Emulsions Structure Stability and Interactions

Nitrojenzene H+ 337 T • + 398 358 Na* K+ 252 Rb* 206 Cs + 161 289 Ag + Tl+ 176 Mg 2+ 370 Ca 2+ 354 Sr+ 348 Ba 2+ 328 284 NH 4 + acetylcholine + 49 choline 80 Me 4 N + 30 EUN* -67 Pr 4 N + -170 9 Me 4 P + Me 2 V 2+ -15 Et 2 V 2+ -43 Pr 2 V 2+ -58 -250 Bu 4 N + Pe 4 N + -408 Hex 4 N + -472 -372 Ph 4 As + -454 Cl" -316 Br" -295 -195 r CN" -393 -253 NO3" Pi' 47 -83 C1O4" -75 IO4-121 BF4" 2,4-DNPh" -76 372 Ph4B" -270 CIO3" BrO 3 ' -340 NO," Cr,O 7 2 " MnO 4 " -16 C1O2" -350 -189 SOT

1,2-Dichloroethane

1,1-Dichloroethane

Oichloromethane

580 493

Chloroform Propylene carbonate 518 247 154 55 -10 -73 195 114

490 477 445,246 364 459 269

299 307 299

182 44 -91

181 112 -23

195 44

-114 -134 -228

-226 -360 -494 -364 -581 -481 -400 -274

-121

-230 -377 -455

-321

-330 -60 -178 -170 -190 364 -330 -390 -410 -320 -410 -264

-283

-440 -317

-481 -408 -273

-330 -210

-373 -580 -412 -310 -142 -373

-290 -69 -221

-190 -190 -240

-62 -31

373 -310 -370 -370 -310 -370 -73

98

A.G. Volkov and V.S. Markin

A;tf+A^ +RTlti±C 2

2F

(14)


The first term reflects the individual characteristics of the solute ions in the standard state, while the second depends on the activity coefficients and concentration effects. However, in practice, the activity coefficients of cations and anions can be assumed to be approximately equal, y^ ~ yaB and y^ «y p s , so that a simplified expression for the interfacial potential can be written: A"^0 + Aa(K° 2

J?T

Pa/P

IF

Pa/P '

In this case the interfacial potential difference is equal to the arithmetic mean of the standard distribution potentials of individual ions. The potential difference between two phases with a binary electrolyte in both phases is called the distribution potential. In this approximation the distribution potential does not depend on the concentration of the electrolyte BA or the interface structure, but is determined only by the thermodynamic properties of the bulk phases. This has been confirmed by experimental studies, although under certain conditions deviations have been observed. At low concentrations of the salt BA, the interfacial potential measurements can be affected by HCO3" if they are performed in air rather than in an inert gas. Distribution potentials also depend on the concentration of ionic solutes when more than two ionic species are present. The structure of the interphase between two immiscible electrolyte solutions is determined by the bulk properties of the contacting phases. When a monolayer of surfactant that is practically insoluble in both phases is adsorbed at the interface, the surfactant should not alter the interfacial potential determined from Eq. (14). However, due to surface blocking the thermodynamic equilibrium can be affected; and as a result the measured potential difference will be a nonequilibrium quantity that differs from the true distribution potential. One may consider the general case of an asymmetric electrolyte Bk*A'~ partitioning between phases a and |3:

a,Bk;4~ |^ + 4'-,p

(16)

In this case the interfacial potential determined by the distribution of electrolytes is described by the expression

Electric Properties of Oil/Water Interfaces

99

*AB.

Fig. 3. The distribution potential of AB2 electrolyte depends on concentration in the nonaqueous phase.

vw-,A-tf+

RT ^

(17)

For an asymmetric electrolyte, the partition potential depends on the salt concentration. The second term on the right-hand side of Eq. (17) is given by the following expression according to the Debye-Huckel theory:

where Ja is the ionic strength of the solution in the phase a, N a is Avogadro's number, e0 is the electric constant, and ea is the dielectric constant of the phase a. In terms of the Debye approximation, the distribution potential is independent of concentration only for a symmetrical electrolyte, where zA = -ZBIn Figure 3 the dependence of the distribution potential on concentration for a 2:1 electrolyte, for example MgCl2, is presented. For this illustration we

100

A.G. Volkov and V.S. Markin

chose s a = 10 and sp = 80, which corresponds to the common experimental system of water/1,2-dichloroethane. In this situation a hydrophilic electrolyte MgCl2 is dissolved in a two-phase system so that the term Jp/28p/2 can be neglected. As one can see from Fig. 3, the distribution potential of a nonsymmetrical electrolyte strongly depends on concentration. 1.4. Incomplete dissociation of the salt We will next describe how incomplete dissociation of the electrolyte can affect the interfacial potential distribution (Fig. 4). Suppose that the reaction B+ + A'

o

BA

(19)

occurs in phase a, where KgA denotes the dissociation constant of the BA molecule. This means that in equilibrium the equality aaBa°

= K"BAaaBA

(20)

is fulfilled. The dissociation constant KgA is related to the chemical potentials of the corresponding species by the equation

RTinKaBA=^-vT+-vT-,

(21)

Fig. 4. The partition of incompletely dissociated binary electrolyte A'B in a system of two immiscible liquids.

Electric Properties of Oil/Water Interfaces

101

Similar equations can be written for dissociation of the electrolyte in the phase (3. The undissociated BA molecules are distributed between the phases and the partition coefficient Ps"/p is defined by the equation:

RT\nP^

= v.2-vYA •

(22)

Incomplete dissociation of the molecules does not affect the value of an interfacial potential, and the distribution potential determined from Eq. (6) is established at the interface. The standard distribution potentials of the individual ions (see Eq. 5) remain unchanged in exactly the same way. However, owing to the dissociation equilibrium and the interfacial partition of undissociated molecules, the constants of these processes are related to the difference between the Gibbs standard energies of the ions (and hence between the standard potentials):

RT

^-£%n* =A*A +AlGl =FAX- ~FAX+ •

(23)

"•BA'BA

1.5. Complex formation in one of the phases Suppose that the binary electrolyte BA is distributed between the phases a and (3 and that phase a contains in addition a neutral compound T which can combine with the cation B+ to form the neutral charged complex BT+ (Fig. 5). The reaction A

+

B (a) + T(a)

/ff+

o

BT+(a)

(24)

takes place in phase a and the dissociation constant K\T+ is defined by the equation: RTlnKaBT+ = VTT+ - VT+ ~ VT •

(25)

Apart from the homogeneous dissociation constant KaBT+ of the complex BT+, it is also possible to introduce the constant of the heterogeneous or surface dissociation K^+ referring to a process in which the complexing agent T and the complex BT+ are in the phase a while the ion B+ is in the phase (3. This constant is defined by the equation

102

A.G. Volkov and V.S. Markin

P T l n Val^ — n O a , ,°.P Kl [nKBT+-[iBT+-[iB+-[iT

..".a .

(")£.\ (26)

A simple relation between the two dissociation constants is thereby introduced, and the partition coefficient of the ion B+ is - ^ BT+ ~ •**• BT+ ' rB+

\LI)

•

The equation for the heterogeneous dissociation equilibrium is cM=^^ r+ exp(^).

(28)

The case where the complex BT+ is very stable (i.e. its homogeneous dissociation constant KgT+ is small) while the ion B+ is only slightly soluble in the a-phase (the partition coefficient Pfia+/p is small) has special interest. One can then assume that, apart from the charged complex BT+, the a-phase contains only one charged species, the counterion A". It is possible to find the interfacial potential difference for this case:

I

1

"

,

-,1/2

f cp V 2K

\ B+ cacom denotes

)

F

ca

cp

K

2K

A- BT+\

BT+

where the overall concentration of the free and bound complexing agent in phase a and the activity coefficients for simplicity are assumed to be unity. We will consider two limiting cases arising from Eq. (29). Suppose that the concentration of the A"B+ electrolyte is low. Then, by expanding Eq. (29) as a series, we obtain

A

^«fl ta -~fefA r

r

(30)

A- ^BT*

Eq. (30) can be written more conveniently by introducing the interphase partition coefficient of the ion B+ taking into account complex formation: p«/p _ a /J^a/P

r

B+

~ '-con, ' IS-BT+ •

f3 1 \

K-1 y)

Electric Properties of Oil/Water Interfaces

103

Figure 5. The distribution of a binary electrolyte B+A" in two immiscible liquids in the presence of a complexing agent T that selectively binds B" ions.

The expression for the interfacial potential is then RT P a / p A°(h = — l n ^ - .

(32)

This is a typical expression for the distribution potential which does not include the complexing agent concentration. The form of the relation is determined by the valences of the electrolyte, and Eq. (32) is valid only for a 1:1 electrolyte. For arbitrary valences, a more general expression is: F 1 ca — A"d> = — - — I n — T ^ - T T K 1

Z

B~ZA

^A

(33)

A

BT+

The dependence of FA^fy/ RT on lnc"om is described by a straight line with a slope of 1/2 for a 1:1 electrolyte, 1/3 for 1:2 electrolyte, and 1/4 for 2:2 electrolyte. We will now proceed to another limiting case. Suppose that the electrolyte concentration is so high that the concentration of the complexing agent in the phase a is close to saturation. We then obtain from Eq. (31) ca

RT

A"<|) = — l n - ^ V r

r

A-

L

A-

(34)

104

A.G. Volkov and V.S. Markin

Fig. 6. The distribution potential versus electrolyte concentration in phase p when complex formation occurs in phase a.

Fig. 6 illustrates the dependence of FA^/RT on lnc"_, which consists of two linear sections. The slope of the second section is -1 for a univalent anion A" or (ZA)"1 in the arbitrary case. The point of intersection (Fig. 6) of two sections yields 0.5\n(c"omKg^+/P^), which makes it possible to determine the heterogeneous dissociation constant of the complex BT+. 1.6. The Donnan potential The Donnan potential [6]is the name given to the interfacial potential difference that arises when certain ionic solutes cannot cross the interface between two immiscible electrolyte solutions while the remaining ions are free to move reversibly from one phase to the other [1-4]. Suppose, for example, that an ion R with a charge number zR has such a high partition coefficient / ^ t h a t it is almost totally concentrated in the phase a. Apart from the ions R, the system contains a binary electrolyte B+A" with a concentration in the p-phase of

a, i?-\B+A" | B+A",

p

(35)

Electric Properties of Oil/Water Interfaces

105

This type of equilibrium was first considered by Donnan for two aqueous (a/(3) electrolyte solutions separated by a semipermeable membrane. By definition, the partition coefficient of a salt is given by Q

AB

which defines at the equation relating the activities of ions B+ and A" in both phases: U

B+UA~

\rAB

U

A-UB+

!

•

\-> I)

Using this equation and the two electroneutrality conditions for the phases zRcaR+caB+-caA_=0

(38)

cl-cA=0

(39)

one readily obtains the Donnan potential [3]: RT

f P a/p v p v a ^ B+ B+JA

Ag4> = — I n p

IF *"r

J

a

p

P Y v \rA-

(

zaca^\

+ — I n l +^ s - .

-

a/(!

RT

IB+lA-J

\ IF *-r

c V

(40)

a

L

B+ J

Equation (40) is similar to Eq. (15), but differs by the salt distribution potential given by the first term on the right-hand side of Eq. (40). The Donnan potential can also be written as a function of the concentration of the dissolved electrolyte AB: Aa6-—In p IF

B+ a/

+ — I n 1+ ZRC^BA v" ) 2F V 2Pa p a p

1B+1A

~

P K

a

+

Z C

R RYBA

2Pa pap

(4l)

As shown in Eqs. (40) and (41), the Donnan potential is the sum of the distribution potential and the term dependent on the concentration of the impermeable ion R. If this concentration is low, the second term vanishes, and Eqs. (40) and (41) are reduced to the expression for the distribution potential. If the concentrations of R ions is high, the potential tends to another limit. If we setzR> 0 then

106

A.G. Volkov and V.S. Markin

RT z cava Ag4> = — In ™ - .

(42)

In this case the concentration of counter-ions A" in phase a is no longer dependent on the partition coefficient and interfacial potential, and approaches the limiting value caA_*zRcaR.

(43)

In this case the co-ions B+ are almost completely expelled from phase a: a

M^W Z Ca

(44) Y"

1.7. The Nernst potential We now consider an especially important particular case of the Donnan potential where only one ion B~B can pass across the interface. The interfacial potential difference arising in such a system is called the Nernst potential [7]: RT

n^

zBF

aB

A^ = A^°fi+—-In-f-.

(45)

If the phases a and [5 are identical but are separated by a membrane permeable to B+, then the potential difference is given by RT

/J' 3

zzBrF

aaB

A"d) = — l n ^a . P

(46)

Sometimes the potential difference determined from Eq. (45) is also referred to as the Nernst-Donnan potential, because this equation was first obtained by Nernst, but corresponds to a special case of the Donnan equilibrium [1-4]. 1.8. Gibbs free energy of electron and ion transport coupling Suppose that each of the two phases, a and p, contains its own oxidationreduction couple Red/Ox (Fig. 7a). The exchange of n electrons across the interface results in the redox reaction

Electric Properties of Oil/Water Interfaces

v.Red, +v2Ox2 <^v3Red2 + v4Oxl.

107

(47)

The following relation holds under equilibrium conditions: —a

—p

— EJ5

—a

v,^ Red , +v 2 |i O ; t 2 =v 3 ^ Rerf2 +vA\iOxl

(48)

from which the interfacial redox potential can be calculated:

(49)

A^ = AX 0 +^ln^4-^T\uOx\)

\uRed2)

Here kffiRI0 denotes the standard interfacial redox potential of reaction (47), which is equal to

A»K/0 = ( v , ^ . +v 2 ^ 2 -v 3 nL°£ -v 4 ^,)/«f,

(50)

where n is the number of electrons transferred from Red] to Ox2. This is the interfacial potential for equal activities of the oxidized and reduced solutes in each solvent phase. It should be noted that the interfacial redox potential of the reaction (47) is defined by equations (49) and (50) even though one particle is neutral in each of the redox couples. If the pH of one of the phases changes during reaction (47), vlRed] +v2Ox2
(51)

then from the equilibrium condition -a lhterf,

V

+V

-p -EP -a -a 2^O,2 = V 3 h ^ 2 +V4^O,l +™VH+

(52)

one can find the interfacial redox potential [8]:

(

\ VI /

r.

\ V2

^ _ m — a ^ ) ya°f— \UOx\)

\UKtd2)

\UH+)

t

(53)

where ApK/0 = ( v , ^ , + V 2 LC 2 - v 3 ^ 2 - v 4 ^« - m^InF.

(54)

108

A.G. Volkov and V.S. Markin

Fig. 7. The distribution of redox components in a system of two immiscible liquids.

The influence of the liquid interface on the standard redox potential of an electron-exchanges reaction has been studied by Samec [9] and Volkov [10]. 1.9. The mixed potential If each phase of two immiscible liquids contains common ions and redox couples, then the interface acquires a mixed potential, which is contributed by both the interfacial redox potential and the ion distribution between the phases. A two-phase system allows various combinations of electron-exchange reactions. For example, if each of the phases a and (3 contain a redox couple and one of the couples is soluble in both phases (Fig. 7) electron-exchange reactions can take place both at the interface Reda + Oxp <-> Red\ + Oxa 1

2

l

1

(55) V

'

Electric Properties of Oil/Water Interfaces

109

and in the bulk phase P: Rerff + Ox[ <-> Rec/2p + Obcf.

(56)

The equilibrium condition implies that

hLl " hLl =0 -P

(5V)

- a

Ho*l - Maxl =

0

(58) and it follows that

Ag(fr = — h ^ +—In ^"T* 1 . " ^

/

Rerfl

"

r

(59)

"Orl"Re
The standard mixed potential for reactions (55) and (56) is given by A

^ ° - o ^ l ^ .

(60)

Equation (59) is equivalent to Eq. (11) for two-electron reactions ifzRedi = -1, zOxi = +1, and aledl'aMa = a^d^a9M . The last condition can be true only at a single point for specified activities of all the ingredients. This condition does not follow from phase electroneutrality because both phases contain not only the redox couples but also other ions. Equation (59) does not explicitly contain the second redox couple Red2/Ox2, but it appears in this equation in the second term on the right-hand side. The second redox couple must match the first couple because equilibrium in the system is described by the four equations

uLi^Li n L = M-Li -a -p -a -p ^ReJl + Vo*2 = M-0.1 + I^Rerfl

(61) (62) (63)

110

A.G. Volkov and V.S. Markin

-P -P ~P -P ^Rerfl + V-Oxl = Hart + h ^ 2 •

,,.. (64)

If a common electrolyte BpAt is introduced into both phases, this will not affect the general form of Eq. (59). However, the electroneutrality condition for each of the phases

X « = O;I^ = o '

(65)

J

will lead to a change in the distribution of the redox component concentrations in each phase, which in turn will contribute significantly to the second term on the right-hand side of Eq. (59). Most redox reactions occurring at water-oil and water-biomembrane interfaces proceed with a change in pH, for example NADPH o NADP+ + 2e" + H+ .

(66)

Consider a system of two immiscible liquids a and p (Fig. 7). Let each of the phases have its own redox couple, and let one of the couples be common to the two phases: Re
(67)

Rerfp + Ox[ <-» Red\ + Ox" + mH* .

(68)

If in the course of the reaction n electrons are transferred from Red| to OX2 and if m •*• n, then from the equilibrium condition one obtains as expression for the mixed potential A a (b-A a (f)°

—

In

{n-m)F

Rerfl

M

(69^1

a R e d l <,

where the standard potential is given by A«<|»« /o =—^

In^-.

(70)

Electric Properties of Oil/Water Interfaces

111

From the condition that the phases are in thermodynamic equilibrium and from the electroneutrality condition, one can write for reaction (63) 4 + 2 equations for two redox couples, which exceeds the number of independent variables in the system. If in the course of reaction (63) proton equilibrium is established between phases a and f5, the mixed potential can be written in the form ^-Ap^o-—In—

p — + —\pH,

(71)

.

(72)

where A"<|)°/o = — I n " " ^ " ^

}

r

Ked\

Another example of a redox reaction at the oil/water interface has the form Red <-> Ox + we" + mH+

(73)

The electrons which are the products of reaction (73) can be accepted at the interface by a second substance dissolved in one of the two phases. The standard Gibbs energies of the reaction (73) for each phase, a and p, are: AGa° = X e r f - a\x°Ox - n\i°e - myH+ A

G p = P Kerf " p V°ax - nK

-

m

(74) (75)

P V°H+ •

Subtraction of Eq. (74) from Eq. (75) gives the change of the standard Gibbs energy at the interface if the electron acceptor is located in one phase only, or localized at the phase boundary:

AG;-AG: = (^ld-

yKed)-(^Ox-

tt[x°Ox)-m(yH+-

yH+)

(76)

or AG°=RT\n—^—, P (P Y

(77)

112

A.G. Volkov and V.S. Markin

where P{ is the distribution coefficient of the z'-th ion: RT\nP, = ^ , - y ,

(78)

In the case of a /7-electron reaction, the standard redox potential A£° at the interface is determined by: AEo =

_^ln_?krf

(79)

It is possible to shift the redox potential scale in a desired direction by selecting appropriate solvents, thereby permitting reactions to occur that are highly unfavorable in a homogeneous phase. If the resolvation energies of substrates and products are very different, the interface between two immiscible liquids may act as a catalyst [11-14]. 1.10. The tetraphenylborate hypothesis and interfacial potentials It is known from thermodynamics that the absolute value of a potential difference can be measured only in conductors of identical composition. Therefore, the difference of Galvani potentials between a point in an aqueous phase and a point in the bulk of an organic solvent cannot be measured experimentally, so that non-thermodynamic assumptions must be made. It is a simple matter to measure the interfacial potential difference by varying one ion of the dissolved salt. It is also possible to measure the partition coefficient for each salt, P^Jf". By virtue of Eq. (5), the interfacial potential can be represented in the form Ap^A^L-^lnP^.

(81)

In this way a scale of interfacial potentials can be constructed, but this scale will include an unknown constant term, Ap(|)°K, the standard distribution potential of the invariant ion. The problem, therefore, is to choose the origin of the scale correctly. The origin cannot be chosen by thermodynamic methods, so auxiliary methods are required. The historical aspects of the problem are considered in a review by Kolthoff [15].

Electric Properties of Oil/Water Interfaces

113

The idea of the tetraphenylborate method is to choose a cation and an anion with equal standard energies of transfer between two solvents, so that the interfacial distribution potential will be zero. In this case the ion partition coefficients will equal the salt partition coefficient which is easily determined experimentally. Once the latter coefficient is found, we can find the standard free transfer energy and the standard potential distributions of individual ions. These potentials are equal in magnitude but have opposite signs: A^:=-A«(|)°.

(82)

In one application of this approach, Koczorowski [16] worked with tetraethylammonium picrate (Et)4NPi, while others used tetraphenylarsonium tetraphenylborate (Fig. 8). The ions comprising the latter salt, TPhB" and TPhAs+, are very much alike. They are symmetric, almost spherical particles of sufficiently large radius, that can be assumed to have the same hydrophobic properties. Their charges are equal in magnitude, are located at the centers of the spheres, and are screened from the solvent, so that the electrostatic contributions to the free resolvation energy are equal. Such ions have a relatively small polarizability, a low surface charge density, and do not participate in a specific interaction with solvent molecules. For these reasons the standard transfer energies of these ions will be nearly equal for any two solvents. The tetraphenylborate method allows one to determine the standard distribution potentials using cyclic voltametry, chronopotentiometry, polarography at a dropping electrolyte electrode, chronoamperometry, and solubility and extraction data. Since

we can choose a zero point on the potential scale and pass from the scale of applied potentials to the scale of Galvani potentials or distribution potentials. The ion standard distribution potential depends both on the nature of solvent (primarily on the optical and static permittivities) and on the ionic radius (Fig. 2). Figure 2 shows the distribution potentials versus the ion radius for water-nitrobenzene, water-1,2-dichloroethane and water-dichloromethane system. The radii of inorganic ions are taken in the Gourary-Adrian scale [17] and those of tetraalkylammonium salts are found according to Robinson and Stokes [18]. As can be seen from Fig. 2, the ion distribution potential drops sharply with increasing ion radius and the permittivity of the non-aqueous solvent. Solid curves in Fig. 2 are from theoretical calculations. For ions of radius less than 0.3 nm the main contribution to A°<> | " is due to the electrostatic

114

A.G. Volkov and V.S. Markin

portion of the free resolvation energy, and for ions of radius greater than 0.3 nm the main contribution is due to the solvophobic effect [1-4].

Fig. 8. Structural formula of tetraphenylarsonium tetraphenylborate.

1.11. Hung's method of Galvani-potentials calculation Interfacial potential values for systems with small volumes were analyzed by Hung [19-21], who considered the distribution of charged particles between phases and the corresponding interfacial potentials. In particular, the finite volumes of contacting phases which affect the distribution of charged particles were taken into account. Such relationships are complex and generally do not permit analytical solutions. However, solutions can be obtained with the help of simplifying nonthermodynamic assumptions or by numerical methods. Hung considered equilibrium distributions of ions /, between two liquid phases a and P: a

/;•

/;• p

(84)

where Va is the volume of phase a. From the mass conservation law it follows that

Vac« + Fpcf = m,

(85)

where mx is the amount of ion I\ in both phases. Hung also assumed electroneutrality of each phase

:>>.<=°

(86)

Electric Properties of Oil/Water Interfaces

2>,cf=0

115

(87)

i

and it follows from Eq. (85) - (87) that

ZZ'W'=0

(88)

Generally speaking, the total system consisting of phases a and p is electroneutral, but electroneutrality of a single phase (86) and (87) cannot be assumed if one of the phase's volume is small. According to Kakiuchi [22] if the diameter of a given phase is less than 10"6 m, the Poisson equation can be used to correlate local ionic concentrations with the interfacial potential, both of which vary with the distance from the phase boundary and depend on the shape of the interface. Combining Eqs. (85)-(88) and (7) gives Hung's equation: y

z

-£i

+

t r i + (Y«/vyf)exp[(z,F/i?rXA^-A^;)] j

y

7r P.o

^

'

( 8 9 )

^=o

^ l + v(Y;VY?)exp[(z,F//?rXA^-A^)] where v = y°IV® and c° denote the initial concentration. Equation (89) has no analytical solution when the number of ions in the system is more than 3 and must be solved numerically. From Hung's equation (89), it follows that the value of Ap<(> depends on temperature, the volume ratio of the two phases, initial concentrations of components, activity coefficients and standard electrical potentials of ion transfer between phases a and p\ If the system contains only one binary electrolyte A+B", Eq. (89) gives the distribution potential (14). Detailed analysis of Hung's method can be found in reviews [4, 8, 16, 21-24]. 1.12. Distribution potential in small systems As described above, the distribution potentials were very well studied in the systems where both phases had macroscopic dimensions. The condition of electroneutrality is obeyed in the bulk of both phases and the value of the potential is given by equation (14). However, this is not the case in the systems where one of the phases has microscopic dimensions like in microemulsions. If the size of the droplets is comparable with Debye length neither the value of distribution potential nor the solute concentration in the microphase can be

116

A.G. Volkov and V.S. Markin

found from Eqs. (6) or (7). Importance of this problem was underlined in a number of papers [1-4, 8, 16, 21-23], but the solution of the problem was not yet given. In this chapter, we shall analyze this issue and derive appropriate equations. Let us consider a microemulsion with droplets of oil (D) in water (W) (Fig. 9). Let the radius of the droplet is equal to R and the aqueous phase contains a uni-univalent electrolyte with concentration in the bulk (far away from the droplet) equal to cm. The electrolyte can partition into the oil droplet with partition coefficients for cations and anions equal to PCD/W and Pa /w , respectively. If these values are not equal to each other then the distribution potential builds up at the interface. To investigate the effect of geometry we shall use the Poisson-Boltzmann equation for electrical potential \\i, which is a function of radius r. Due to the spherical symmetry of the system the angles are not involved. In the aqueous phase (r > R) the equation is: 1 d_( 2 dy_\ = e^J fe_Mr)>| _ 2 r dr{ dr) zozD[ \ kT ) \

f # ) | kT )\

In the oil droplet (r < R) it has a similar form, but also includes the distribution coefficients:

These equations imply that concentrations of anions and cations in the aqueous phase are

c.(r) = « p [ S ^ ) ) and C , W = e x p ( - 2 ^ ) ,

(92)

correspondingly, while in the oil they are

«.
(93)

We would like to obtain an analytical solution of the problem. To do this, we assume that potentials are small and the exponentials in the PoissonBoltzmann equations can be expanded into series:

Electric Properties of Oil/Water Interfaces

r dry

117

dr J

J_ d( tdyjA^i r dry dr J

(r)

]t

ifr<^

(95)

Here we provided potential \\> in the water or oil with subscripts W or D respectively and introduced the Debye parameters in the water and oil:

£Q Syp Kl

S Q S Q Kl

Additional parameter v|/o that appeared here after expansion into series represents the distribution potential in the macroscopic phases. As was shown above it is equal to O

kT PDIW =—In-^TF-

(97)

However, because we assumed that potentials are small, it can also be expanded into series taking into consideration that the difference between partition coefficients is not very large. Then expression (97) reduces to ( pD/W

pDIW\

M>o-(p/w+r,)eo.

