 − R12

R2

I rΨ z=h dr,

 dΨ  = 0, dz z=d

(16)

where 2d is the distance between the plates. Using the boundary conditions (2)–(4), (13), (14), and (16), the Poisson–Boltzmann equations (1) and (15) can be solved numerically to obtain the electrical potential profile. Using the electrical potential in the middle between the two plates, the force per unit area can be calculated using the Langmuir expression [20]


eΨ0 F = 2kT (c0 + ch )N0 cosh −1 , (17) kT where Ψ0 is the electrical potential at the middle between the two plates.

3. Results and discussion 3.1. The effect of the electrolyte concentration on the force The electrolyte concentration affects the repulsive force between the plates through the screening effect. As the electrolyte concentration increases, the magnitudes of the electrical potential in the brush and the outside regions decrease (Fig. 2). Therefore, as expected, the force decreases with increasing electrolyte concentration (Fig. 3). For comparison purposes, the figure also contains the force between two plates free of polyelectrolyte chains but with the same surface charge density σp (−0.01 C/m2 ) on the plates as when the chains are present. Figs. 4 and 5 present the chain surface electrical potentials Ψs and the chain surface charge density σ at various electrolyte concentrations, and pH0 = 7, respectively. For the conditions employed, the charge of the polyelectrolyte

(13)

R1

  R2 dΨ II 2 ∂Ψ I = r dr, dz z=h R22 − R12 ∂z z=h

(14)

R1

where I and II represent the brush and the outside regions, respectively. In reality, there is a transition region in which the potential in region II satisfies a two-dimensional Poisson–Boltzmann equation before becoming one-dimensional. However, for the sake of simplicity, the thickness of that region is considered small. In the outside region, the Poisson–Boltzmann equation has the form d 2Ψ 2(c0 + ch )eN0 sinh(eΨ/kT ) = . 2 dz εε0

(15)

Because of the symmetry of the system, there is a boundary condition in the middle between the two plates,

R Fig. 2. The average electrical potential Ψ¯ = R 2 2Ψ dr/(R22 − R12 ) for var1 ious electrolyte concentrations c0 . The parameters used in the calculation are pKa = 3.4, pKb = 12, Na = 8 × 1013 cm−2 , Nb = 4 × 1013 cm−2 , pH0 = 7, h = 40 Å, R1 = 5 Å, R2 = 25 Å, d = 51 Å, σp = −0.01 C/m2 . The dashed lines are for two bare plates.

Polymer brushes

663 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 275 (2004) 548–554

Fig. 3. The repulsive force between two plates at various electrolyte concentrations c0 . The parameters used in the calculation are pKa = 3.4, pKb = 12, Na = 8 ×1013 cm−2 , Nb = 4 × 1013 cm−2 , pH0 = 7, h = 40 Å, R1 = 5 Å, R2 = 25 Å, σp = −0.01 C/m2 . The dashed lines are for two bare plates.

551

Fig. 5. The chain surface charge density for various electrolyte concentrations. The distance between the two plates is 100 Å. The other parameters are as in Fig. 2.

Fig. 6. The local pH near the chain surface. The distance between the two plates is 100 Å. The other parameters are as in Fig. 2. Fig. 4. The chain surface electrical potential at various electrolyte concentrations. The distance between the two plates is 100 Å. The other parameters are as in Fig. 2.

chains and the sign of the surface potential of the chains are negative. As expected, the same sign of the chain charge as that of the plate increases the repulsive force between the two plates. For higher electrolyte concentrations, the chains have less negative surface electrical potentials. Because in our calculations the surface charge density of the plate σp is negative, the potential at the end of the chain near the plate is more negative than at the free end. The electrical potential being more negative near the plates, the concentration of H+ ions near the chains in the vicinity of the plate are higher (i.e., lower local pH; see Fig. 6) than further from the plate. Through the dissociation equilibria, this provides lower dissociations of Ga H and Gb H+ near the plates than near the free end. As a result, the chain surface charge density σ is not uniformly distributed, being more negative near the free

end. This variation is stronger for low electrolyte concentrations. 3.2. The effect of the bulk pH (pH0 ) The pH0 changes the surface charge density on the chain through the dissociation equilibria. Fig. 7 shows that when pH0 increases from 2 to 7, the force between the two plates first decreases, passes through a minimum, and then increases. Figs. 8 and 9 allow to explain this behavior. Fig. 8 shows that the surface charge density σ on the chain is positive for pH0 = 2.0 and pH0 = 3.0, but becomes negative for pH0 = 5.0 and 7.0. This occurs because the surface charge of the chain is generated through the dissociation of surface sites. As the pH0 increases, the number of hydrogen ions dissociated from the sites increases. As a result, the surface charge becomes negative at high values of pH0 and posi-

Nanodispersions

664 552

H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 275 (2004) 548–554

Fig. 7. The repulsive force between two plates for various values of pH0 . The parameters used in the calculations are pKa = 3.4, pKb = 12, Na = 8 × 1013 cm−2 , Nb = 4 × 1013 cm−2 , R1 = 5 Å, R2 = 25 Å, h = 40 Å, c0 = 0.01 M, σp = −0.01 C/m2 . The dashed lines are for two bare plates.

Fig. 8. The surface charge density along the chain for various values of pH0 . The distance between the two plates is 100 Å. The other parameters are as in Fig. 7.

tive at low values (see Eq. (12)). Consequently, the surface charge density σ of the chain is positive for both pH0 2.0 and 3.0 (but larger for pH0 = 2.0 than for pH0 = 3.0) and becomes negative for pH0 = 5.0 or 7.0. Because the charge of the plate is assumed negative, the force between the plates decreases at low pH0 values and increases at larger values, as pH0 increases. The repulsion force between two bare plates of charge density −0.01 C/m2 is located between those for pH0 = 2.0 and pH0 = 5.0 for plates covered with brushes. The local pH (Fig. 10) along the chain near the plate is affected by the surface charge of the plates, which being negative attracts H+ ions. The latter ions can associate to the chain which thus becomes less negatively charged near the plate when the chains are negatively charged (high pH0 ),

Fig. 9. The surface electrical potential along the chain for various values of pH0 . The distance between the two plates is 100 Å. The other parameters are as in Fig. 7.

Fig. 10. The local pH near the chain surface. The distance between the two plates is 100 Å. The other parameters are as in Fig. 7.

or more positively charged when the chains are positively charged (low pH0 ). When the chains are negatively charged, the electrical potential at the free end of the chain is less negative than near the plate. This leads to a lower H+ concentration and to a more negative surface charge density on the chains near the free end than near the plate. It should be also noted that at high pH0 the chain surface charge density decreases and the chain surface electrical potential increases along the chain monotonously. However, at low pH0 values, the surface charge density on the chains first decreases and then increases, and the surface electrical potential first increases and then decreases. This occurs because the surface charge of the chains has the same sign as the plates at high pH0 and the opposite sign at low values. For low pH0 values, the electrical potential of the chains is positive, and repulses the hydrogen ions. As a result, the local pH near the chain surface is larger than the bulk pH0

Polymer brushes

665 H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 275 (2004) 548–554

553

Fig. 11. The effect of the thickness of the polyelectrolyte brush h on the repulsive force. The parameters used in the calculations are pKa = 3.4, pKb = 12, Na = 8×1013 cm−2 , Nb = 4×1013 cm−2 , pH0 = 7, R1 = 5 Å, R2 = 25 Å, c0 = 0.01 M, σp = −0.01 C/m2 . The dashed line is for two bare plates.

Fig. 12. The effect of the coverage density on the repulsive force. The parameters used in the calculations are pKa = 3.4, pKb = 12, Na = 8 × 1013 cm−2 , Nb = 4 × 1013 cm−2 , pH0 = 7, R1 = 5 Å, h = 40 Å, c0 = 0.01 M, σp = −0.01 C/m2 .

for pH0 = 2.0 or 3.0. For pH0 = 5.0 or 7.0, the local pH near the chain surface is lower than the bulk pH0 , because the surface electrical potential is negative.

Table 1 The chain surface charge density σa used in Eq. (19) to obtain the same force as the one calculated from the two-dimensional model

3.3. The effect of the thickness of the polyelectrolyte brush Fig. 11 presents the effect of the thickness of the polyelectrolyte brush on the repulsive force for pH0 = 7 and c0 = 0.01 M. For large thicknesses of the brush, the force is larger than for small thicknesses. The calculations also indicated that, for h > 15 Å and pH0 = 2 or 9, the force increases as the thickness of the brush increases. 3.4. The effect of the chain density of the polyelectrolyte chains on the plates

pH0

103 σa (C/m2 )

7.0 5.0 3.0 2.0

−69 −36 −7.5 0.98

the chains will collapse on the plate surface. The present model is no longer valid under such conditions. 3.6. Comparison with the traditional treatment

The effect of the density of the polyelectrolyte chains on the force is presented in Fig. 12 for pH0 = 7. As expected, the force between the two plates increases with increasing coverage density. This occurs because the decrease of R2 increases the number of chains per unit area and the magnitude of the average electrical potential due to the interaction between the chains in the brush.

In the traditional treatment, the electrical potential is assumed unidimensional and a uniform charge density is introduced into the Poisson–Boltzmann equation to account for the presence of the charged chains. If the chain surface has an average charge density σa , the uniform charge density ρun can be evaluated as the total surface charge of a chain divided by the brush volume of the cylinder of radius R2 and height h,

3.5. Collapse of the polyelectrolyte brush

ρun =

Under certain conditions the polyelectrolyte brush has the tendency to collapse on the plate surface. In Figs. 8 and 9, one can see that the chain surface charge has a positive value near the plate for pH0 = 2.0 or 3.0; however, because the plate has a negative charge, the electric field in this region is negative. This means that the polyelectrolyte segments are attracted by the plates. If the repulsion between the monomers of the chain is weaker than the attractive force between the oppositely charged plate surface and monomers,

2πR1 hσa πR22 h

=

2R1 σa R22

.

(18)

Using this expression for the uniform charge density, the Poisson–Boltzmann equation in region I acquires the form 2(c0 + ch )eN0 sinh(eΨ/kT ) + 2R1 σa /R22 ∂ 2Ψ = . εε0 ∂z2

(19)

By equating the force provided by the two-dimensional model to that obtained from Eq. (19), the values listed in Table 1 have been obtained for σa as a function of pH0 .

Nanodispersions

666 554

H. Huang, E. Ruckenstein / Journal of Colloid and Interface Science 275 (2004) 548–554

Table 2 The average chain surface charge density σ¯ for various pH0 values and various distances between the two platesa 103 σ¯ (C/m2 ) 2d (Å)

pH0 = 2.0

pH0 = 3.0

pH0 = 5.0

pH0 = 7.0

86 88 90 92 94 96 98 100 102

40.455 40.631 40.766 40.871 40.956 41.024 41.079 41.124 41.160

16.074 16.147 16.202 16.244 16.278 16.304 16.325 16.342 16.355

−28.176 −28.394 −28.585 −28.755 −28.908 −29.047 −29.174 −29.290 −29.397

−62.257 −62.281 −62.301 −62.318 −62.332 −62.345 −62.356 −62.366 −62.376

a The parameters used in the calculations are c = 0.01 M, pK = 3.4, pK = 12, N = 8 × 1013 cm−2 , N = 4 × 1013 cm−2 , h = 40 Å, R = 5 Å, a a 0 b b 1 R2 = 25 Å, σp = −0.01 C/m2 .

Let us now compare the value of σa to the average values σ¯ provided by the expression σ¯ =

1 h

ness of the polyelectrolyte brush and the coverage of the plate, the larger the repulsive force.

h σ (z) dz,

(20)

References

0

where σ is the surface charge density on a chain calculated on the basis of the two-dimensional model. Table 2 lists the values obtained. Comparing the two tables, one can see that σa is somewhat different from σ¯ . This indicates that the one-dimensional model approximation can be unsatisfactory. In the extreme case in which the sites are completely dissociated, one can easily evaluate σa , which becomes equal to e(Nb − Na ).

4. Conclusions In this paper, a two-dimensional model is employed to describe the repulsive interactions between two parallel plates to which polyelectrolyte chains are grafted. The chains are assumed to be completely extended. The charge is generated through the dissociation of the ionizable sites of the chains. The force between the two plates at distances larger than twice the thickness of the brush of chains was calculated. The calculation revealed that with increasing bulk pH0 , the force can pass through a minimum, and the larger the thick-

[1] D.H. Napper, Polymeric Stabilization of Colloidal Dispersions, Academic Press, New York, 1983. [2] H. Casanova, J. Chen, E. Dickinson, B.S. Murray, P.V. Nelson, M. Whittle, Phys. Chem. Chem. Phys. 2 (2000) 3861. [3] N.S. Bell, J. Sindel, F. Aldinger, W.M. Sigmund, J. Colloid Interface Sci. 254 (2002) 296. [4] Y.L. Ong, A. Razatos, G. Georgiou, M.M. Sharma, Langmuir 15 (1999) 2719. [5] S. Alexander, J. Phys. 38 (1977) 983. [6] P.G. de Gennes, Adv. Colloid Interface Sci. 27 (1987) 189. [7] E. Ruckenstein, B. Li, J. Chem. Phys. 107 (1997) 932. [8] E. Zhulina, O. Borisov, V. Priamitsyn, J. Colloid Interface Sci. 137 (1990) 495. [9] M. Muthukumar, J. Ho, Macromolecules 22 (1989) 965. [10] S. Misra, S. Varanasi, P.P. Varanasi, Macromolecules 22 (1989) 4173. [11] S.T. Milner, T.A. Witten, M.E. Cates, Macromolecules 21 (1988) 2610. [12] S. Alexander, J. Phys. 38 (1977) 983. [13] P.G. de Gennes, J. Phys. 37 (1976) 1443. [14] P.G. de Gennes, Macromolecules 13 (1980) 1069. [15] S. Misra, S. Varanasi, W. Mattice, Macromolecules 29 (1996) 6055. [16] H. Ohshima, Colloid Polym. Sci. 277 (1999) 535. [17] D.A. Hoagland, Macromolecules 23 (1990) 2781. [18] D. Dean, J. Seog, C. Ortiz, A.J. Grodzinsky, Langmuir 19 (2003) 5526. [19] D. Prieve, E. Ruckenstein, J. Theor. Biol. 56 (1976) 205. [20] I. Langmuir, J. Chem. Phys. 6 (1938) 893.

Polymer brushes

667

Advances in Colloid and Interface Science 112 (2004) 37 – 47 www.elsevier.com/locate/cis

The bridging force between two plates by polyelectrolyte chains Haohao Huang, Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, 303 Furnas Hall, Buffalo NY 14260, USA Available online 14 August 2004

Abstract The interaction between particles in a colloidal system can be significantly affected by their bridging by polyelectrolyte chains. In this paper, the bridging is investigated by using a self-consistent field approach which takes into account the van der Waals interactions between the segments of the polyelectrolyte molecules and the plates, as well as the electrostatic and volume exclusion interactions. A positive contribution to the force between two plates is generated by the van der Waals interactions between the segments and the plates. This positive repulsive contribution plays an important role in the force when the distances between the plates are small. With increasing van der Waals interaction strength between segments and plates, the force between the plates becomes more repulsive at small distances and more attractive at large distances. When the surfaces of the plates have a constant surface electrical potential and a charge sign opposite to that of the polyelectrolyte chains, the force between the two plates becomes less attractive as the bulk polyelectrolyte concentration increases. This behavior is due to a higher bulk counterion concentration dissociated from the polyelectrolyte molecules. At short distances, the force between plates is more repulsive for stiffer chains. A comparison between theoretical and experimental results regarding the contraction of the interlayer separation between the platelets of vermiculite clays against the concentration of poly(vinyl methyl ether) was made. D 2004 Elsevier B.V. All rights reserved. Keywords: Bridging interaction; Polyelectrolyte; Self-consistent field theory; van der Waals interactions; Repulsive force

Contents 1. 2. 3. 4.