(98)

The set of equations (94) and (95) should be supplemented with two boundary conditions. The first one is the condition of continuous potential at the water-oil interface: Vw(R) = yD(R).

(99)

The second one is the condition of electroneutrality of the total system, which can be presented by the integral: \cc(r)-ca(r)] o

r2dr = 0.

(100)

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A.G. Volkov and V.S. Markin

Solution of equations (94) and (95) can be found as a combination of the modified Bessel functions of the order 1/2: = r- | / 2 [4/ 1 / 2 (K w r) + ^ I / 2 ( K w r ) ]

(101)

v|/o(r) = r- 1/2 [5!/ 1/2 (K w r) + 52Jf1/2(Kwr)] + v|/0.

(102)

¥^(r)

Both potentials should be limited and this immediately gives Af= B2 = 0. The remaining coefficients A2 and B\ should be found from the boundary conditions (99) and (100). Substituting functions (101) and (102) in (99) and (100) and performing integration, one obtains the following system of equations for coefficients A2 and B\.

e-K»* = Rf2-

\h^-

sinh{KDR) +WoR

(103)

2A2 p i ^ ± I e - - « + V r

(r+r)s,J-

(104)

1

AC0Sh(Koi?)_^_Sinh(K^)

=0

Solving this system and substituting coefficients A2 and B\ in (101) and (102) one finds the potential profile in the oil droplet and in surrounding aqueous media:

(105)

(106)

Electric Properties of Oil/Water Interfaces

119

The most interesting parameter is the potential in the center of the droplet. It can be found from Eq.105:

(107)

The expression in the square brackets is always less than 1. Therefore, the potential in the oil droplet is always less than \\i0 and the value of the distribution potential would be established if the oil was present in macroscopic amount. Dependence on radius. The potential (107) depends on the droplet radius R. Let us consider two limiting cases for very small and very large radii. For

(108) Therefore, in the very small droplet the potential disappears and becomes zero. Then again for R —> 0

(109)

In a very large droplet the distribution potential approaches its macroscopic value \|/0. The total dependence of the distribution potential on the droplet radius is presented in Fig. 10. There are two parameters that define this function. In this example, they are selected as: (110) The potential is normalized by \\i0 and the droplet radius is normalized by Debye parameter, KD . One can see the quadratic dependence of the potential on KDR at

120

A.G. Volkov and V.S. Markin

small radii. The half of the maximum potential is achieved when KDR « 2 and then the potential asymptotically approaches the maximum value of v|/0. Dependence on electrolyte concentration. In macroscopic phases the distribution potential does not depend on l:l-electrolyte concentration if one neglects the difference in activity coefficients of ion. The same is true for small droplets. It can be easily seen from Eq. (107) if one introduces a dimensionless radius p = KDR .

(111)

Debye parameters KD and KW depend on electrolyte concentration as 4c. However, these parameters appear in the equation only as a ratio of one to another. This ratio does not depend on concentration, and hence the distribution potential also does not depend on electrolyte concentration.

Fig. 9. Oil droplet (D) in aqueous solution of electrolyte (W). Radius of the droplet is R.

Electric Properties of Oil/Water Interfaces

121

Fig. 10. Distribution potential in the center of the droplet.

Potential at the surface of the droplet. The value of the potential at the droplet surface \\iw(R) = yD(R) = v|/(7?) can be found from Eq. (106):

(112)

Of course, it is only a fraction of the total distribution potential \|/ D (0). We have analyzed the case of small potentials, which is certainly justified for small droplet. We have also obtained analytical solution of the problem. If the droplet is not small and the potential exceeds kT/e0, then the solution can be found numerically.

122

A.G. Volkov and V.S. Markin

2. SURFACE POTENTIALS Mechanical and electrical properties of all interfaces - from pure water to elaborate surfaces of cellular membranes - depend on ion adsorption at these surfaces [25]. This phenomenon plays a very important role in colloid and physical chemistry, biology and medicine. The structure and stability of large biomolecules and membranes depend on the distribution of counterions in the aqueous phase. The transport of ions through ion channels in cellular membranes is determined by the surface potential generated by adsorption of ions from aqueous phase. Conduction of nerve impulses and effect of anesthetics strongly depend on this adsorption. Therefore, it is very important to know the structure of aqueous interfaces and distribution of ions in them. Other fields in which this knowledge is indispensable are seen in environmental aquatic chemistry and chemistry of the atmosphere. Chemical reactions involving aerosolized particles in the atmosphere are derived from the interaction of gaseous species with the liquid water associated with aerosol particles and with dissolved electrolytes. For example, the generation of HONO from nitrogen oxides takes place at the air/water interface in seawater aerosols or in clouds. Clouds convert between 50 and 80% of SO2 to H2SO4 which contributes to the formation of acid rain [26]. Another example is the release of chlorine atoms from sea salt aerosols reacting with the gaseous components. Molecular chlorine, a photolabile precursor of Cl atoms, is a product arising from a heterogeneous reaction of ozone with Cl" ions. Measurements of inorganic chlorine gases indicate the presence of reactive chlorine in the remote marine boundary layer; reactions involving chlorine and bromine can affect the concentrations of ozone, hydrocarbons and cloud condensation nuclei. The heterogeneous reaction between HOBr and HCl converts significant amounts of inactive chlorine (HCl) into reactive chlorine (CIO). These are only a few examples of global processes in environmental chemistry occurring at the air/water interface. In all these cases, the adsorption of simple inorganic ions is especially interesting. In the beginning of the last century, Heydweller [27] found that surface tension of water increases with addition of inorganic electrolytes having relatively small ions. Some organic molecules (sodium formate, glycine, amino acids at the isoelectric point when both the amino and the carboxyl groups are ionized in the form of zwitterion) also increase surface tension. The surface tension of solutions increases almost linearly with increasing concentration of salts, and for inorganic salts there is an empirical equation: y = y0 + be, where b = 1.7 mNttV'M"1 for LiOH, 1.8 niNn^'M"1 for NaOH and KOH [28], 1.63 mNm' 'M"1 for NaCl and 1.33 m N m ' V for KC1 [29]. For organic tetraalkylammonium salts the constant b decreases with increasing of cation

Electric Properties of Oil/Water Interfaces

123

radius: for tetramethylammonium chloride b is positive, for tetraethyl ammonium chloride b is about zero, and for bigger cations b is positive. This phenomenon was explained as a result of negative adsorption of small ions. Wagner [30] was the first to describe this phenomenon as an electrostatic effect of image forces. Onsager and Samaras [31] derived equations, which predicted the increase of surface tension with increase of electrolyte concentration, but only for the very dilute solutions. Later Buff and Stillinger [32] presented the statistical mechanical formulation of the same problem by also employing the conventional model for very dilute solutions. They calculated the molecular distribution functions from a generalized form of Kirkwood's integral equation and found the surface tension directly from the molecular theory for this thermodynamic parameter. These approaches were quite successful, although they used a simple theory for point charges and did not take into account effects of ion radii. It was Frumkin [33,34] who indicated that the increase of surface tension and surface potential of aqueous solutions of inorganic salts depended on ionic radii. However, it was believed for a long time that equivalent concentrations of different salts gave very similar variations of surface tension^ at least at small concentrations [35]. Later it was found that the differences are rather pronounced even at small concentrations [36]. At higher concentrations, different ions exhibit specific differences: the higher the ion hydration, the higher the surface tension increment. The contributions of two ions of the electrolyte to the surface tension appear almost additive. There was a series of attempts to advance the theoretical explanation of the surface tension given by Onsager and Samaras [31], but without much success. A critical discussion of these attempts can be found in Randies [37]. Krylov and Levich [38] considered the discreetness of the charge of adsorbed ions, but found this effect to be rather small. Having in mind all these improvements, we still have to stress that the original works of Wagner [30] and Onsager and Samaras [31] correctly identified the major players determining the surface tension of the aqueous electrolyte - image forces at the interface and negative adsorption of ions. Using the well-known Gibbs adsorption equation, they calculated the value of surface tension without ascribing too much importance to the difference between anions and cations [30, 31]. The surface tension increment was explained at least for the small concentration of electrolytes. In contrast to this, the situation with the surface potential was completely hopeless. In Onsager-Samaras approach, if there is no difference between anions and cations (beyond the charge sign), there is no reason to expect that they would generate any surface potential because they would equally be negatively adsorbed at the interface and their charges completely compensate each other. Therefore, the whole effect of change of the surface potential of water in the presence of electrolytes originates from the

124

A.G. Volkov and V.S. Markin

difference between anions and cations. In this sense, this effect is more subtle and elusive than surface tension. The theoretical explanation of this phenomenon was published recently by Markin and Volkov [39]. The explanation of the surface potential can be found in the difference in ionic radii and their manifestations at water surface. Because of finite radii, different ions can interact differently with interfaces. This results in a different interfacial distribution of energy, concentration and charge density of these ions. We shall use the model of ions with finite radii to describe both surface tension and surface potential at the wide range of concentrations. 2.1. The model: Ions of finite radii at the air/water interface Let us consider an air/water interface as depicted in Fig. 11. Coordinate x is directed from the interface to the bulk of water. The energy of an ion with radius a at the interface was calculated by Kharkats and Ulstrup. They accounted for the Born solvation energy and the energy of interaction with the image. When x > a, the energy of an ion can be presented as WKU{x) = (zeof \ 327te0E(ra|

Uw-zA\la

Uw-zA~J\

\KEW+EA)X

\EW+EA)

2 [_\-{2xlaf

^a^lx 2x

+ a^ 2x-a_\

(H3)

The first term in curls in (113) represents the Born solvation energy, and the last two terms give the interaction with the image. In the region 0 < x < a:

327iEoSw,a [V

\\ + xla){\-2xla) \ + 2xla

|

a)

a

f

\zw+zA)\ |

2x \

2xVjj a)\\

a) |

{EW+EAJ

(zeof f 2s, Y ^ x~\ \6%zazAa\zw +e 4 J ^

a)

When the center of the ion is right at the interface (x = 0), the electrostatic Gibbs energy is equal to WKU(x = 0) =

(ze Y K -^

4mo(Ew+sA)a

.

(115)

Electric Properties of Oil/Water Interfaces

125

Fig. 11. Distribution of water and ion concentration at distance* from the air/water interface; XG is the position of the Gibbs dividing surface.

When the ion is located in the air, the electrostatic free energy is obtained from equations (113) and (114) by exchanging Zw <=> %AThe image energy can be derived from eqs. (113) - (115) by subtracting Born energy W^om = JFKu(+GO) • One can see an important difference: the energy of a point charge becomes infinite at the interface, while the Kharkats-Ulstrup function eliminates this feature and makes the curve continuous. Equations (113)-(114) give the energy of a lone ion interacting only with its image. However the presence of other ions in the solution will modify this interaction due to Debye screening of electrical field. Therefore the ion/image interaction should decay much faster than in equations (113) - (114). To take this effect into consideration, we shall introduce the screening function J[x) as follows: (116)

126

A.G. Volkov and V.S. Markin

with Debye constant K and ion radii aan and acat. For binary electrolyte the Debye constant is given byK= jegCg^zf /zoekT . This definition consistently takes into consideration the model of ions as hard spheres so that cations (cat) and anions (an) cannot approach each other closer than at aan + acal. Therefore the image energy can be presented as

Wbnv(x) = f(x)[WKU{x)-WKU(^)].

(117)

This function would work rather well for calculations of the surface tension. However, for calculation of the surface potential one would need to take into account more subtle effects that help to distinguish between cations and anions. One of them is the solvophobic effect that takes into consideration the work of creating the cavity to place the ion into the solvent. If one assumes that there is surface tension ycavily in this cavity, then in the bulk of the solvent the solvophobic energy is 4na2ycavity. If the ion crosses the interface from right to left (Fig. 11) the area of the cavity decreases and with it the solvophobic energy. If the ion center is positioned at x, then in the range -a < x < a the energy is 2na(a + x)ycmily; at x<-a, it is 0. If the reference point is selected in the bulk of water, then these quantities should be decreased by 4na2ycmjty, and one finally obtains the relative solvophobic energy as 2a, if x < -a Wsolvophoblc = -2%aycavily-a-x,if

-a<x
(118)

0, if x>a The surface tension in the cavity is not very well known; it is believed to be close to the surface tension of water, but is probably less than that due to the small radius of curvature as was argued by Tolman [42]. We shall estimate this value in the range of 40 - 50 mN/m. One can see that the solvophobic effect decreases the energy at the interface. The total energy of ion i with charge number z, includes image and solvophobic components and electrical potential of the electrical double layer cp(x): W, (x) = Wwmge (x) + WsohophohK (x) + z,e0 [q,(x) - 9(00)] .

(119)

Electric Properties of Oil/Water Interfaces

127

Fig. 12. Surface tension of water in the presence of NaCl: (dots) experimental data; (dashed line) Onsager limiting law; (solid line) according to Markin and Volkov [39].

Notice that the image energy and the solvophobic energy also depend on the type of ions. To calculate the spatial distribution of ions we used standard methods for the electrical double layer. The Poisson-Boltzmann equation can be presented in the following way:

£2=-S&2> l O J-S&l. dx

sos i

\_

(120)

kT

where dielectric constant 8 assumes the value eA or ew depending on coordinate x. We assume that the aqueous phase contains a binary 1:1 electrolyte with bulk concentration c0. Energy W\ is given by equation (96) and summation includes both cations and anions. For boundary conditions, we assume that the electrical potential

^

=0.

(121)

Equation (97) with boundary conditions (98) can be solved numerically and both surface tension and surface potential can be found.

128

A.G. Volkov and V.S. Markin

2.2. Concentration dependence of surface tension In the first theoretical description of the surface tension of electrolyte solutions Wagner and Onsager and Samaras [31] presented the energy of interaction of an ion with its image at the distance x from the interface as

W(x) =

^

exp(-2Kx)>

(122)

l6m0Ewx where e0 designates the charge of the ion, s0 - the permittivity of free space, sw - the dielectric constant of water, K - Debye constant. This function is presented in Fig. 12 by the dashed line. The surface excess of ions (/) with respect to water (w) was calculated as CO

,(W) = o J W h W(XV kT] - l)dx .

r

c

(123)

o Using surface excesses of ions, Onsager and Samaras calculated the surface tension as a sum of a series and tabulated this sum [31]. For very dilute solutions of 1:1 electrolytes, they found an analytical expression for the surface tension, which they called the limiting law: Ay = y-y 0 =1.012clog(1.467/c)-

024)

This equation is presented in Fig. 12 (dashed line), together with experimental data for NaCl. As one can see, the limiting law coincides with the experimental data only at concentrations smaller than 0.1 M, after which the discrepancy becomes very large. The tabulated values improve the situation only a little. And of course, the limiting law does not distinguish between different ions describing them by the same equation. The main drawback of previous approaches was the absence of an appropriate model of ions at the air-water interface. To obtain a radical improvement of the theory of surface tension in the presence of simple electrolytes, we shall use the new expression for energy (119) that accounts for both finite radius of ions and solvophobic effect. We shall begin with calculation of the surface tension, because surface potential at low electrolyte concentrations has some peculiarities and cannot be found directly from the standard Poisson-Boltzmann equation. Fortunately, this does not influence the calculation of surface tension because at low

Electric Properties of Oil/Water Interfaces

129

concentrations the potential is small and has no effect on the distribution of ions that are determined by image forces and solvophobic effect only. Let us calculate surface excesses of ions i, with respect to water Fj(W) . The Gibbs dividing surface corresponding to zero-excess of water (Fw = 0) does not necessarily coincide with the surface of water as demonstrated in Fig. 11, where its coordinate is designated xG . If the bulk concentration of ions in the air and water are correspondingly 0 and CQ, then the ion surface excesses [39] are

F, ( w ) = c ;|exp[-^lj dx + c 0 + |[exp[-^ll-l \dx.

(125)

Position of the Gibbs dividing surface, xG, was found in the following manner. Water concentration in solutions of simple salts depends virtually linearly on salt concentration as one can see in Fig. 13 for solutions of NaCl and KC1. The data for these examples were taken from CRC Handbook of Chemistry and Physics [29]. The trend lines of these dependences can be presented as cwaler = 5 5 . 6 - a csah.

(126)

The slope a was found for NaCl to be equal to 1.18 and for KC1 equal to 1.73. One has to have in mind that at the far right end of the concentrations scale (about 3 M) the data taken from CRC Handbook of Chemistry and Physics [29] probably are not very reliable because of the saturation effect. However, at smaller concentrations, the data ideally fit a straight line and this is the only result we take from these data. Besides, as indicated later in the text, the effect of the displacement of the Gibbs dividing surface is rather small. Therefore, any small inaccuracy resulting from saturation effect will become negligible in the final result. Because concentrations of anions and cations are slightly different at the interface, we shall define the salt concentration as the mean of the two. Then the spatial distribution of water concentration in the solution can be presented by

c,^(x) = 55.6-^S> jexp " ^ ^

+«xP - ^ ^

j.

(127)

Assuming that the actual surface of water is located at coordinate xD = 0, the water excess at this dividing surface is

130

A.G. Volkov and V.S. Markin

r».,,,« ) =|]{2-expf-2^W] + e x p [ - ( ^ ] } ,

(12S)

and the position of the dividing surface corresponding to zero water excess is

Fig. 13. Water concentration in aqueous solutions of NaCl and KCI. Experimental points are taken from ref. [29].

Fig. 14. Dependence of the surface tension of aqueous solution of KCI on concentration. Experimental points are taken from ref. [29]. The solid line is calculated using Eq. 131.

Electric Properties of Oil/Water Interfaces

131

However, position of this Gibbs dividing surface is different from 0 only at very high salt concentrations. In the isothermal case the surface tension is related to the surface excesses by the Gibbs adsorption equation:

rfy = - t r w 4 .

(130)

In the simple case of a uni-univalent electrolyte the increment of the surface tension can be presented as C

j

Ay = y - Yo = - j(r flmon(w) + rcalmniw))—. o

(131) c

To calculate this quantity we have to solve the Poisson-Boltzmann equation (123) and find the integral in Eq. (131). We carried out numerical integration for a number of different electrolytes and compared them with experimental data. Results for KC1 are presented in Fig. 14. One can see that the present model provides radical improvement comparative to the Onsager limiting law, although at very high concentrations there is a certain discrepancy between experimental and theoretical data. 2.3. Surface potential The elegant theory by Onsager and Samaras for surface tension predicted a zero surface potential because it did not envision any difference between the way cations and anions interacted with the interface. However, experimental observations clearly demonstrated rather pronounced surface potential depending on electrolyte concentration. The sign of this potential with respect to the bulk of water very often is negative, although some salts give positive potential. In the absence of a rigorous theory, a number of semi-empirical attempts were made [37, 43]. The increase of the surface tension with electrolyte concentration was interpreted as the presence of a solute-free layer 4-5 A thick at the surface. Randies [43] assumed that this surface layer is completely inaccessible to cations in solutions of all alkali metal salts, but anions can penetrate more closely to the surface. In other words, Randies postulated two different planes of closest approach for anions and cations. The anions built up negative charge at the surface that was balanced by positive charge of cations deeper in the bulk and hence a negative surface potential was established. Using this simple model and the standard kinetic theory of the electrical double layer,

132

A.G. Volkov and V.S. Markin

Randies calculated concentration dependence of the surface potential. However, comparing his theoretical results with experimental data he had to admit that his model was quite unable to account for the observed surface potentials. The problem was not simply in numbers: theoretical predictions were qualitatively wrong. The Randies theory predicted that the surface potential at low concentrations should increase as a square root of concentration, while experimentally this dependence was either linear or even superlinear. Therefore the theoretical and experimental lines had opposite curvature which was unacceptable. This finding seemed rather strange because the model qualitatively cannot be wrong: there is definitely spatial separation of positive and negative ions at the interface that creates surface charge and this charge should be balanced by the other part of the electrical double layer. At low concentrations the electrical potential is small and does not influence the ion distribution at the interface. Therefore one can conclude that the surface charge Q should grow linearly with the electrolyte concentration: Q~c0 . If the double layer capacitance is CDL , then the surface potential is (? = Q/CDL. According to Gouy-Chapman theory, at low concentrations the capacitance is equal to the capacitance Cdiffuse of the diffuse part of the layer and hence is proportional to the square root of the electrolyte concentration: CDL

~ Cd,ffuse ~ V C 0 •

(132)

That is why Randies came up with the result


(133)

However, there is an interesting anomaly in the capacitance of the double electric layer: at low electrolyte concentrations it does not obey the theory of Gouy-Chapman. This phenomenon named the Parsons-Zobel effect [44] was discovered by Samec et al. [45] and later studied by Wandlowski et al. [46]. These authors analyzed the relationship between inverse capacitances CD£X and Cdiffuse'X- According to Gouy-Chapman-Stern or Vervey-Nissen theory this

Electric Properties of Oil/Water Interfaces

133

Fig. 15. Parsons-Zobel (PZ) plots for the nitrobenzene/water interface. The solid line was calculated according to the Gouy-Chapman theory with the concentration independent capacitance of compact layer equal to 0.74 F/m~. Experimental points are taken from Samec etal. [45].

relationship is given by a straight line with cut-off at the ordinate determined by the compact layer capacitance [47]. However, experimentally observed, the total capacitance at low concentrations of electrolyte (large Q , ^ / 1 ) strongly departs from the straight line and displays the tendency to become constant (Fig. 15). We shall discuss this peculiarity later but now just accept this fact 'at face value'. If relationship (132) does not hold and the double layer capacitance is rather constant, then relationship (133) should rather transform to cp~c 0 .

(134)

This corresponds much better to the experimental observations. Following this line of arguments, we calculated the surface potential for different salts. First, using previous equations we calculated surface excesses of cations and anions and found surface charge as Q = F(rcatl0HK)-ranwniw)),

(135)

134

A.G. Volkov and V.S. Markin

Fig. 16. Theoretical estimation of the surface potential on concentration for NaCI aqueous solution based on Gouy-Chapman (GC= - 4.77-y/c) model, with Parsons-Zobel (PZ= - 10c) effect at liquid interfaces, and the actual curve (Approximation= x(0.0001 + 0.00193x2 J

).

Fig. 17. Dependence of the surface potential on concentration. Theoretical lines for KF (triangles), KC1 (filled circles), and NaCI (diamonds). Experimental points are taken from Frumkin and Randies [33, 34, 37]. The curves were drawn according to the following equations: KF = |"(2.092V^ +0.261c)' 4 +cT4 /1296~| NaCl= -c(0.0001 + 0.00193c 2 )"" 4 .