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-consistent field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutral polymer between two plates . . . . . . . . . . . . . . . . . . . . . . . . . Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The effect of the van der Waals interactions between plates and segments . . 4.2. The effect of salt concentration . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The effect of the bulk polymer concentration . . . . . . . . . . . . . . . . . 4.4. The effect of the stiffness of the chains. . . . . . . . . . . . . . . . . . . . 4.5. The effect of the charge density of the chains . . . . . . . . . . . . . . . . 4.6. The effect of the plate surface charge density . . . . . . . . . . . . . . . . 4.7. Polyelectrolyte chains generate their charge through dissociation equilibrium 5. Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* Corresponding author. Tel.: +1 716 645 2911x2214; fax: +1 716 645 3822. E-mail address: [email protected] (E. Ruckenstein). 0001-8686/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2004.06.003

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

38 38 40 41 41 42 43 43 44 45 45 45 46 46

Nanodispersions

668 38

H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

1. Introduction Polymers are often used to stabilize colloidal systems by grafting them on the particle surfaces to provide steric repulsion [1,2]. Polymers can also induce flocculation due to either bdepletionQ or bbridgingQ interactions [3]. The bridging interaction occurs because monomers of the same polymer chain are adsorbed on different particles and draw them together [4,5]. The bridging was extensively studied experimentally [6–8], theoretically [9–11] and by Monte Carlo simulations [12–16]. It was exploited practically for water purification by adding high molecular weight polyacrylamides to flocculate the small particles suspended in water [17]. However, the strength of the bridging could not be quantitatively expressed. Recently, it was noted that the addition of poly(vinyl methyl ether) to a clay–salt–water system induced the contraction of the interlayer separation between the clay platelets [18]. The n-butylammonium vermiculite used in those experiments provided an ideal structure of regularly spaced and parallel charged colloidal platelets in a salt solution [19,20]. Swenson et al. [21–23] studied the effect of the addition of poly(ethylene oxide) (PEO) on the distance between the vermiculite layers and observed that, with increasing PEO concentration, the distance between the vermiculite layers decreased. They estimated that the bridging force per bridge was 1.4 pN. Polyelectrolyte molecules affect in a more complicated manner than the neutral ones the bridging interactions between colloidal particles because they influence through their dissociation the electrical potential of the electrical double layer [24–27] which, in turn, affects the conformation of the chains and the overall electrostatic interaction. A number of theoretical approaches which can describe the segment concentration distribution of polymers adsorbed on the surface of a plate or the surfaces of two interacting plates [28–44] were proposed. Edwards [28] employed a continuous self-consistent field theory to obtain the segment distribution in a solution of flexible neutral polymers adsorbed on a plate. Scheutjens and Fleer [32,33] developed a lattice model to examine the polymer adsorption on a plate. Skau et al. [42] compared the two approaches and obtained quantitative agreement for the segment density profile in a good solvent. Li and Ruckenstein [43] employed a lattice model to calculate the interaction between two grafted polymer brushes with the emphasis on the interdigitation between the chains of the two brushes [44]. Monte Carlo simulations were also carried out to investigate the effect of grafted polyelectrolyte on the interaction between two colloidal particles [45]. Using a self-consistent field theory, Varoqui et al. [46] calculated the segment density profile of polyelectrolyte molecules adsorbed on one plate and Podgornik [9] extended the procedure to the interaction between two plates.

Borukhov et al. [11] investigated the effect of polyelectrolyte adsorption on the intercolloidal forces. They assume that the adsorption of polyelectrolyte molecules on the surface of the plates was solely due to the electrostatic interactions between the negatively charged plates and the positively charged polyelectrolyte segments. However, there are additional interactions between the surfaces of the plates and the segments, such as the van der Waals interactions. When the polyelectrolyte chains form bridges between two plates, the plates are pulled towards each other by the chains. If the van der Waals interactions between the segments of the chains and the surfaces are sufficiently strong, the increased polymer concentration in the gap between the plates can significantly affect the bridging force between them. A self-consistent field theory was employed by us to calculate the bridging force between two plates immersed in a polyelectrolyte solution, following Borukhov et al.’s [11] approach. However, in the calculations, the van der Waals interactions between plates and segments were also taken into account. It was shown that the effect of the latter interactions is important.

2. Self-consistent field theory Two infinite-size plates are immersed in a semidilute solution of polyelectrolyte in a good solvent which also contains the small ions of a salt. One of the plates is located at x=0 and the other one at x=D. The system is considered to be in contact with a large reservoir, which contains a polyelectrolyte/salt solution. In addition to the electrostatic interactions, the segments of the polymer have a van der Waals interaction
UkT with the plates. In the mean field approach, the intra- and interchain interactions together with the electric field induced by the surface charges of the plates and polyelectrolyte molecules are expressed as an external potential. Within the mean-field approximation, the free energy of the system with respect to that in the reservoir can be expressed as the sum of three contributions, the polymer contribution F pol, the salt ions contribution F ion and the electrostatic field contribution F ele, F¼

Z

D

hdx ¼ Fpol þ Fion þ Fele

ð1Þ

0

where h is the free energy density. The polymer contribution F pol includes the effect of the polymer chain connectivity and the short-range excluded volume interaction 2  Z D 2  a du 1 4 4 2 2 2 bFpol ¼ þ vðu
ub Þ
ðu
ub Þlp
U u dx 6 dx 2 0 ð2Þ

Polymer brushes

669 H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

where a is the Kuhn segment length of the chain, b=1/kT, k is the Boltzmann constant, T is the absolute temperature, v is the excluded volume parameter, u is the bpolymer order parameter,Q which is related to the local concentration c p(x) of the segments of the polymer chains through c p(x)=u 2(x), kTl p is the chemical potential of the segments of the chains and the subscript b indicates the bulk. The first term in Eq. (2) is due to the connectivity of the segments of the polymer chain, the second term to the excluded volume interaction between segments, the third term to the concentration difference between a point in the gap and the bulk and the fourth term to the van der Waals interaction between plates and segments. A larger a means a stiffer chain. For a good solvent, v has a positive value, while for a poor solvent, v has a negative value, i.e., the interaction between segments becomes attractive. In the h condition, v is zero. In the following calculations, v is assumed a positive constant, independent of the concentration of the salt in solution. The chemical potential kTl p can be obtained from the kTlp =vkTu b2. The ionic contribution to the free energy of the system is due to the entropic increase of the small ions between the plates compared to that in the reservoir.

bFion ¼

2

Z

D 0

Z

D

c

xi 6 X ci ln 4 x i;b i X

0

i

!

1
þ cw ln

1


 ci ci ln
ci þ ci;b dx ci;b

X i X

xi

xi;b

3 7 5dx

1 ee0 2

Z

D 0



ð3Þ

ð4Þ

dw dx

2 dx
2rw0

Fele

1 ¼ ee0 2

Z D 2 Z dw dx
ee0 0 dx

¼
ee0

Z

D 0

"

d2 w 1 w 2 þ dx 2



D 0

dw dx

  d2 w dw 2 w 2 þ dx dx dx

2 # dx

ð6Þ

Because of the Poisson equation, d2 w q ¼
dx2 ee0

ð7Þ

where q is the charge density in the solution, Eq. (6) becomes   # Z D"  1 dw 2 þ
2 Fele ¼ ew ðc
c þ au
dx ð8Þ 2 dx 0 where c + and c
are the concentrations of the cations and anions, respectively. If the surface charge density changes with the distance between the plates through equilibrium, the chemical contribution to the free energy is given by [47] Z

rð DÞ

W0 dr

ð9Þ

rðlÞ

where r is the surface charge density. Therefore, the electrostatic free energy contribution can be expressed as

Fele ¼

Using the Gauss law and the Green theorem, the electrostatic contribution can be rewritten as

Fchem ¼
2rð D ¼ lÞWð D ¼ lÞ
2

i

where c i is the concentration of ion i, cw is the concentration of water and x i is the mole fraction of ion i. In the following calculations, two kinds of small ions are present in solution: the ions of the salt and the counterions dissociated from the polyelectrolyte molecules which are assumed to be the same as one of the ions of the salt. The electrostatic free energy contribution is the sum of the electrostatic energy of the solution and the chemical contribution of the surface, if the potential determining ions are in thermodynamic equilibrium with those in solution. At constant surface electrical potential W 0, the chemical contribution is given by Fchem ¼
2rw0

39

ð5Þ

where e is the dielectric constant, e 0 is the vacuum permittivity and w is the electrical potential.

The second term on the right-hand side of Eq. (9) represents the variation of the chemical contribution to the free energy when the distance between the plates is changed from infinity to D. It was obtained by decomposing the pathway from infinity to D into infinitesimal steps at constant charge, for which the chemical contributions to the free energy are zero, each followed by an infinitesimal step at constant potential with a change dr of the surface charge density, providing each a contribution
2W 0dr. By minimizing the free energy with respect to w, and u, one obtains   a2 uW ¼ ðabew
U Þu þ v u3
uu2b 6

ð10Þ

 d2 w 1  ¼ 2cesinhðbewÞ
aeu2 þ aeu2b expðbewÞ dx2 ee0

ð11Þ

Eq. (10) represents the self-consistent field equation for the local segment density of the polymer chains subject to an external electrical potential w, a van der Waals interaction with the plates
UkT and an excluded volume interaction. Eq. (11) is a modified Poisson–Boltzmann equation in which the first term accounts for the charges of the small ions of the salt, the second term for the charges of the polyelectrolyte chains and the third one for the charges of the ions dissociated from the polyelectrolyte molecules.

Nanodispersions

670 40

H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

Combined with the appropriate boundary conditions, Eqs. (10) and (11) can be solved to obtain the distributions of the electrical potential and of the chain segment density. Both quantities are required to calculate the interaction force between plates. If one assumes that no polymer segments are adsorbed onto the surfaces, the boundary condition commonly used [48,39,11,49] for the polymer order parameter at the surface is

at such distances. However, at large distances between the plates, the contribution of the third term is not important because U D/2 drastically decreases with increasing distance between plates, and the first term is negative and dominant because the electrical potential at the middle is positive at large distances. If only the small ions are present in the system, the above equation reduces to the Langmuir expression

uð0Þ ¼ 0

f ¼ 2ckT fcosh½ewð D=2Þ=kT 
1g

ð12Þ

This boundary condition implies that the chain segments are depleted in the region near the plates. This is an arguable boundary condition [50,51]. When the surfaces of the plates are assumed to be kept at constant surface potential w 0, wð0Þ ¼ w0

At the midway between plates, the symmetry of the system provides two boundary conditions  du  ¼0 ð14Þ dx x¼D=2 ð15Þ

The force per unit area between the plates can be calculated as the derivative of the free energy with respect to the distance between plates f ¼


d½ F ð DÞ
F ðD ¼ lÞ dD

For the van der Waals interactions between segments and plates, one can write the expression [52] " # AH aV3 1 1 U¼ þ ð19Þ 3kT x3 ðD
xÞ3

ð13aÞ

If the surfaces of the plates are kept at constant charge density, the corresponding boundary condition becomes  dw  r ¼
ð13bÞ dx x¼0 ee0

 dw  ¼0 dx x¼D=2

ð18Þ

ð16Þ

Consequently, the force can be expressed as

where A H is the Hamaker constant and paV3/6 is the volume of a segment assumed spherical and of radius aV. Very close to the surface, there is a Born repulsion between plates and the polyelectrolyte chains, which will be included in the calculations by considering an excluded region of 3.5 2 near the plates.

3. Neutral polymer between two plates If the plates and the chains are not charged, Eq. (10) becomes   a2 uW ¼
U u þ m u3
uu2b 6

ð20Þ

The second term on the right-hand side of Eq. (20) is negligible when the distance between plates is not too large. As an approximation, u between two plates can be obtained from the superposition of the profiles of u near the single 3 2 H aV u plates. For a single plate, the equation a6 uW ¼
A3kT x3

i2 mh bf ¼
abu2D=2 ewD=2
u2D=2
u2b þ UD=2 u2D=2 2 n h i o 2 þ aub exp ewD=2 =ðkT Þ
1 n h i o þ 2c cosh ewD=2 =kT
1 ð17Þ where W D/2 is the electrical potential at the middle between plates, U D/2 is the van der Waals interaction between plates 2 and segments at the middle and u D/2 is the segment concentration at the middle. When the electrical potential at the middle distance is positive, the first two terms of Eq. (17) are negative, and the last two terms are positive. It should be noted that, although the plates and the segments have an attractive interactions
U, the third term is positive because U is positive. At small distances, the third term has a significant positive contribution to f because U D/2 is large

Fig. 1. The force between two plates with no charge immersed in a neutral polymer solution. a=5 2, u 2b=1 10
6 2
3, v=50 23, aV=5 2. (1) A H=0.5 10
20 J; (2) A H=0.25 10
20 J; (3) A H=0.1 10
20 J.

Polymer brushes

671 H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

41

combined with the boundary conditions u(0)=0 and du/ dx(l)=0 has the solution uð xÞ ¼ c1 x1=2 J1 ð2C 1=2 x
1=2 Þ

ð21Þ

where C=2A HaV3/(a 2kT), J 1 is thepffiffiffifirst-order Bessel ffi function of the first kind and c1 gub = C is obtained from the segment concentration at infinite distance u(l)=u b. By assuming that the profiles for the two plates are additive, one obtains uð xÞ ¼ c1 x1=2 J1 ð2C 1=2 x
1=2 Þ þ c1 ðD
xÞ1=2 J1 ð2C 1=2 ð D
xÞ
1=2 Þ By expanding the Bessel function in series and neglecting higher order terms, u at the middle between two plates can be expressed as   uD=2 ¼ 2c1 ð D=2Þ1=2 J0 23=2 C 1=2 D
1=2     C C i2ub
1 þ1 ð22Þ D 3D Therefore, the force between two plates is given by m bf ¼
½u2D=2
u2b 2 þ UD=2 u2D=2 2     2 2 mu4 C C ¼
b 4
1 þ1
1 2 D 3D    2 64AH aV3 u2b C C þ
1 þ 1 3kT D3 D 3D

Fig. 3. The force between two plates at various salt concentrations. The values of the parameters used in the calculations are: a=1, a=5 2, u 2b=1 10
6 2
3, eW 0/kT=
2, v=50 23, A H=0.5 10
20 J, aV=5 2. (1) c=0.001 M; (2) c=0. 1 M; (3) c=1 M.

segments, the latter are attracted into the gap between the 2 plates and the segment concentration u D/2 is larger than that in the bulk. However, the magnitude of the second term in Eq. (23), which provides a repulsive contribution to the force, is in most cases larger than the first term. Consequently, the force between the plates is repulsive. As the van der Waals interaction strength increases, the repulsive force increases. Numerical calculations based on Eq. (20) led to results which coincide with those obtained with Eq. (21).

ð23Þ 4. Numerical results

The force is repulsive for neutral polymers (Fig. 1) because the second term in Eq. (23) is dominant. Because of the van der Waals interactions between the plates and the

If the polymers are charged, the segment concentration and the force between plates can be calculated numerically only. 4.1. The effect of the van der Waals interactions between plates and segments

Fig. 2. The force between plates for various van der Waals interaction strengths. a=1, a=5 2, u 2b=1 10
6 2
3, eW 0/kT=
2, v=50 23, c=0.001 M, aV=5 2. (1) A H=1 10
20 J; (2) A H=0.5 10
20 J; (3) A H=0.25 10
20 J; (4) A H=0.05 10
20 J.

Because of the van der Waals interactions, the plates attract a larger number of polymer segments into the gap between the plates than when there are only electrostatic interactions with the chain segments. As a result, the interaction between plates, with the latter at constant electrical potential, becomes more attractive at distances larger than about 12 2 than when the van der Waals interaction is neglected (Fig. 2). This behavior occurs because u 2 increases and hence the number of bridges increases. The force between plates at distances smaller than 12 2 becomes, however, more repulsive when the van der Waals interaction is taken into account than when it is neglected. This increased repulsion can be attributed to the third term in Eq. (17), which provides a positive contribution to the force and becomes dominant at short distances.

Nanodispersions

672 42

H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

Fig. 4. The segment concentration distribution between plates for various salt concentrations. Only half space from one plate to the midway is plotted. D=12 2, and the other parameter values used in calculations are as for Fig. 3. (1) c=0.001 M; (2) c=0.01 M; (3) c=0.1 M; (4) c=1 M.