, KC1 = -c(o.OO666 + O.O393c 2 )~"\

135

Electric Properties of Oil/Water Interfaces

Fig. 18. Surface potential of an aqueous solution of KI. Experimental points are taken from Frumkin [33, 34] (V) and Jarvis and Scheiman [36] ( • ) . Radius of I" was taken 0.206 nm (dashed line, equation = radius

P

of

0.22 V4

-[~(l8.839Vc - 2 . 5 1 4 C ) " 4 +
(thin

continuous

line,

), or crystallographic equation

=

-1-1/4

30.42Vc-5.429cJ + c~4/83521 ), or estimated as 0.25 nm (solid line, equation = -x [2.744* 10~6 +0.000138x 2 (9.23-4.724cVc ) * 1

where F is a Faraday. Using the data for the interface nitrobenzene-water [45] we assumed that at low concentration the limiting value of the double layer capacitance is about 0.1 F/m2. The results for NaCl are presented in Fig. 16 by the straight line marked PZ. Calculations based on the Gouy-Chapman theory, as described in the section 'Surface tension', are presented by the line marked GC. One can see that the Parsons-Zobel effect works only at small concentrations where it predicts bigger capacitance and hence the smaller magnitude of the surface potential than Gouy-Chapman theory. Later on, this relationship changes and the Gouy-Chapman theory determines the limiting value of capacitance. Therefore the actual curve should smoothly change from PZ to GC as presented by the 'Approximation' line. Similar calculations were performed for a number of salts and the results are presented in Fig. 17. Notice that the surface potential for KF is positive while all the others are negative in total agreement with experimental data. At the separate Fig. 18 we presented data for KI. Iodide salts often display a number of peculiarities that put them aside from other simple salts of alkali metals. For example, iodide in aqueous solutions in the presence of oxygen has the tendency to form complexes like IO3~ and/;. Therefore, it came as no surprise that calculated curves predicted much smaller (in magnitude) surface

136

A.G. Volkov and V.S. Markin

potential than experimental data. To account for the presence of complexes I3 , we took the radius of anions in this case as a fitting parameter and found that the data can be nicely described with r_ = 0.250 nm. We have calculated both surface tension and surface potential for aqueous solutions of simple electrolytes. For this purpose we used a new model of ions at the interface. It considered the solvent in the classical way as a dielectric continuum. The model took into consideration finite radii of ions, which permitted to eliminate the divergent function, describing the energy at the interface for point ions and providing a way to distinguish between different types of ions. By applying standard methods of the electrical double layer for ions of finite radii and solving the ensuing equations, we were able to find the surface tension in a wide range of concentrations in agreement With experimental data. Previous investigations could explain the surface tension up to centimolar concentrations, while in the present work we extended this range to molar concentrations. The surface potential was explained for the first time both qualitatively and quantitatively. In this model [39], the sign of the surface potential was determined by the larger ion that was less repelled from the surface than the smaller one. However, if only the finite radii of ions were taken into account, then the calculated potentials were much smaller in amplitude than experimental data. This discrepancy happened because the surface potential is much more sensitive to the subtle details of the model than is the surface tension. While the latter is determined by the sum of anion and cation adsorptions, the former is determined by their difference. This is a classical case of a small difference of big numbers: even a very small inaccuracy of any of these big numbers brings a large error in their difference. Therefore we looked for additional effects that could help to distinguish between cations and anions. This happened to be the solvophobic effect, which does not change much the adsorption of each ion but considerably contributes to their difference. As a result, the amplitude of calculated potentials increased significantly and became closer to the observations. Calculation of the energy of the solvophobic ion-solvent interaction is based on phenomenological or statistical mechanical molecular models in which the parameters describing both the ion and solvent are not always known. To avoid uncertainty, the solvophobic contribution to the solvation energy is sometimes estimated using the semi-empirical solvophobic formula [48]. The main idea was formulated and discussed in a number of papers [49-53]. Surface energy at the interface of cavity in water created to accommodate the ion with radius a, when expressed in terms of surface tension at the air/water interface Ycavity, is equal to ApG°(svp/!) = 47ra2ya[isgn(Ya - y p ). This relation infers that the molecules of the two solvents water and air do not mix or react

Electric Properties of Oil/Water Interfaces

137

chemically with each other. The hydrophobic contribution to the Gibbs free energy of ion or molecule is obviously greater for particles with larger radii, a. One also has to take into account that ions are very small entities and the surface tension in cavities should be smaller than at the planar interface. The effect of curvature on the surface tension at a molecularly sized sphere was estimated by Tolman [42]. The surface tension at radius a can be written as

Jcav^ = YP/awar(l + 25/ a ) > where S according to Tolman, is the distance from the surface of tension to the dividing surface for which the surface excess of fluid vanishes. Parameter S can vary between 0 and a few angstroms. We found that this effect has optimum contribution to the adsorption of ions at ycavity = 50 mN/m. From a theoretical point of view, the limits of applicability of the solvophobic formula are not quite clear. Nonetheless, it works surprisingly well for calculation of the partition coefficients of a system of two immiscible liquids [54]. The only problem that remained with surface potentials was the very fast growth of calculated potentials at small concentrations: they grew as a square root of concentration, meaning that it is infinitely fast, while experimentally potentials changed rather linear with concentration. This last problem was resolved by taking into consideration the Parsons-Zobel effect [45, 46] according to which at small electrolyte concentrations the capacitance of the double layer did not vary like Jc^, but rather became constant in an obvious contradiction with the classical point of view. We modified our calculations at small concentrations taking into account the constant capacitance and providing the smooth transition from one limiting case to another. The results are presented in Figs. 17 and 18. So, our analysis was based on the model that accounts for finite radii of ions, the solvophobic effect and the Parsons-Zobel effect. The model worked rather nicely for simple ions like Na+, K+, Cl" and F ' as one can see in Fig. 17. However, with ions having the tendency to redox and photoredox reactions with oxygen (air), like iodide that is known to form IO3" and I3" ions, the situation is different. Following the same method we have calculated the surface potential for the solution of KI, assuming the radius of I" equal to 0.205 nm according to Gourary and Adrian or the crystallographic value of 0.220 nm. The theoretical curves are presented by the broken and thin continuous lines in Fig. 16. As one can see, both of them predict much smaller potential than was actually observed. We believe that it is due to the fact that a fraction of iodide ions exists in the form of I3" or IO3" complexes and has a bigger radius. We did not try to estimate its value independently but rather considered it as an effective radius and a curve-fitting parameter to achieve the agreement between the theory and

138

A.G. Volkov and V.S. Markin

experiment. We found that it can be done with effective radius of 0.250 nm. This number does not seem unrealistic and hence it provides an estimate of an important parameter for this salt. We found a reasonable agreement between the theory and experiment, although not an ideal one. There are a few reasons for certain discrepancies. First, the experimental data on the surface potential is not very numerous and often are contradictory. Measuring the surface potential is a rather delicate procedure and it is also method-dependent. Surface potentials at the air/water interface can be measured by the dynamic capacitor method, the radioactive probe method, and the jet electrode method [8]. Each method gives the opportunity to measure surface potential at the air/water interface with the accuracy about ±1 mV, but difference in measured values between different authors can reach more than ten percent. There can be a problem with elimination of diffusional potential in the salt bridge between the measuring cell and the reference electrode [33, 34] or effect of charge generation by the radioactive probe. For example, for 0.5 M KI aqueous solution surface potential was measured to be equal to 22 mV [33, 34], 10 mV [37], or 12 mV [36]. Second, the present theory is based on a rather simple (although improved with respect to previous ones) model of interface. The early publications [55, 56] merely assumed that there is a layer of a few water molecules at interface completely devoid of ions. This naive approach could explain qualitatively the increments of the surface tension, but was unable to explain the surface potential. For this purpose, Randies [43] arbitrarily postulated that this layer is accessible to only one type of ion, and hence the surface potential can build up. However, even in this case, results were qualitatively unsatisfactory. Our model overcomes these difficulties. It is formulated in a classical way where the solvent is modeled as a continuum dielectric characterized by its macroscopic dielectric properties. This approach invokes a simple, easily interpretable physical picture that does not involve many adjustable parameters. However, it applies the concept of the macroscopic dielectric response to microscopic distances where it may not have a clear physical meaning. There is a number of other aspects where our model may be vulnerable to criticism. It is obviously naive to visualize the air/water interface as a flat and sharp dielectric border between two phases. Fluid interfaces are usually treated in electrochemistry like that, but they are known to be never ideally flat because of the thermal excitation of capillary waves. Capillary waves induce roughness of fluid interfaces and may influence their electrochemical properties [57]. So, the surface is dynamic. Its dielectric constant changes with distance and so on. One could not expect from this simple model to provide an ideal agreement with the experiment. It is rather surprising that we were able to go this far predicting the surface tension with discrepancy of only about 10% and about the same for the surface potential. It is rather amazing because the model

Electric Properties of Oil/Water Interfaces

139

completely ignores many important properties of the system like the discrete nature of the solvent, dynamic variability of the interface and many other features. Any future development should include all these features and account for the detailed structure of the interface. Unfortunately, this structure is not very well known. There are a number of publications where authors try to elucidate the structure of water interface and the manner in which ions interact. One interesting question is, are the water molecules oriented at a certain degree at the interface? If they are, then they could create a permanent dipole potential at the interface that could influence the distribution of ions at the interface, and hence generate the increment of the surface potential in the presence of electrolytes. However, opinions about this issue are controversial. There are no reliable experiments to determine this orientation. Computational methods of molecular dynamics might be helpful, but different authors come to different conclusions. Wilson and Pohorille [58], as well as other authors cited in this paper, found that the molecular dipoles of water tend to lie parallel to the surface with a slight asymmetry of this distribution. This leads to a net dipole moment of the interfacial layer of water pointing to the liquid. Wilson and Pohorille [58] studied Na+, Cl" and F" ions at the water-vapor interface. They came to the rather unexpected conclusion that the free energy curves of two anions at the interface are qualitatively similar to one another, and qualitatively different from the free energy curve of the cation, which is more strongly repelled from the interface. Since Na+ and F" are about the same size, while Cl" is larger than F", Wilson and Pohorille concluded that the free energy required to move these simple, monovalent ions to within one molecular layer of the interface depends primarily upon the sign of their charges, and not upon their sizes. They ascribed this effect to the asymmetric orientational distribution of water molecules at the pure water liquid-vapor interface. Another interesting conclusion is related to the solvation of ions at the interface. The ions at the surface were found to retain their solvation shells. As a consequence of this, a bulge is formed in the surface above the ion. The size of this bulge was found to depend more on the sign of the charge of the ions rather than on their relative sizes. 3. ELECTRIC DOUBLE LAYERS 3.1. Polarizable and nonpolarizable liquid interfaces The electrical double layer at the oil/water interface is a heterogeneous interfacial region that separates two bulk phases of polarized media and maintains a spatial separation of charges. Electrical double layers at such interfaces determine the kinetics of charge transfer across phase boundaries, stability and electrokinetic properties of lyophobic colloids, mechanisms of

140

A.G. Volkov and V.S. Markin

phase transfer or interfacial catalysis, charge separation in natural and artificial photosynthesis, and heterogeneous enzymatic catalysis [8, 59-70]. The interface between two immiscible electrolyte solutions (ITIES) can be either polarizable or non-polarizable, depending on permeability to charged particles. If the interface is relatively impermeable it is called polarizable. Otherwise it is called non-polarizable or reversible. Planck [71, 72] introduced a rigorous thermodynamic concept called a completely polarizable interface, defined as "an interface whose state is completely determined by the charge that has passed through it beginning from a given instant." Planck's definition assumes that direct current cannot flow through the interface [71]. There can only be the transitive current in the boundary layer. However, the definition says nothing about how charged particles from different phases are distributed in this transitional layer. The distribution of particles in the boundary layer can have any configuration and the tales of this distribution for different particles can even overlap in the boundary layer (Fig. 19 a,c). At a completely polarizable interface charge transfer between bulk phases becomes impossible. A different concept of an ideally polarizable interface was formulated by Koenig [73] and represents a particular case of a completely polarizable interface (Figure 19 b,d). Two homogeneous phases a and p are separated by a transitional layer whose properties gradually change from the properties of phase a to those of phase p. A Koenig's surface, which is impermeable to all charged particles (both ions and electrons), is drawn in the transitional layer. This surface represents an infinitely thin and infinitely high-energy barrier for all the charged particles involved. It is this assumption that distinguishes an ideally polarizable interface from a completely polarizable surface, which makes it irrelevant that charged particles are contained in each phase. Koenig's definition cannot be described thermodynamically, because it suggests a definite structure of the interfacial region and employs non-thermodynamic assumptions about the existence of an infinitely high-energy barrier for all charged particles. The term "ideally polarizable interface" was introduced by Grahame and Whitney [74], whereas Koenig used Planck's term. Frumkin demonstrated that a completely polarized electrode, unlike an ideally polarizable one, can include systems with local adsorption charge transfer. Although local charge transfer exists at the interface and Koenig's surface cannot be drawn, there is no actual charge transfer between the phases, so the electrode is completely polarizable according to Planck, but non-ideally polarizable according to Koenig. If phases a and P contain at least one common ion that can freely pass across the interface, the interface may be called reversible or nonpolarizable (Fig. 20). Although the latter term is commonly accepted, it does not correctly describe the state of the system. A reversible (non-polarizable) interface can be partially polarizable or completely non-polarizable.

Electric Properties of Oil/Water Interfaces

141

Figure 19. The structure of a perfectly polarizable interface between two immiscible liquids a and p (a, b, c, d) and an ideally polarizable interface.

142

A.G. Volkov and V.S. Markin

Fig. 20. Scheme of the structure of a nonpolarizable ITIES (a) and the distribution of the concentrations of ions (b) capable of passing from one phase to another.

Fig. 21. Distribution of the electric potential at ITIES.

A completely non-polarizable interface containing at least one common ion can pass a high current in either direction without causing a deviation of the

Electric Properties of Oil/Water Interfaces

143

interfacial potential difference from the equilibrium value. Although in practice we encounter neither ideally polarizable or completely nonpolarizable interfaces, under certain conditions the properties of some interfaces are very close to being ideal. 3.2. Verwey-Niessen model Verwey and Niessen first described the electrical double layer at ITIES as two non-interacting diffuse layers, one at each side of the interface [75]. Both solvents were assumed to be structureless media with macroscopic dielectric permittivities, and potential distribution in the electrical double layer was defined by Gouy-Chapman theory [76-78]. Gavach et al. [79] extended the Verwey-Niessen model by introducing an ion free transition layer at the interface between two immiscible electrolyte solutions. This is directly analogous to the compact layer (or the inner Helmholtz layer) in classical electrochemistry. Stern theory was extended to ITIES, and the final model is referred to as the modified Verwey-Niessen model (MVN). In the MVN model, the electrical double layer consists of two diffuse ion layers back to back, which produce a compact inner layer between the two phases (Fig. 21). Dielectric permittivity of the medium at any point in the diffuse layer is assumed to be constant and equal to the bulk phase value s. The compact layer or inner Helmholtz layer is located between -8" and +8P. In a more detailed analysis the dielectric permittivities in both parts of the compact layer are s^ and z\. In the MVN model, the boundary of the compact layer (called the outer Helmholtz plane) is at the distance of closest ion approach to the interface. For ions with different radii, a few outer Helmholtz planes can be introduced as necessary. In the absence of specific adsorption the compact layer located between two nearest outer Helmholtz planes disposed on different sides of an ideal interface is usually called the ion-free layer. The maximum electrical potential in the compact layer (A"<|>.) includes a dipolar potential (Apg), which is shown schematically as a narrow region at the sharp interface. A dipolar layer can be located not only in the compact layer but can also occupy part of the diffuse layer. The amplitude and sign of Apg can differ from the total interfacial potential. Figure 21 illustrates four possibilities for potential distribution at ITIES. Generally speaking, the dipolar potential depends on the total interfacial potential, A°<j). The interfacial potential difference consists of the sum of the potential drops:

144

A.G. Volkov and V.S. Markin

£$ - P = $ - K + ^ h

•

(136)

Charge density at each side of the electrical double layer is usually called the interfacial free charge density, and depends on the model chosen for the interphase. In an ideal polarized electrode, the free charge equals the total thermodynamic charge in the Lippmann equation of electrocapillarity if all components of an interface have the same charges as they have in the bulk phases. We will denote surface charge density in the diffuse layers as qd, qd, and in the compact layer as q^. From electroneutrality of the interphase we have: <£+<7*+?5 = O.

(137)

At the potential of the free zero charge (fzcp) the compact layer has not adsorbed ions and the charges of the diffuse layers are equal to zero: < j ) X = 0 and A ^ = A « ^ .

(138)

The dipolar potential A^h also needs the index fzcp due to its dependence on total interfacial potential. Usually the dependence is weak and equation (137) is used to determine the dipolar potential. From the Gouy-Chapman theory for a 1:1 electrolyte it follows that 2z0RT q.= Hd

F

. F$d SKSinh—— ,

(139)

2RT

where K, the Debye length, is: J2F2c° K= J ^ -

(140)

\ se o i?r and c° is the bulk electrolyte concentration in the corresponding phase. If a Helmholtz layer does not have charges, potential drops in the two diffuse layers have a simple relation: saKasinh^ + 8pKpsinh^ =0 . 2RT 2RT

(141)

Electric Properties of Oil/Water Interfaces

145

The relation between potentials of two diffuse layers depends on a parameter a/p _

S K

_

£

C

which is the reverse ratio of the diffuse layer capacitance. There is a direct proportional dependence between ^ and <j>^ with coefficient r|a/p when the potential drops are small:

4>5 / 4>5 = - n a / p .

(143)

From equations (142) and (143) it follows that the potential drop (<j)d) in the diffuse layer will be less if the dielectric permittivity (s) or electrolyte concentrations (c°) are increased or, if the diffuse double layer capacitance (Cd) is increased. At the contact between aqueous and organic phases virtually the entire drop of the potential occurs in the organic phase. However, with increasing interfacial potential the situation changes dramatically. When potentials are large enough equation (141) can be presented in a different approximation: RT C = " ^ - — sgn^.lnTf /p .

(144)

Here the coefficient of proportionality is equal to one. From equations (136) and (141) it is possible to find the dependence of the potential drop in both diffuse layers Ap<j>- A^<> | A on potential drops ^ and ^ in each diffuse layer: , ^ and

=

RT F

(s V n

/ s a K«) + exp[F(A"p4> - A ^ , ) / 2RT]

( s P K 3 / s a K a ) + exp[-F(A a p (t)-A a p^)/2i?r]

(145)

146

A.G. Volkov and V.S. Markin

=

"

RT (£ K K a /E p K p ) + e x p [ F ( A ^ - A > J / 2 / ? r ] _ F n (E a K a /s [ 5 K p ) + exp[-F(Aap<|)-Aap(t)J/2JRr]~

_

RT CM

V K )+

ln F

(B p K p /(s°K°) + e x p [ F ( A ^ - A ^ J / 2 J ? r ] • ( 1 4 6 ) ( E P K P/( 8 « K ") + exp[-F(Aap<|)-Aapfi)/2J??r]

The spatial distribution of potential near the interface is accompanied by changes in electrolyte concentrations that produce surface excesses of ions. In each phase the surface excess T\ of an ion i can be divided into two components in the inner layer F* and in the diffuse layer Ff :

r; = r*+r^

(HV)

The charge of each contacting phases is equal to:
.

(148)

For a binary electrolyte the ion excess in the diffuse layer can be written as

r

'-V [o ^-Mr ) - 11 = ^ [ J 1 -fei? + iidfe y " 11 - (149)

where c° is the electrolyte concentration in the bulk phase. The interfacial capacitance C consists of the capacitance of the compact layer, C/,, and capacitance of two diffuse layers Q and Cf,. Differentiating equation (148) with respect to charge qa of phase a, and assuming that the drop of the potential in the compact layer Aa^h does not depend on an electrolyte concentration, we have: ^ =^ ' . +^ + C(q,c) Ch{q) Cad(q,c) Cpd{q,c)

(150)

Diffuse layer capacitance depends on potential or surface charge q and electrolyte concentration: |2s n sc Ffo, F I rCOSh = y^U 2RT 2RT^°ERTc

Cd = F

+q

7 •

(151)

Electric Properties of Oil/Water Interfaces

147

3.3. The electrocapillary equation A comprehensive thermodynamic theory of the electrocapillary phenomena at polarizable and nonpolarizable liquid interfaces was developed by Markin and Volkov [64-69] using Hansen's method [80]. Electrochemical processes occurring at the interface between two immiscible liquids are traditionally described based on Gibbs thermodynamics of surfaces. However, Hansen's method, by extending and generalizing Gibbs method gives us a better understanding of the nature of interfacial phenomena and provides us with an improved method for describing them. Consider two phases in contact, a and P, with rh neutral components, whose chemical potentials are designated by \ih, rc types of cations and ra types of anions with chemical potentials Uj. Using Hansen's reference system and choosing the solvents a and p as reference substances one can write down Hansen's rendition of Gibbs adsorption equation: d

J = S(«fi)dT

+r

Ufi)dP- Z r *(a, P )4^ - Z r / ( « . P ) J ^ . fcta.p

(152)

/

Here ^ a P ) , t*ap) and rh(<Xip) are the surface excesses of the entropy, volume and substance h in the reference system, when the absolute excesses a and p are zero. The terms corresponding to different components of the system are divided into two summations, one for neutral and the other for charged particles, the reference substances a and P being excluded from the sum of neutral particles. Individual components of the system, both charged and neutral, can be present either in both phases or in only one. The electrocapillary equation in its final form, which includes the electromotive force of the measuring cell can be written as:

dy = -(ss(afi)-(VzrF)Qas'Ra

-(Vzm,F)Q\)dT

+

+ (i; r (i/z j ,F)e"v;-(i/z,,F)^;)*-^r %pl rf f i r hzafi

-X(i/v;,jr t(a , P) 4i /t - Za/vy )r,
}*}'

1/v

" E ( rjr m(ap) <% m -X(1/vL')r/(a,p)^/,,,' -

-(l/^,v:,r)(^ril(aP) + 2z/,
zJ^»M*-QadE$&

148

A.G. Volkov and V.S. Markin

Note the symmetry of the last term with respect to the transposition of the a and (3 indexes: QadE;H] = &dl%$

.

(154)

Equation (153) contains rh + rc + ra - 1 independent differentials of intensive variables, whose number is equal to the number of the degrees of freedom of the system. The impedance or electrocapillary properties of the interface between two immiscible electrolyte solutions can be measured with the following cell:

RE1

1 H2O(a) M+C1" a, AAga4>REi

2 A + B"(p) oil a2 A >

3 A+C1" (a) H2O a3 A$D

RE2

(I)

-A A g a KE2

Silver/silver chloride electrodes are usually used as the reference electrodes (RE), water as phase a , and nitrobenzene or 1,2-dichloroethane as phase p . T h e interface between phases 1 and 2 is the polarized oil/water interface serving as a working electrode (interface). T h e common cation A + is usually the terabutylammonium ion, and B" is tetraphenylborate or dicarbollylcobaltate. The potential difference measured in the cell (I) consists of the s u m of two interfacial potential drops:

-E = A»4> + AM> W1 - A > f f £ 2 + A(»D .

(155)

A<|>D is the Nernst-Donnan

potential difference between the nonaqueous fraction and the aqueous solution of RE2 with the common ion A+: RT

A^=-A

a

o

pC+7Tln^,

(156)

and Aap<|>°, is the standard potential difference for A+ known from literature data or calculated theoretically. By suitable selection of the Cl" concentrations and ionic strengths of the electrolytes into which Ag/AgCl - electrodes are immersed, we obtain:

Electric Properties of Oil/Water Interfaces

RT

149

n

A?4>™-Aj
(157)

Substituting (156) and (157) into (155) we obtain

d£?$ = -dE-

a

dA ^D

HT n +-—dln^ r ax

(158)

With equation (158) the electrocapillarity equation at the flat oil/water interface can be written as follows:

dy=-Yjrid»,-QadE + Qa^dln^^ i

r

(159) ay

where Q is the charge which must be supplied to each side (a or p, correspondingly) of the polarizable interface when expanding it by a unit area in order to maintain the potential difference between the phases [81]. At the ideally polarizable interface, the ions of each group are separated by the Koenig barrier, which rules out a possible overlap between the groups (Fig. 19). In this case the free charges qa and qp of the phases are equal to the thermodynamic charges: q a = Q a = -q p = - Qp.

3.4. Zero-charge potentials Potentials of zero charge of the interface can be found reliably by the same independent methods that are used at the metal/water interface. These include finding the differential capacitance minimum of the electric double layer from electrocapillary curves, with a flowing-electrolyte electrode, vibrating boundary method, radiotracers, or by measuring the second harmonic generation. Potentials of the zero free charge at nitrobenzene/water and 1,2dichloroethane/water interfaces, obtained from the differential capacitance minimum of the electric double layer in solutions of a surface-inactive electrolyte do not necessarily correspond to the potentials of thermodynamic zero charge. They can depend on electrolyte concentration when the capacitance of the compact layer is affected by surface charge as a result of nonlinear double-layer properties.

150

A.G. Volkov and V.S. Markin

The potential difference between two immiscible electrolyte solutions can be written as the sum:

Aap4> = A-S+4>5,

(160)

where Aap(j) and <|)d are the potential drops across the compact and diffuse layers, respectively. In the linear approximation and in the absence of specific adsorption, the electric double layer is equivalent to three capacitors in series:

d^ dq

_ dAa^h dq

d$l +

dq

d^ +

(161)

dq •

One can use the following approximation when modelling the electric double layer:

dq1

dq2

dq2

dq2 '

where j-p

Z

i ( ^ ) = — [y{x + d)+\y{x1

rp

d)\-——

-. X + d

\\[>(x')dx' l U

(163)

x-d

We find the position of the minimum in the C vs. Aap<)) curve as:

A>"=Aap(t);'+(t)«"+tf"=O.

(164)

It follows from the Gouy-Chapman diffuse-layer theory that if qa= q$= 0, then((|>°)" = (" = 0 applies only when Aa^" = 0. That is, when the compact layer capacitance is independent of surface charge. It has been shown that Cc generally depends on potential and charge, and Aap<j)i" ^ 0. As the value of Aap(j)i" in the region of the potential of zero free charge increases, the value of q corresponding to the minimum in the C vs. Aap<)) curve will also increase [81]. For the water-nitrobenzene system, one observes a strong dependence of capacitance of the compact double-layer on surface charge, even in the absence of specific adsorption. This leads to a dependence of the potential minimum in

Electric Properties of Oil/Water Interfaces

151

the capacitance curve on electrolyte concentration. Therefore, a correct determination of the zero free charge potential for the nitrobenzene-water system in the presence of a binary surface-inactive electrolyte is possible only at base-electrolyte concentrations less than 0.01 M. For the water-nitrobenzene system, this quantity Awnb<j> pzfc = -0.29 V and for the water-1,2-dichloroethane system AWde<)>pZfc= -0.27 V. For many systems the maximum of the electrocapillary curve is located in the region of ideal polarizability of the electrode, and the maximum potential corresponds to the potential of zero total (thermodynamic) as well as zero free charge. At the mercury/water interface the region of ideal polarization is a few volts when soluble mercury salts are absent (2.2 V), but at the interface between two immiscible liquids the region is about a tenth of that value. Under conditions where current flow does not significantly alter the compositions of the liquid phases and where the equilibrium at the interface is not disturbed by the current, the electrocapillary curves yield the potential of zero total charge. However, under realistic conditions where the region of ideal polarization is narrow, the maximum of the electrocapillary curve corresponds to the potential of zero total charge, rather than zero free charge. This closely resembles systems consisting of amalgams and liquid electrolyte solutions that are described by Frumkin [82]. The structure of the electric double layer at ITIES has been investigated using such common electrochemical methods of capacitance measurements as impedance, galvanostatic and potentiodynamic techniques. More recently, it has become possible to measure electric double-layer capacitance at the interface between two immiscible electrolyte solutions using a four-electrode or twoelectrode potentiostat. Commonly studied systems are nitrobenzene/water and 1,2-dichloroethane/water, in which the organic phase has a relatively high dielectric permittivity and a high dissociation constant. In typical investigations, cyclic current-potential curves are usually determined before impedance measurements in order to find the potential range where the contribution of faradaic impedance is low. The electric double-layer capacitance is measured in the potential window between extremes in the cyclic voltamogram. Over this range of potentials, the water/dichloroethane interface has properties close to those of an ideally polarizable electrode. A "diffuse" picture of the compact layer has been considered [83]. According to this hypothesis adsorbed ions can penetrate into the compact layer, which consists of solvent dipoles, analogous to nonlocalized electron gas penetration from a metal electrode to an aqueous electrolyte solution. Here a penetrating ion can act as a hydrophobic anion in the organic phase and potential distribution §(x) near to the point of zero charge can be calculated using the Poisson-Boltzmann equation:

152

A.G. Volkov and V.S. Markin

If X<X2W,E = EW: f = (K,) 2 (*-Ay)

(165)

If x " Z < x < x ° 2 r g , z = z h :

(166)

If x>x^,z

(|)" = ( K J 2 ( ) )

= zorg: V = (Karg)\
(167)

In each of these three areas, the solution can be written as: <j)(x) = A+e" + A_e-",

(168)

where six coefficients A+ and A. can be determined from the six boundary conditions:

K-oo) = AWOJ (j)(oo) = 0

'(^-0)

(169)

= S/i(t)'(x2w+0)

8^'(xr-0) = 8o^(x^+0). As a result the equation for the capacitance of the electric double layer is:

, (170)

fl + _E*!S*_>|exp(KA6) + fi__!*!S*_l eX p(_ KA5 ) V

E

orgKorg

)

^

E

orgKorg

)

fl + ^ ^ Y l + -^ 1 ^]exp(K / ,5) + |l-^ K ^lfl-^ K ^lexp(-K / ,5) where 8 = xfx - x j . It should be noted that the size of a hydrophobic penetrating anion is large in relation to the compact part of the electrical double layer and it is not clear if the Poisson-Boltzmann equation can be used in this case.