As the distance between plates increases, the third term drastically decreases, and the first term, which provides a negative contribution when the electrical potential at the midway is positive, becomes dominant at larger distances. Hence, the van der Waals interaction between the plates and the segments of the chains plays an important role. As the van der Waals interaction between segments and plates becomes stronger, the force between plates becomes more attractive at moderate distances (DNabout 12 2) between them and more repulsive at Db12 2.

Fig. 6. The force between plates for various bulk polyelectrolyte concentrations. a=1, a=5 2, c=0.001 M, eW 0/kT=
2, v=50 23 , A H=0.5 10
20 J, aV=5 2. (1) u 2b=1 10
6 2
3; (2) u 2b=1 10
4 2
3; (3) u 2b=1 10
3 2
3.

The effect of salt concentration on the force is presented in Fig. 3 for a Hamaker constant of 0.5 10
20 J. As the salt concentration increases, the force between plates at constant surface potential becomes more attractive at large distances.

The segment concentration profiles is presented in Fig. 4 for u b2=1 10
6 2
3. In the gap between plates, there are two concentration peaks, each located near one of the surfaces. The concentrations of the monomer in the gap can be by 3 orders of magnitude larger than its bulk concentration. These segment concentrations are much larger than the bulk salt concentration. As a result, salt concentrations lower than 0.1 M have a small effect on the force. For u b2=1 10
6 2
3, the electrical potential and the segment concentration at the midplane are almost independent of the salt concentration when the latter is lower than 0.1 M (Fig. 5). However, for salt concentrations higher than 0.1 M, the segment concentration at the midway increases and the electrical potential at the midway decreases with increasing salt concentration. As a result, the force becomes more

Fig. 5. The electrical potential distribution between plates for various salt concentrations. Only half space from one plate to the midway is plotted. D=12 2, and the other parameter values used in calculations are as for Fig. 3. (1) c=0.001 M; (2) c=0.01 M; (3) c=0.1 M; (4) c=1 M.

Fig. 7. The segment concentration distribution between the plates for various bulk polyelectrolyte concentrations. D=20 2, and the other parameter values used in calculations are as for Fig. 6. (1) u 2b=1 10
6 2
3; (2) u 2b=1 10
4 2
3; (3) u 2b=1 10
3 2
3.

4.2. The effect of salt concentration

Polymer brushes

673 H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

Fig. 8. The force between plates for various stiffnesses of polyelectrolyte chain. The parameter values used in calculations are: a=1, c=0.001 M, u 2b=1 10
6 2
3, eW 0/kT=
2, v=50 23, A H=0.5 10
20 J, aV=5 2. (1) a=4 2; (2) a=5 2; (3) a=8 2; (4) a=10 2.

repulsive at both short and large distances with increasing salt concentration. 4.3. The effect of the bulk polymer concentration The bulk segment concentration affects the distributions of the electrostatic potential and osmotic pressure in solution. Fig. 6 shows that as the bulk segment concentration increases, the force becomes less attractive and at large distances and, for high bulk polymer concentrations, it becomes slightly repulsive. The concentration of the segments is plotted in Fig. 7 for D=20 2. As expected, the polymer segment concentration at the middle is larger for higher bulk concentrations. A higher segment concentration can lead to an increased attraction between plates because it increases the number of bridges.

Fig. 9. The segment concentration distribution between plates at a small distance for various stiffnesses of the polyelectrolyte chains. D=11 2, and the other parameter values used in the calculation are as for Fig. 8. (1) a=4 2; (2) a=5 2; (3) a=8 2; (4) a=10 2.

43

Fig. 10. The electrical potential distribution between plates at small distances for various stiffnesses of the polyelectrolyte chain. D=11 2, and the other parameter values used in calculation are as for Fig. 8. (1) a=4 2; (2) a=5 2; (3) a=8 2; (4) a=10 2.

However, it also leads to a higher concentration of counterions dissociated from the polyelectrolyte that generate repulsion [the fourth term on the right-hand side of Eq. (17)]. This higher concentration can generate a higher repulsion between plates for the same electrical potential at the middle. 4.4. The effect of the stiffness of the chains The  stiffness of the chains, represented by the term du 2 in Eq. (2), together with the electrical and van der dx Waals interactions determine the conformation of the polyelectrolyte chains and hence their charge and segment concentration distributions in solution. The chains become more extended with increasing stiffness. As a result, at short distances, the force between plates is repulsive for a2 6

Fig. 11. The electrical potential distribution between plates at large distances for various stiffnesses of the polyelectrolyte chain. D=28 2, and the other parameter values used in calculation are as for Fig. 8. (1) a=4 2; (2) a=5 2; (3) a=8 2; (4) a=10 2.

Nanodispersions

674 44

H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

distances, providing a repulsive contribution to the force, but becomes positive at large ones (Fig. 11), generating an attractive contribution. The flexible chains are more easily attracted by the plates and generate a concentration peak near each of the plates. The segment concentration at the middle between plates is low. However, at large distances between plates, the stiff chains provide a higher segment concentration at the middle than the flexible ones (Fig. 12). Consequently, the force becomes more attractive for the stiffer than flexible chains when the distances between plates become sufficiently large. 4.5. The effect of the charge density of the chains Fig. 12. The segment concentration distribution between plates at a large distance for various stiffnesses of the polyelectrolyte chains. D=28 2, and the other parameter values used in the calculation are as for Fig. 8. (1) a=4 2; (2) a=5 2; (3) a=8 2; (4) a=10 2.

a=8 and 10 2 and decreases with decreasing stiffness (Fig. 8). For flexible chains, a larger number of chains are present between plates at small distances between them (Fig. 9) because they can be more easily located there than the stiffer chains. The increase of the electrical potential with the distance from the plates is larger for the flexible than stiffer chains (Fig. 10). For a=8 or 10 2, the electrical potential at the middle is negative and leads to a positive contribution of the first term of Eq. (17) to the force. However, the positive electrical potential at the middle for flexible chains provides a negative contribution of the first term of Eq. (17) to the force. Consequently, the force is more repulsive for the stiffer than flexible chains at small distances between plates. The electrical potential at the middle between plates increases with increasing distance between them. For stiffer chains, the electrical potential is negative at short

Fig. 13 presents the effect of the charge density per segment, which is assumed uniform along the chains, at constant surface potential of the plates, on the force. The surface of the plates is considered negatively charged and the chains are considered to be positively charged. As the charge density of the chains increases, the location of the minimum of the attractive force is displaced to a smaller distance. At distances larger than 16 2, the smaller the charge density of the segments a, the stronger the attraction between plates. As the charge density of the chains increases, the peaks of the segment concentrations are displaced towards the plate surface because of the stronger electrostatic attraction between segments and plates. At the middle between plates, the segment concentration is lower for higher charge densities of the segments because of the positive electrical potential. A lower segment concentration at the middle for high charge densities of the segments leads to a small negative contribution of the first term of Eq. (17), which is dominant at large distances. Therefore, the force at large distances is more attractive for lower charge densities of the segments.

Fig. 13. The force between plates for various charge densities of the polyelectrolyte segments. The parameter values used in calculations are: a=5 2, u 2b=1 10
6 2
3, eW 0/kT=
2, v=50 23, A H=0.5 10
20 J, c=0.001 M, aV=5 2. (1) a=2; (2) a=1; (3) a=0.5.

Fig. 14. The force between plates for various surface charge densities of the plates. The values of the parameters used in calculations are: a=1, a=5 2, u 2b=1 10
6 2
3, eW 0/kT=
2, v=50 23, A H=0.5 10
20 J, c=0.001 M, aV=5 2. (1) r=
0.002 C/m2; (2) r=
0.005 C/m2; (3) r=
0.01 C/m2.

Polymer brushes

675 H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

45

concentration of sodium ions is provided by the sum of the electrolyte concentration in the bulk and of the counterion concentration of the polyelectrolyte in the bulk. 2 cþ b ¼ c þ hsub

The force can be therefore expressed as bf ¼ bsu2D=2 ewD=2 K=fK þ cþ b exp½
ewD=2 =ðkT Þg
mðu2D=2
u2b Þ2 =2 þ UD=2 u2D=2 þ su2b K½expð
ewD=2 =kT Þ
1=ðK þ cþ bÞ þ 2c½coshðewD=2 =kT Þ
1

Fig. 15. The force between plates immersed in the solution of a polyelectrolyte which can generate charges through dissociation equilibrium. The values of the parameters used in calculations are: s=1, a=5 2, u 2b=1 10
6 2
3, eW 0/kT=2, v=50 23, A H=0.5 10
20 J, c=0. 1 M, aV=5 2. (1) K=1 10
2 2
3; (2) K=5 10
4 2
3; (3) K=1 10
4 2
3; (4) K=5 10
5 2
3.

4.6. The effect of the plate surface charge density In Fig. 14, the force between plates is plotted against the distance D for various constant plate surface charge densities. The force is repulsive at short distances but becomes attractive at large ones. This behavior occurs because the first term on the right-hand side of Eq. (17), which is dominant, is positive at small distances because the electrical potential at the middle between plates is negative. At large distances, the electrical potential at the middle becomes positive, and the force becomes attractive because of the negative contribution of the first term on the righthand side of Eq. (17), which is dominant. 4.7. Polyelectrolyte chains generate their charge through dissociation equilibrium

ð27Þ

Fig. 15 presents the effect of the equilibrium constant K on the force. As K increases, the force between plates can change sign from repulsive to attractive. When the dissociation constant of the polyelectrolyte ionizable groups increases, the concentration of polyelectrolyte between the two plates decreases. Therefore, the dominant term for short distances, the third term of Eq. (17), becomes larger as the dissociation constant decreases, and the force less repulsive. At large distances, the first term of Eq. (17) is dominant and the higher the dissociation constant, the less attractive the force between plates.

5. Comparison with experiment Jinnai et al. [18] measured by neutron-scattering the interlayer separation between the butylammonium vermiculite platelets as a function of the poly(vinyl methyl ether) (PVME) volume fraction in an aqueous butylammonium chloride solution. The n-butylammonium vermiculite swelled in the n-butylammonium chloride solution, and the interlayer separation of the platelets increased from 2 to 12 nm at 8bTb20 8C. When the PVME was added to this

If the polyelectrolyte chains generate their charge through the dissociation equilibrium, SNa X S
þ Naþ

ð24Þ

where SNa is an ionizable group of the polyelectrolyte chain and S
is a dissociated group. The fraction h of the dissociated polyelectrolyte ionizable groups is given by Ku

h½Naþ  1
h

ð25Þ

where K is the dissociation equilibrium constant and [Na+] is the sodium concentration near the surface of the chains. Consequently,   a ¼
sh ¼
sK= K þ cþ ð26Þ b expð
ew=kT Þ where s is the number of ionizable groups of each segment and c +b is the bulk concentration of Na+ ions. The bulk

Fig. 16. The interlayer separation of n-butylammonium vermiculite against the volume fraction of PVME. The experimental data are from Fig. 4 of Ref. [18]. The values of the parameters employed in the calculation were: T=8 8C, the salt concentration is 0.1 M, a=1.88 nm, the Hamaker constant between the plates A H V =1 10
19 J, the Hamaker constant between the segments and plates A H =1 10
20 J, r=
0.035 C/m2, v=300 23, e=83.

Nanodispersions

676 46

H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

clay–salt–water system, the interlayer separation decreased with increasing PVME volume fraction. The latter behavior can be attributed to the bridging of the vermiculite layers by PVME. The experimental and theoretical results are compared in Fig. 16. The molecular weight of PVME is 18 000, value which satisfies the condition for the chain to be sufficiently long imposed by the theory. In the calculations, the total free energy of the system is the sum of the free energy F and the van der Waals interaction energy F vdw between plates, which is given by " # AHV 1 1 2 Fvdw ¼
þ
48p D2 ð D þ dÞ2 ð D þ d=2Þ2 where A HV is the Hamaker constant between the platelets and d=1 nm is the thickness of the platelets. The position of the secondary minimum of the total free energy plus the thickness of a platelet (1 nm) was considered as the equilibrium separation between layers. The surface charge density was assumed to be constant. The Kuhn segment length was evaluated using the expression a ¼ hR2 i=Rexe

ð28Þ

2

where hR i is the mean square end-to-end distance and R exe is the fully extended chain length, which is 77.5 nm for PVME with a molecular weight of 18 000. For hR 2i in solution, one can use the expression [53] hR2 i1=2 ¼ 900M1=2 10
4 nm

ð29Þ

which together with Eq. (28) leads to a=1.88 nm. The following values of the parameters were employed in the calculation: the Hamaker constant between platelets A HV =1 10
19 J, the Hamaker constant between the segments and plates A H =1 10
20 J, the charge density of the clay platelets r=
0.035 C/m2, the excluded volume v=300 23, the temperature T=8 8C and the salt concentration c=0.1 M. For the dielectric constant, the value 83 was employed in the calculations for T=8 8C. The satisfactory agreement between experiment and calculations suggest that the contraction of the separation between the clay platelets can be the result of polymer bridging interactions.

6. Conclusions The van der Waals interactions between the polyelectrolyte segments and plates are included into the meanfield approach together with the excluded volume and electrostatic interactions. A repulsive contribution of the force between plates is generated by the van der Waals interactions between the segments and the plates. With increasing van der Waals interactions between the segments and the plates, the force between the plates at constant surface potential becomes more repulsive at small distances between the plates and more attractive at large

distances because of a large segment concentration in the gap. As the bulk concentration of the segment concentration increases, the force becomes less attractive because of the higher counterion concentration in the bulk. When the polyelectrolyte chains generate their charge through a dissociation equilibrium between their ionizable group and counterions, as the equilibrium constant increases, the concentration of the polyelectrolyte molecules between the two plates at constant potential increases and the force decreases. The experimental results of the contraction of the interlayer distance between layers of the n-butylammonium vermiculite induced by the poly(vinyl methyl ether) are in satisfactory agreement with the theoretical calculations. References [1] D.H. Napper, Polymeric Stabilization of Colloidal Dispersions, Academic Press, New York, 1983. [2] P.G. de Gennes, Adv. Colloid Interface Sci. 27 (1987) 189. [3] R.R. Netz, D. Andelman, Phys. Rep. 380 (2003) 1. [4] R.K. Iler, J. Colloid Interface Sci. 37 (1971) 364. [5] E. Dickinson, L. Eriksson, Adv. Colloid Interface Sci. 34 (1991) 1. [6] P.F. Luckham, J. Klein, J. Chem. Soc., Faraday Trans. 80 (1984) 865. [7] P.M. Claesson, B.W. Ninham, Langmuir 8 (1992) 1506. [8] K. Lowack, C.A. Helm, Macromolecules 31 (1998) 823. [9] R. Podgornik, J. Phys. Chem. 96 (1992) 884. [10] R. Podgornik, J. Chem. Phys. 118 (2003) 11286. [11] I. Borukhov, D. Andelman, H. Orland, J. Phys. Chem., B 103 (1999) 5042. [12] T. Akesson, C. Woodward, B. Jonsson, J. Chem. Phys. 91 (1989) 2461. [13] J. Jimenez, J. de Joannis, I. Bitsanis, R. Rajagopalan, Macromolecules 33 (2000) 7157. [14] S. Misra, W.L. Mattice, Macromolecules 27 (1994) 2058. [15] S. van Giessen, I. Szleifer, J. Chem. Phys. 102 (1995) 9069. [16] T. Holzl, M. Wittkop, S. Kreitmeier, D. Goritz, J. Chem. Phys. 106 (1997) 7792. [17] D.H. Everett, Basic Principles of Colloid Science, Royal Society of Chemistry, London, 1988. [18] H. Jinnai, M.V. Smalley, T. Hashimoto, S. Koizum, Langmuir 12 (1996) 1199. [19] M.V. Smalley, Langmuir 10 (1994) 2884. [20] J. Swenson, M.V. Smalley, R.K. Thomas, R.J. Crawford, L.F. Braganza, Langmuir 13 (1997) 6654. [21] M.V. Smalley, H.L.M. Hatharasinghe, I. Osborne, J. Swenson, S.M. King, Langmuir 17 (2001) 3800. [22] J. Swenson, M.V. Smalley, H.L.M. Hatharasinghe, Phys. Rev. Lett. 81 (1998) 5840. [23] J. Swenson, M.V. Smalley, H.L.M. Hatharasinghe, J. Chem. Phys. 110 (1999) 9750. [24] H. Ohshima, Colloid Polym. Sci. 277 (1999) 535. [25] D.A. Hoagland, Macromolecules 23 (1990) 2781. [26] D. Dean, J. Seog, C. Ortiz, A.J. Grodzinsky, Langmuir 19 (2003) 5526. [27] H.H. Huang, E. Ruckenstein, J. Colloid Interface Sci. 275 (2004) 548. [28] S.F. Edwards, Proc. Phys. Soc. 85 (1965) 613. [29] F.W. Wiegel, J. Phys., A 10 (1977) 297. [30] P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, 1979. [31] P.G. de Gennes, Macromolecules 15 (1982) 492. [32] J.M.H.M. Scheutjens, G.J. Fleer, J. Phys. Chem. 83 (1979) 1619. [33] J.M.H.M. Scheutjens, G.J. Fleer, Macromolecules 18 (1985) 1882.