Electric Properties of Oil/Water Interfaces

153

3.5. Specific adsorption at liquid-liquid interfaces Ions can be adsorbed specifically if the main contribution to their interaction with the interface (ions, dipoles) is caused by non-coulombic shortrange forces. Specific adsorption cannot be explained using only the theory of diffuse double layer. Specifically adsorbed ions penetrate to the compact layer and form a compact or loose monolayer. The surface passing through the centers of specifically adsorbed ions is usually called the inner Helmholtz plane. If several kinds of specifically adsorbed ions are present, each ion can have its own inner Helmholtz plane. It was found while studying the mechanism of electron transfer across the interface between two immiscible liquids, that specific anion adsorption occurs at the octane/water interface in the presence of metalloporphyrins. This adsorption increases in the order of Cl", Br", I", and is caused by coordinative bonding of the anions as ligands of the porphyrin metal atoms. Specific ion adsorption at the polarizable nitrobenzene/water interface containing monolayers of phosphatidylcholine, phosphatidylserine, octaethylene glycol monodecyl ethers, tetraethylene glycol monodecyl ether, and hexadecyltrimethylammonium (HTMA+) were studied in detail [84]. HTMA+ exhibited no specific adsorption in the potential range where the aqueous phase was positive. However, a strong adsorption occurred in the potential range, where the electric potential in the aqueous phase was negative with respect to the nitrobenzene. Another example of specific ion adsorption was discussed in terms of the formation of interfacial ion pairs between ions in the aqueous and the organic phase. The contribution of specific ionic adsorption to the interfacial capacitance can be calculated using the Bjerrum theory of ion-pair formation. The results show that a phase boundary between two immiscible electrolyte solutions can be described as a mixed solvent region with varying penetration of ion pairs into it, depending on their ionic size. The capacitance increases with increasing ionic size in the order Li+ < Na+ < K+ < Rb+ < Cs+. Yufei et al. [85] found that significant specific ion adsorption occurs at the interface between two immiscible electrolytes and the potential dependence of the capacitance is strongly influenced by the adsorption isotherms due to the interfacial ionic association. When there is specific adsorption of ions dissolved in phase a the condition of electroneutrality can be written as: q*=-qa=-{q*+qad)

,

(171)

where q, is the charge of the inner Helmholtz plane. The separation of qa from g° and qad cannot be done without introducing a model of the interface.

154

A.G. Volkov and V.S. Markin

Although capacitance of the interface can be calculated as before using the equation:

dq

dq

dq

dq

^

'

The second term in the right side of the equation is not the diffuse layer capacitance since the charge of a diffuse layer in phase a is equal to

€ =-%-Yu(lTJ .

(173)

j

The diffuse layer capacitance is equal to:

c

<174>

«'-%-

and one can write

C-j=(C')-l+{C%)-1

1+ V

d

,,

+(C*)~'.

(175)

/

The drop of a potential in the compact layer depends on the surface charge qa and the charge of the inner Helmholtz plane q"-J. It shows that

or

dq> = crWl,

Hw-'iw

d

^ •

(177)

Here the index q"J means that all q"J are constant except one. In the absence of specific adsorption the capacitances of the interface, compact and diffuse layers are always positive. The capacitance of the compact layer in the presence of specific adsorption can be either positive or negative.

Electric Properties of Oil/Water Interfaces

155

3.6. Adsorption isotherm Traditional models for calculation of adsorption isotherms are based on the assumption that surface-active compounds at the interface can substitute for adsorbed molecules of one solvent, but cannot penetrate the second phase [34, 61, 86, 87]. Although these models are useful for metal/water interfaces, recent interest has focused on the surface chemistry of amphiphilic compounds which can penetrate both phases and replace adsorbed molecules of both solvents, for example water and oil. We present here a theoretical analysis of the generalized Frumkin adsorption isotherm for amphiphilic compounds [61]. The interface between two immiscible liquids may be considered to be a surface solution of surfactant in a special kind of solvent. In order to calculate the entropy of such a solution, we will adopt a simplified lattice model and use lattice statistics, a widely used method for describing surface solutions. The transition from three-dimensional (3-D) to two dimensional (2-D) geometry may cause errors in statistical formulas, if some peculiarities of 2-D solutions are overlooked.The main difficulty when dealing with a monolayer of a surfactant, one can consider this monolayer as a 2-D system. The solvent molecules do not form a monolayer, but rather a multilayer. Therefore the transition from 3D- to 2D-geometry should be specified. Consider molecules of both solvents which are substituted by a surfactant (Fig. 22). Suppose that these molecules can be assembled into columns consisting of m 0 molecules of oil and m w molecules of water. Suppose that one column of oil molecules matches the n w molecules of water. This match of 1 oil column and n w water columns will be considered in what follows as a quasi-molecule of solvent Q. These quasi-molecules constitute a "monolayer" of solvent. They consist of m 0 oil molecules and n w niw water molecules. Designate the molecules of surfactant in the bulk as A, and in the monolayer as B. At the interface aggregation of surfactant molecules can take place, rA <=> B, such as dimerization of porphyrin or pheophytin molecules at the octane/water interface. Let the surfactant B replace p quasi-molecules at the interface. Therefore one can write pQ + rA = B+p (oil) +/?« w (water). The chemical potentials for (178) are

(178)

156

A.G. Volkov and V.S. Markin

Fig. 22. The structure of the oil/water interface with adsorbed monolayer of amphiphilic surfactant. Taking the 2-D solution as ideal, we have:

y£> =u°Q + R T l r i $

(180)

and

ufs = u RS + R T lriXS.

(181)

In the bulk phase we have

A = u ? + R T l n XhA,

(182)

uh0 = u ? + RTln X^ ,

(183)

u^ = u° w b +RTln X^ .

(184)

Electric Properties of Oil/Water Interfaces

157

In all these equations X designates the mole ratio of corresponding substances. Substituting these equations into (175), one obtains: p& + ry?* - n 0 / - p\x°f - pnywb + RT In

/

X

^

= RT In -%-

(185)

Using the standard Gibbs free energy of adsorption:

AIG° = ft - rtf + ptf + pny* - n%

(186)

one obtains the adsorption isotherm: J^=(^l exp(-^) (X^)" (X*)"(Xt)'"1" RT

(187)

We considered the 2-D solution of surfactant B in the solvent of quasi-particles Q, in which the mole ratios were defined as N

Y .«

N

.y»

B

X =

Q

X

' V7iF-' °-N-B+Nl

0 88) (188)

Some authors prefer another set of definitions when real particles in the interface are considered. The equation for this state with real particles A, O, W becomes:

X

>^fe'

Xs

K

Q

N'A+N'O

+

K'

(190) x,

_

" (191)

K N'A+N^+N'W'

and we can obtain:

O89)

158

A.G. Volkov and V.S. Markin

X;=—^—;X!;=—^—.

(192)

The adsorption isotherm can then be presented in the form

-^MX*

+XT1

/X'^

=

s

b

{^ Qy

exp(-^^)

b

(x onx H,r»

pv

RT

(193)

In the past the adsorption isotherm was presented in terms of the fraction 6 of the surface actually covered by the adsorbed surfactant. If we introduce r| as the ratio of areas occupied in the interface by the molecules of surfactant and oil, the mole fractions in surface solution can be presented as follows: B

© 0 + n(i-0)

s =

°

nO-Q) ©+Ti(i-©)

(194) V ;

The adsorption isotherm takes the form: 0 p

n C\ —Pi\p V \l U 7

[Q + , ( l - e ) r

=

/ ^ ^ (Y \P(Y V"' \^o) y-^w)

exp(-^ RT KI

(195)

In this isotherm the mole fractions X*,X*,X* of the components in the bulk solution are presented. In the general case, they must be substituted with activities: 0

npC\—CV\p

[0+ ,(l- Q ) r- = y*\

exP(-^ RT

K1 J \ao> \aw) (196) If the molecules B can interact as pairs in the adsorbed layer and the energy of each new particle is proportional to its concentration, then their chemical potential, \x*B, instead of equation (181), should be presented as:

V U

U

(nb\p(nb\p"'-

VSB = \xf + RT In X-2aRTX^

( 197 )

where a is so called attraction constant. Then after some algebra we obtain, instead of Eq. (196), the isotherm [61, 88]:

Electric Properties of Oil/Water Interfaces

F

^(l-©)"

'

(x*)"(x*f)p"«

159

i?r

v

'

Recall that ri was introduced as the ratio of areas occupied in the interface by the molecule of surfactant to the same of oil and p was introduced as the number of columns of oil, which could be supplanted with one molecule of surfactant. Therefore, p is a relative size of the surfactant molecule in the interfacial layer. It is reasonable to suppose that: T\=P

.

(199)

If the concentration of surfactant in the solution is not high and the mutual solubility of oil and water is low, then we can use the approximation x ! = x * = I s o that the general equation (196) simplifies to: 0 [ / 7

-^"l)0]

ex P (-2a0) = (X*)' e x p ( - ^ - )

(200) This is the final expression for the isotherm that we will call the amphiphilic isotherm. It is straightforward to derive classical adsorption isotherms from the amphiphilic isotherm (200): 1. The Henry isotherm, when a — §,r=\,p=\,

0 « I:

© = X*flexp(-^). 2. The Freundlich isotherm, when a = 0, p= I,® 0 =(X*)rexp(-^).

(201) «l: (202)

Kl

3. The Langmuir isotherm, when a = 0, r = \,p= I: - ^ - =X*exp(-^). 1-0 RT 4. The Frumkin isotherm, when r— l,p= 1:

(203)

160

A.G. Volkov and V.S. Markin

-^— exp(-2a©) = X"a exp(- ^ - ) .

(204)

Therefore, the amphiphilic isotherm (200) could be considered as a generalization of the Frumkin isotherm, taking into account the replacement of some solvent molecules with larger molecules of surfactant. Of course, the amphiphilic isotherm includes all the features of the Frumkin isotherm and displays some additional ones. To elucidate them, it will be convenient to change the variable x* to the relative concentration y= X* / X ^ ( 0 = 0.5), where x*(0.5) is the concentration corresponding to the surface coverage 9 = 0.5: y=

:

exp(a-2a&)

(205)

This equation gives the coverage fraction 8 as a function of relative concentration y, while a and p are the parameters of this isotherm - the first being the attraction constant and the second, the size of surfactant. These parameters play an important role because their effect on the shape of amphiphilic isotherm is very strong. Amphiphilic isotherm (200) analysis can be used for the determination of the interfacial structure. An amphiphilic molecule, which consists of two moieties with opposing properties such as a hydrophilic polar head and a hydrophobic hydrocarbon tail, should be used as an analytical tool located at the interface. Pheophytin a is a well-known surfactant molecule that contain a hydrophobic chain (phytol) and a hydrophilic head group. The value of p less than 1.0 indicates that adsorbed molecules of w-octane are parallel to the interface between octane and water. Substitution of one adsorbed octane molecule requires about 4-5 adsorbed pheophytin or chlorophyll molecules. These are supported by molecular dynamic studies in the systems decane/water, nonane/water and hexane/water. The structure of both water and octane at the interface is different from the bulk. Adsorbed at the interface octane molecules have a lateral orientation at the interface. 3.7. Image forces Any charge near phase boundary interacts with the interface due to the different polarization of two phases. The force of this interaction can be found by solution of the Laplace equation, but it is more convenient to use the method of images. This method employs a fictitious charge {image) which, together with

Electric Properties of Oil/Water Interfaces

161

the given charge creates the right distribution of electric potential in the phase [8, 90]. All ions interact with all images giving rise to the so called image forces. The image forces are largely responsible for both positive and negative adsorption. Important contribution to this effect is given by the change of the size of solvation shells of ions in the boundary layer. This has an impact on the planes of closest approach of ions and dipolar molecules to the phase boundary. The electrostatic Gibbs free energy for an ion in the vicinity of a boundary between two liquids with dielectric constants sj and E2 is determined by the Born ion solvation energy and by the interaction with its image charge [8]. Dealing with the energy of image forces and with the interactions of charges at the oil/water interface, approximate models of the interface are often employed, which are based on the traditional description of the interface between two local dielectrics. In the oil/water system, the force of attraction (or repulsion) of charge in the organic (P) phase with its image in the aqueous phase (a) sitting on the same axis perpendicular to the dividing plane is given by: F(h) = -^^S.-^-j,

(206)

where h is the distance from the interface. If ea > sp, the charge in the nonpolar phase is attracted to its image, but if za < 8p there is repulsion between the charge in an organic phase and its image. From equation (206), it follows that charges in the organic phase are attracted to the oil/water interface. Image forces attract the diffuse layer on the organic side, making it thinner, and repel, thus thickening the aqueous diffuse layer. The Kharkats-Ulstrup model [40, 41] incorporates the finite radius of an ion a, which is assumed to have a fixed spherically uniform charge distribution and can continuously pass between the two phases. For a point charge at long distances from the interface, electrostatic Gibbs free energy can be written as

AG =

j £ ^ L f 4 + 5iZft2£l 327ieoeaa^

(207)

ea+sp h )

where a is an ionic radius, h is the distance from the interface and ze is the charge of an ion. The potential <> j for the spherically-symmetrical charge distribution is:

162

A.G. Volkov and V.S. Markin

Z6

°

+—^

4jiE0ear1

£g l

~£p

(in region a)

4jre 0 E o #' 1 e a + Ep

• —^ 4TIS 0 S^

^L_ 8a+sp

(208)

(in region P)

Using standard procedures of calculations, Kharkats and Ulstrup obtained equations for the electrostatic part of the free Gibbs energy of a finite size ion [40, 41]. When h> a AG

&of \ 327i808aa|

UaSi\2a Ua-z^\ 2 [ s a + e j / ? [e a +E p J \\-(2hlaf

^ ^ 2 / z + alj- (209) 2h 2/z-aJj

The first term in (209) is the Born solvation energy, and the second is the interaction with the image. For a charge located in region p, the electrostatic free energy is obtained from (209) by exchanging s a <^> ep Kharkats and Ulstrup [40, 41] obtained the expression for the electrostatic free energy in region 0 < h< a:

AG=325T£ ^ £ a {[V f 2a)3 l Vs + ^+EYJV4 - ^a)+ 0 a

a

(za-zA2U\+h/a)(l-2h/a) E

Ua+ JL

\ + 2h/a

p

^ a Ju2hVJ 2h V

a) \\

(ze0)2 f 2sp Yf

hV

167i£0Epa l v s a +E p J V a)

If h - 0 and center of an ion is at the interface, electrostatic Gibbs energy will be equal to (ze \2 AG(h = 0) = ^ ^ . 47i8 0 (s a +s p )a

(211)

Image forces play a significant role in electric double layer effects. The excess surface charge density is: K

q = ec0 ]{exp[-*g(h)/kBT]-l}dh, K

(212)

Electric Properties of Oil/Water Interfaces

163

where Ag(h) = AG{h)--^—.

(213)

AG{h) is the electrostatic contribution to the Gibbs energy of solvation, and hd and hc are the distances of closest approach of the anion and the cation. If image forces are taken into account, the diffuse layer capacitance can be calculated by the equation Cd = Cf exp[Ag(h,)/2kBT),

(214)

where C'f is the Gouy-Chapman diffuse layer capacitance without image terms. As noted by Kharkats and Ulstrup [40], equation (214) always gives diffuse layer capacitance corrections toward higher values. The first step to the statistically correct Gouy-Chapman theory for the diffuse double layer uses a restricted primitive electrolyte model. This model considers ions to be charged hard balls of identical radii in a structureless dielectric continuum with constant dielectric permittivity. There are three main approaches to creating a statistical theory with this model. The first approach uses the Gouy - Chapman theory, there the average electrical field in the diffuse layer is calculated from the Poisson equation. The structure and physical properties of the double layer are then calculated from the restricted primitive electrolyte model. As a result, the modified Poisson-Boltzmann theory (MPB) was developed. The second statistical approach incorporates correlation functions and integral equations from the theory of liquids. In this case, the well-known uniform Ornstein-Zernike equations were modified to calculate the ion distribution function near a charged interface. Modified in this way, the Ornstein-Zernike equations became nonuniform and were solved using hyperneted chain approximations (HCA) incorporating a mean spherical approximation for correlation functions (HCA/MSA). A third statistical approach used to describe the electrical double layer was achieved by using the integral Born-Green-Ivon equations. These correlation function approaches resulted in the theory of ionic plasma in semi-infinite space. The next advance in describing the double layer was achieved by upgrading the restricted primitive electrolytes model to a "civilized" or "non-primitive" model. This was accomplished by addition of hard spheres with embedded point dipoles to the restricted "primitive" model.

164

A.G. Volkov and V.S. Markin

This theory takes into account the presence of long-range Coulomb interactions, image effects and external electrical fields. The new "non-primitive" model became the basis for the theory of ion-dipole plasma. The serious mathematical difficulties encountered in describing the "non-primitive" models necessitated the development of less statistically rigorous intermediate models. In these intermediate models, the first monolayer of solvent molecules at the charged interface was considered as a complex of discrete particles and an electrolyte outside this layer was described by the Gouy-Chapman theory. This intermediate approach was able to provide good theoretical agreement with experimental results. The concept of constant dielectric permittivity in the double layer is also of concern. Different factors such as high ionic concentration or strong electric fields can affect the dielectric permittivity. In addition, description of the dielectric properties of solutions by a single parameter, the dielectric permittivity, also came under criticism and led to the development of nonlocal electrostatics. Non-local electrostatics takes into consideration discrete properties of solvent molecules. It assumes that fluctuations of solvent polarization are correlated in space. This means that the average polarization at each point is correlated with the electric displacement at all other points and, therefore, uses the solvent dielectric function which couples polarization vectors throughout the solvent. 3.8. Modified Poisson-Boltzmann (MPB) model The popular Poisson-Boltzmann equation considers the mean electrostatic potential in a continuous dielectric with point charges and is, therefore, an approximation of the actual potential. An improved model and mathematical solution resulted in the modified Poisson-Boltzmann (MPB) equation [91]. This equation is based on a restricted primitive electrolyte model that considers ions as charged hard spheres with diameter d in a continuous uniform structureless dielectric medium of constant dielectric permittivity s. The sphere representing an ion has the same permittivity s. The model initially was developed for an electrolyte at a hard wall with dielectric permittivity ew and surface charge density a. The charge is distributed over the surface evenly and continuously. This theory takes into account the finite size of ions, the fluctuation potential, and image forces in the electrolyte solution next to a rigid electrode, but it still an approximation. The MPB theory begins with the Poisson equation for the mean electrostatic potential \\> in solution:

Electric Properties of Oil/Water Interfaces

d2y

-*

r =

1

v

165

0

-wZ**»,g«W.

(215)

where s0 is the electrical constant, z\ is the charge number of ion i, e0 is the elementary charge, n° is the bulk number density, and go\(x) is the wall-ion distribution function representing the ratio of the local ion density to the ion density of the bulk electrolyte solution. The distance x is measured from the electrode so that the distance of closest approach of an ion / is d/2. According to Kirkwood the distribution function is given by the following equation:

go,X*) = S,X*)exp[- —z,.e o v|/(*) + T|,-].

(216)

The factor tfa) accounts for the excluded volume of the ion and r\, is the fluctuation potential that takes into consideration the ionic atmosphere around an ion created by other ions. The fluctuation potential can then be presented as n =

' ~~kf -T W * ' -^.b)d(z,eo),

(217)

where

*:=i-7

£ l 5 L

-,

(218)

<> | j is the fluctuation potential created by the ion / and its atmosphere, r is the distance from the center of the ion. The difference <>|* is the potential created by the ionic atmosphere only and *tf is the value of this potential in the bulk electrolyte solution. There are several methods for calculating §; , and if the image effect is taken into account, it can be presented as ,. = fl0 - e x p ( - K r ) + — e x p ( - K r * )

}

ifr>^,x>0,

(219)

where/is the image factor determined by the difference between two dielectric constants of the electrolyte solution and the electrode:

166

f

A.G. Volkov and V.S. Markin

£

~ Sw

/ =

.

(220)

The local Debye-Huckel parameter at a given point is:

£Q£/C-/

,

In equation (219), 5 0 is a constant and r* is the distance from the mirror image in the interface. Hence, equation (219) takes into account both the ionic atmosphere and image forces. If r < d and x > 0, the equation for the fluctuation potential should be written as: d2\\>

vy2,

Ib^'

r0.

(222)

The term Cfe) can be presented as a power series expansion in the bulk density:

;,(x)=i+5(x)+^-5>;,

(223)

where X+

\[(x'-X)2-d2]goj(x<)dX',

D/

\

V

^<x<^,

0 rf/2

5(x) = ^ ^ L ,

(224)

Now we can combine these equations and obtain the modified PoissonBoltzmann equation for the diffuse layer:

^dx = —EL Ex ;^ o « , ° C , W e x[p j kT - ^ U[_ ( V ) + -87t£ ^ - eaf ( F - F o ) jJJ l x>± (225) 2 O

o

Electric Properties of Oil/Water Interfaces

167

Since the distance of the maximal approach of ions to the interface cannot be less then d/2, the usual linear dependence of potential in the compact layer can be introduced: ¥(*)

= V(0)

+

* « .

(226)

Additional functions in equation (225) are:

C,(*) = l+ ^ & 0 + * 2 > , 0 '•

m^)

"{ \(x<-x)2-a>}goi(x')dx\

(227)

(d A max I — ,x-d I

= —[w(x + d) + \i(x-d)]—-— 2

1

j\\i(x')dx'. "

(228)

x-d

a

2(l + KX)exp(-Kti) + 2(l - Kt/)exp(-icc) + 2K266C jr" 1 exp(-Kr)t/r + <

F=

X

+ — [exp(-K
>-,

-<x
— [2o: exp(-KJc) + (1 + Krf)(exp(-2icc) - 1 ) exp(-Kd)] .

[KX

F=

(iJrtW-fsy)>

52 — 1

K^

ifx< d

~-

exp[K.(d - 2x)]sinhKc/

(K J cosh Kd - sinh K J ) 5 2 ~ (l + K^)sinhKJ • = lim K2 = - ^ _ V z,2«°.