Polymer brushes

677 H. Huang, E. Ruckenstein / Advances in Colloid and Interface Science 112 (2004) 37–47

[34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

M. Muthukumar, J. Chem. Phys. 86 (1987) 7230. X. Chatellier, J.-F Joanny, J. Phys. II Fr. 6 (1996) 1669. J.-F. Joanny, Eur. Phys. J., B 9 (1999) 117. R. Varoqui, J. Phys. II Fr. 3 (1993) 1097. E.M. Blokhuis, K.I. Skau, J.B. Avalos, J. Chem. Phys. 119 (2003) 3483. I. Borukhov, D. Andelman, H. Orland, Macromolecules 31 (1998) 1665. I. Borukhov, D. Andelman, H. Orland, Euro. Phys. J., B 5 (1998) 869. A. Shafir, D. Andelman, R.R. Netz, J. Chem. Phys. 119 (2003) 2355. K.I. Skau, E.M. Blokhuis, J. van Male, Macromolecules 37 (2004) 1969. B.Q. Li, E. Ruckenstein, J. Chem. Phys. 106 (1997) 280.

47

[44] E. Ruckenstein, B.Q. Li, J. Chem. Phys. 107 (1997) 932. [45] M.K. Granfeldt, B. Jfnsson, C.E. Woodward, J. Phys. Chem. 95 (1991) 4819. [46] R. Varoqui, A. Johner, A. Elaissari, J. Chem. Phys. 94 (1991) 6873. [47] M. Manciu, E. Ruckenstein, Langmuir 19 (2003) 1114. [48] E.A. Di Marzio, J. Chem. Phys. 42 (1965) 2101. [49] R.J. Allen, P.B. Warren, Langmuir 20 (2004) 1997. [50] S.J. Miklavic, J. Phys. Chem. 97 (1993) 3924. [51] R. Podgornik, J. Phys. Chem. 97 (1993) 3927. [52] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, New York, 1985. [53] J. Brandrup, E.H. Immergut, Polymer Handbook, Wiley, New York, 1989.

Nanodispersions

678

3174

Langmuir 2006, 22, 3174-3179

Steric and Bridging Interactions between Two Plates Induced by Grafted Polyelectrolytes Haohao Huang and Eli Ruckenstein* Department of Chemical and Biological Engineering, State UniVersity of New York at Buffalo, Buffalo, New York 14260 ReceiVed October 17, 2005. In Final Form: January 3, 2006 If colloidal particles are grafted with a polymer, then the grafted chains can provide steric repulsion between them. If some of the grafted polymer chains are also adsorbed to a second particle, then a bridging force is generated as well. For uncharged plates and polymer, the following contributions to the free energy of the system have been taken into account in the calculation of the interaction force: (i) the Flory-Huggins expression for the mixing free energy of the grafted chains with the liquid; (ii) the entropy loss due to the connectivity of the polymeric segments; (iii) the van der Waals interactions between the segments and the plates; and (iv) the free energy of adsorption of the polymer segments of the grafted chains on the other plate. For charged plates, the electrostatic free energy as well as the free energy of the electrolyte are included in the total free energy of the system. By minimizing the free energy with respect to the segment concentration and, when it is the case, with respect to the electrical potential, equations for the segment number density distribution and for the electrical potential are obtained, on the basis of which the interactions between two plates grafted with polymer chains that can be also adsorbed on the other plate were calculated. The interaction thus obtained includes steric and bridging forces.

1. Introduction When two plates with polymer chains grafted to their surface approach each other, a steric repulsion is generated between the two brushes as soon as the tips of the grafted chains begin to contact each other.1,2 The repulsive steric interactions can increase the stability of a colloidal system. When the distance between the plates becomes less than twice the thickness of the polymer brush, the polymer chains on the two plates begin to interdigitate. The increase of the segment concentration in the interdigitation domain leads to repulsion. When opposite polymer brushes contact each other, the brushes become compressed in addition to their interdigitation. Therefore, the steric repulsion includes entropic and elastic contributions. However, if some of the grafted chains are also adsorbed on the second plate, then the polymers draw the plates toward each other, resulting in an attractive bridging force.3,4 The bridging force may induce the coagulation of the colloidal particles. The bridging was extensively examined experimentally,5-7 theoretically,8-10 and by Monte Carlo simulations.11-14 Of course, the steric and bridging forces are simultaneously present between the plates, and their effects are not expected to be additive. Many theories were developed to describe the distribution of segments between two polymer brushes. On the basis of a lattice * To whom correspondence should be addressed. E-mail: feaeliru@ acsu.buffalo.edu. Tel: (716) 645-2911 ext. 2214. Fax: (716) 645-3822. (1) de Gennes, P. G. AdV. Colloid Interface Sci. 1987, 27, 189. (2) Roan, J. R; Kawakatsu, T. J. Chem. Phys. 2002, 116, 7283. (3) Iler, R. K. J. Colloid Interface Sci. 1971, 37, 364. (4) Dickinson, E.; Eriksson, L. AdV. Colloid Interface Sci. 1991, 34, 1. (5) Luckham, P. F.; Klein, J. J. Chem. Soc., Faraday Trans. 1984, 80, 865. (6) Claesson, P. M.; Ninham, B. W. Langmuir 1992, 8, 1506. (7) Lowack, K.; Helm, C. A. Macromolecules 1998, 31, 823. (8) Varoqui, R. J. Phys. II 1993, 3, 1097. (9) Podgornik, R. J. Phys. Chem. 1992, 96, 884. (10) Borukhov, I.; Andelman, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (11) Akesson, T.; Woodward, C.; Jonsson, B. J. Chem. Phys. 1989, 91, 2461. (12) Jimenez, J.; de Joannis, J.; Bitsanis, I.; Rajagopalan, R. Macromolecules 2000, 33, 7157. (13) Misra, S.; Mattice, W. L. Macromolecules 1994, 27, 2058. (14) van Giessen, S.; Szleifer, I. J. Chem. Phys. 1995, 102, 9069.

model, Scheutjens et al. developed a discrete self-consistent theory to describe the steric repulsion.15-18 A similar model involving interdigitation was employed by Li and Ruckenstein.19,20 The assumption of free jointed chains used by Scheutjens et al. was replaced by the latter authors with the assumption of a correlation between neighboring bonds. Another kind of self-consistent theory used a continuum diffusionrepresentationtodescribethedistributionofsegments.21-23 The segments were assumed to be subjected to an external potential of a mean field. An analytical approximation of the latter self-consistent theory was suggested by Milner et al. (the MWC model)24 on the basis of the observation that at high stretching the partition function of the brush is dominated by the classical path as the most probable distribution. Under this assumption, it was found that the self-consistent field is parabolic and leads to a parabolic distribution of the monomer density. Similar theories for polyelectrolyte brushes25-27 also adopted the parabolic distribution approximation. The polyelectrolyte molecules affect in a more complex manner than the neutral ones the steric and bridging interactions between colloidal particles because they affect the electrical potential of the electrical double layer,28,29 which in turn affects the conformation of the chains and the overall electrostatic interactions. (15) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (16) Leermakers, F. A. M.; Sheutjens, J. M. H. M. J. Chem. Phys. 1988, 89, 3264. (17) Leermakers, F. A. M.; Sheutjens, J. M. H. M. J. Chem. Phys. 1988, 89, 6912. (18) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. Langmuir 1994, 10, 4514. (19) Li, B. Q.; Ruckenstein, E. J. Chem. Phys. 1997, 106, 280. (20) Ruckenstein, E.; Li, B. Q. J. Chem. Phys. 1997, 107, 932. (21) Edwards, S. F. Proc. Phys. Soc. 1965, 85, 613. (22) Borukhov, I.; Andelman, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (23) Huang, H. H.; Ruckenstein, E. AdV. Colloid Interface Sci. 2004, 112, 37. (24) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (25) Miklavic, S. J.; Marcelja, S. J. Phys. Chem. 1988, 92, 6718. (26) Misra, S.; Varanasi, S.; Varanasi, P. P. Macromolecules 1989, 22, 4173. (27) Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107, 5952.

10.1021/la0527947 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/01/2006

Polymer brushes

679

Steric and Bridging Interactions between Two Plates

Langmuir, Vol. 22, No. 7, 2006 3175

A statistical model by Manciu and Ruckenstein,30,31 which assumed that the chain distributions are generated by all possible random walks and employed an average free energy for all of the chain distributions that end at a certain distance x from the interface, showed that by accounting for the most likely chain distribution and discarding all other chain distributions one arrives at the same scaling law as does the MWC model. However, when all possible distributions (with their appropriate Boltzmann weights) have been taken into account, a departure from the MWC model results, most notably, at high grafting density, when the brush becomes more step-like than parabolic-like.30 With the assumption of a step function for the distribution of the segment density and using the ground-state dominance approximation, Alexander32 and de Gennes1 suggested a scaling analysis to calculate the interaction between two brushes. An equation is derived in the present article for the free energy of the system by using various contributions to the free energy. For uncharged plates and polymer, the following contributions were considered: (i) the Flory-Huggins expression for the mixing free energy of the grafted chains with the liquid; (ii) the entropy loss due to the connectivity of the segments of the chains; (iii) the van der Waals interaction between segments and plates; and (iv) the free energy of adsorption of the segments of the grafted polymers on the other plate. For charged surfaces and polymers, the electrostatic free energy and the free energy of the electrolyte were also added to obtain the total free energy of the system. The minimum free energy with respect to the segment density and, when it was the case, with respect to the electrical potential provides two differential equations for the segment density and electrical potential. The minimum of the free energy with respect to the segment density of the polymer near the surface provided the concentration of the segments that are subject to adsorption. Because both the steric and bridging interactions occur simultaneously, these two interactions are coupled and should not be calculated separately. Some polymer chains are grafted on only one plate and contribute to the steric repulsion; the chains that are also adsorbed on the interacting plate contribute to bridging. The goal of the present article is to calculate the interaction between two plates grafted with polymer chains, the segments of which can be also adsorbed on the interacting second plate.

[

βFp ) ∫0 D

( ) ] dx

1 2 2 1 3 3 l2 dφ τV n + wV n Un + 6 dx 2l3 6l3

U)

(1)

The polymer contribution contains the Flory-Huggins mixing free energy of the segments with the solvent molecules, the van der Waals interaction -UkT between segments and the two plates, and the entropy loss caused by the connectivity of the segments 8,23 (28) Ohshima, H. Colloid Polym. Sci. 1999, 277, 535. (29) Hoagland, D. A. Macromolecules 1990, 23, 2781. (30) Manciu, M.; Ruckenstein, E. Langmuir 2004, 20, 8155. (31) Manciu, M.; Ruckenstein, E. Langmuir 2004, 20, 6490. (32) Alexander, S. J. Phys. 1977, 38, 983.

[

]

AHl′3 1 1 + 3kT x3 (D - x)3

(3)

where AH is the Hamaker constant and l′ is the diameter of a segment that is assumed to be spherical. Close to the surface, there is a Born repulsion that we consider that generates an infinite repulsion over a distance of 3.5 Å. The double-layer contribution Fd contains the entropic Fent of the ions and the electrostatic contributions Fele. The entropic contribution of the electrolyte to the free energy of the system is due to the change in the entropy of ions between the plates compared to that in the reservoir.

βFent ) ∫0

D

[

(∑ ) xi

ci ln

i

xi,b

1 - ∑ xi + cw ln

(

i

i

ci

∫0D ∑ ci ln c i

]

1 - ∑ xi,b

dx ≈

)

- ci + ci,b dx (4)

i,b

where ci is the concentration of ion i, cw is the concentration of water, xi is the mole fraction of ion i, and the subscript b indicates the concentration in the bulk. The electrostatic free-energy contribution is the sum of the electrostatic energy of the system and the chemical contribution of the surface. For constant surface potential ψ0, the chemical contribution is given by

Fchem ) -2σψ0

Two plates grafted with polymer chains on their surface are immersed in an electrolyte solution. The two plates are considered to be located at x ) 0 and D, respectively, and the system is considered to be in contact with a large reservoir containing an electrolyte solution free of polymer. Compared to the concentration of the ions of the electrolyte, the concentration of the counterions dissociated from the grafted polyelectrolyte is neglected. The free energy F of the system contains a polymer (Fp), a double layer (Fd), and an adsorption (Fa) contribution.

(2)

In eq 2, n and V are the local number density and the volume of a polymer segment, τ is the dimensionless excluded volume parameter, w is the second virial coefficient, φ ) xn, l is the Kuhn length of a segment, β ) 1/kT, k is the Boltzmann constant, and T is the absolute temperature. For the van der Waals interactions between plates and segments in the solution, one can write the expression

2. Theoretical Framework

F ) Fp + Fd + Fa

2

(5)

where σ is the surface charge density. Therefore, the electrostatic free-energy contribution can be expressed as D dψ 2 1 0 ∫0 dx - 2σψ0 2 dx

( ) 1 dψ dψ )  ∫ ( ) dx + 2 ψ | 2 dx dx

Fele )

2

D

0

0

0

0

(6a)

x)0

Using Green’s theorem, eq 6a can be rewritten as

( ) ∫[

[

( ) ] dx

2 D dψ D 1 d2 ψ dψ Fele ) 0 ∫0 dx - 0 ∫0 ψ 2 + 2 dx dx dx

) -0

D

0

ψ

( ) ] dx

d2ψ 1 dψ + dx2 2 dx

2

2

(6b)

where  is the dielectric constant, 0 is the vacuum permittivity, and ψ is the electrical potential. The optimization of the total free energy of the system with respect to the electrical potential provides the following relation

Nanodispersions

680

3176 Langmuir, Vol. 22, No. 7, 2006

Huang and Ruckenstein

between the electrical potential and the charge density (Poisson equation)

dψ F )2  dx 0 2

(7)

where the charge density in the solution F includes the charges from the electrolyte ions and the polyelectrolyte +

-

F ) e(c - c + Rφ ) 2

D

[

1 dψ 2 dx

2

( ) ] dx

(9)

The adsorption contribution to the free energy is a result of the replacement of solvent molecules on the surface by adsorbed polymer segments. The adsorption contribution is given by

Fads )

2Γ∆Gads N0

(s0 - s)Kadsns 1 + Kadsns

(11)

where s0 is the total number of adsorption sites on the surface per unit area, s is the grafting density of the polymer chains. ns is the segment concentration near the plates and Kads is the adsorption constant (RT ln Kads ) -∆Gads, R being the gas constant and T the temperature in K taken 300 K). All of the segments between the two plates belong to the grafted polymer chains, and the plates are in contact with an electrolyte solution free of polymer. Consequently, the total number of segments in the solution between the two plates is constant. If the grafting density of the polymer chains is denoted by s, then the total number of segments in the solution is 2sN, where N is the number of segments per chain. This constraint must be taken into account in the minimization of the free energy with respect to the segment density. Because the number of segments in solution is much larger than the number of segments adsorbed, the number of the latter segments can be neglected. Hence, one can write

2sN ≈ ∫0 φ2 dx D

(12)

The minimization of the free energy with respect to φ under the (33) Farrokhpay, S.; Morris, G. E.; Fornasiero, D.; Self, P. J. Colloid Interface Sci. 2004, 274, 33.