(230)

(232) (233)

(234)

168

A.G. Volkov and V.S. Markin

The boundary conditions for these equations are: 1) \\i(x), \\i'(x) -> 0, if x —» oo, 2) \|/'(0+) = -5/S 0 E,

(235)

3) \\i(x), \\i'(x) are continuous at x > 0. This set of equations completes the mathematical formulation of the problem, but the equations are very cumbersome and can only be solved numerically. The major difficulty is related to the finite ion size. If we consider the limiting case of point charges, the equation for potential in the double layer would be:

^Y = - s— I»!Vo exp{-^Lx) + - ^ { , c 0 -K +f exp(-2Kx)}lj . (236) dx e , [ kT [_ 87ie e [ 2x JjJ o

0

In comparison with the Gouy-Chapman theory, the MPB equation for point charges contains two improvements, in that the Debye-Huckel parameter K depends on distance and the ion image is screened. In solving the MPB equations, different estimates of C, \{x) were considered. Depending on the type of C,\(x), the equations were called MPB1, MPB2, MPB3, and so on. The MPB theory overcomes certain limitations of the Gouy-Chapman model by taking into account ionic volume, ionic atmosphere and image forces. Results of both theories coincide at low electrolyte concentrations and small surface charges, but beyond this limiting case there is a very significant qualitative and quantitative discrepancy. For instance, the MPB theory predicts a thinner diffuse ion layer and higher capacitance than the Gouy-Chapman model. Near a charged interface MPB predicts considerable adsorption of co-ions. The major difference is that the decaying potential distribution of the GouyChapman model transforms into the decaying-oscillating solution of the MPB theory, with oscillations beginning at high electrolyte concentrations when Kad> 1.24. This oscillation was also predicted by Stilinger and Kirkwood [92] who estimated a different value KOd> 1.03. Other modern theories predict oscillatory behavior of the electrostatic potential, but under different conditions. In the hyperneted chain approximation (HCA) theory the condition is K.od > 1.23, which is very close to MPB theory. The Bogolyubov-Born-Green-Ivon theory puts the condition at K0 1.72. Therefore, oscillatory behavior is a consistent finding of all theories. It is worthwhile to mention that damped oscillation in the

Electric Properties of Oil/Water Interfaces

169

MPB theory is due to a fluctuating potential and its existence does not depend on the excluded volume. One of the major drawbacks of MPB theory is that it does not take into account the discreteness of a solvent. In the diffuse layer the discreteness of solvent molecules strongly influences ion-ion interactions at small distances. Furthermore, an understanding of the structure and properties of the compact layer is impossible without an adequate model of solvent molecules and their interactions with the electrode and between themselves. Application of MPB model to ITIES was made first by Torrie and Valleau [93]. Cui et al. [94] applied the modified Poisson-Boltzmann theory (MPB4) to the interface between two immiscible electrolyte solutions and found that MPB4 describes the experimental results at the nitrobenzene/water and dichloroethane/water phase boundaries. It reproduces the Monte Carlo calculation more accurately than the MVN theory for 1:1 electrolytes over a wide range of conditions including variations in the image forces, ion size, the inner layer potential distribution, electrolyte concentration, surface charge density and solvent effects. They used the equation for electrostatic potential from the MPB4 model for the inner layer at ITIES when x < all: V|/(x) = V|/(0) + xv|/' (0) - xkc\ qaq |"2 qaq I eh ,

(237)

where \\i(x) is the electrostatic potential at coordinate x, i|/(0) is the electrostatic potential at the surface, vj/'(O) is the derivative of y(0), c is the electrolyte concentration, qaq is the surface charge density of the aqueous phase, A, is the parameter determined by fitting the experimental values of the Galvani potential differences and Eh is the dielectric permittivity of the compact layer. The third term in equation (237) represents the dependence on electrolyte concentration, surface charge density, and the solvent effect on the inner-layer potential distribution. These effects can be ascribed to ionic penetration into the opposite phase and ion-ion correlations across the interface. Cui et al. [94] obtained good agreement between theoretical calculations and experimental data and came to the conclusion that the structure of the 1,2-dichloroethane/water interface is similar to that of the water/nitrobenzene interface, except that the effects of the image force and ion size are more pronounced and the inner-layer potential drop plays a more important role. 4. ELECTROCHEMISTRY OF EXTRACTION In recent years, there has been a great surge of interest in the experimental determination and theoretical calculation of the standard Gibbs free energy of ion transfer for individual ions moving between two solvents (Gibbs energy of

170

A.G. Volkov and V.S. Markin

resolvation). The Gibbs free energy of ion resolvation is a key concept intimately related to the electrochemistry of the interface between two immiscible electrolyte solutions, ion transport across biological and artificial membranes, the mechanisms of interfacial and surface catalysis, the kinetics of ion transfer across an oil-water interface, the coupling of heterogeneous reactions in bioenergetics, extraction, and the design and manufacture of ionselective electrodes [95-97]. By solvation, we mean the sum of all structural and energy changes occurring in a system when ions pass from the gaseous phase into solution. It has become customary to divide the interaction between ion and solvent and the corresponding energy, into several components. To understand the energies involved in transferring an ion or dipole between two solvent phases, one must take the following effects into account: (i) electrostatic polarization of the medium; (ii) production in a medium of a cavity to accommodate the ion also known as the solvophobic, hydrophobic or neutral effect; (iii) changes in the structure of the solvent that involve the breakdown of the initial structure and the production of a new structure in the immediate vicinity of the ion; (iv) specific interactions of ions with solvent molecules, such as hydrogen bond formation, donor-acceptor and ion-dipole interactions; (v) annihilation of defects: a small ion may be captured in a "statistical microcavity" within the local solvent structure, releasing energy of this defect; (vi) the correction term for different standard states. This sub-division is purely conditional, since many of these different effects may overlap: for example, electric polarization of the medium may significantly influence its structure. Assigning such a division does permit analysis of the individual components. However, sometimes, components can be grouped into "blocks". For example, one may speak about the solvophobic effect which combines the formation of a cavity and structural changes of the solvent in the vicinity of the new particle. This is quite justifiable because the solvophobic effect, together with the electrostatic effect, provides the major contribution to the solvation energy [48, 98]. On the other hand, sometimes it becomes necessary to consider each component of a given effect. For example, the total solvation (or resolvation) energy can be divided into electrostatic and non-electrostatic parts:

AapG,° = AapG°(e/) + A ° G , W 0 .

(238)

Electric Properties of Oil/Water Interfaces

171

To find the Gibbs standard free solvation energy, one needs to compare the Gibbs free energies for a given ion in each media. If one of the phases ((3) is a vacuum {vac), the difference: vacG, = Gi

+G ,

(239)

is called the solvation energy of the ion / in the a phase. The resolvation energy can be represented as the difference of two solvation energies: 101O A

G

P ,

=

— = AvacG,

~ KacGi

,

(240)

z:r where A"p<> | ° is the standard potential for transfer of ion / from a to (3 phase. Table 2 lists the standard free energies of ion transfer from water into different organic solvents obtained by cyclic voltammetry, chronopotentiometry, polarography and spin-lattice relaxation of quadrupole nuclei of ions, using solubility and extraction data. As can be seen from Table 2, the standard free energy of ion transfer from one solvent to another strongly depends on the nature of both ion and solvent. If two immiscible liquids a and p are mutually saturated, a certain amount of one solvent will be dissolved in the other solvent and vice versa. The corresponding energy of resolvation between two mutually balanced solvents is called the free partition energy. For most ions and solvents the standard free energies of transfer and partition coincide within experimental error. However, for some solvents and ions these quantities may differ significantly. When considering the standard free resolvation energy attention should be paid to whether one deals with ion transfer between pure or mutually saturated solvents. 4.1. Electrostatic contribution to the solvation energy Two sets of models exist to calculate the electrostatic portion of the solvation energy. In the first set the medium is considered as a structureless continuum, while in the second set it is represented by a set of individual particles having either realistic or simplified properties. In early works the solvent structure was taken into account by directly calculating the energies of particular configurations of solvent molecules in the vicinity of the ion. The configuration and the number of molecules in it were chosen with a certain degree of arbitrariness, proceeding from some physical considerations, which ensured an excellent fit between experimental and theoretical data. Such models completely ignored the statistical properties of the solution which are very important for obtaining a correct description of solvation. The modern approach

172

A.G. Volkov and V.S. Markin

lies in developing a statistical theory of ion-dipole plasma which describes both the energy and the statistical aspects of the phenomenon. There is also an intermediate approach, in which, while remaining within the framework of continuum theories, attempts are made to take account of the influence of the discrete nature of the solvent on the effective parameters of the model. For this purpose, account is taken of non-linear dielectric effects, and the mutual correlation of the polarization vectors of the solvent molecules, situated at short distances from one another, is analyzed using the theory of non-local electrostatics. Each of these approaches, reflecting the role of different effects, has its own advantages. 4.2. The Born model The first continuum model, developed by Born [99], considered the ion as a hard charged sphere of a given radius a immersed in a continuous medium a of constant dielectric permittivity s". This was a macroscopic theory, for it used macroscopic laws to describe the properties of a solvent at a small distance from the ion. In the Born theory, the solvation energy (its electrical part, to be exact) is given by the well known relation:

where e0 is the electric constant and s a , the permittivity of the medium. Equation (241) gives the energy per particle. Proceeding from this relation, the free energy of resolvation during the transfer of an ion from medium a to medium p or the difference between the energies in media p and a can be represented in the form:

^•^-iS^?-?'-

< 242 >

The solvation energies calculated by the Born formula differ noticeably from experimental values [48, 98]. Since, in most of the cases, the resolvation energy is the difference between two comparatively large solvation energies, even a relatively small error in each energy may give rise to considerable error in the resolvation energy, even an incorrect sign. For example, equation (219) implies that if ea > sp there is a higher probability for the ion to reside in solvent a, irrespective of ion size. In practice this is not always the case. It is known that small ions of radius a < 0.2 nm reside mainly in a polar solvent of high permittivity, while large organic ions reside preferably in the hydrophobic

Electric Properties of Oil/Water Interfaces

173

phase. Data on the partition coefficients, extraction, solubility, and currentvoltage characteristics show that the standard free energy of ion transfer from water into a less polar solvent is positive for small radius ions and negative for large radius ions, while equation (242) implies that the sign of this energy does not depend on the ion radius. 4.3. The Non-local electrostatics method Another rapidly developing semi-macroscopic approach is to calculate the electrostatic part of the solvation energy using the non-local electrostatics method. Non-local electrostatics takes into account that fluctuations of solvent polarization are correlated in space, since liquid has a structure caused by quantum interaction between its molecules. This means that the average polarization at each point depends on the electric displacement at all other points of the space correlated with a given point [98]. In non-local electrostatics the electric displacement D and electric field E are related by the tensor emn(r):

£ m (r) = ZJ*'£ o e m n (r-r')£"(r'), (m,n = x,y,z).

(243)

n

It should be noted that, although, this relation is spatially complicated, it is linear. Further calculations are carried out in terms of the Fourier transform of the tensor emn(r), which is called the static dielectric function e(k):

eflO = YMr\ m,n

d(r - V )e~lk^\

K

mn(r -

r').

(244)

The potential produced in a medium by a charged sphere of radius a at a distance r from its center is given by:

ze0

"t dk sin kr sin ka

Hence, we can easily find the electrostatic contribution to the solvation energy:

174

A.G. Volkov and V.S. Markin

Polarization of the medium can be divided into three main groups: optical, vibrational and orientational. To evaluate equation (246), one has to specify the function s(k). This can be done, for example, by subdividing fluctuations of the medium polarization into three modes relating to different degrees of freedom: namely, (1) electronic or optical; (2) vibrational or infra-red; (3) orientational or Debye. If the radius of the correlation of the zth mode of a fluctuation is Xx, then: j . ^ )

j =

1

" S

_ O

/

i (

B

O

i ^ B

i )

2

1

+

^

i _ i + ( 2

S 2

" B

i )

3

1 + ^

2

3 -

( 2 4 7 )

In this expression the correlation length of the electronic mode is set equal to zero. The exact values of the correlation lengths X2 and X,3 cannot be determined a priori, but they can be approximated from physical considerations. In the infra red region, the length ^3 depends on liquid type. In the case of non-associated liquids the correlation length for orientational vibration is approximately equal to the intermolecular distance, while for associated liquids (water for example) A,3 is equal to the characteristic length of the hydrogen bond chain, i.e. 0.5 - 0.7 nm [48, 98]. Integration of equation (246) with (1 - l/s(k)) expressed by equation (247) yields:

A—cm=T^ ( • - - • ( - - - ) #)•{ Snsoa[ where: \\)(x) = l-(l-e~x)-

sopt

.

\sopl

sj

\A2J

L \s2

- -Mr)L <*»> s3j

V/Vj'v

(249)

The agreement between theory and experiment obtained by Kornyshev and Volkov is more than satisfactory [48]. Yet, the question arises whether the results obtained are sensitive to the fitting parameters Xt and whether such a choice of X\ is justified. In addressing this question, one can conclude that the results are most sensitive to parameter A,2 and depend very little on A,3. A detailed analysis of this situation is presented in [48]. The non-local electrostatics method refers to the continuum models, but the effective parameters needed for calculation are chosen by analyzing the solvent structure. This method gives rather accurate values for the solvation energy for small ions, and also permits calculation, by virtue of equation (248), of the resolvation energy. For large ions

Electric Properties of Oil/Water Interfaces

175

there remains considerable discrepancy between theory and experiment, which makes it necessary to take into account other effects. In our case, these are the work done in the formation of a cavity in the solvent and the solvophobic effect. The largest value for the solvation energy is obtained in the Born limit A.2 = A,3 = 0, i.e. in a structureless solvent. The major disadvantage of the continuum and semi-continuum approaches to the solvation problem lies in the solvent model itself. An allowance for dielectric saturation or dipole correlation is an attempt to partially describe the discrete properties of the solvent within the framework of the continuum model. Other analogous approaches attempt to take into account the unknown effect of the solvent molecular structure on the thermodynamic properties of a system. However, the problem can only be solved correctly using a statistical model of the solvent and taking into account its discrete structure. 4.4. Statistical solvent models A consistent approach to the solvation problem is to develop a statistical theory that will consider both ions and solvent molecules as discrete species, though such an approach demands the introduction of some model potentials for particle interaction. Unlike "primitive" electrolyte models which use continuum descriptions of the solvent, the modern statistical analysis employs a "nonprimitive" or a so-called ion-dipole plasma model in which the electrolyte solution is considered as a system of hard spheres of radius a and aQ each carrying at their center a point electric charge or a constant point dipole, respectively. Ordinary electrostatic interactions are assumed to exist between the charges and dipoles. The system is described in terms of many-particle correlation functions satisfying the corresponding equations [100, 101]. To solve these equations, one has to introduce various simplifications, the mean-spherical approximation being the most widely used [102]. The ion-dipole plasma model permits calculation of the ion solvation energy, and also the properties of the pure dipole liquid. Wertheim [101] calculated the permittivity of the dipole liquid in the mean-spherical approximation: e = q(2Q/q{-Q,

(250)

where: <7(x) = (l+2x) 2 /(l-x) 2 ,

(251)

and the parameter C, is a solution of the equation: q(2Q-q(Q=nd]x2/3E0kT.

(252)

176

A.G. Volkov and V.S. Markin

Here nA is the number of solvent particles per unit volume and \x is the solvent molecule dipole moment. This result is a considerable improvement over the well known Onsager relation. For highly associated liquids there exists a noticeable discrepancy between theory and experiment. This is not surprising because the model ignores a number of intricate forces which act between real solvent molecules. For example, for a dipole liquid with density and dipole moment (1.8 D) equal to the corresponding parameters of water, equation 250 yields, at room temperature, the permittivity value 48. To obtain a value of 80, the dipole moment should be set equal to 2.62 D. Naturally, a system of hard, constant-dipole spheres can hardly seem as a proper model for water. Yet, such a system may be a reasonable model of an organic solvent. In any case, it is the first step for constructing a relevant model of water. In particular, the application of the model to studying solvation and the interaction of ions has revealed a number of essential effects caused by solvent discreteness. The energy of ion solvation (the "Born" energy) was calculated by Chan et al. using the mean spherical approximation:

KacG° = -^—r^—TTV^1"")' 87ieo(a + ad IX)

(253) e

where A, is found using the equation: A. 2 (l-A,) 4 =16e. (254) The parameter A, ranges from 1 to 3; for E = 80 it is equal to 2.6. Together with the radius of the solvent dipole moment, this parameter determines a new characteristic length for the problem, aj'k which naturally arises in the solution of the equations. This characteristic length is solely determined by the properties of the dipole liquid and depends on two parameters: the molecule radius and permittivity. Formally, equation (253) is analogous to the Born energy with the renormalized radius increased by aJX. It is interesting to note that an empirical relation exists, which is identical in form to equation (253) and which can be used to describe the solvation energy and the free resolvation energy [103]. If the solvent molecule radius is set equal to 1.5 A, the aforementioned relation gives the value 0.58 A for aJX. An apparent increase of the ion radius in equation (253) is caused by solvent discreteness. It is obvious that the permittivity cannot have the bulk

Electric Properties of Oil/Water Interfaces

177

value immediately at the ion "surface". Equation (253) implies that this value is effectively attained at a distance of a + aJX from the ion center. One should, however, bear in mind that this idea of a stepwise change in the permittivity is only an approximate description of sufficiently rigorous physical results obtained in the discrete model. Furthermore, an analysis of the model shows that at small distances the concept of local permittivity is simply invalid and a more correct concept of medium polarization must be used. If a dielectric is treated macroscopically, its polarization at a distance r from the ion is given by the well known relation: PnJir) = (e - 1 W47re 0 r 2 .

(255)

The results obtained in the mean spherical approximation model are mainly of a qualitative character, since the model is approximate. Nevertheless, they provide a more accurate physical representation of solvation and provide another way with which one can evaluate other theories and understand how these theories can agree with experimental results. 4.5. Contribution of the solvophobic effect to the resolvation energy Modern approaches to the calculation of the energy of the solvophobic ion-solvent interaction are based on phenomenological or statistical mechanical molecular models in which the parameters describing both the ion and solvent are not always known. To avoid uncertainty, the solvophobic contribution to the resolvation energy is sometimes estimated using the semi-empirical solvophobic formula [48, 99]. The main idea is as follows. Surface energy, when expressed in terms of surface tension at the ionsolvent interface y0 a , is equal to Ana y0a and the difference in the surface energies in media a and P is Ana (y0 a - y0 p ). According to Antonov's rule [104] Yo.a " Yo,p = YaPSgn(Ya " Yp),

(255)

where sgn(ya - yp) = +1 for ya > yp and sgn(ya - yp) = -1 for ya < yp, y ap is the interfacial tension at a flat boundary between solvents, and y aj yp are the surface tensions at the boundaries of solvents a and p, respectively. With air under the same pressure and temperature, we obtain the formula for the solvophobic contribution to the resolvation energy: AapG°(svph)

= -47w 2 y a p sgn(y a - y p ) .

(256)

178

A.G. Volkov and V.S. Markin

Relation (256) infers that the molecules of the two solvents w and m do not mix or react chemically with each other. The hydrophobic contribution to the Gibbs free energy of ion or molecule (dipole, multipole) transfer from water to hydrocarbon is negative and the absolute value is obviously greater for particles with larger radii, a. The effect of curvature on the surface tension at a molecularly sized sphere was calculated by Tolman [42]. The surface tension at radius a can be written as

1+ — a where 8 is a parameter, which according to Tolman is the distance from the surface of tension to the dividing surface for which the surface excess of fluid vanishes. Parameter 5 can vary between 0 and a few angstroms. From the theoretical point of view, the limits of applicability of the solvophobic formula are not quite clear. Nonetheless, it works surprisingly well for calculation of the partition coefficients of a system of two immiscible liquids. Some deviation from the solvophobic formula is observed if the interfacial tension of two pure immiscible solvents is less then 10 mNm"1 and if the size of the dissolved particle is less than 0.2 nm. An important advantage of the solvophobic formula is that it does not involve fitting parameters because yaP is determined experimentally. 4.6. The total resolvation enegy In the preceding sections we considered different effects which contribute to the energy of ion resolvation. However, for most solvents which are of practical interest the greatest contribution is due to the electrostatic and solvophobic effects. Therefore, the resolvation energy can be represented as a sum of all contributions: AG(tr) = AG(e/) + AG(solv) + AG(si),

(258)

where AG(el) is electrostatic contribution, AG(solv) is the hydrophobic effect and AG(si) is caused by specific interactions of the transferred particle (ion, dipole) with solvent molecules, such as hydrogen bond formation, donoracceptor and ion-dipole interactions. The calculation results are in excellent agreement with experimental data. The discrepancy observed for small radius anions may be attributed to the formation of hydrogen bonds between anion and solvent or to defect

Electric Properties of Oil/Water Interfaces

179

annihilation. The energy corresponding to this discrepancy is on the same order of magnitude as the energy gain obtained in the formation of a weak hydrogen bond between anion and solvent. It should, however, be noted that for small anions resolvation energies reported by different authors differ considerably. 4.7. Dipole resolvation Bell [105] calculated the electrostatic part of Gibbs energy of dipole molecule transfer from a vacuum to a medium of dielectric constant SJ:

^G{el) = --^-[^-\

(259)

12ne0/ l ^ . + l j where (j. is the dipolar moment and / is the effective dipole size. Now one can calculate the electrostatic ("Born") Gibbs energy of dipole transfer from phase w to phase m using a thermodynamic cycle [8]:

A'Giel) = -^-J ^^- - -^±) = - J ^ J

^ >

]-

(260)

Generally speaking, the dipole moment, |^, can vary from solvent to solvent. The concentration of dipole molecules in an oil phase can be calculated from the estimated free Gibbs energy of a dipole molecule transfer from water to the hydrophobic phase: cot, = cwcxp(-AGlr/RT).

(261)

The partition coefficient of an ion or dipole Kj can be also calculated from the free Gibbs energy of transfer: Kt = exp(-AG? (tr) IRT)

(262)

Equations (260) - (262) are very useful in the theory of extraction, partitioning and passive transport of small neutral molecules across liquid membranes. The partition coefficient Kj is also important in the formulation and delivery of pharmaceutical agents because it describes relative lipophilicity and solubility in lipid/water systems. For many drugs the inequality can be written: 0 < log Kj < 4.

(263)

180

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The choice of a given drug delivery method can be guided by the partition coefficient of the agent. For instance, a drug with a low partition coefficient will likely be given by injection, while a skin patch or ointment could be used for drugs with high partition coefficients. Oral consumption would be appropriate for drugs with coefficients in the intermediate range. REFERENCES [I] V.S. Markin and A.G. Volkov, J. Colloid Interface Sci, 131 (1989) 382. [2] V.S. Markin and A.G. Volkov, Russ. Chem. Rev., 57 (1988) 1963. [3] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 23 (1987) 1405. [4] V.S. Markin and A.G. Volkov, Adv. Colloid Interface Sci., 31 (1990) 111. [5] E. Lange and K.P. Miscenko, Z. Phys. Chem., 149 (1930) 1. [6] F.G. Donnan, Z. Electrochem., 17 (1911), 572. [7] W. Nernst, Z. Physik. Chem., 4 (1889) 129. [8] A.G. Volkov, D.W. Deamer, D.J. Tanelian and V.S. Markin, Liquid Interfaces in Chemistry and Biology, J. Wiley, New York, 1998. [9] Z. Samec, in: Liquid-Liquid Interfaces. Theory and Methods (A.G. Volkov and D.W. Deamer, eds.) pp. 155-178. CRC-Press, Boca Raton, New York, London, Tokyo, 1996. [10] A.G. Volkov, J. Electroanal. Chem., 205 (1986) 245. [II] A.G. Volkov and Yu.I. Kharkats, Chem. Phys., 5 (1986) 964. [12] A.G. Volkov and Yu.I. Kharkats, Kinetics and Catalysis, 26 (1985) 1322. [13] Yu.I. Kharkats and A.G. Volkov, J. Electroanal. Chem., 184 (1985) 435. [14] Yu.I. Kharkats and A.G. Volkov, Biochim. Biophys. Acta, 891 (1987) 56. [15] I.M. Kolthoff, Pure Appl. Chem., 25 (1971) 305. [16] Z. Koczorowski, in: Liquid-Liquid Interfaces. Theory and Methods (A.G. Volkov and D.W. Deamer, eds.) pp. 375-400. CRC-Press, Boca Raton, 1996. [17] B.S. Gourary and F.S. Adrian, Solid State Physics, 10 (1960) 127. [18] R.A. Robinson and R.H. Stokes, Electrolyte Solutions, Butterworths, London, 1959. [19] L.Q. Hung, J. Electroanal. Chem., 115 (1980) 159. [20] L.Q. Hung, J. Electroanal. Chem., 149 (1983) 1. [21] L.Q. Hung, in: Interfacial Catalysis, A.G. Volkov (Ed.), pp.83-112, M. Dekker, New York, 2003. [22] T. Kakiuchi, in: Liquid-Liquid Interfaces. Theory and Methods (A.G. Volkov and D.W. Deamer, eds.) pp. 1-18, CRC-Press, Boca Raton, 1996. [23] T. Kakiuchi, in: Liquid Interfaces in Chemical, Biological and Pharmaceutical Applications, A.G. Volkov (Ed.), pp. 105-121, M. Dekker, New York, 2001. [24] A.G. Volkov and D.W. Deamer (Eds.) Liquid-Liquid Interfaces: Theory and Methods, CRC-Press, Boca Raton, 1996. [25] A.G.Volkov, Sov. Electrochemistry, 23 (1987) 90. [26] A.G. Volkov, J. Mwesigwa, A. Labady, S. Kelly, D.J. Thomas, K. Lewis and T. Shvetsova, Plant Sci., 162 (2002) 723. [27] A. Heydweller, Ann. Phys., 33 (1910) 145. [28] J.J. Bikerman, Physical Surfaces, Academic Press, New York, 1970. [29 D.R Lide, Ed., CRC Handbook of Chemistry and Physio?, 77th Edition, CRC-Press, New York, 1996. [30] C. Wagner, Physik. Zeitschr., 25 (1924) 474.