(13)

where λ is a Lagrange multiplier, which is provided by eq 12. The minimization of the free energy with respect to ψ leads to the equation

d2ψ 1 ) [2ce sinh(βeψ) - Reφ2] 2  dx 0

(14)

Equation 13 provides the segment density of the polymer chains subjected to the external electrical potential, ψ, and the van der Waals interaction with the plates, -UkT. Equation 14 represents a modified Poisson-Boltzmann equation in which the first term on the right-hand side accounts for the charges of the small ions of the salt and the second term accounts for the charges of the polyelectrolyte chains. When the polymer is not charged, eqs 13 and 14 can be solved independently. Because of symmetry, there are two boundary conditions at the middle distance between them

(10)

where Γ is the number of segments adsorbed per unit area, ∆Gads is the molar adsorption free energy of the polymer, and N0 is Avogadro’s number. Typical values for ∆Gads are -13 kJ/mol for poly(acrylic acid) on alumina and -10 to -16 kJ/mol for polyacrylamide on talc.33 Here the adsorption free energy of a polymer molecule is assumed to be the same as the adsorption free energy of a segment. The number of segments adsorbed per unit area can be calculated using the Langmuir adsorption isotherm

Γ)

l2 d2φ 1 ) -τl3φ3 + wl6φ5 - Uφ + Rβeψφ + λφ 6 dx2 2

(8)

c+ and c- are the concentrations of the cations and anions, respectively, and R is the charge of each segment. Substituting eq 7 into eq 6b, one obtains

Fele ) ∫0 eψ(c+ - c- + Rφ2) -

constraint (eq 12) and considering that V ) l3 leads to

dψ (x ) D/2) ) 0 dx

(15)

dφ (x ) D/2) ) 0 dx

(16)

At the surface of the charged plates, the boundary condition is

ψ ) ψ0

(17)

for a constant potential of the plates. The segment concentrations at z ) 0 and D, which are the same, are selected by trial and error such as to minimize the free energy. The Lagrange multiplier is determined also by trial and error using eq 12. The profiles of the segment concentration and the electrical potential for the minimum free energy allow one to calculate the interaction force f between the two plates, which is given by the expression

f)-

δF δD

( { [ ]}

)

1 1 ) kT l3τφ4D/2 - wl6φ6D/2 + kTUD/2φ2D/2 + 2 6 eψD/2 ∂(φ2D/(1 + Kadsφ2D)) 2c cosh + 2s0KadsRT ln Kads kT ∂D (18) 3. Calculation of the Steric Interaction If the grafted polymer is not adsorbed on the second plate, then the grafted polymer provides only steric interaction between the plates. This repulsive steric interaction increases the stability of the colloidal system. To calculate the steric interaction between two plates, the adsorption constant Kads in eq 11 should be taken to be zero in this case. Figure 1 presents the steric interaction between two plates for various graft densities. As the grafting density increases, the segment density at the middle between plates increases. Therefore, the steric interaction between the two brushes increases. As shown in Figure 2, the steric repulsion increases as the polymer chain becomes longer. As the length of the polymer

Polymer brushes

681

Steric and Bridging Interactions between Two Plates

Figure 1. Steric interaction between two plates for various grafting densities. τ ) 0.3, w ) 1, N ) 200, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, and s0 ) 0.8 nm-2. s/s0 is equal to the following values: (1) 0.01; (2) 0.02; (3) 0.03; (4) 0.04; and (5) 0.05.

Figure 2. Steric interaction between two plates for various chain lengths. τ ) 0.3, w ) 1, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, and s0 ) 0.8 nm-2. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. N is equal to the following values: (1) 150; (2) 200; (3) 300; and (4) 500.

Langmuir, Vol. 22, No. 7, 2006 3177

Figure 4. Steric interactions between two cross cylinders for τ ) 0.3, w ) 1, l ) 7.77 Å, N ) 430, s0 ) 1.66 nm-2, R ) 0, ψ0 ) 0, and s/s0 ) 0.00779. (1) de Gennes’s model; (2) the present model; (3) Ruckenstein and Li’s model. The experimental data (o) are from ref 34.

concentration of the segments against the distance to the surface of the plate can be well approximated by a step function. As the distance between the two plates increases, the concentration of the polymer segments decreases. Our model confirms one of the basic assumption of the Alexander-de Gennes theory, namely, that the segment distribution can be approximated by a step function. For comparison purposes, the expressions for the interaction between two uncharged plates grafted with uncharged chains suggested by de Gennes1 and Ruckenstein and Li20 are given below and compared with results of the present calculations. Two effects, the osmotic and the elastic contributions, were taken into account in the de Gennes analysis for the interaction between two parallel plates1

f=

{( ) ( ) }

kT 2L0 d3 D

9/4

-

D 2L0

3/4

(19)

where d is the average separation between two graft sites and L0 is the brush thickness. Using a lattice model, Ruckenstein and Li derived a more rigorous expression for the steric interaction between two parallel plates.20 The interaction force p between two crossed cylinders of radius R, for which experimental data are available,34 can be obtained from the interaction free energy F between two parallel plates using the Derjaguin approximation

p ) 2πF R

(20)

A comparison between the three expressions for the steric interaction p is made in Figure 4, which shows that for polystyrene the present model is in better agreement with the more rigorous approach of Ruckenstein and Li than the equation of de Gennes. Figure 3. Distribution of polymer segments between two plates at various distances. τ ) 0.2, w ) 1, l ) 5 Å, ψ0 ) 0.005 V, R ) 2e, N ) 200, s0 ) 4 nm-2, and c ) 0.01 M. The ratio of the number of grafting sites to the number of adsorption sites is 0.1. Only half the distance is plotted. (1) 200; (2) 300; (3) 400; and (4) 600 Å.

increases, the number of segments between the two plates increases, and this results in an increase in the free energy and in the repulsive force. Figure 3 presents the distribution of polymer segments between the two plates for various distances. The distribution of the

4. Steric Bridging Interactions Induced by a Polyelectrolyte If the grafted polymer molecules are also adsorbed on the second plate, then the force between the two plates includes both bridging and steric interactions. In this case, the adsorption contributions to the total free energy must be included among the free-energy contributions. When the polymer is charged, eqs (34) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Macromolecules 1990, 23, 571.

Nanodispersions

682

3178 Langmuir, Vol. 22, No. 7, 2006

Figure 5. Interaction between two plates at various grafting densities. τ ) 0.3, w ) 1, N ) 200, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, s0 ) 0.8 nm-2, and ∆Gads ) -13.25 kJ/mol. s/s0 is equal to the following values: (1) 0.01; (2) 0.02; (3) 0.03; (4) 0.04; and (5) 0.05.

Figure 6. Distribution of polymer segments between two plates for various grafting densities. τ ) 0.2, w ) 1, l ) 5 Å, ψ0 ) 0.005 V, R ) 2e, c ) 0.01 M, N ) 200, and ∆Gads ) -13.25 kJ/mol. s/s0 is equal to the following values: (1) 0.05; (2) 0.1; (3) 0.2; and (4) 0.3 and s0 ) 4 nm-2.

13 and 14 are not independent and have to be solved simultaneously. The segment concentration ns at the surface was selected by minimizing the free energy of the system with respect to ns. 4.1. Effect of the Grafting Density of the Polymer. The grafting density of the polymer affects the total number of segments and the adsorption of the segments on the plate surface. Figure 5 shows that as the grafting density of the segments increases the force becomes more repulsive. At relatively small values of the grafting density, the force decreases with increasing distance and passes through a minimum, after which it increases. Figure 5 shows that there are conditions under which the overall force becomes negative, and hence the bridging becomes dominant. Figure 6 shows that the shape of the segment concentration distribution can be described by a step function and that the uniform concentration becomes larger for higher grafting densities. Consequently, higher bridging is generated, and the attractive bridging force increases. However, the higher segment concentration also increases the steric interaction. Because the latter dominates, the overall interaction that includes both the bridging and the steric forces becomes more repulsive with increasing grafting density, at least for the parameters employed in the present calculations. 4.2. Effect of the Stiffness of the Chains. The stiffness of the chains, represented by the term l2/6 (dφ/dx)2 in eq 2, together

Huang and Ruckenstein

Figure 7. Interaction between two plates for various segment lengths l. τ ) 0.3, w ) 1, N ) 200, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, s0 ) 0.8 nm-2, and ∆Gads ) -13.25 kJ/mol. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. l (nm) is equal to the following values: (1) 0.3; (2) 0.4; (3) 0.5; and (4) 0.8.

Figure 8. Interaction between two plates for various chain lengths. τ ) 0.3, w ) 1, l ) 5 Å, ψ0 ) 0.01 V, R ) 1e, c ) 0.01 M, s0 ) 0.8 nm-2, and ∆Gads ) -13.25 kJ/mol. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. N is equal to the following values: (1) 150; (2) 200; (3) 300; (4) 500; and (5) 800.

with the electrical and van der Waals interactions determines the conformation of the polyelectrolyte chains and hence their charge and segment concentration distributions in solution. A larger value of l means a stiffer polymer chain. The chains become more extended with increasing chain stiffness, and the repulsive force between plates increases with increasing stiffness (Figure 7). For stiffer chains, the contribution of bridging is smaller than for flexible chains because the concentration of segments near the plate surface decreases with increasing stiffness. 4.3. Effect of the Length of the Polymer. As the length N of the chain increases, the repulsive force becomes stronger (Figure 8). Obviously, the steric repulsion is expected to increase with increasing N. The adsorption of the polymer segments is expected to increase as well. However, for the conditions employed, the steric repulsion increases more than the bridging attraction. 4.4. Segment Charge through Dissociation Equilibrium. If the polyelectrolyte chains generate their charge through dissociation equilibrium, then

SM T M- + S+

(21)

where SM is an ionizable group of the polyelectrolyte chain, Mis the anion of the electrolyte and chain, and S+ is a dissociated site.

Polymer brushes

683

Steric and Bridging Interactions between Two Plates

Langmuir, Vol. 22, No. 7, 2006 3179

The fraction θ of the dissociated ionizable groups is given by

Kd ≡

θ[M-] 1-θ

(22)

where Kd is the dissociation equilibrium constant and [M-] is the anion concentration.

(eψ kT )

[M-] ) c exp

(23)

where c is the electrolyte concentration in the reservoir.

R ) ξeθ )

ξeKd Kd + c exp(eψ/kT)

(24)

where ξ is the number of ionizable groups per segment. Figure 9 presents the effect of the equilibrium constant Kd on the force. As Kd increases, the force becomes more repulsive. This happens because as the dissociation constant of the polyelectrolyte ionizable groups increases the segments of the polyelectrolyte become more positively charged. Therefore, the segment adsorption on the surfaces that are also positively charged becomes smaller. As a result, the bridging contribution becomes smaller, and the interaction becomes more repulsive with increasing dissociation constant.

5. Conclusions The interaction between two plates grafted with charged polymer chains was calculated by taking into account both the bridging and steric interactions. The free energy of the system was calculated by accounting for a number of contributions: (i) the Flory-Huggins free energy for the grafted chains and liquid; (ii) the van der Waals interactions between the segments and plates; (iii) the connectivity free energy of the segments; and (iv)

Figure 9. Interaction between two plates grafted with a polyelectrolyte for various dissociation equilibrium constants. τ ) 0.3, w ) 1, l ) 5 Å, ψ0 ) 0.01 V, c ) 0.01 M, s0 ) 0.8 nm-2, ∆Gads ) -13.25 kJ/mol, and ξ ) 1. The ratio of the number of grafting sites to the number of adsorption sites is 0.02. Kd is equal to the following values: (1) 1; (2) 10; (3) 100; and (4) 1000 mol/m3.

the adsorption free energy. For charged polymers and surfaces, an entropic contribution due to the electrolyte ions and an electrostatic contribution were included as well. The overall interaction becomes more repulsive with increasing grafting density of the polymer on the surface and with increasing length of the polymer chains. When the plates and the segments have the same sign of charge, the interaction is more repulsive for higher charge densities of the segments because both the steric interaction and the double-layer force are stronger and the bridging weaker. The interaction between two plates is more repulsive for flexible charged chains than for stiff charged chains. There are conditions under which the bridging attraction dominates the steric repulsion. This occurs at sufficiently low grafting densities as well as sufficiently low segment charges (when the charges of the segments and the plate surfaces have the same sign). LA0527947

Nanodispersions

684

Langmuir 2006, 22, 4541-4546

4541

Effect of Steric, Double-Layer, and Depletion Interactions on the Stability of Colloids in Systems Containing a Polymer and an Electrolyte Haohao Huang and Eli Ruckenstein* Department of Chemical and Biological Engineering, State UniVersity of New York at Buffalo, Buffalo, New York 14260 ReceiVed January 20, 2006. In Final Form: March 1, 2006 Experiments carried out by Stenkamp et al. [Stenkamp, V. S.; McGuiggan, P.; Berg, J. C. Langmuir 2001, 17, 637.] have shown that polystyrene latexes can be restabilized at sufficiently high electrolyte concentrations in the presence of an amphiphilic block copolymer [poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) (PEOPPO-PEO)] At even higher electrolyte concentrations, the systems can again be destabilized. The present paper attempts to explain the restabilization through the dominance of steric interactions and the destabilization through the dominance of depletion interactions. Because of salting out, as the concentration of electrolyte increases, the polymer molecules are increasingly precipitated onto the surface of the latex particles and, at sufficiently high electrolyte concentrations, form, in addition, aggregates. The precipitation onto the latex particles generates steric repulsion, which is responsible for the restabilization, whereas the formation of aggregates generates depletion interactions, which are responsible for destabilization.

1. Introduction The stability of a colloidal dispersion generally decreases as the electrolyte concentration increases.2 At low electrolyte concentrations, the electrostatic repulsion is responsible for the stability of the colloidal system. However, at sufficiently high electrolyte concentrations, the thickness of the double layer is significantly decreased, and the electrostatic repulsion no longer contributes to the stability of the system. The steric stabilization, which is imparted by polymer molecules grafted onto the colloidal particles, is extensively employed.3 Amphiphilic block copolymers are widely used as steric stabilizers. The solvent-incompatible moieties of the block copolymer provide anchors for the polymer molecules that are adsorbed onto the surface of the colloidal particles, and the solventcompatible (buoy) moieties extend into the solvent phase. When two particles with block copolymers on their surface approach each other, a steric repulsion is generated between the two particles as soon as the tips of the buoy moieties begin to contact, and this repulsion increases the stability of the colloidal system.4-6 Polymers can also induce aggregation due to either “depletion”7-11 or “bridging” interactions.12-15 * To whom correspondence should be addressed. Telephone: (716) 6452911 ext. 2214. E-mail: [email protected]. Fax: (716) 645-3822. (1) Stenkamp, V. S.; McGuiggan, P.; Berg, J. C. Langmuir 2001, 17, 637. (2) Verwey, E. J.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (3) Napper D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983. (4) Alexander, S. J. Phys. 1977, 38, 983 (5) Roan, J. R.; Kawakatsu, T. J. Chem. Phys. 2002, 116, 7283. (6) Bell, N. S.; Sindel, J.; Aldinger, F.; Sigmund, W. M., J. Colloid Interface Sci. 2002, 254, 296. (7) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (8) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (9) Vrji. A. Pure Appl. Chem. 1976, 48, 471. (10) Joanny, J. F.; Leibler, L.; de Gennes, P. G. J. Polym. Sci., Polym. Phys. E. 1979, 17, 1073. (11) Rao, I. V.; Ruckenstein, E. J. Colloid Interface Sci. 1985, 108, 389. (12) Dickinson, E.; Eriksson, L. AdV. Colloid Interface Sci. 1991, 34, 1. (13) Podgornik, R. J. Phys. Chem. 1992, 96, 884. (14) Borukhov, I.; Andelman, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (15) Huang, H.; Ruckenstein, E. Langmuir 2006, 22, 3174.

Stenkamp et al.1 experimentally determined the stability ratio of polystyrene latexes as a function of electrolyte concentration for various electrolyte solutions in the presence of the amphiphilic block copolymer poly(ethylene oxide)-poly(propylene oxide)poly(ethylene oxide) (PEO-PPO-PEO). As the electrolyte concentration increases, the stability ratio of the colloidal system first decreases as a result of the decrease of the electrostatic stabilization. However, at sufficiently high electrolyte concentrations, the stability ratio increases with increasing electrolyte concentration. In some of the colloidal dispersions, Stenkamp et al. observed that the restabilization was followed by destabilization at higher electrolyte concentrations. Stenkamp et al. expressed the opinion that neither the additional adsorption of the block copolymer caused by decreased polymer solvency with increasing electrolyte concentration, nor ion binding, but rather the presence of micelles or aggregates is responsible for the restabilization. Another possible and more plausible explanation is that the steric interaction is responsible for the restabilization at sufficiently high electrolyte concentrations because of the salting out of the polymer onto the surface of the colloidal particles. The solubility of the polymer in water decreases as the electrolyte concentration increases. As a result, the hydrophobic PPO chains of the block copolymer are adsorbed upon the surface of the polystyrene particle, and the PEO chains stretch out, forming a brush that stabilizes the colloidal dispersion. Whereas, for not too large electrolyte concentrations, the polymer molecules precipitate upon the surface of the particles, at large concentrations, they generate, in addition, aggregates. The latter particles are responsible for the attractive depletion interactions and hence for the destabilization, which follows the maximum in the curve stability ratio against electrolyte concentration. The above explanation of the behavior of the complex dispersions investigated by Stenkamp et al. can be accurate only if, for not too low electrolyte concentrations, the concentration of the block copolymer is lower than the critical micelle concentration (CMC) and, for larger ones, greater than the CMC. The dependence of the CMC on the electrolyte concentration is not known for the present case. However, in pure water, at 25 °C, the CMC is very

10.1021/la0602057 CCC: $33.50 © 2006 American Chemical Society Published on Web 04/11/2006

Polymer brushes

685

4542 Langmuir, Vol. 22, No. 10, 2006

Huang and Ruckenstein

large (25.5 g/L for Pluronic L64 and 24.7 g/L for Pluronic L35 16), and at large electrolyte concentrations, intuition suggests that it is much smaller. Consequently, the proposed explanation might be accurate. In the present paper, calculations are carried out to explain the restabilization followed sometimes by destabilization, as the electrolyte concentration increases, observed by Stenkamp et al. In these calculations, double-layer, steric, and depletion interactions are taken into account. The steric interaction at various electrolyte concentrations is calculated using the scaling theory. The surface density of the polymer chain was evaluated by using the Sechenov equation for the polymer solubility as a function of electrolyte concentration.

the polymer chains present on the surface constrain the orientation of the water molecules nearby. As a result, in the region near the surface, the dielectric constant is lower than in the bulk. The region near the surface (region I) of thickness δ is considered to have a dielectric constant I different from that in the remaining region between plates (region II), II, because of the lower dielectric constant of the polymer and the constraint induced by the chains on the water molecules. For simplicity, the dielectric constants in both regions are assumed to be constant, with that in region II being equal to the dielectric constant of water (80) and that in region I being equal to 60. The electrical potential distribution between the two plates is given by the Poisson equations

2. Theoretical Framework

∂2Ψ F ) - I (for region I) ∂x2  0

(4)

∂2Ψ F ) - II (for region II) ∂x2  0

(5)

The colloidal stability can be expressed as the stability ratio, which is given by the expression17

exp(Vtot/kT) ∞ exp(VvdW/kT) ds/∫2 ds 2 s s2

W ) ∫2

∞

and

(1)

where W is the stability ratio, Vtot is the total interaction potential between two spherical particles, k is the Boltzmann constant, T is the absolute temperature, s is the ratio between the center distance of two colloidal particles and the radius of the particles, and VvdW is the van der Waals interaction potential. Because the block copolymer is uncharged, the electrostatic repulsion, the steric interaction, and the depletion repulsion are independent quantities and are hence additive. The total interaction potential Vtot is therefore given by the sum

Vtot ) Vd + Vsteric + Vdep + VvdW

(2)

where Vd is the double-layer potential between two spheres, Vsteric is the steric potential, Vdep is the depletion potential, and VvdW is the van der Waals interaction potential. When the radius of the spherical colloidal particles a is much larger than the shortest distance between the surfaces of two colloidal particles, the free energy between two identical spherical particles, due to double-layer, steric, and depletion interactions, can be calculated using the Derjaguin approximation:2

Vi ) πa∫H [Fi(H) - Fi(∞)]dH i ) d, steric, dep ∞

(3)

0

where Fi(H) is the free energy of interaction between two parallel plates, H is the distance between plates, and H0 is the shortest approach between the two spheres. The solubility of a block copolymer in water decreases as the concentration of electrolyte increases. When the concentration of the polymer is larger than its solubility, the polymer molecules precipitate onto the surface or form aggregates that remain dispersed into the colloidal system or deposit on the wall of the vessel that contains the colloidal dispersion. Let us start from two parallel plates, and then calculate the stability ratio of the system for spheres using the Derjaguin approximation. 2.1. Double-Layer Interaction. When the copolymer PEOPPO-PEO precipitates onto the surface of the particles, the PPO segments are attached to the surface, and the PEO segments extend into the solution. The dielectric constant near the surface of the plates is changed by the presence of the polymer chains because the latter have a lower dielectric constant. In addition, (16) Lopes, J. R.; Loh, W. Langmuir 1998, 14, 750. (17) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, England, 1986; Vol. 1.

where Ψ is the electrical potential, F is the local charge density, I and II are the dielectric constants in regions I and II, respectively, and 0 is the vacuum permittivity. At the boundary between the two regions, the electric potentials should be the same: II Ψ|Ix)-d+δ ) Ψ|x)-d+δ

(6)

where 2d is the distance between the two plates, and x is measured from the middle of the two plates. If there is no charge accumulation, the Gauss law provides the equation

I

∂Ψ I ∂Ψ | ) II |IIx)-d+δ ∂x x)-d+δ ∂x

(7)

where the superscripts I and II refer to regions I and II, respectively. The boundary condition at the surface has the form

dΨ σ )dx 0

(8)

where σ is the surface charge density. The symmetry condition provides an additional boundary condition,

∂Ψ | )0 ∂x x)0

(9)

Using boundary conditions 8 and 9, the Poisson equations 4 and 5 can be solved to calculate the distribution of the electrical potential for a given surface charge density. For two plates, the double-layer free energy Fd of the interaction between two plates is the sum of three contributions:18

Fd ) Fele + Fentropy + Fchem

(10)

where Fele, Fentropy, and Fchem are the electrostatic, entropic due to the small ions, and chemical contributions to the free energy, respectively. (18) Overbeek, J. T. G. Colloids Surf. 1990, 51, 61.

Nanodispersions

686

Restabilization and Destabilization of Colloids

Langmuir, Vol. 22, No. 10, 2006 4543

The electrostatic contribution comes from the electrical potential energy of the charge in the field:

Fele ) ∫-d d

I 0 ∂Ψ 2 -d+δ  0 ∂Ψ 2 dx ) 2∫-d dx + 2 ∂x 2 ∂x II d-δ  0 ∂Ψ 2 dx (11) ∫-d+δ 2 ∂x

( )

( )

( )

Because the two layers have different dielectric constants, the electrostatic contribution is the sum for those two layers. The entropic contribution to the free energy due to the electrolyte ions is given by18

()

Fentropy ) kT∑ ∫-d ni ln d

i

ni

n0i

- ni + n0i )dx

(12)

where ni is the number of ions of species i per unit volume, and n0i is its value in the bulk. The chemical contribution ∆Fchem is 0 at constant surface charge density. 2.2. Steric Interaction. The steric interaction force fsteric between two plates containing attached polymer chains can be evaluated on the basis of the scaling theory19

fsteric =

[( ) ( ) ]

kT hc D3 2d

9/4

-

2d hc

3/4

(13)

where D is the average distance between two attachment points, and hc/2 is the length of the buoy chain in the solvent phase. (It is assumed to be equal to δ.) This equation is valid only for 2d < hc. For 2d > hc, the steric force is zero. Therefore, the steric interaction free energy between two plates is given by

Fsteric = -∫h fstericd(2d) ) 2d c

[( ( ) )

4kThc 1 hc 5/4 1 + 2d D3 5 1 2d 7/4 1(14) 7 hc

( ( ) )]

The steric interaction energy increases as the average distance between the attachment points decreases. The electrolyte concentration affects the steric interaction through the distance D between the attachment points. To evaluate the distance between the attachment points at different electrolyte concentrations, we consider that the polymer begins to precipitate when the polymer concentration becomes larger than the solubility of the polymer in water. As the electrolyte concentration increases, the solubility of the polymer in the solution decreases. Therefore, the polymer chains dissolved in the solution are salted out onto the surface of the particles and/or form aggregates with increasing electrolyte concentration. As a result of the first effect, the steric interaction free energy increases. The second effect is responsible for depletion interactions. 2.3. Depletion Interaction. When the distance between two plates is smaller than the diameter l of the small particles (micelles), which, in the present case, are uncharged, the depletion force fdep acting between two plates is given by7,20,21

fdep ) -nmkT

(15)

(19) de Gennes P. G. AdV. Colloid Interface Sci. 1987, 27, 189. (20) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168, 485. (21) Huang, H.; Ruckenstein Langmuir 2004, 20, 5412

where nm is the number of aggregates (micelles) per unit volume. When the distance between the two plates is larger than the diameter of the small particles, the depletion force becomes zero because the small particles can be located in the gap. Consequently, the depletion free energy between two plates is given by

Fdep ) -∫l fdepd(2d) ) nmkT(2d - l) 2d

(16)

To calculate the surface density of the polymer on the particles and the number of micelles at various electrolyte concentrations, the solubility of the polymer in the electrolyte solution is required. The Sechenov equation can be employed to calculate the solubility of a polymer in an electrolyte solution. This equation was initially proposed to calculate the solubility of a gas,22 but it is valid for a polymer solution as well.23 It has the form

C log ) -kccsalt cw

(17)

where C is the solubility of the polymer in an electrolyte solution of concentration csalt, cw is the polymer solubility in pure water, and kc is the Sechenov constant, which depends on both the electrolyte and solute. Tadros and Vincent24 have observed that, by decreasing the polymer solvency, one can increase the adsorbed amount. When the polymer concentration is larger than its solubility, the excess polymer is assumed to precipitate on the surface of the particles and/or to form micelles. This assumption allows us to write for the polymer surface density η the expression

η)R

C0 - C 4MNπa2

+ η0

(18)

where N represents the colloidal particle concentration, 2 × 1012 particles/L in the experiments considered;1 C0 (g/L) is the initial concentration of the block copolymer in solution; η0 (mol/m2) is the surface density of polymer before the addition of the electrolyte; R is the fraction of precipitated polymer adsorbed onto the surface; and M is the molecular weight of the polymer. The average distance D between the attachment points decreases with increasing electrolyte concentration due to the precipitation caused by the salting out, and can be calculated from the surface density of the polymer:

D)

1

x2ηNA

(19)

where the factor 2 accounts for the two PEO blocks of the copolymer chain, and NA is Avogadro’s number. The ratio of the polymer deposited on the surface of the particles to that generating micelles was determined by fitting the experimental results. If the number of polymer chains in each micelle is m (which is taken as 50 in the present calculations1), the number density of the micelles is

C0 - C nm ) (1 - R)NA mM

(20)

The size l of an aggregate can be easily calculated. For m ) 50, one obtains that l ≈ 100 Å. The above equation will be used to calculate the depletion free energy between particles. In some (22) Sechenov, M. Z. Phys. Chem. 1889, 4, 117. (23) Borukhov, I.; Leibler, L. Phys. ReV. E 2000, 62, R41. (24) Tadros, T. F.; Vincent, B. J. Phys. Chem. 1980, 84, 1575

Polymer brushes

687

4544 Langmuir, Vol. 22, No. 10, 2006

Huang and Ruckenstein

Figure 1. The stability ratio vs electrolyte concentration for various polymer lengths. The data are from ref 1. 0: L64 (M ) 2900); 4: L35 (M ) 1900). The initial polymer concentration is 60 × 10-3 g/L, a ) 110 nm, AH ) 1 × 10-21 J, hc ) 2 nm, σ ) -0.028 C/m2. (a) The electrolyte is LiCl. Parameters used in the calculation: for Pluronic L35, kc ) 5 × 10-7 M-1 and η0 ) 4.15 × 10-8 mol/m2; for Pluronic L64, kc ) 6 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2. (b) The electrolyte is NaCl. Parameters used in the calculation: for Pluronic L35, kc ) 1 × 10-6 M-1 and η0 ) 3.8 × 10-8 mol/m2; for Pluronic L64, kc ) 3 × 10-6 M-1 and η0 ) 5.1 × 10-8 mol/m2. (c) The electrolyte is KNO3. Parameters used in the calculation: for Pluronic L35, kc ) 1 × 10-6 M-1 and η0 ) 3.5 × 10-8 mol/m2; for Pluronic L64, kc ) 1 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2. The dotted lines represent the calculation results in a range for which no experimental data are available. (d) The electrolyte is KSCN. Parameters used in the calculation: for Pluronic L35, kc ) 5 × 10-7M-1 and η0 ) 4 × 10-8 mol/m2; for Pluronic L64, kc ) 1 × 10-7 M-1 and η0 ) 4.5 × 10-8 mol/m2.

cases, the stability ratio decreases with increasing electrolyte concentration after a minimum followed by a maximum. To explain this behavior, the depletion interaction was taken into account. In some cases, the stability ratio does not pass through a maximum after the minimum; the fraction of polymer that forms micelles is assumed in those cases to be zero. 2.4. van der Waals Interaction. The van der Waals interaction between two identical spherical particles of radius a has the form16

VvdW ) -

[

AH 2a2 2a2 + + 6 H0(4a + H0) (2a + H )2 0 ln

]

H0(4a + H0) (2a + H0)2

(21)

where AH is the Hamaker constant, and H0 is the closest distance between the surfaces of the two spheres. Adding all interaction free energies together, the stability ratio can be calculated for various electrolyte concentrations.

3. Fitting the Experimental Results In Stenkamp et al.’s experiments,1 the particles consisted of polystyrene latex containing sulfate functional groups. In the calculations, we assumed a constant surface charge density. Complex dispersions were generated by the addition of electrolyte solutions to polystyrene latex dispersions containing the amphiphilic block copolymer PEO-PPO-PEO. The stability of such a dispersion was quantified by determining the stability ratio through photon correlation spectroscopy of the incipient aggregation. The stability ratio first decreases and then increases with increasing electrolyte concentration. However, in some cases, at sufficiently high electrolyte concentrations, the stability ratio decreased again after passing through a maximum with increasing electrolyte concentration. The restabilization and destabilization concentrations depend on the nature of the electrolyte. In the present paper, we attempt to explain the restabilization and destabilization using the above theoretical framework. Our explanation implies that the excess polymer precipitates onto the surface of the particles when the polymer concentration becomes larger than its solubility, which decreases with increasing electrolyte concentration in the solution.