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181

[31] L. Onsager and N.N.T. Samaras, J. Chem. Phys., 2 (1934) 528. [32] F.P. Buff and F.H. Stillinger, J. Phys. Chem., 11 (1955) 312. [33] A.N. Frumkin, Sbornik rabot po chistoy i prikladnoy khimii, Scientific Chemical Technical Publishing House, Petrograd, pp. 106-126, 1924 (in Russian). [34] A.N. Frumkin, Z. Phys. Chem., 109 (1924) 34. [35] R.M. Suggitt, P.M. Aziz and F.E.W. Wetmore, J. Amer. Chem. Soc, 71 (1949) 676. [36] N.L. Jarvis and M.A. Scheiman, J. Phys. Chem., 72 (1968) 74. [37] J.E.B. Randies, in: Advances in Electrochemistry and Electrochemical Engineering, P. Delahay, Ed., Vol. 3, pp. 1-3 0, Interscience, New York, 1963. [38] V. S. Krylov and V.G. Levich, Doklady Acad. Nauk SSSR, 159 (1964) 409. [39] V.S. Markin and A.G. Volkov, J. Phys. Chem. B, 106 (2002) 11810. [40] Yu.I. Kharkats and J.J. Ulstrup, J. Electroanal. Chem., 308 (1991) 17. [41] J.J. Ulstrup and Yu.I. Kharkats, Russ. J. Electrochem., 29 (1993) 299. [42] R.C. Tolman, J. Chem. Phys., 17 (1949) 333. [43] J.E.B. Randies, Discuss. Faraday Soc, 24 (1957) 194. [44] R. Parsons and F.G.R. Zobel, J. Electroanal. Chem., 9 (1965) 323. [45] Z. Samec, V. Marecek and D. Homolka, J. Electroanal. Chem., 187 (1985) 31. [46] T. Wandlowski, K. Holub, V. Marecek and Z. Samec, Electrochim. Acta, 40 (1995) 2887. [47] A.G. Volkov, D.W. Deamer, D.I. Tanelian and V.S. Markin, Progress Surf. Sci., 53 (1996) 1. [48] A.A. Kornyshev and A.G. Volkov, J. Electroanal.Chem., 180 (1984) 363. [49] B. Sisskind and J. Kasarnowsky, Zh. Fiz. Khim., 4 (1933) 683. [50] H.H. Uhlig, J. Phys. Chem., 41 (1937) 1215. [51] V.S. Markin and A.G. Volkov, J. Electroanal. Chem., 235 (1987) 23. [52] L.S. Bartell, J. Phys. Chem. B, 105 (2001) 11615. [53] T.V. Bykov and X.C. Zeng, J. Phys. Chem. B, 105 (2001) 11586. [54] V.S. Markin and A.G. Volkov, Electrochim. Acta, 34 (1989) 93. [55] W.D. Harkins and E.C. Gilbert, J. Am. Chem. Soc, 48 (1926) 604. [56] W.D. Harkins and H.M. McLaughlin, J. Am. Chem. Soc, 47 (1925) 2083. [57] L.I. Daikhin, A.A. Kornyshev and M.I. Urbakh, J. Electroanal. Chem., 483 (2000) 68. [58] M. Wilson and A. Pohorille, J. Chem.Phys., 95 (1991) 6005. [59] V.S. Markin and A.G. Volkov, Progress Surf. Sci., 30 (1989) 233. [60] V.S. Markin, A.G. Volkov, Electrochim. Acta, 34 (1989) 93. [61] V.S. Markin and A.G. Volkov, in: Liquid-Liquid Interfaces. Theory and Methods (Eds.: A.G. Volkov and D. W. Deamer), CRC-Press, Boca Raton, pp. 63-75, 1996. [62] V.S. Markin and A.G. Volkov, in: The Interface Structure and Electrochemical Processes at the Boundary between Two Immiscible Liquids (V. E. Kazarinov, Ed.) pp. 94-130, Vol. 28. VINITI, Moscow, 1988. [63] A.G. Volkov, M.I. Gugeshashvili and D.W. Deamer, Electrochim. Acta, 40 (1995) 2849. [64] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 318. [65] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 325. [66] V.S. Markin and A.G. Volkov, Progress Surf. Sci., 30 (1989) 233. [67] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 478. [68] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 579. [69] V.S. Markin and A.G. Volkov, Russ. Chem. Rev., 57 (1988) 1963. [70] V.S. Markin and A.G. Volkov, Electrochim. Acta, 35 (1990) 715. [71] M. Planck, Ann. Phys., 44 (1891) 385. [72] M. Planck, Ann. Phys. Chem.N. F., 40 (1890) 561.

182

A.G. Volkov and V.S. Markin

[73] F.O. Koenig, J. Phys. Chem., 38 (1934) 111. [74] D.C. Grahame and R.W. Whitney, J. Amer. Chem. Soc, 64 (1942) 1548. [75] E.J.W. Verwey and K.F. Nielsen, Phil. Mag., 28 (1939) 435. [76] G. Gouy, Compt. Rend. Acad. Sci., 149 (1910) 654. [77] D.L. Chapman, Phil. Mag, 25 (1913) 475. [78]. A.G. Volkov (Ed.) Liquid Interfaces in Chemical, Biological, and Pharmaceutical Applications, M. Dekker, New York, 2000. [79] C. Gavach, P. Seta and B. d'Epenoux, J. Electroanal. Chem., 83 (1977) 225. [80] R.S. Hansen, J. Phys. Chem., 66 (1962) 410. [81] A.G. Volkov, Langmuir, 12 (1996) 3315. [82] A.N. Frumkin, Zero Charge Potentials, Nauka Publ., Moscow, 1979. [83] Z. Samec, V. Marecek and D. Homolka, J. Electroanal. Chem., 187 (1985) 31. [84] T. Kakiuchi, M. Kobayashi and M. Senda, Bull Chem. Soc. Jpn, 60 (1987) 3109. [85] C. Yufei, VJ. Cunnane, D.J. Schiffrin, L. Murtomaki and K. Kontturi, J. Chem. Soc. Faraday Trans., 87 (1991) 107. [86] Freundlich, H. (1926). Colloid and Capillary Chemistry, Methuen, London. [87] A.N. Frumkin, Electrocapillary Studies and Electrode Potentials, Saposhnikov Publ. House, Odessa, 1919. [88] V.S. Markin and A.G. Volkov in: Encyclopedia of Electrochemistry, Eds. E. Gileadi and M. Urbakh, Vol. 1, pp. 162-187, Wiley-VCH, Weinheim, 2002. [89] A.R. Vanbuuren, S J. Marrink and H.J.C. Berendsen, J. Phys. Chem, 97 (1993) 9206. [90] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, 2nd Ed, Pergamon, New York, 1984. [91] C.W. Outhwaite, L.B. Bhuiyan and S. Levine, J. Chem. Soc, Faraday Trans. II, 76 (1980) 1388. [92] F.H. Stilinger and J.G. Kirkwood, J. Chem. Phys, 33 (1960) 1282. [93] G.M. Torrie and J.P. Valleau, J. Electroanal. Chem., 206 (1986) 69. [94] Q. Cui, G.Y. Zhu and E.K. Wang, J. Electroanal. Chem, 383 (1995) 7. [95] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 23 (1987) 1105. [96] V.S. Markin and A.G. Volkov, Russian Chem. Rev, 56 (1987) 1953. [97] V.S. Markin and A.G. Volkov, J. Electroanal. Chem, 235 (1987) 23. [98] A.G. Volkov and A.A. Kornyshev, Sov. Electrochemistry, 21 (1985) 814. [99] M. Born, Z. Phys, 1 (1920) 45. [100] D. Henderson, Progress Surf. Sci, 13 (1983) 197. [101] M.S. Wertheim, Ann. Rev. Phys. Chem, 30 (1979) 471. [102] D.J.C. Chan, D. Mitchell and B.W. Ninham, J. Chem. Phys, 70 (1979) 2946. [103] A.K. Govington and K.E. Newman, in: Modern Aspects of Electrochemistry, Eds. J.O'M. Bockris and B.E. Conway, Vol. 12, pp. 41-129, Plenum Press, New York, 1977. [104] G. Antonow, J. Chim. Phys, 5 (1907) 372. [105] R.P. Bell, J. Chem . Soc, 32 (1931) 1371.

Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 5

Deformation of fluid particles in the contact zone and line tension V. Starov Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK 1. TWO DROPS/BUBBLES AT EQUILIBRIUM IN A SURROUNDING LIQUID The hydrostatic pressure in thin liquid films intervening between two drops/bubbles differs from the pressure inside the drops/bubbles. This difference is caused by the action of both capillary and surface forces. The manifestation of the surface forces action is disjoining pressure, which has a special S-shaped form in the case of partial wetting (aqueous thin films and thin films of aqueous electrolyte and surfactant solutions). Disjoining pressure solely acts in thin fiat liquid films and determines their thickness. If the film surface is curved then both disjoining and the capillary pressure act simultaneously. Disjoining pressure (DP), Tl(h), is a manifestation of colloidal forces action. In Fig. 1 S-shaped DP isotherm is presented, which correspond to a sum of three component: dispersion, electrostatic and structural [1]. DP pressure isotherms can be directly measured, however, not in the whole range of film thickness but only those where Tl'(h)<0, that is, at htmax (equilibrium /?- films). Theory of dispersion and electrostatic components of DP is well developed and referred to as DLVO theory [1]. However, in aqueous electrolyte solutions an additional DP component, the structural component, is equally important. The latter component is determined by the orientation of water dipoles in a proximity to liquid-solid and liquid-air interfaces [1]. This component considerably influences the shape of DP isotherm of aqueous solution. Theory of the structural component is to be developed [3]. That is why below a general form of S-shaped DP isotherms is used. The sum of dispersion, electrostatic and structural components gives S-shaped isotherm of DP in Fig. 1 [1, 2]. Below based on the consideration of a combined action of capillary pressure and DP a theory is developed enabling one to calculate equilibrium position and shape

184

V. Starov

Fig. 1. S-shaped isotherm of disjoining pressure.

of two drops/bubbles immersed in a liquid. The theory is based on a consideration of the shape of a liquid interlayer between emulsion droplets or between gas bubbles of different radii under equilibrium conditions taking into account both the local DP of the interlayer and the local curvature of its surfaces. The model of solid non-deforming particles is frequently used, when carrying out an analysis of the forces acting between colloidal particles. However, real droplets/bubbles and even soft solid particles within the contact zone can deform. The latter changes the conditions of their equilibrium. In this case, interaction is not limited only to the zone of a flat contact, but is extended onto the surrounding parts within the range of action of surface forces. For elastic solid particles, such a problem was treated in [4-6]. Below, the case of droplets/bubbles is considered (e.g., emulsions, gas bubbles in a liquid), whose shape changes especially easily under the influence of surface forces (Fig. 2). For the first time the equilibrium position of droplets in a surrounding liquid was studied earlier in [7-10]. However, the finite thickness of a liquid interlayer between droplets/bubbles and variation in its thickness in the transition zone between the interlayer and the equilibrium bulk liquid phase have not been taken into account in [7-10]. The effect of the surface forces in a thin interlayer was taken into account only as a change in the magnitude of the interfacial tension [8, 10]. The problem in using this approach arises if we try to utilize Navier-Stokes equation for description of flow/equilibrium in thin liquid films: a thickness dependent surface tension results in an unbalanced tangential stress on the surface of thin films.

Deformation of Fluid Particles in the Contact Zone and Line Tension

185

Fig. 2. Two identical drops/bubbles, 1, at equilibrium in a surrounding liquid, 2. h(x) is the thickness of the liquid film between two drops/bubbles.

Below a different approach is used, which takes into account the interlayer thickness and the effect of the transition zone between the thin inter- layer and the bulk liquid. A low slope approximation and constant surface tension approximations are used. Then, as was shown earlier in [2, 11] it is possible to use the equation taking into account both the disjoining pressure and the capillary pressure in the interlayer.

Fig. 3. The equilibrium profile of interlayer, h(x), between two cylindrical droplets of the same radius, R.

186

V. Starov

1.1. Two identical cylindrical drops/bubbles At first, let us consider a simplified case of two identical cylindrical drops/bubbles (Fig. 3). It is assumed for a simplicity that the radius of action of surface forces is limited to a certain distance, tsf. Beyond this distance, the surface of droplets retains a constant curvature radius, R, and is not disturbed by surface forces. The interacting droplets are considered as being surrounded by a liquid with constant pressure, Pt. The thickness of the liquid interlayer, 2h(x), varies from 2h0 on the axis of symmetry at x=0 to 2h=tsj at x=xn (Fig. 3). For the drop profiles not disturbed by the surface forces we use the designation hs at yB (Fig. 3). The liquid in droplets is assumed incompressible, which leads to the condition of the constancy of the volume of droplets per unit length: V = 4 } (Hs~hsjdx = nRQ = const , where Ro is the 0 radius of an undisturbed isolated droplet prior to contact. Note, that in general the system of two droplets is thermodynamically unstable with regard to coalescence. Therefore, the following calculations give the conditions of the meta-stable equilibrium of droplets separated by a thin interlayer of the surrounding liquid. The excess free energy of the two cylindrical drops (per unit length of cylindrical droplets), F, is equal to F = yS + F + PV , where S is the total interfacial area per unit length, y is the interfacial tension, FD is the excess free energy per unit length determined by the surface forces action, P is the excess pressure and V is the volume. In the case of cylindrical drops/bubbles S, FD and V are as follows:

S = 4^\+H'/dx 0 x

+ 4 ^l+tifdx x0

+4 \ 0

i\+h'ldx

o[~°°

F = 2 | \H{h)dh dx 0 I2h V = 4 J \Hs-hs}dx x0

+4 \{Hs-h)h= n RQ = const 0

(1)

where h(x), hs (x), Hs (x), x0 and radius, R, are determined in Fig. 3; tsfis the radius of surface forces action. Substitution of the latter expression into the excess free energy results in

Deformation of Fluid Particles in the Contact Zone and Line Tension

187

R I rRI XQ / ^ 1 X 0l~°° l F = 4y j^l+H'fdx+4 l^l+h'f dx+4 J i\+h' dx + 2 \ \U(h)dh dx + [O x0 0 J 0 I2h + P 4 \{Hs-hs)dx+4 xQ

\{Hs-h)k 0

A variation of the excess free energy with respect to h(x), hs (x), Hs (x) and two values xn and R results in the following equations: r oi-3/2 yh"\\+h'l\ +U{2h)=-P

?r3/2

r

^[1+Vj r

=- P

?r

(2)

3/2

where the last two equations give the equation of a circle of radius R. The latter y

immediately determines the unknown excess pressure as P=-—. After that the first Eq. (2) takes the following form r

T~I~3/2

yh"\ l+h'z

y

+U(2h)=L

(3)

where the first term in the left hand site is due to the capillary pressure, the second term is determined by the local value of DP, n(2/z). The boundary conditions for Eq. (3) are as follows: h{xOhhs{xo)

h{Xo)=hs{xo\

h{xo) = tsf/2

(4)

hs{R)=Hs{R), hs{R)=-H^{R)=cc,

(5)

A1(0)=//jl(0)=0.-

(6)

Reminder, hs describes the profile of the droplet at 2hs > tsf .

188

V. Starov

Fig. 4. Partial wetting. S-shaped disjoining pressure isotherm (left side, a) and the liquid profile in the transition zone (right side, b).

If the droplets are located at distance, 2h§>tsf, then n(2/z)=0; and in this case, the solution of Eq. (3) gives the profile hs (x), which corresponds to the circular form of the cross-section of non-deformed droplets that do not interact with one another. At2fiQ=2 \Il(2h)dx. 0

(7)

Substituting the expression for H(2h) from Eqs. (2) and (3) into Eq. (7) and then carrying out integration, we obtain:

Deformation of Fluid Particles in the Contact Zone and Line Tension

« = 2 r L- r ^I-|

189

(8)

In view of the boundary condition (4), at the point x=x0 both values of h(x)=hs(x) and their derivatives, h'=h's, are equal. This enables one to express h (XQ) through the central angle cp (Fig. 3) and values xn and R: h'(xo)=tm
.

(9)

Substituting the latter expression for h (XQ) into Eq. (8), we obtain 0 = 0 , as should be at the equilibrium. Thus, as one should expect that the conditions of equilibrium preset by Eq. (3) and by the boundary conditions (4)-(6), correspond to zero interaction forces between the droplets. It should be noted that angle cp has a value, which is very close to that of the contact angle 0, to be determined at the point of intersection of the continuation of the undisturbed profile of a droplet with axis x (Fig. 3). The values of 8 and cp practically coincide when the interlayer thickness is small as compared with R, and when XQ is not too small. This enables one to use Eq. (9) for calculation of the contact angles. Thus derived values of x0, B, 0, and function h(x) give the full solution of the problem, where the distance between the centers of droplets, B, (Fig. 3) is: B=tsf+2^R2-x^

=t^+{2xo/taa
(10)

1.2. Interaction of cylindrical droplets of different radii Let us now consider a more complicated case of interaction of cylindrical droplets of different radii R2 >R/ (Fig. 5a). Applying he same method of minimization of the excess free energy as above, we obtain the following equations:

7\

r1+

2

T3/2

(^)

r

+n(t) = y/Rl;

(11)

2r3/2

yh2 U + {h2)

-U(t)=-r/R2

(12)

190

V. Starov

Fig. 5. The equilibrium profile of interlayer t(x) = h/.(x) - I12 (x) between two droplets of different radii Ri, and R2 in the general case (a); and in the simplified case (b).

where h]{x) and h2ix) are measured from an arbitrary plane perpendicular to the axis of symmetry, t(x)= hi(x)-h2(x), is the thickness of the interlayer. The latter two equations enable one to determine two profiles, hj (x) and h2(x). These equations with the boundary conditions

/i|(0)=^(0)=0;40)=y; = tan

^= x o ^i-*oJ

h

2(xo)=tan(P2=xolR2-xo)

(13)

; (14)

and with the conditions of constancy of volumes give a solution to the problem. After addition of Eqs. (11) and (12), we obtain

{r?RiHr/R2}=M>k-<

(15)

The latter means that the curvature of the whole interlayer equalizes the capillary pressure drop between the droplets. In a similar way may be solved the same problem for two droplets of different composition with different interfacial tensions, /jand y2, of the first and the second droplet, respectively. In this case, even for the droplets of the same radius, the interlayer on the whole proves to be curved owing to the appearance of a capillary pressure drop, AP/c={/-^-y2)/R •

Deformation of Fluid Particles in the Contact Zone and Line Tension

191

In the case of not strongly curved interlayer between droplets the term ( h '>

2

may be neglected as compared with 1. Then, we obtain from Eqs. (11),

(12), and (15), taking into account that t{x) = h,(x) -h2(x) and t"(x) = h, \x) h2"(x): r-t"+2-u(t)=r

-j-+^-

(16)

This equation should be subjected the following boundary conditions

/(0)=0; 4 o R / ; A^O^^O [R\-4]

+ R

{ 2~4)

(17)

and coupled with the conditions of constancy of volumes, which determine the unknown radii R: and R2. The latter conditions and (16)-(17) enable one to calculate t(x), determining the variable thickness of the interlay er. Thereafter, on substituting the known dependence n[?(x)]=n(x) into Eq. (12), it is possible to obtain from y-h2-n(x)=-y

IR2

(18)

the profile h2(x) of the lower surface of the interlayer. The boundary conditions for Eq. (18) are as follows:

h2(0)=0; h2(x0)=Rcos
h'2(xQ)=tan
(19)

It has been taken into account in the latter expression that angle cp2 is determined from Eq. (14). The sum h2(x) + t(x) gives the profile of the upper surface of the interlayer. Calculation of the interaction of cylindrical droplets of different radii (R2 > Ri) can be simplified, if we assume that the interlayer is of a constant thickness. This means that the effect of a transition zone is neglected, which is justified only at xQ»t. In this case, the curvature of each surface of the interlayer is constant (Fig. 5b), and Eqs. (11) and (12) may be rewritten in the following way:

192

V. Starov

{y I r)+U(t)=y I Rx

(20)

I//(r+t)]-Yl(t)=-y / R2

(21)

This system of equations enables one to determine two unknown values: t, the interlayer thickness; and r, the radius of curvature of its surface on the side of the smaller droplet. Summing and subtracting by terms Eqs. (20) and (21), we obtain (at r»t):

r-lRfo/fa-Rx) Il(t)~y IRy-yl r~y{R2 +RX )l 2RXR2 •

(22) (23)

If the shape of disjoining pressure isotherm, Il(t), is known, then Eq. (23) determines the equilibrium thickness / = const of the curved interlayer. At R2 » R, Eqs. (22) - (23) results in: r~2Rl

(22')

Tl(t)~y/2Rl

(23')

Eq. (22') coincides with that derived earlier by Princen and Mason [8]. Let us note that Eqs. (22) and (23) may be derived by another method using the concept of disjoining pressure [1]: K = Pd-P,

(24)

where Pd is the pressure just under the interlayer surface, and Pt is the pressure in the bulk phase, which the interlayer is at the equilibrium with. On the side of a smaller droplet, we have Pd = P, +(y IRt)-(y Ir); on the side of a larger droplet,Pd = P, +(y/R2)+ y(r + h). As the disjoining pressure does not depend on which side of the interlayer it is determined, then from Eq. (24) it follows: n = ( / / Rx )-(y I r)={y IR2)+[/ l(r+t)\

(25)

which coincides with Eqs. (20)-(23). However, determination of r and t does not solve completely the problem, because the position of the center of the interlayer curvature remains unknown. Its

Deformation of Fluid Particles in the Contact Zone and Line Tension

193

position can be determined by minimizing the value of the free energy of the system

F=2yR2{7t-(p2y2yRx{7r-(piy

2r

L

(26)

t

and by taking into account the condition of the constancy of the volume of droplets:

i( 1 \ l( \ \ 2 R\ n-(py\—sin2^j \+r (p—sin2^ \=n-R^Q-const R2\7t-(p2^—sin2
\-r \cp—sm2q> \=n-R2Q=const

(27)

where Rt0 and R2Q are the radii of the undisturbed droplets (at/?Q>^). It is possible to express all the values via XQ: XQ =R\ sin
i+(/,;)

, i+[*;)

\Ah2j

r l-^frjJ

194

V. Starov

The boundary conditions for Eqs. (28) and (29) are given by (13)-(14), where x0 should be replaced by r0. In this case, conditions of the constancy of the volume of droplets can be written as:

V =2n \ {Hls-hls)dx +2n \{Hls-h)ix= -nR^const 3 rQ 0 V = In f(H2s-h2s)dx rQ

+27T ](H2s-h)ix= 3 0

\^=const

At (h1)2 « 1, Eqs. (28) and (29) can be simplified. On summing these equations we obtain:

(3O)

where t(r) = h\(r) -h2{r). Solution of Eq. (30) describes the variations in the thickness, t{r), of the interlayer between the droplets. The distance between the centers of the droplets can be obtained using the same methods as are used in the case of cylindrical droplets. Thus, the use of the isotherms of disjoining pressure of thin interlayers enables one to solve the problem of the equilibrium of droplets when they are in contact with one another, and to calculate the shape of the deformed droplets and of the interlayer. 1.3. Shape of a liquid interlayer between interacting droplets: critical radius As has been shown above, the shape of a liquid interlayer between interacting droplets depends on their size, interfacial tensions, and the shape of DP. The calculations presented below describe the shape of an interlayer between droplets under equilibrium conditions, until its possible rupture. In carrying out further calculations, we assume that the interlayer thickness, h(r), varies within the contact region but not very sharply, that is, approximation (h )2« 1 can be used. This enables one to use the isotherm of disjoining pressure of flat interlayers, TJh), in equations of equilibrium, as well as to simplify the expression for the local curvature of the interlayer surfaces. Let us consider the interactions of two identical spherical droplets (Fig. 3). In this case, the equation of the interlayer profile, h(r), can be derived by solving Eq. (30)

195

Deformation of Fluid Particles in the Contact Zone and Line Tension

Fig. 6. The disjoining pressure isotherm. n(h) as used in calculations.

Y

- /?"+— \+n(h)=2y / R=P

2{

(31)

r)

where h' = dh/dr ; h = d 2h/dr2; y is the interfacial tension, and R the radius of droplets. For carrying out quantitative calculations, we use below the simplified isotherm of disjoining pressure, IJ(h), in the following form:

K

' \a(to-h),

0
(32)

Such an isotherm (Fig. 6a) provides a possibility for obtaining an analytical solution of the problem, and, at the same time, it possesses the main properties of real isotherms (Fig. 6b), corresponding to the attraction of droplets at large separations, i.e., t0 < h < tsy, and to their repulsion at small distances, at h < toHere, the thickness tsf corresponds to the radius of action of surface forces. Beyond its limits, at h > ts/, the interaction forces vanish: 77= 0. Parameter a=(rii-ri2)/^/- determines the slope of the isotherm. Solution of Eq. (31) together with isotherm (32) has the following form:

196

V. Starov

h(r)=to-(2y/aR)+A-Io(z),

(33)

where z = r(2a/y)'/2, and Io is the Bessel's function of an imaginary variable. The constant A is determined from the boundary condition, h(rn) = tsf, where rn is the radius of the zone of deformation (Fig. 3), and z^-r^al' A=[tsf-t0+(2r/aR)\/I0{z0).

yf (34)

Substituting Eq. (34) into Eq. (33), we obtain an equation determining the profile of the interlayer between droplets:

Accordingly, the minimum distance between the surfaces of droplets is equal to:

In Eqs. (35) and (36), the value of r0 remains still to be determined. For this purpose let us use the condition (7):

0= ]n{h}2xr-dr=0 0

(37)

Substituting into Eq. (37) the equation of isotherm (32), and replacing in the latter h(r) by its expression from Eq. (5), we obtain the following expression: ? AaR(

°

r\sf0

2rY"0

Ift(z)

aR)]Q 1 ^ )

In the case whenzg>5 (sufficiently large drops) the integral in Eq. (38) is equal to TQ (// 2a) . In this case Eq. (38) results in the following expression for the radius of the contact zone:

r O =<2«r)" 2 [^ + i]

(39)

Deformation of Fluid Particles in the Contact Zone andLine Tension

197

Using Eq. (39) the relative extension of the contact zone, where the effect of the surface forces is pronounced, can be expressed as:

|Wto + i] R

\

2y

At 1/aR«\t s r-tQJ/2/,

,40)

aR\ or R»2y Ia(tss-tQ) the ratio rJR tends to the value

(tsf-to){al2y) , which is independent of the radius of the drop, R. This means the geometric similarity of all the large fluid droplets that are deformed by the contact interaction. For the droplets of small radius, R, one may use another approximation for Io(z), which is valid at z < 1: I0(z) = 1 + (z2/4). In this case, integration in Eq. (38) yields: 2_

2#^/-'oJ

r =

(4i)

° I-M/2;>(V-,0)

Let us compare the latter expression with the solution for solid spheres. Their equilibrium state (Fig. 7b) can also be characterized by the interaction region of radius rn, within which the surface forces exert their effect. The profile of the solid sphere can be represented as h=hQ+2R- \—^l-(r/R.)

. At r0 «R,

its

approximate form may be used: h = ho+{r1IR)

(41')

The minimum distance, h0, between the solid particles can be determined from the same condition (7) and the known Derjaguin's approximation [1]:

<&(ho)=x-R]n{hydh

(42)

Replacing the lower integration limit by 4/and using the model DP isotherm (32), after integration in Eq. (42) we obtain the following expression: h0 - 2ta -tx. On substituting this expression into Eq. (41) and taking into account that h(r0 )=?s/we get:

198

V. Starov

Fig. 7. The schematic representation of the contact interaction of the fluid (a, c) and the solid (b) particles.

r:=2R{tsf-t0)

(43)

The same value of r0 is also obtained by solving Eq. (41) for "small" fluid droplets. The latter means that very small emulsion droplets and gas bubbles practically behave themselves as solid particles: they practically do not deform in the contact zone. Let us call the critical size of fluid droplets, R*, such that at R« R* their interaction does not differ from that of solid spheres. Comparing Eqs. (41) and (43), note that these coincide under the condition that the second term in the denominator of Eq. (41) is much smaller than unity. This condition allows one to determine R * as R'=2r/{t^-to)-a.