Nanodispersions

688

Restabilization and Destabilization of Colloids

Langmuir, Vol. 22, No. 10, 2006 4545

Figure 2. The stability ratio vs electrolyte concentration for various initial polymer concentrations. The data are from ref 1. Pluronic L64 is employed. a ) 110 nm, AH ) 1 × 10-21 J, hc ) 2 nm, σ ) -0.028 C/m2, l ) 10 nm. (a) The electrolyte is LiCl. Parameters used in the calculation: for C0 ) 60 × 10-3g/L, kc ) 6 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2; for C0 ) 0.6 × 10-3 g/L, kc ) 6 × 10-7 M-1 and η0 ) 4.8 × 10-8 mol/m2. (b) The electrolyte is KNO3. Parameters used in the calculation: for C0 ) 60 × 10-3 g/L, kc ) 1 × 10-7 M-1 and η0 ) 5.1 × 10-8 mol/m2; for C0 ) 0.6 × 10-3 g/L, kc ) 1 × 10-7 M-1 and η0 ) 4.2 × 10-8 mol/m2. The dotted lines represent the calculation results in a range for which no experimental data are available. (c) The electrolyte is KBr. Parameters used in the calculation: for C0 ) 0.01 × 10-3 g/L, kc ) 5 × 10-6 M-1 and η0 ) 1.2 × 10-8 mol/m2; for C0 ) 0.6 × 10-3g/L, kc ) 5 × 10-6 M-1 and η0 ) 4.9 × 10-8 mol/m2. (d) The electrolyte is KSCN. Parameters used in the calculation: for C0 ) 60 × 10-3g/L, kc ) 1 × 10-7 M-1 and η0 ) 4.5 × 10-8 mol/m2; for C0 ) 0.6 × 10-3g/L, kc ) 1 × 10-7 M-1 and η0 ) 4.2 × 10-8 mol/m2.

The effect of the molecular weight of the polymer is presented in Figure 1a-d. The increase in the molecular weight of the polymer increases the stability ratio. In some cases, restabilization occurs at sufficiently high electrolyte concentrations. The lowest molecular weight Pluronic L35 does not induce restabilization, while Pluronic L64 induces restabilization for the electrolytes LiCl, NaCl, and KNO3. For Pluronics L35 and L64 in KSCN, the stability ratio decreases to 1 at high electrolyte concentrations without passing through a minimum, or a minimum and a maximum. Another experimental aspect investigated by Stenkamp et al. was the effect of the concentration of the polymer initially dissolved in solution (Figure 2). In general, by decreasing the initial concentration of the polymer, the probability for restabilization to occur decreases. For LiCl and KNO3, a decrease from 60 × 10-3 to 0.6 × 10-3 g/L of the initial concentration of the polymer causes the loss of restabilization. KCl had a stronger propensity for restabilization and could restabilize the colloidal dispersion for a 0.6 × 10-3 g/L initial polymer concentration, but not for a 0.01 × 10-3 g/L concentration. The hydrodynamic diameter of the latex particles was also determined by Stenkamp et al. The results revealed that the average diameter of the restabilized latexes was close to that of

the polymer-coated latex particles in distilled water. However, this does not disprove that a significant additional adsorption is one of the possible causes of restabilization because the increase in the polymer coverage of the surface with increasing electrolyte concentration does not mean that the hydrodynamic diameter changed appreciably. Because the hydrophilic moiety of a polymer chain extends into the water phase, more adsorption of the hydrophobic moieties onto the particle surface due to the salting out will not lead to an appreciable increase in the thickness of the adsorption layer. On the contrary, the thickness of the adsorption layer has the tendency to decrease because of the poor solvency of the polymer at high electrolyte concentrations. The more than qualitative agreement between the experimental data and the calculations shows that the steric interaction can indeed contribute to restabilization at high electrolyte concentrations below the CMC. Of course, if the salting-out effect of the electrolyte is not large enough to precipitate enough polymer on the surface (small kc), the restabilization alone or restabilization followed by destabilization will not occur.

4. Conclusions The stability of the polystyrene latex in electrolyte solutions containing an amphiphilic block polymer first decreases because

Polymer brushes

689

4546 Langmuir, Vol. 22, No. 10, 2006

of the decrease in the electrostatic repulsion. However, at higher electrolyte concentrations, the polystyrene latex could be restabilized in the presence of amphiphilic block copolymers (Pluronics), which, because of salting out, precipitate onto the particles. This occurs because, with increasing surface coverage of the polymer, the steric interaction between two latex particles increases. When, because of salting out, aggregates are also

Huang and Ruckenstein

formed, an attractive depletion force is also generated, which decreases the stability ratio at very high electrolyte concentrations. The calculations suggest that the steric interaction and the depletion force can explain the restabilization followed sometimes by destabilization with increasing electrolyte concentration. LA0602057

Nanodispersions

690

Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

Estimation of the available surface and the jamming coverage in the Random Sequential Adsorption of a binary mixture of disks Marian Manciu, Eli Ruckenstein∗ Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 11 August 2003; accepted 2 October 2003

Abstract A simple analytical expression is proposed for the area available to a disk on a surface for a Random Sequential Adsorption (RSA) of binary mixtures of disks. The expression was obtained by combining the low-order terms of the density expansion of the available area with the asymptotic behaviour of the surface coverage near the jamming point. Comparison with Monte Carlo simulations shows that this approach provides a fair estimation of the jamming coverage for both kinds of disks. © 2003 Elsevier B.V. All rights reserved. Keywords: Random Sequential Adsorption; Jamming; Disk mixture

1. Introduction Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1]. However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3]. The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this

∗ Corresponding author. Tel.: +1-716-645-2911/2214; fax: +1-716-645-3822. E-mail address: [email protected] (E. Ruckenstein).

0927-7757/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2003.10.001

low-density expansion are known [4]; they do not provide sufficient information for the high density behaviour. A typical example is a fortunate continuation of the virial expansion of a hard sphere gas, the Carnahan–Starling equation of state [5], which predicts that the divergence of the compressibility occurs when the total volume of spheres is equal to the total volume available, while at close packing the spheres occupy only about 74% of the total volume. Since the density at which the compressibility diverges is known for monodisperse rigid spheres, it can be incorporated “a posteriori” in the equation of state [6], or one can test which one of the Padé analytic continuations of the virial series provides a better agreement [7]. However, there is no reliable procedure to predict “a priori” the compressibility divergence using only the first virial coefficients. As a consequence, there is no procedure which can predict even roughly the divergence (phase transition) for a binary mixture of hard spheres. While accurate empirical equations of state are known at low densities for binary mixtures [8], they predict divergence of the compressibility only when the total volume is occupied. The situation is similar for a 2D binary mixture of hard disks. The non-equilibrium problem is even more complicated. The large particles can have surface binding energy much larger than kT and in this case they neither diffuse nor desorb from the surface. The Random Sequential Adsorption (RSA) model [9] assumes that a particle, which arrives at a random location on a surface, is adsorbed only if there is

Polymer brushes 2

691 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

sufficient room for it around that particular location (that is, it does not overlap with any other adsorbed particles). Only one continuum RSA model, the 1D random parking of cars, received an analytical solution [10]. This problem implies that points are selected randomly along a line of length L. If there is enough space around the point to fit a “car” of length l, the car is parked; otherwise, another random point is selected, and so on. For a L → ∞ line, when the density of cars is about NRSA l/L ∼ = 0.76 there is no more room to place a car. This result can be contrasted to the “equilibrium” problem of parking cars, for which cars can be parked until close compaction NE l/L = 1 (the 1D gas of hard rods does not have a phase transition [11]). Widom [9] realized the importance of this problem for statistical mechanics and showed that the centers of the particles of a hard disk gas, in an equilibrium position, are not uniformly random distributed. The available area for a new particle Φ(θ) can be written as a power series in particle density θ = Nπr2 /A, where N is the number of adsorbed particles, r their radius and A the total area of the surface. The coefficients of the series terms are identical up to the second power of θ for the equilibrium and the RSA models. The differences in the higher powers coefficients lead for RSA to jamming for θC = 0.76, 0.547 and 0.38 for the 1D (segments on a line), 2D (disks on a surface) and 3D (spheres on a volume), respectively, while for the equilibrium configurations the close-packing occurs at θ = 1, 0.91 and 0.74, respectively. The RSA model received renewed attention after Feder [12] observed that the adsorption on the surface of apoferritin molecules (large iron-storage proteins with a diameter of about 10 nm), which adsorb irreversibly, reached saturation at a coverage θC = 0.518. Monte Carlo simulations of Random Sequential Adsorption of disks on a surface last prohibitively long in the vicinity of the jamming point; however Feder [12] noted that in the vicinity of the jamming coverage, θ has a power-law dependence on time: 1 θC − θ ∝ √ . (1) t By extrapolating this scaling law, an accurate value for the jamming coverage, θC = 0.547 was obtained. The power-law (1) was later demonstrated theoretically, and it was shown that it is accurate not only in the immediate vicinity of jamming, but in a broader θ range [13,14]. Schaaf and Talbot [15] continued Widom’s analytical approach by calculating the available area for the RSA of disks and obtained the coefficient of θ 3 , which is different in RSA and equilibrium models. The next coefficient was obtained independently by Dickman et al. [16] and Given [17]. The first five terms of the series are not, however, very useful for the calculation of the jamming point θ C . Indeed, using the five known terms, there is no jamming, because Φ(θ) = 0 has no solution for 0 < θ < 1. Furthermore, almost all analytical continuations based on Padé approximants P[i, j] pro-

vided the same conclusion, that no jamming occurs (except P[1, 4], which predicted an unsatisfactory value θC ∼ 0.4). A modality to employ the scaling law (4) in the analytical continuation of the series was proposed by Dickman et al. [16]. They calculated the first terms of the time expansion of the coverage θ(t), transformed it to a new variable with an √ appropriate asymptotic behavior y = 1 − (1/ 1 + bt) and found, when the Padé [3,2] approximant was employed, an excellent agreement for the jamming coverage (within about 0.15%) by selecting b = 3. The use of a different value for b or of another Padé approximant deteriorated, however, dramatically the agreement. Later, Wang [18] suggested to estimate b from the convergence of the jamming coverages predicted by various Padé approximants. While the method seems to work well for the RSA of oriented squares (for which the first nine terms of the low-density expansion are known), Wang concluded than there are not sufficient known terms in the RSA expansion for disks to reach an accurate prediction [18]. The estimation of the jamming coverage for the RSA of monodisperse disks is not an important issue, because its value is already accurately known from Monte Carlo simulations [12]. However, it is of interest to develop a procedure that can predict the available area and the jamming coverage for a mixture of disks, for which much less information is available. Even at equilibrium, for which reasonable accurate equations of state for binary mixtures of hard disks are known for low densities [19,20], the available area vanishes only for the unphysical total coverage θ = θS + θL = 1 (where the subscripts S and L stand for small and large disk radii, respectively), hence there is no “jamming”. Exact analytical expressions are known only for the first three virial coefficients of a binary mixture of disks [21]. The fourth and fifth coefficients were computed numerically for some diameter ratios and molar fractions for an equilibrium gas [22]. However, there are no such calculations for the RSA model. Let us first briefly review the recent conclusions about the jamming in a RSA model of binary mixtures. For the 1D model, an exact solution for the random parking of “cars” of two different sizes predicted that the coverage is always larger for a binary mixture than for unisized cars [23]. However, another solution for the 1D model predicted a smaller coverage for binary than for unisized cars in a continuum model, but a larger coverage for binary in a lattice model [24]. For the 2D model, a Monte Carlo simulation showed that a binary mixture of disks always covered the surface better than monodisperse disks [25]. However, another Monte Carlo simulation of deposition of larger spheres on a surface randomly precovered with smaller ones indicated that the total coverage is always smaller for binary mixtures than for unisized disks [26]. These puzzling results can be understood qualitatively. In a mixture of large and small disks, if the large disks are adsorbed first, they can cover up to a fraction θ C of the surface. The jamming then is reached for the large disks,

Nanodispersions

692 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

but the small disks can still be adsorbed, and the binary mixture covers a larger fraction of the surface than the monodisperse disks. In the limit of very large and very small disks, the total coverage is clearly θC + (1 − θC )θC ∼ 0.795. However, if some small disks are adsorbed first without reaching jamming, there is enough room for the small disks, but might not be for the larger ones adsorbed later. In this case, the coverage is lower for the binary mixture. Therefore, the available area and the jamming depend not only on the concentration of the particle already adsorbed on the surface, θ S and θ L , but also on the order of their adsorption. The only approximate analytical solution for the RSA of a binary mixture of hard disks was proposed by Talbot and Schaaf [27]. Their theory is exact in the limit of vanishing small disks radius rS → 0, but fails when the ratio γ = rL /rS of the two kinds of disk radii is less than 3.3; its accuracy for intermediate values is not known. Later, Talbot et al. [28] observed that an approximate expression for the available area derived from the equilibrium Scaled Particle Theory (SPT) [19] provided a reasonable approximation for the available area for a non-equilibrium RSA model, up to the vicinity of the jamming coverage. While this expression can be used to calculate accurately the initial kinetics of adsorption, it invariably predicts that the abundant particles will be adsorbed on the surface until θ = 1, because the Scaled Particle Theory cannot predict jamming. The evaluations of the jamming coverage and of the available area for a binary mixture is of interest in many practical applications, such as protein and cell separation (affinity chromatography), biocompatibility of biomaterials and separation of toner and ink particles [29]. A simple approach to estimate the available area to a disk and the jamming points for binary mixtures of disks will be proposed in what follows. The approach combines the power-law dependence of the available area in the vicinity of the jamming point with the known dependence at low coverage densities. For monodisperse disks, an excellent approximate expression can be constructed in this manner by employing information about the behaviour near the jamming point provided by Monte Carlo simulations. However, reasonable results can be obtained without using the latter information. To calculate the area available to a large disk on a surface for binary mixtures ΦL (θ S , θ L ), it will be assumed that the small disks are deposited first until θ S is reached and then the large disks are adsorbed until jamming. To calculate the area available to small disks, “large” and “small” should be interchanged. The available area ΦL (θ S , 0) can be calculated using one of the available approximations, that are accurate at low densities. The approximations for ΦL (θ S , 0) are in most cases accurate, even when the same approximation for ΦL (θ S , θ L ) fails. The available area ΦL (θ S , θ) is a function of θ alone and can be continued analytically taking into account its asymptotic behavior, which is the same as that corresponding to monodisperse disks. The predictions of the jamming

3

points, obtained from ΦL (θS , (θL )(θS ))C ) ≡ 0 are compared with the Monte Carlo simulations available in [25,26].

2. RSA of monodisperse disks The rate of adsorption of monodisperse disks on a surface, dθ/dt, is proportional to the total surface available to a disk, Φ: dθ = Φ(θ) (2) dt where the surface coverage θ = Nπr2 /A, N being the number of adsorbed disks, r the radius of a disk, A the total area and t a dimensionless time. Monte Carlo simulations indicated that, in the vicinity of jamming, K θ = θC − √ (3) t where θC = 0.547 and K = 0.236 [12]. From Eq. (3) one obtains that dθ K = √ dt 2 t3 which combined with Eqs. (2) and (3), leads for the available area in the vicinity of the jamming point to the expression: 1 ΦF (θ) = (θC − θ)3 (4) 2K2 The first terms of the series expansion of the available area for low surface coverages are also known [15–17]:
Φ(θ) = ci θ i i=0

= 1 − 4θ + 3.307973θ 2 + 1.406876θ 3 + 0.720565θ 4 (5) An approximate expression, which can interpolate between Eqs. (4) and (5), has the form [15]: Φi (θ) = (θC − θ)3 (a1 + a2 θ + a3 θ 2 + a4 θ 3 + · · · )

(6)

where the coefficients ai are determined from the conditions to obey the limit law (4) near θ C : 1 a1 + a2 θC + a3 θC2 + a4 θC3 + · · · = (6a) 2K2 and to match the terms of the θ expansion (5): θC3 a1 = 1

(6b)

θC3 a2 − 3θC2 a1 = −4

(6c)

θC3 a3 − 3θC2 a2 + 3θC a1 = 3.308, . . .

(6d)

The best agreement with the limiting laws Eqs. (4) and (5) is obtained using only four terms in Eq. (6). Then, Eq. (6a)–(6d) lead to 1 a1 = 3 = 6.107 (7a) θC

Polymer brushes

693

4

a2 = a3 =

M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

3 − 4θC = 9.059 θC4 3.308θC2 − 12θC + 6 θC5

(7b) = 8.669

(7c)

(1/2K2 ) − 3.308θC2 + 16θC − 10 = −28.56 (7d) θC Only the first three terms of expansion (5) were employed in Eq. (7a)–(7d). These terms coincide to those of the equilibrium expansion. The use of the next coefficients does not a4 =

improve, however, the accuracy, a situation that is not uncommon in the analytical continuation of a virial expansion [4]. The interpolating approximation (6) is compared with the limiting laws, namely the series expansion Eq. (5) and Feder law Eq. (4) in Fig. 1a. The accuracy of the interpolation can be better seen in Fig. 1b, where the ratios between Eq. (6) and the limiting laws are plotted. The interpolating function (6) satisfies both limiting laws (in their ranges of validity) within a few percents, practically representing the Monte Carlo simulations within their numerical error.

Fig. 1. (a) The area available to a new disk as a function of surface coverage. (b) The ratios of the interpolating function, Eq. (6) to the limiting laws (Eqs. (4) and (5)) as functions of surface coverage. The limiting law (4) is valid at high surface coverages (near jamming), while Eq. (5) is valid at low coverages.

Nanodispersions

694 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

5

Fig. 2. Predictions for the available area Φ(θ).

The jamming point can also be evaluated without using the information provided by the Monte Carlo simulations [30]. While this approach is not useful for monodisperse disks, for which a much better interpolating expression can be obtained using the accurate values for θ C and K predicted by Monte Carlo simulations, it can be employed to estimate the available area and the jamming coverage for binary mixtures of disks, for which in general the values of θ C and K are not known. In Fig. 2, we represent the interpolating function in a natural representation, (Φ(θ))1/3 versus θ, which becomes linear near the jamming point. The interpolating function can be approximated by an expression of the type  
1/3 i ˜ (Φ)(θ)) = (a0 − θ) ai+1 θ (8) i=0

with the coefficients provided by the matching of the first terms of the low-density expansion Eq. (5). The best approximate was obtained by using the first four coefficients, which are given by √ a0 a1 = 3 c0 (8a) √ 3

c1 3c0   2  √ c c 2 1 a3 a0 − a2 = 3 c0 − 3c0 3c0 a2 a0 − a1 =

a3 =

√ 3

 c0

c0

5c13 2c1 c2 − 9c02 81c03

(8b)

(8c)

 (8d)

where ci are the coefficients of the low-density expansion (Eq. (5)). The approximate expression predicts a jamming at

a0 = 0.539, which is not far from the value obtained from Monte Carlo simulations, θC = 0.547. An even simpler approximation can be constructed by matching only the first three terms of expansion (5), with those of expression: 1/3 ˜ (Φ(θ)) = b1 (b0 − θ) + b2 (b0 − θ)2

(9)

whose coefficients are obtained from b0 b1 + b2 b20 =

√ 3 c0

(9a)

√ c1 b1 + 2b2 b0 = − 3 c0 3c0 √ b2 = 3 c0



c2 − 3c0



c1 3c0

(9b) 2  (9c)

providing jamming for θC = b0 = 0.578. It was recently noted by Talbot et al. [28] that an approximate equation derived for thermal equilibrium of disks from the Scaled Particle Theory, namely [19]   3θ θ2 ΦSPT (θ) = (1 − θ) exp − − 1−θ (1 − θ)2

(10)

provided good agreement for the available area in the non-equilibrium RSA model, up to the vicinity of the jamming coverage. In Fig. 2, the interpolating function (6), is compared with the approximations (8) and (9), which are more accurate than the SPT prediction (10), particularly at high θ values.

Polymer brushes 6

695 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

3. RSA of binary mixture of disks When two kinds of disks of radii rS and rL ≡ γrS (with γ > 1) are deposited randomly on a surface, the problem becomes much more involved. It was shown that the Feder law holds for monodisperse disks [12–14]. In contrast, for the RSA of binary mixtures, the jamming of the large particles is reached exponentially and then the density of small particles obeys Feder’s power law (Eq. (4)) until their jamming [31]. However, the Monte Carlo simulations of an alternate deposition of particles, namely: first all the small ones and then the large ones, revealed that the approach to jamming by the large particles also obeys Feder’s law [26]. The jamming coverage θ C and the proportionality constant K depend on the ratio γ = rL /rS and on the initial area occupied by the small particles (the precovered area), θ S . To seek a reasonable accurate analytical approximation for the available area, as a function of θS = NS πrS2 /A and θL = NL πrL2 /A one should have accurate values for a reasonable number of coefficients in the low-density expansion of the binary RSA model, which is not a trivial task. Even for binary mixtures of disks at equilibrium, a problem that received much more attention than RSA, analytical expressions are known only for the first three terms of the virial expansion [21]. The values of the fourth and fifth terms, obtained using laborious numerical calculations, were reported only for a few values of γ and molar fractions of the two types of disks [22]. In the non-equilibrium RSA of binary particles, one should take into account, when calculating the higher terms of the series, not only various γ and molar fractions, but also the order of deposition of particles. Furthermore, as already noted, it is not clear whether the involved calculations needed to obtain the next unknown terms of the low-density expansion would improve much the accuracy of estimating the jamming coverage. An alternate possibility is to obtain approximations for the coefficients of the low-density expansion using analytical approximations of the available surface and to extend analytically these approximations. The expressions for available area, such us those derived from Scaled Particle Theory, ΦSPT i (θS , θL ) [19]: ΦSPT S (θS , θL ) = (1 − θS − θL )   3θS + (θL /γ)(2+(1/γ)) (θS +(θL /γ))2 × exp − − 1−θS −θL (1−θS −θL )2 (11a) ΦSPT L (θS , θL ) = (1 − θS − θL )   3θL + γ(2 + γ)θS (θS + γθL )2 × exp − − (11b) 1 − θS − θL (1 − θS − θL )2 were already shown by Talbot et al. [28] to be excellent approximations at low densities. However, by predicting a

total final coverage of θS + θL = 1, they fail in the vicinity of the jamming point. Therefore, they fail to provide both the final coverage and the final ratio of adsorbed species at jamming. In the present approach, the area available for large disks, ΦL (θS , θL ), will be approximated as follows (interchange everywhere “S” and “L” for the area available to small disks). It will be assumed as an approximation that only the small disks are deposited until the surface coverage θ S is reached, and then only the large particles are adsorbed until their jamming. Along the path (θS = 0, θL = 0) → (θ S , 0) it is assumed that the initial area available to the large disks, ΦL (θS , 0) can be approximated by ΦSPT L (θS , 0), providing that this it is not in the vicinity of the jamming point (which implies that ΦSPT L (θS , 0) is sufficiently large). Along the path (θS , 0) → (θS , θ), the SPT approximation fails at large θ, in the vicinity of the large disks jamming point. However, ΦL (θS , θ) is a function of θ alone, and obeys the asymptotic behaviour (4) (with the constants θ C and K dependent on γ ˜ L (θS , θ) of the type (8) and θ S ). Therefore, an expression Φ or type (9) can be constructed starting from the first coefficients of the low-density expansion: 
1 ∂k ΦSPT (θS , θ)  L ΦL (θS , θ) = θk (12)   k! ∂θ k k=0

θ=0

˜ L (θS , θL ) will provide an estimate for The approximation Φ the available area for large disks on a surface upon which the adsorbed small and large disks cover the fraction θ S and θ L of the surface, respectively. While no data are available in literature for ΦL (θS , θL ), the accuracy of the estimation ˜ L (θS , θL ) can be verified by comparing the jamming of Φ large disks obtained analytically from the conditions ˜ L (θS , (θL )C ) = 0; ˜ S ((θS )C , θL ) = 0 Φ Φ (13) with the jamming predicted by Monte Carlo simulations. Let us first examine whether Eq. (8) or Eq. (9) provides better results, by calculating the total jamming coverage, θC ≡ θS + (θL )C as a function of θ S in the simple case γ = 1 (monodisperse disks), for which accurate results are known from Monte Carlo simulations (θC = 0.547). The low-θ L expansion is given by 
1 ∂k Φ(θ)  ΦL (θS , θ) = (θ − θS )k (14)  k! ∂θ k  k=0

θ=θS

were we selected for Φ(θ) either the interpolating function (6) or the SPT expression (10). The interpolating function (6) was preferred to expression (5) because the former is accurate over the entire θ range, while the latter only for low θ values. The jamming of the “large particles” estimated in this manner for various initial coverages θ S are compared in Fig. 3 with the exact result, θC = 0.547. As expected, both approximations based on Eqs. (8) and (9) are better at large θ S values, where Feder’s law is obeyed, when the interpolating function is employed for the low-θ L expansion. In

Nanodispersions

696 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

7

Fig. 3. The total jamming coverage for a “mixture” of monodisperse disks (γ = 1) predicted using either Eq. (8) or Eq. (9), combined with the low-θ L expansion terms calculated from either Eq.(6) or Eq. (10), as a function of the “small” disks coverage, θ S , are compared to the accurate jamming point, θC = 0.547.

contrast, when the SPT expression (10) is used in Eq. (14), the jamming point estimation deteriorates at high θ S values, mainly because Eq. (10) ceases to be a good approximation in that region. While both approximations (8) and (9) overpredict systematically the jamming, when the SPT ex-

pression is employed for the low-θ expansion (14), Eq. (9) appears to be a better one and will be used in what follows. Let us compare the approach described above with the existing Monte Carlo simulations. While the values for the available area ΦS (θ S , θ L ) are not available in literature, we

Fig. 4. Predicted jamming point for small disks, as a function of the surface coverage by large disks, θ L , compared with the results of Monte Carlo simulations reported in [25], for γ = 2.0 and 8.0.

Polymer brushes 8

697 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

will use the results obtained by Meakin and Jullien [25] for the RSA of a binary mixture of hard disks of different radii and different concentrations. While both kinds of disks are present in the vicinity of the surface, the large ones reach exponentially the jamming and then practically only the small ones are adsorbed until they reach their jamming point. The values of θ L and (θS )C , which correspond to the jamming of small particles, ΦS (θS , θL ) = 0, derived from Monte Carlo simulations [25] are compared in Fig. 4

with the estimates based on the approach described above, which employs Eqs. (9), (11a), (12) and (13). Except for the systematic overprediction of jamming by about 0.05, due mainly to the use of the SPT approximation, the estimates are in a fair agreement with the simulations. It is of interest to test the accuracy of the present approach by estimating the jamming of large disks in the presence of small ones. When both kinds of disks are present in the vicinity of the surface, the large ones reach rapidly their

Fig. 5. Predicted jamming point for the large spheres as a function of the initial surface available to the large spheres, ΦL (θ S ,0), compared with the results of the Monte Carlo simulations reported in [26]: (a) λ = 2.2; (b) λ = 5.0; (c) λ = 10.0.

Nanodispersions

698 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

9

Fig. 5. (Continued ).

jamming, whereas the small disks continue to be adsorbed on the surface for a much longer time. Because the small disks coverage changes very much in the region in which the large disks are almost at their jamming (see Fig. 2 of [25]), the value of θ S , at which the jamming of large particles occurs, cannot be determined accurately (the precision being of the same order of magnitude as the estimation employed here). Accurate values could be obtained from Monte Carlo simulations of successive depositions of small and large disks. The only simulations reported in literature, however, refer to a related, but different problem, namely the RSA of large spheres adsorbed on a surface precovered with small spheres [26]. As noted by Talbot and Schaaf [27], the surface available for a large sphere of radius aL = λaS on a surface precovered with small spheres is the same, in the limit θL → 0, as the surface available to large disks of radius rL = γrS for √ γ = 2 λ − 1. (15) Unfortunately, since the mapping of spheres into disks fails for θL = 0, the derivatives with respect to θ L cannot be calculated accurately from Eqs. (11a) and (11b). Since these values are needed to approximate Φ, this inaccuracy generates additional errors. An additional difficulty is that the SPT approximation is excellent for low surface coverages, (when Φ(θS , θL ) ∼ 1), but fails at large surface coverages (Φ(θS , θL ) ∼ 0). While the small particles might not occupy a large area on the surface, they can exclude most of the surface to the large ones. Since the SPT approximation fails when the available area is small, we plot in Fig. 5a–c the jamming coverage of large particles, ΦL (θS , (θL )C ) as a function of the initial area

available to them, ΦL (θS , 0), to better identify the range of validity of the approximation. There is a good agreement between the Monte Carlo simulations and the present approach for not too low values of ΦL (θ S , 0).

4. Conclusions Despite the attention received by the Random Sequential Adsorption model of particles, only the one-dimensional problem (Renyi’s car parking problem) was solved analytically. Accurate results could be obtained from Monte Carlo simulations in higher dimensions. Simple and accurate expressions for the available area can be constructed from the low-density expansion, combined with the information provided by Monte Carlo simulations for the behaviour near the jamming point. However, reasonable estimates can be made without using the Monte Carlo results. In the present paper, a procedure was suggested to estimate the available area for a Random Sequential Adsorption of binary mixture of disks, for which the Monte Carlo calculations are prohibitively long. It was assumed that first all the small (large) disks were deposited until a value θ S (θ L ) was reached, then the other type of disks were adsorbed until they reached their jamming. Then, the approximate Eqs. (11a) and (11b) were employed to calculate the low-density expansion coefficients of the available area as a function of θ L (θ S ) alone. Finally, an approximate expression (9) with appropriate asymptotic behaviour near the jamming point was constructed. Whereas this method provided only estimates of the jamming points, they represent a significant improvement over the predictions of the Scaled Particle Theory, Eqs. (11a) and (11b).

Polymer brushes 10

699 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects 232 (2004) 1–10

These estimates were compared to the Monte Carlo simulations available in literature for the Random Sequential Adsorption of a binary mixture of disks and spheres. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

I. Langmuir, J. Am. Chem. Soc. 40 (1918) 1361. M.K. Khoshkbarchi, J.H. Vera, Fluid Phase Equilib. 142 (1998) 131. B. Widom, J. Chem. Phys. 39 (1963) 2808. E.J.J. van Rensburg, J. Phys. A 26 (1993) 4805. N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51 (1969) 635. K.R. Hall, J. Chem. Phys. 57 (1972) 2252. F.H. Ree, W.G. Hoover, J. Chem. Phys. 40 (1964) 939. G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland, J. Chem. Phys. 54 (1971) 1523. B. Widom, J. Chem. Phys. 44 (1966) 3888. A. Renyi, Publ. Math. Inst. Hung. Acad. Sci. 3 (1958) 109. L. Tonks, Phys. Rev. 50 (1936) 955. J. Feder, J. Theor. Biol. 87 (1980) 237. R.K. Swendson, Phys. Rev. A 24 (1981) 504.

[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

Y. Pomeau, J. Phys. A. 13 (1980) L193. P. Schaaf, J. Talbot, J. Chem. Phys. 91 (1989) 4401. R. Dickman, J.S. Wang, I. Jensen, J. Chem. Phys. 94 (1991) 8252. J.A. Given, Phys. Rev. A 45 (1992) 816. J.-S. Wang, Colloids Surf. A 165 (2000) 325. J.L. Lebowitz, E. Hefand, E. Praestgaard, J. Chem. Phys. 34 (1965) 1037. C. Bario, J.R. Solana, J. Chem. Phys. 115 (2001) 7123. J.L. Barrat, H. Xu, H.J.-P. Hansen, M. Baus, J. Phys. C: Solid State Phys. 21 (1988) 3165. E. Enciso, N.G. Almarza, M.A. Gonzalez, F.J. Bermejo, Phys. Rev. E 57 (1998) 4486. M.K. Hassan, J. Schmidt, B. Blasius, J. Kurths, Phys Rev. E 65 (2002) 045103R. B. Bonnier, Phys. Rev. E 64 (2001) 066111. P. Meakin, R. Jullien, Phys. Rev. A 46 (1992) 2029. Z. Adamczick, P. Weronski, J. Chem. Phys. 108 (1998) 9851. J. Talbot, P. Schaaf, Phys. Rev. A 40 (1989) 422. J. Talbot, X. Jin, N.-H. Wang, Langmuir 10 (1994) 1663. Z. Adamczyk, Adv. Colloids Interf. Sci. 100 (2003) 267. J.W. Evans, Phys. Rev. Lett. 62 (1989) 2641. M.C. Bartelt, V. Privman, Phys. Rev. A 44 (1991) R2227.