(44)

The distance between the centers of droplets can be determined using Eq. (10) and simple geometrical considerations: B = tsl +2{R2-r02)"2

(45)

Accordingly, the contact angle, 6, can also be determined at the point of intersection between the non-deformed surface of a droplet and the r axis (Fig. 7a): cos<9 = BI2R = (tsfl2R)+ [l - (r0 /R)1]'1

(46)

Deformation of Fluid Particles in the Contact Zone andLine Tension

199

This expression holds only for relatively large droplets. As has been already pointed out above, small droplets (R < R*) are practically non-deformable, and their undisturbed, circular profile does not intersect the r axis. In the case of small droplets, like that of complete wetting [2], the contact angle is absent. Let us now numerically calculate the profiles of fluid droplets in the contact zone using Eq. (35), and compare them with two other known models: the model of solid non-deformable particles (Fig. 7b) and the model of a flat interlayer between similar droplets (Fig. 7c) (in the latter model the effect of a transition zone between the surrounding bulk medium and the flat portion of the interlayer is not taken into account). In the case of flat interlayers let us use the equations of equilibrium of thin flat films (see below): 2 • /(cos0 -1) = ]n(/j)-dh + 2P- hc

(47)

2/1,

where Of is the equilibrium contact angle determined at the point of intersection between the continuation of the non-deformed part of a sphere and the symmetry plane. In that case, the thickness of the flat interlayer, 2he, is determined using DP isotherm, UJi), as 77 = P, where P = 2ylR is the capillary pressure drop at the spherical interface. The contact area of droplets at he « R is equal lorrr* = nR2 • sin2 6 where r0 is the radius of the contact zone. Substituting Eq. (32) of the disjoining pressure isotherm into Eq. (47), we obtain the following expression: cos9 = \-(a/4y)\(tl

-tj

-{2y/aR)2\+(l/R)-[t0 -(2y/aR)]

(48)

Fig. 8 represents the calculated profiles of droplets in the contact zone for the model of solid particles (curves 3), the model of a flat interlayer (curves 2), and real fluid profiles (curves 1), that is, the profiles, while taking into account the deformation of the profile under the simultaneous effect of both capillary and surface forces within the contact zone. The calculations were carried out for the model DP isotherm, FI(h), (Eq. (32)) preset by the following parameters: y= 30 dyne/cm; ts/= 3 x Iff6 cm; t0 = 2 x Iff6 cm; 77; = 77(0) = 3 x 106 dyne/cm2, and I7min = 77(tsf) = -1.5 x 106 dyne/cm2, which gives a = 1.5 x 10u dyne/cm3. According to Eq. (47) the adopted values correspond to, the contact angle 6 = 9°. As has been shown by Aronson and Princen [10], the emulsion droplets, separated by a thin inter- layer of a dispersion medium, form finite contact angles with the bulk phase within the range of 0 to 90° The left-hand part of the plots in Fig. 8 relates to the spherical droplets having radius R = 10"3 cm; the right-hand part to R = 10"4 cm. As appears from the left-hand part me large-sized droplets deform considerably thus forming the

200

V. Starov

Fig. 8. The profiles. h(r/R), of the contact zone of identical spherical particles, R = lO'Jcm (to the left), .and the R= 10"4 cm (to the right), calculated while taking into account the transition zone (I), and in the approximation of a flat interlayer (2), and solid particles (3).

practically flat contact zone. Profiles I and 2 are close to each other, and the transition zone occupies rather a small region immediately near the contact perimeter. Now, deviations from the profile of solid particles are very large (curve 3). Thus, in the case of R » R* the conditions of the droplets equilibrium may be described with the framework of the theory of flat interlayers, i.e., on the basis of Eq. (47). A decrease in the size of droplets (the right-hand side of the graph in Fig. 8) makes the profiles of solid particles (curve 3) and of droplets (curve I) approach to one another. In the central part of the contact region, the flat interlayer regionreduces, while the differences from the profile of the flat interlayers (curve 2) increase. A further, decrease in the droplet radius causes a still greater approaching of the profiles of the fluid and the solid particles to one another, and disappearance of the flat portion of the interlayer. As has been shown above at R « R* one may use the known solutions for the interaction of solid spheres. In this connection, it is important to evaluate values of R *. This allows one to determine the regions of applicability of different solutions; namely, flat

Deformation of Fluid Particles in the Contact Zone and Line Tension

201

interlayers for R »R*; solid particles for R «R*; and, finally, the approach developed above for the intermediate values of R. As appears from Eq. (44), the critical radius R* depends on the parameters of the disjoining pressure isotherm (a, tsft and tn) and the interface tension. Under otherwise equal conditions, a decrease in the interface tension reduces the values of R*, thus limiting the region of the applicability of the theory of solid spheres. Let us evaluate R*, assuming y = 50 dyne/ cm, and (%-/«) = Iff6 cm. The values of parameter a will be varied, which, in accordance with Eq. (48), is equivalent to a change in the contact angle Of cos^=l-(a/4r)-(/t/-/2)2

(49)

The calculations show that the values of R* decrease as 0 increases. Thus, at small values of 0close to zero, R* = 10"3 cm; at 0 = 5-6°, R* = 10"4 cm; at 0 = 2030°, R* = 5 x Iff6 cm; and at 0 = 90°, R* = 5 x Iff7 cm. In this way, an increase in the contact angle, corresponding to the enhancing of the droplets interaction, causes a decrease in the critical radius R * Now, this means that in the case of a strong interparticle interaction, the approach of solid spheres becomes ever less applicable; yet in the case of weakly interacting droplets, that approach may be used even for relatively large-size droplets. The solutions obtained allow one to evaluate R* and choose corresponding equations for calculation of the equilibrium shape of the droplets within the contact zone. 2. LINE TENSION The presence of the transition zone between the drop/bubble and thin liquid interlayers can be described in terms of line tension r, which has been introduced by Gibbs [16]. In the case of surface tension the transition zone between the liquid and vapour is replaced by a plane of tension with excess surface energy y. In the same way the transition zone between the drop/bubble and the thin liquid interlayer may be replaced by a three-phase contact line with excess linear energy r. In contrast to surface tension y, the value of the line tension, r, may be both positive and negative. In the case of positive value it makes the wetting perimeter to contract but to expand in the case of the negative value of the line tension. When the line tension is introduced, an additional terms should be incorporated in the Neuman-Young equation [13]: my- (cos # - cos # , ) = + — I

U

dr

(50)

o)

where r0 is the radius of the wetting perimeter, m and plus or minus depends on the system geometry: m=l and "-" correspond to the drop on the solid substrate,

202

V. Starov

m=2 and "+" correspond to the two identical drops/bubbles in contact (Fig. 9); contact angle Of in the case of big cylindrical drops (see below). In the case of liquid drops on the flat solid substrate, the positive values of x cause an increase in the values of contact angles 6 , whereas the negative value results in their decrease. In the case of flat films in a contact with a concave meniscus, the influence of the line tension ris inversed, because "-" sign should be used now in Eq. (50). For water and aqueous electrolyte solutions the line tension values are of about 10"6-10"5 dyne [15]. Thus, the terms in the right-hand side of Eq. (50) becomes noticeable at r < 10~4cm. The values of the line tension, r, for drops on solid substrates has been calculated as a difference between the values of y-COSo, calculated in two different ways (i) neglecting the transition zone and (ii) taking it into account [15]. Since the line tension arises due to the existence of the transition zone, it is clear that this difference is just associated with the additional terms in the right hand site of Eq. (50). An expression for x was obtained in the case of a model isotherm of disjoining pressure [15] and the line has been estimated as 10~5 •*-10~6 dyn and negative. De Feijter and Free [17] have considered the transition zone between Newton black film and bulk liquid. According to their estimations the line tension value is also negative and has the same order of magnitude.

Fig. 9. Magnification of the upper part of the transition zone between the drop/bubble and thin liquid interlayer. I- real deformed profile of the drop/bubble, 2 - ideal spherical profile, when the influence of DP pressure has been ignored, 3 - thin liquid interlayer of thickness 2he.

Deformation of Fluid Particles in the Contact Zone and Line Tension

203

T. Kolarov and Z.M. Zorin [18] have measured the line tension value. They have used Sheludko's cell for measurements of properties of free liquid films. Aqueous solution 0.1% NaCl with SDS 0.05% concentration (CMC=0.2%) hasbeen used. They have calculated line tension for this system using the Neuman-Young Eq. (50). The value of line tension has been found -1.7- l(T6dyn. That is, negative and in the good agreement with the theoretical predictions [15]. D.Platikanov et al. [19] have carried out an experiment measurements of line tension dependency on salt concentration and present experimental evidence of line tension sign change. Let us consider two identical drops/bubbles in contact (Fig. 9) under equilibrium conditions. It is convenient to specify the reference state to write down an excess free energy of the system (Fig. 9). This reference state is introduced as "a flat equilibrium liquid film of the thickness 2h". From mathematical point of view it means addition/subtraction of a constant to/from the excess free energy. The latter constant does not influence the final equation, which describes the profile in the transition zone. However, as we see below, it influences substantially the definition of the line tension. The latter means that the choice of the reference state is very important and we use below the same choice of the reference state as in [15], which is the uniform flat equilibrium film of thickness 2he, where he is the half-thickness of the equilibrium film. Using that choice the excess free energy, F, of the system (curve 1 in Fig. 9) has the following form: R

.

GO

OO

2h

2he

F=lx\r 2y(Jl+h'2 -1)+ \Tl{h)dh- \n(h]dh+2P-(h-he) dr 0

(51)

where h(x) is the half-thickness of the liquid layer, he is a half-thickness of the flat equilibrium thin liquid film; P=2ylR is the excess pressure, u(h) is the disjoining pressure, h(R)=H is the position of the drop's end. The lower limit of integration correspond to the end of the transition zone there h=he. We can use infinity as the upper limit of integration instead of R, because at this stage we are not interested in the upper part of the drop/bubble. Under the equilibrium condition the system is at the minimum free energy state, that is two the following conditions should be satisfied: SF = 0

(52)

The latter condition results as above in the equation for determination of the liquid profile in the transition zone:

204

V. Starov

However, the condition (52) is not the only one required for the equilibrium. Let us remind the second condition. If the end of the transition zone can move along a curve
-\® + (
(54)

where cj> = A 2y{^j\ + h'2 -1) + )u(h)dhl_

)n(h)dh + 2P-(h-hc) '

. Substitution -

of the latter expression into condition (54) and taking into account that in the case under consideration (p(r)=he=const, hence,
2

\

h'2

2yr U\ + h' -\)—==

=0

(55)

Condition (55) can be satisfied only if h' -> 0,

(56)

at the end of the transition zone, which means a smooth transition from the transition zone to the flat thin film. Let us introduce an ideal profile of the liquid interlayer in the transition zone, 2hid, which is a spherical part up to the intersection with the equilibrium liquid interlayer of thickness 2he (2 in Fig. 9). The excess free energy of such ideal profile differs from the exact excess free energy given by Eq. (51) because the presence of the transition zone is ignored in the case of the ideal profile. The latter means that the line tension, r, should be introduced to compensate the difference:

F = 2n\r 2y{j\ + (h'J - 1 ) - )u{h)dh + IP• (hid -he)dr

+ Inr.r (57)

Under equilibrium conditions the following two conditions should be satisfied: 5^=0

(58)

Deformation of Fluid Particles in the Contact Zone and Line Tension

205

which gives an equation for the ideal liquid profile in the transition zone

f

L± , rdr

K

=p

r

2^

(59)

\ + \h.

v L

v

J

\ )

and the transversality condition which in the case of ideal liquid profile with excess free energy given by Eq. (57) is as follows:

-**-O^l dhid \r_r

+ ^ =0

(60)

dr

°

where cDrf = r 2y(^l + hut -1) - "\U{h)dh + IP • (hd - hj

. Substitution of the

2/1,

latter expression into condition (60) results in the following equation at r=r0 and h=he:

2r{j^hJ-1)- )n(h)dh--pL

-{*L+ L]=o.

After a rearrangement the latter condition becomes 2 r (cos<9-cose / )= - + - ^ 1 Vro vr0 J

(61)

where 2/ cos 0 =2y+ ]u (h)dh.

(62)

2/1,

Eq. (62) gives the expression for the contact angle of a big cylindrical drop (see below). Note that integration in the right hand site is over h from he to infinity, which does not depend on the profile (real or ideal profile) but only on the

206

V. Starov

integration limits. Eq. (61) coincides with Eq. (50) for the system under consideration (Fig. 9). Excess free energy given by Eq. (51) and (57) should be equal. The latter gives the following definition of the line tension, x: )r 2yUl + (h',d)2-l]-)n(h)dh

+ 2P-(h,d-he) dr + rox =

r

(63)

i 1

= JV 2y (VlT/7 -1) + ^U{h)dh- \ll(h)dh + 2P-(h-he) 0

2*

dr

2b,

The latter equation is an exact definition of the line tension, r in contrast to Eq. (61), where the value of line tension is unknown. In Eq. (63) the real liquid profile, h{r), is the solution of Eq. (53) and the ideal liquid profile, hic/(r), is the solution of Eq. (59). Dependency of the line tension on the radius r0 has been investigated in [15] in the case of a model DP isotherm. Below we focus on the absolute value of the line tension and a possible comparison with experimental data. For this purpose let us consider the line tension in the simplest possible case: contact of two identical cylindrical drops/bubbles. In this case the corresponding excess free energies (51) and (57) take the following form OC

~Z

I

F= | 2/(i\+h 0

00

00

-1)+ \U{h)dh- \U(h)dh+2P-(h-he) dx Ih 2he

(64)

and F = ]\2y{jl

+ (h'J-\)-

)n{h)dh + IP • (hld - ht)\dx + r ,

(65)

where now F is an excess free energy per unit length. From Eq. (64) we conclude rh

"

[i+ (h')'}

+n(2h)=p.

(66)

Deformation of Fluid Particles in the Contact Zone and Line Tension

207

Eq. (66) describes the whole range of the liquid profile including the lower bulk part of the drop/bubble, the thin flat liquid interlayer in front of it and the transition zone in between. The boundary conditions for Eq. (66) are h{R)=H,

h-+he, ti

h'{R)=v

(67)

x^-0

= 0,

(68)

x^O

h=\,

We can integrate Eq. (66) once using boundary condition (67), which yields:

r

,

v = w,

(69)

[i + w} where L(h) = P(H-h)-

^U(h)dh. Eq. (69) can be rewritten as: 2*

h=

' iWf~

y

<70)

The left hand side of Eq. (69) is always positive and less than y. That means the same should be true for the right hand site of Eq. (69):

0
(71)

where L(h) = y, if h = he and L(h) = 0, if h = H . Condition (68) results in:

p-(H-he)=r+)u(h)dh.

(72)

The capillary pressure can be expressed as before: P =

L R' (73)

208

V. Starov

where R is the radius of the curvature of the cylindrical drop. Simple geometrical considerations show: R = -^-r

(74)

cos 9 With the help of the latter condition we can conclude that:

/•cos 0 P

=

(75)

H Using Eqs. (72) and (75) we conclude:

1 + — • )n(h)dh cos

# =

" Y

J.

(76)


H <-——

= n{2ht)

{ii)

ti

If he/H«l the contact angle in Eq. (76) is refereed to as d^and coincide with that given by Eq. (62). In the case of partial wetting contact angle is in the following range 0 < 0 < — , o r O < COS 6 < 1. Using condition (71) the latter unequally can be rewritten as

oo

y-h \U{h)dh<-—^.

-y<

H

2he

Hence, the integral, \Tl(h)dh, should be negative in the case of partial wetting. 2/1,

In the case of the ideal profile we should use Eq. (65), which results in

r

i

, 2 f= P

2

1 + fc)

(78)

Deformation of Fluid Particles in the Contact Zone and Line Tension

209

The boundary conditions for Eq. (78) are hid{R) = H,

h'id{R) = «>

(79)

Eq.(78) can be integrated once using boundary conditions (79), which yields:

h id=

' i~P~~l

(80)

V '-'id

where Lld(hld) = P(H-h,d). Line tension r can be expressed using Eqs. (64) and (65) as:

r = J ly^X + h'1 -1) + ]n(h)dh - )ll{h)dh+ 2P-(h-hJ\ dxo|_

2/i

Ih,

J

- f 2 ^ 1 + fc)2 -\]-"\n{h)dh+2P\hd -h^dx Using Eqs. (70) and (80) we can rewrite the latter equation as

T =2)[yJl + h'2 -L(h)]dx-2)[r^

+ {h'J -4,(AjJfc

(81)

x,,

0

Using Eqs. (70) and (80) we can switch from integration over x to integration over thickness. The latter transformation of Eq. (81) gives:

x = 2{(Vy2-Z2(/z) - V Y 2 ~ 4 W ) dh

(82)

K

The similar equation was obtained earlier [20]. Eq. (82) can be rewritten as \2

/„ m 2P(H

t =- 1

,

- h) \u{h)dh -

\l\(h)dh 2

. ^"

^ dh

(83)

In the case of h«H inside the whole transition zone expressions for L(h) and Lid(h) can be rewritten as

210

V. Starov

/ \ L(h) = r\cos0f-e(h)\

L(h)

1 °° e(h) = - \U(h)dh

= ycosd Using the latter expression Eq. (83) can be rewritten as 2 2 cos 6,e{h)-E y{h) ' y ' ' dh 2 2cosdfs(h)-s (h)

2y " r =- = 2 — { sin#, K •

i

i+

+

(84)

1— 2

"y

sin 6f

In the case of small contact angles s ( h ) « l and the latter equation takes the following form: r =- ^ - f tan 0f I

,

£{k)

dh

(85)

2cos$fe(h) l+ + 2

f

sin ^

It is possible to show that 0.5 < 1+

, <1 in the case under I , 2cosd,E{h) "\( sin 2 ^

consideration. Let the mean value of the latter expression be co, where 0.5
J

J()u(h)dh\ih 0f

ih\ih

(86)

)

Eq. (62) can be rewritten as c o s ^ = 1 + s(hj, or sintf. * 0 = 2 ^ - e(h ) = 2 I- - ]n(h)dh V Ylh-

(87)

Combination of the latter expression and Eq. (86) results in

r=

2C

°

] (]u(h)dh\lh =

2C0 Y

^

J f ]n{h)dh\h

(88)

Deformation of Fluid Particles in the Contact Zone and Line Tension

211

We can compare the experimental data by D. Platikanov, M. Nedyalkov [19] to the theory predictions according to Eq. (88) (at a>=l). We used below only dispersion and electrostatic components of the disjoining pressure. The following expressions for different component of disjoining pressure are used: for dispersion component: n,=--^,

(89)

where A ~ 10~udyn • cm is the Hamaker constant and for electrostatic component n , ; = 6 4 -c-R -T -tan/2 2 ((^/4)-exp(-K- • h),

(90)

where c, R, T.F, \u = ——, K-J are concentration of the electrolyte, the RT V ERT universal gas constant, temperature in °K, Faraday number, dimensionless zeta potential of the film surfaces, and the inverse Debye length, correspondingly; 8 is the dielectrics constant of water. Hence, the total DP is:

Fig. 10. DP according to Eq. (90) with cut-off thickness, ' * .

212

V. Starov

U(h) = ~

+ 64-c-R-T-

Xwh2{y7lA) • exp(-/c • h)

(91)

h Unfortunately the DP in the form given by Eq. (91) does not allow any equilibrium liquid films at low thickness (a - films). To overcome this problem we introduce a cut-off thickness, K (Fig. 10). According to this choice the equilibrium thickness 2he does not depend on the pressure inside the drops/bubbles and is always equal to K . D. Platikanov and et al. [19] have determined both contact angle and line tension on the NaCl concentration in the range of concentrations 0.2-0.45 mol/1. These two dependencies are used below. 2.1. The comparison with the experimental data [19] and discussion Details of experimental measurement and system under consideration are given in [19]. Line tension of free liquid film between two bubbles was investigated in the range of NCI concentration 0.2-0.45 mol/1. The film and bubble surfaces were stabilised by surfactants [19]. Zeta-potential of the film surface was \\i= 17 mV or (//=0.68 according to [19].

Fig. 11. Calculated dependency of line tension on electrolyte concentration.

Deformation of Fluid Particles in the Contact Zone and Line Tension

213

The cut-off thickness, tt, was used as a fitting parameter. We used experimental values of contact angle from [19] on salt concentration to determine the cut-off thickness tt according to Eq. (87). A reasonable agreement between experimental dependency of the contact angle on the salt concentration and the calculated according to Eq. (87) has been attained. The fitted dependency of the contact angle on the salt concentration was much weaker than the original experimental data [19]. However, we tried to compare our calculation of line tension according to Eq. (88) and the corresponding experimental data of line tension from [19] using already calculated cut-off thickness tt. Calculated dependency of line tension on the electrolyte concentration is shown in Fig. 11. The conclusions are as follows: • line tension dependency on the salt concentration is in a qualitative agreement with experimental dependency in [19], that is line tension goes from positive to negative values at the increase of the electrolyte concentration; • the absolute values of the calculated line tension were found considerably different from the corresponding experimental values, the calculated electrolyte concentration (0.022 mol/1) at which line tension switches from positive to negative values does not match the experimental value (0.36 mol/1); • the calculated line tension remain almost constant in the range of electrolyte concentrations used in [19]; • line tension decreases much faster with electrolyte concentration (Fig. 11) than experimental values [19]. The discrepancy between the measured and calculated line tension dependencies can be caused by one or both of the following reasons: • only dispersion and electrostatic components of DP pressure have been used to compare with the experimental data. It looks like these two components are not enough to describe adequately behaviour of thin liquid films and the transition region in the system under consideration. Influence of both structural component (caused by the water dipoles orientation in a vicinity of free film surfaces) and steric (caused by the direct interaction of head of the surfactant molecules on the film surfaces) can not be ignored. The theory of these components of DP is to be developed; • if the experimental system used in [19] was really an equilibrium one. According to the definition of line tension (83) it is determined by the equilibrium liquid profile in the transition region from the thin flat liquid interlayer to the bulk surface of bubbles. It is difficult to prove that under experimental conditions used in [19] the liquid profile in the transition region was equilibrium one in spite of the efforts undertaken to reach the equilibrium [19]. Note that in the case of partial wetting, which is under consideration, the contact angle hysteresis [2] is unavoidable. The latter phenomenon can completely mask non-equilibrium state of the system. Unfortunately except for [19], where considerable efforts have been undertaken to reach the equilibrium

214

V. Starov

state, and other few publications, the possible deviations from the equilibrium state of the system under consideration usually even does not discussed. We would like to emphasise again: if the liquid profile in the transition region from thin liquid interlayer to the bulk drop/bubble interface is not at the equilibrium then an arbitrary values of line tensions can be measured, which have nothing to do with the equilibrium value of line tension. 3. ACKNOWLEDGEMENTS This research has been supported by the Royal Society, UK.

REFERENCES [I] Derjaguin, B.V., Churaev, N.V., and Muller, V.M. "Surface forces", Plenum Press, New York, 1987. [2] Starov, V.M. Adv. Colloid Interface Sci., 39, 147 (1992). [3] Churaev, N.V., Sobolev, V.D. Adv. Colloid Interface Sci., 61, 1 (1995). [4] Derjaguin, B. Y., Colloid J.(USSR Academy of Sciences), 69, 155 (1934). [5] Derjaguin, B. Y., Muller, V. M., and Toporov, Yu. P., Colloid J.(USSR Academy of Sciences), 37, 455, (1975); 37, 1066 (1975). [6] Muller, V. M., and Yushchenko, Y. S., Colloid J.(USSR Academy of Sciences), 42, 500 (1980). [7] Princen, H. M., J. Colloid Sci, 18, 1978 (1963). [8] Princen, H. M., and Mason, S. G., J: Colloid Interface Sci. 20, 156 (1965); 20, 246 (1965). [9] Princen, H. M., J: Colloid Interface Sci. 71, 55 (1979). [10] Aronson, M. P., and Princen, H. M., Nature (London) 286, 370 (1980). II1] Churaev, N.V., and Starov, V.M. J. Colloid Interface Sci., 103 (2), 301 (1985). [12] De Feijter, J. A., and Vrij, A., J. Eleclroanal. Chern. Interface Eleclrochem, 37, 9 (1972). [13] Starov, V. M., and Churaev, N. V., Colloid J. (USSR Academy of Sciences), 42, 703 (1980). [14] Churaev, N. Y., Starov, Y. M., and Derjaguin, B. Y., J. Colloid Interface Sci, 89, 16 (1982). [15] Derjaguin, B. Y., and Churaev, N. Y., J. Colloid Interface Sci, 38, 438 (1976); 66, 389 (1978). [16] Rowlinson, J.F., and Widom, B. "Molecular Theory of Capillarity", Clarendon, Oxford, 1984. [! 7] de Feijter, J.A., and Vrij, A. J. Electroanal. Chem., 37, 9 (1972). [18] Kolarov, T., and Zorin, Z.M. ColloidPolym. Sci., 257, 1292 (1979). [19] Platikanov, D, Nedyalkov, M., and Nasteva, V. J. Colloid Interface Sci., 75, 620 (1980). [20] Dobbs, H.T., and Indekeu, J.O. PhysicaA, 201, 457 (1993).

Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 6

Hydrodynamic interactions and stability of emulsion films E. Mileva" and B. Radoevb "Department of Colloid and Interface Science, Bulgarian Academy of Sciences "Acad. G. Bonchev" Str., bl. 11, 1113 Sofia, Bulgaria E-mail: [email protected] b

Department of Physical Chemistry, Sofia University 1 "J. Bourchier" Str., 1126 Sofia, Bulgaria E-mail: [email protected]

1. INTRODUCTION The hydrodynamic interactions are among the most important factors that determine the properties and the stability of emulsion systems. Diverse aspects of this topic might be found in numerous publications (see e.g. Refs. 1-6). One specific feature that distinguishes emulsions from other complex disperse systems, like foams and suspensions, is the coupling of fluid motion in the contiguous phases. This interdependence of the flows is closely related to the mobility of the fluid/fluid interfaces and to the presence of surfactant species. The interfacial mobility is the most prominent characteristics: due to viscosity of the adjoining fluids, the flow in one liquid phase is transferred through the boundary into the neighboring phase and the fluid motions mutually influence one another. Consequently, the mathematical model of an emulsion system inherently comprises the coupling of the hydrodynamic equations that describe the flow fields in the contiguous phases, and they all have to be solved simultaneously. The kinetic stability of emulsions is particularly related to the hydrodynamic interactions at close approach of two fluid particles. If the gap between them becomes of the order of their dimensions or smaller, the fluid motion is mostly localized in the narrowest parts of the system. An anizodimetric region might be distinguished where the major dissipation of energy takes place [5,7-10]. This zone generates the leading term of the drag

216

E. Mileva and B. Radoev

force between the two droplets and is termed a thin liquid layer or an emulsion film. Therefore, the close distance hydrodynamics of two axially approaching droplets presents the basic flow patterns that could outline the governing effects correlating flow and stability in emulsion systems. Within this model, two aspects are of particular interest: the drainage peculiarities of the emulsion film, and the flow pattern inside the droplets as caused by the squeezing outflow in the film. In most of the studies it turns convenient to assume two major regimes of interacting droplets: constant approach velocity (e.g. Refs.[11,12]) and constant driving force (e.g. Refs. [13-15]). The specific emphasis in this chapter is related to the second case. The emulsion film investigations came out as a further development of the close-distance hydrodynamics of foam films. The latter was vigorously investigated in the seventies and the eighties of the last century [16-22] and is very much allied to the film microinterferometric technique, which is one of the most popular instrumentations in thin liquid layer studies [18,23,24]. The leading idea is that at small gap widths between the particles, the fluid motion in the film is retarded and the governing flow equations are, as a rule, of creepingflow type. The anizodiametry of the gap often allows additional simplifications, like a lubrication flow mode, named sometimes Reynolds law, or similar [1621]. Insofar as the flow inside the droplets is concerned, various simplifications of the general Navier-Stokes equations have been proposed ranging from lubrication-type relations [12,25], through full creeping-flow model [4,7,11,15, 26-30] and, to hydrodynamic boundary layer presentation [31]. Another very important factor for the behavior of the emulsion systems is the presence of surfactants. The most popular observation is the so-called Bancroft rule: "the phase in which the stabilizing agent is more soluble will be the continuous phase" [see e.g. Ref.32]. There are different interpretations of this rule, related either to energy considerations [32], or to some hydrodynamic reasoning [3,31,33-37]. In the latter case, the presence of surfactant is always interpreted in view of its impact on the interfacial mobility of the fluid interfaces. This point of view stems from the physicochemical hydrodynamics of Levich [38] and was developed for the specific interactions in emulsion films by Lee and Hodgson [3]. The key idea is that the surfactant mass transfer additionally couples with the hydrodynamics in the emulsion systems. The geometrical anizodiametry of the emulsion films modifies further on both the flow and the mass transfer regime [31,33-37]. One substantial property of the emulsion systems is their coalescence stability. The stability of the thin liquid layer, most generally presented by the so-called 'thickness of rupture of the film', plays a decisive role. The theory of film rupture was first formulated for the case of foam systems [17,39]. With a slight modification, it is also applicable for emulsion films [31]. The studies

Hydrodynamic Interactions and Stability of Emulsion Films

217

show that the true stability of the emulsion films is related to their thermodynamic properties (surface tension, disjoining pressure). As for the intrinsic coupling of hydrodynamics and mass transfer of surfactants, it concerns the so-called kinetic stability, related to the time evolution of the draining emulsion film. It may be expressed by the interfacial mobility of thin liquid layer alone (see Section 4). The basic result is that while for some characteristics (such as lifetime of films) the interfacial mobility is of utmost importance, the thickness of rupture is actually independent on mobility factors. The aim of the present chapter is to give a survey on some particular aspects of the influence of interfacial mobility on the hydrodynamic interactions in emulsion films, as well as to follow the effect of the mass transfer of surfactant species on these interactions. A special attention is paid on the possibility to develop a systematic scaling procedure for the analysis of the general dynamic equations which model the fluid motion in the contiguous liquid phases. The procedure offers universal approach to estimate the relative importance of the possible coupling cases of hydrodynamics and mass transfer in emulsion systems. It permits the proper outline of the key asymptotic cases that are most important for experimental practice and could be solved explicitly. In Section 2 is presented an overview on the surfactant-free case of closedistance hydrodynamics of two fluid particles. The basic steps in the scaling procedure are traced. Two important asymptotic cases: (i) a solid sphere and tangentially mobile flat interface, and (ii) a solid sphere and a droplet, are analyzed. In Section 3, the scaling procedure is modified to account for the effect of the surfactant fluxes on the flow coupling in the adjacent phases. In Section 4 are summarized the basic elements of the theory of thickness of rupture. The influence of the interface mobility and the presence of surfactant are discussed, as well as the impact of film radius on film stability. A special attention is paid on the random character of the perturbations and on its consequences for the behavior of the emulsion systems. 2. EMULSION HYDRODYNAMICS The major element in modeling of hydrodynamic interactions in emulsion systems is the study of the axial approach of two emulsion droplets in an incompressible viscous medium (see Fig. 1). In this case a mutual deformation of their hydrodynamic fields occurs, as compared to the motion of a single droplet in otherwise undisturbed infinite fluid. This deformation results in additional forces acting on the particles. The two-droplet model presents the 'primitive model of emulsion hydrodynamics' and gives the basic relations that govern the peculiarities of the fluid motion and the coupling of flows in the adjacent phases.

218

E. Mileva and B. Radoev

Let us start with the surfactant-free case. The flows exterior and interior the fluid particles obey the steady Navier-Stokes equations and the continuity equation [40]: p'(v' • v)v' = -Vp' + n ' V V V-v'=0

(2.1a) (2.1b)

The following notations are used: p', v' - for the pressure and the velocity fields; u ' - f o r the viscosity and p ' - for the density of the fluid; V,V-,V~ are the gradient, divergence and the Laplacian operators. From now on all quantities referring to the droplet will be supplied with superscript (rf), and those concerning the film (continuous phase) - with (/). The basic boundary conditions that represent the coupling of the contiguous phases are [4]: • the velocity vectors outside and inside the droplets are continuous on the interface yf = \d

(2.2a)

• the tangential component of the normal stress vectors of the fluids interior and exterior the droplets are continuous through the interface:

K=K

(2-2b)

with n,\ being normal and tangential coordinates in a local coordinate system, bound to the droplet's surface. The normal components of the normal stress vectors have discontinuities but for simplicity, we shall regard the fluid particles as spherical and this boundary condition won't be necessary. 2.1. Scaling concepts As already stated, if the separation between the liquid particles becomes of the order, or thinner than their dimensions, the flow is mostly localized in this gap anizodimetric region, which generates the leading term of the drag force for two droplets and is called a thin liquid layer (an emulsion film). Due to the inherent interfacial mobility of the fluid particles, the flow outside causes a fluid motion inside them. At small gap width, the 'degeneration' of the flow pattern in the thin layer leads to the formation of a respective 'effective zone of interaction' inside the droplets, occupying regions adjacent to the thin layer outside [7-9]. With the help of this basic scaling concept, and by means of dimensional and similarity theories [41], the general

Hydrodynamic Interactions and Stability of Emulsion Films

219

hydrodynamic equations are rendered dimensionless and the range of several physically reasonable simplifications in the statement of the interaction problem are outlined. Based on the flow equations and the boundary conditions (2.1-2), the idea about the existence of this 'effective zone of interaction' is specified by introducing the following scaling parameters (Fig. 1) [8-10]: • Hf,Rf - characteristic dimensions of the liquid film, s = HJ /RJ « 1; • Hd,Rd ~ Rf - characteristic dimensions of the effective zone inside the droplets, adjacent to the thin liquid layer; • Uf - characteristic radial velocity of the film outflow; • Ud - characteristic radial velocity of the interface; Besides, due to (2.1b) the relative approach velocity of the droplets is f V =-dl/dt, and Vf ~zUf

(2.3)

Thus V1 is the characteristic drainage velocity of the film fluid motion.

Fig. 1. Two emulsion droplets of radii R at small gap widths: / - distance of closest approach of the droplet surfaces; hd(r,t) - generant curve of the droplets; (r,vf,z) - cylindrical coordinate system.

220

E. Mileva and B. Radoev

Insofar as the pressure should balance the inertia and the viscous terms in the Navier-Stokes equations, the scaling of this quantity in the film and droplet phase is [9,10]:

,/.^[,+J1,/f*l]\^r*Vl f

f

R[

f

{H )

(2.4)

f

U {H )

pd = VdK l + ReJ + {^j) f

P

(2.5)

d

R

[H ) J

AU Uf-U" , Here — - = — , and f f U U

HfRf

\if

\xd

are the Reynolds numbers for the film and the droplet phase. These scaling parameters are used for rendering dimensionless the respective quantities in the differential equations and the boundary conditions (Eqs. (2.1-2)). In a cylindrical coordinate system, the governing flow equations acquire the form of [9]: ReJ vrJ—— + —fT v J — — = dr U * dz J 1 dpf\ - ~~ J

2

n Jf

8 +Re

L

d?

AC/1 d 1 d ( , \J AUd2vf + — - + s2 [rv + — r—

&

J

2 dpf\ 2 D , AC/1 41 d _dv? 25 v/ J - ^ — 8 +Re + —j- + s r—— + s fdz L Uf \ r dr dr dz2 for the emulsion film, and

(2.7a)

(2.7b)

221

Hydrodynamic Interactions and Stability of Emulsion Films

Red\vd^

dr

+

v

d

^ dz

z

=

dp'l, nc (Rd)2] -— l + Re+\—dr [Hd)

d 1 d id\ (Rd)2d2v* + [rvf + — T \ dffdrK ' {Hd j dz2

J

{Hd) r

z

' dr

d dp" D d (R ) -J— 1 + Re +\—T dz [Hd)

,1

r

-

dz (2-8b)

,

(Hd) 1 d _dvd d2vd + \—r\ r-^ + =\Rd ) r dr dr dz2

for the droplet phase. All quantities captioned by a short line are dimensionless. The particular goal here is to track the flow coupling in the adjacent phases. The basic result is that this coupling is incorporated in an interface mobility scaling parameter. As is seen from Eqs. (2.7-8), this mobility may be expressed via the already introduced dynamic and geometrical scaling parameters, namely [8-10] Ud=-^ 1 + fn

with

^ = ^ Uf l + fh

(2-9)

where fh is a hydrodynamic factor related to the viscosity in the film and the droplet, the geometric scaling parameters of the layer and of the 'effective zone of interaction' interior the droplets, and it reflects the coupling of the flows in the contiguous phases. In the specific case of tangentially mobile but undeformable spherical droplets [hd(r)=l + R-R^jl acquires the form of [9,10]: f,=^ Jh

H

+ E2\ l + ^+u + uH\ \ H * * )

with

E = ^T V

and

-(r/R)2

], this factor

H=^— HJ

(2.10a)

For the simple case of creeping flow inside the droplets (see further (2.16)) fh specifies into a characteristic complex, involving only the geometrical dimensions of emulsion film and the ratio of the viscosities in both phases. fh=ZV

(2.10b)

222

E. Mileva and B. Radoev

Instead of fh, it is often very convenient to work with the so-called mobility of the fluid interface (see also Section 4): Mo = —

(2.10a')

Jh

Then the scaling of the interface velocity becomes:

u" = v^-

with

^L = _L_ UJ

l + Mo So, Ud =0, AU/Uf =l,ifMo

(2 . 90

1 + Mo

= 0;andUd =Uf, AU/Uf = 0, if Mo ->oo.

2.2. Thin liquid layer The mutual approach of the droplets in a regime of a constant driving force results in an overall slowdown of the flow motion in the emulsion layer and therefore, to Ref < 1 in the set (2.7). Hence, the fluid motion in the thin layer is adequately modeled by the full Stokes equations [40], which in dimensionless form are expressed as [9]: dpf\ 2 At/1 AU d2vf 2 d 1 d (__r\ J —~ £ + — r = £ \rv;)+—r T df L UJ \ dffdr' Uf dz2 f 2 dp A t / 1 e 4 1 d _dvf 2 2d v! J \ r - — 8 + —Jr = ~^- + s fdz I U \ f df dr dz2

..... (2.11a) /O11U,

(2.11b)

The structure of this set of equations allows the introduction of a general criterion for the appraisal of the lubrication-theory approximation as a model for the emulsion film outflow, namely [9,35-37]: ^ >

E

'

(2.12)

Provided the latter relation is valid, (2.11) might be reduced to the well-known lubrication-theory equations:

SLJ-!iL . ^-o or

dz

dz

(2.H)

Hydrodynamic Interactions and Stability of Emulsion Films

223

The set (2.13) has been traditionally used to model the flow in thin liquid layers within the general presumption of their geometric anizodiametry and of decreased tangential mobility of the film surfaces [3,11-12,16-22,27-28]. Note that if (2.12) is valid, the lubrication model is always adequate, no matter what actually causes the tendency for effective immobilization of the fluid interfaces. The reason might be either the higher viscosity of the droplets compared to that of the continuous phase, or the presence of surfactants (e.g. termed 'controlled slip' in Ref.[3]), or the combined effect of both (see also Section 3). If Eqs. (2.9) and (2.10b) are inserted in Eq. (2.12) the criterion for the validity of the lubrication model becomes fh >e2 [8,9]. Two main subcases are of particular interest [8]: (i)Hd[Red <1J~ RJ (see also Eq.(2.16)) - a creeping flow in the droplet, and the expression (2.12) acquires the form of pT > s , or (ii) Hd(Red >l)~RJ

/^fRe7

(see Eq.(2.16)) - a hydrodynamic boundary layer

interior the fluid particles, and (2.12) is reduced to [x>z/^Red . These relations show that the larger the viscosity ratios and the thinner the layers, the more adequate are the lubrication theory equations (2.13) as a model of the flow pattern in the emulsion film. It is also obvious from case (ii) that these equations might be used even if hydrodynamic boundary layers are formed in the contiguous phases, as is the case e.g. in Ref.[31]. What is more, higher Reynolds numbers in the droplets lead to a relative widening of the viscosity-ratio values for which Eqs. (2.13) are asymptotically correct. The lubrication theory cannot represent correctly the fluid motion in the layer only when \AU/s2Uf )< 1. The appropriate asymptotics then is presented by the following set of equations [36]:

df

dz2

d f r d r '

^L = ^lL dz

dz

dpf

did

df

,

f

Uu/e'U')*!

(2.14a)

( A £// E '£/')< 7

(2.14b)

\

df f df

^Lj^lL dz

for

dz

for

224

E. Mileva and B. Radoev

2.3. Flow inside the droplets Generally the scaling of the fluid flow inside the droplet in the direction normal to its surface might be presented as [9]: H d ~ . R" ~ ,Rf ; d 4l + Re 4l + Red Hd(Red
and

~Rf

(2.15)

or

Hd (Red > l)~-^=

(2.16)

Upon substitution of Eq. (2.15) in (2.8), the dimensionless form of the Navier-Stokes equations interior the droplets acquire the form of [9]:

ffi L d? J -®L(l + Re<)+°-L°-(rv')+(l+ Re'pl-

(2.17a)

Re^^^^L5F

L dz

V

5z

^ ;

"J

(2.17b)

r 5r

5r

v

;

dz2

This system reduces to the full Stokes equations for the case of Red < 1 [7-9]:

^1 = A Z A ( ^ ) + ^ df f dfv

df

®L = L°-F*L dz

r dr

dr

r

'

(2.18a)

dz2

+ ?*L

(2.18b)

dz

and to the hydrodynamic boundary equations, for Red > 1 [9]: ^

+

dr ^ dz

=0

v - ^ =- ^ dz dr

+

^ dz

(2.19a) (2.19b)

Hydrodynamic Interactions and Stability of Emulsion Films

225

It is evident from the dimensionless Eqs. (2.17), that the anizodiametry of the flow in the thin layer does not directly affect the order of magnitude of the various terms of the hydrodynamic equations inside the droplets. It is the Reynolds number alone, which determines the flow pattern inside the droplet phase. This is true for any interface mobility insofar, as the source of droplet's motion is exterior the fluid particles, in the thin gap between them. 2.4. Important asymptotic models The scaling analysis shows that for a broad range of cases the fluid flow in the thin layer might be modeled by the lubrication theory equations (2.13). Therefore, the essence of the effect of film drainage on the overall coupling of flows in adjacent phases might be adequately traced by analyzing simpler models, with one of the particles being just a solid sphere. Here are presented the major steps in constructing an asymptotic model for the flow interior the droplet, when it is at close proximity of another particle (solid sphere) and a thin liquid layer is formed between them [7,29-30]. The onset of this thin layer might be regarded as a restricted 'stirring' of the droplet's surface [7,8]. Insofar as at small gap widths the perturbation of the interface covers only a small portion of its surface (~ Rf), the droplet phase does not set any other geometric constraint to the 'natural propagation' of the flow pattern inside it. The realm of this perturbation is insignificant as compared to the dimensions of the droplet itself [Rf «R). Hence, the motion inside the fluid particle might be asymptotically modeled as a flow in a liquid semi-infinite space [7,8,29]. So, the basic characteristics of the flow coupling might be adequately described by the results of the first simple model: a solid sphere moving towards a flat tangentially mobile interface bordering a semi-infinite space (see Fig. 2a) [7,29]. In order to account for the effects of 'opening of the gap', another limiting case is also examined: two spheres of equal radii, one solid and the other liquid, approach along their common axis at close separations [30] (see Fig. 2b). The method of solution of these two models is based on the potential theory approach. The classical potential theory is a general methodology for finding solution of boundary-layer problems involving differential equations of elliptic type [42,43]. Odquist [44] and Ladyzhenskay [45] have specified this approach for the case of steady creeping flows, by constructing the so-called hydrodynamic potentials. The rapid development of the computer techniques has revived the use of this method [26-28,46]. Following this procedure, the flow model defined by Eqs. (2.1-2), is reformulated into velocity and pressure fields, given by integrals, whose densities are velocities and shear stresses on the mobile thin layer interfaces [45]. Hence, the problem of flow coupling is reduced to the solution of integral equations involving these densities.

226

E. Mileva and B. Radoev

Fig. 2. Asymptotic models: (a) a solid sphere and a flat tangentially mobile interface, (b) a solid sphere and a droplet.

The particular case of the motion of a particle towards a fluid interface, and of two particles along their common axis, additionally simplifies the mathematical formulation of the hydrodynamic interaction problem. Because of the axial symmetry of the model, the Stokes equations may be expressed as a biharmonic equation for the stream function [7,29-30]. In particular, for undeformable spheres and plane surfaces, the general solution might simply be presented via the usual harmonic potentials [7,28-30]. In this case, the issue of hydrodynamic interactions at small gap widths is reduced to solving integral equations for the tangential mobility on the fluid interfaces. 2.4.1. Sphere and flat interface (Fig. 2 a) The outflow of the thin liquid layer is modeled by the lubrication theory equations (2.13), while the creeping motion inside the semi-infinite liquid phase is presented as a biharmonic equation for the stream function *¥ d(r,y,z) [7,29,36,37,40]:

V 2 V 2 f—cosy} = 0 { r

with

V 2 = - ^ - + - — + ^- + J-^dr2

)

rdr

dz2

(2.20)

r2 5cp2

The velocity components are related to the stream function via the relations [40]:

*=l?f. , t = -L*f. r oz

r or

(2.21)

Hydrodynamic Interactions and Stability of Emulsion Films

227

The general solution of differential equation (2.20) might be written as [7,47]: d

XL/

cosip = zud

Aud = 0

where

(2.22)

r The specification of the respective boundary conditions is [29,36]:

(a)

vf[r,h'(r)]=0

(e)

P,{ (r ,0) = P,'(r ,0)

(b)

v{[r,h'{r)]=V/ = - —

(f)

Vd(r,6)=0

(c)

v{(r,0)=U(r)

(g)

p'(co,z) = 0

(d)

v{(r,0)=0

(2.23)

where hs(r) = -l - Rfl - ~Jl - (r/R)2 ] is the generant curve of the solid sphere. Most often in thin film studies, this curve is reduced to a parabolic profile [29,30]: hs(r) = -l-r2/2R . Equation (2.20) and the conditions (2.23) result in the following presentation of the harmonic functions in (2.22) [7,29,36]:

«'(r,q,,z) = -M"f T 2n

^ 2

^

^

V/2

2

(2.24)

2

o o [r + r - 2rr, cosfa -
as a double layer potential or, equivalently u"(r Q

A -

l

Tf\U°(r>)-U(n)] \™ Sol h V')

J [r

cosyf.dr.dy, > ~ > COS(V " ( P/)+ z J

+r

2rr

as a single layer potential. Here the notation is used:

U°{r) = -^f-

(2.26)

4h (r) The last expression is the radial velocity on a liquid/gas interface (jl = 0,z = 6).

228

E. Mileva and B. Radoev

The coupling problem as formulated by Eqs.(2.13) and (2.20), together with boundary conditions (2.23), result in the basic expression for the tangential velocity on the flat interface U{r) [29]:

£/(,)_ ir^feir^ii

=uo{r)

(2.27)

2 cos ip{_ dz ) z = n

According to the potential theory [42], the latter equation might be written in two equivalent forms [29]: —i \

hHrYc'cK 2K

\u{r,)\cos(n .r.dr.dy,

"2V

U(r) + eii^n\l

V n

V"

o o [r + ff - 2rr,

_.. .

'Z'2=U0{f)

(2.28)

cosy,)

and

U(r)=~—W^J^X

C0S9 F d d(?

' ' "' '

v/2

(2.29)

Here the effective parameters, used in rendering the respective quantities dimensionless, are as follows: Rf =^2Rl;z - •yJ2l/R;UJ =\VJ/s). Once either of the two integral equations (2.28-29) has been obtained, the problem of hydrodynamic interaction is solved, at least in principle. Any flow property might then be found by substituting the solution for U(r) in the respective expressions. Below the steps in calculating the drag force are traced. The integral equations (2.28-29) are of Fredholm type [43]. Most often similar equations are solved numerically [26-28,46]. The scaling concepts that have been introduced in the subsections 2.2-2.3 however, allow the application of an iterative procedure for finding asymptotic solutions in several important cases. Thus, at low and high viscosities in the contiguous phase, the results are presented as: (1) sfT < 1 (low viscosity in the semi-infinite space z>0). Here more convenient is the form (2.28) and the solution is developed as a power series in the hydrodynamic factor fh (as defined by (2.10b)):

t7(r)=XW^)

(2.30)

Hydrodynamic Interactions and Stability of Emulsion Films

229

and more explicitly:

C7(r)=t7»(r)-Bp:^y/(r)+^r)

(2-3D

with J/(r) shown in the Appendix (Eq.(A.l)). In the above expression, the notation O{°) is used, showing the order of magnitude of the accuracy of the asymptotic solution. (2) ejl > 1 (high viscosity in the semi-infinite space z>0). The integral equation (2.29) turns to be the more convenient initial form and the solution is given in power series of the mobility factor Mo (as defined by (2.10a')):

U(r)=YWuXr)

U(r)=-^J,(r)+ci

-4

or

U(F)=YJ(MO)UI(F)

(2.32)

or

U(r) = MoJ3(r)+a([Mo]2) (2.33)

[1^] J

en

where the explicit form of J3(r) can be found in the Appendix (Eq.(A.2)) For creeping flows with axial symmetry, the resultant action of the shear stress on a moving particle is represented by one integral property - the drag force, acting parallel to the axis of symmetry [40]. In the present model (Fig. 2a) the expression becomes [29,36]: °°J~

yfr

1

2

<234)

*--«*'fcn+wm$* The insertion of Eqs.(2.31) and (2.33) in (2.34) results in [36,37]: Fplane=-2it\ifVfIj-{l

+ 3.57zvi)

for

efi < /

(2.35)

The factor outside the brackets is the drag force for a solid sphere, approaching a flat gas/liquid interface. This is the minimal drag force value of the model system on Fig. 2a. The second term inside the brackets concerns the influence of tangential mobility of the interface. Eq.(2.35) shows that in this particular case (low viscosity in the adjacent phase), the effect of nonzero \xd leads to a relative increase in the minimal value of the drag force.

230

E. Mileva and B. Radoev

As for the case of s|d > 1, one has [36,37]:

Fplam = 6n»fVf ~{l-0.39-L)

or Fplane = 6n^Vf ^-(l-0.39Mo)

(2.36)

The factor outside the brackets is the drag force for the solid sphere, approaching a flat solid interface. This is the maximal drag-force value of the model system on Fig. 2a. The second term inside the brackets again concerns the influence of the tangential mobility of the fluid interface. It accounts for the decrease in the maximal drag force when the solid phase is replaced by a liquid semi-infinite space of finite viscosity \id. These results are in accordance with calculations of other authors as well (e.g. Refs.[26-28,48]). These references concern the surfactant-free case for similar systems and the numerical coefficients in the respective cases are comparable with our results. One should also note that the method of calculations proposed here is universal for axially symmetric hydrodynamic interactions in emulsion systems and with slight modifications might readily include additional effects, like deformation of the fluid interface, etc. 2.4.2. Solid sphere and a droplet (Fig. 2b) The goal here is to propose the correct estimation of the impact of the finite dimensions of the droplet on the flow pattern inside it. Two spheres, a solid and a fluid one, approach each other at small separations. It is important to have them with identical radii, so that the effect of the curvature to be maximal. The general procedure is completely analogous to that in 2.4.1 and details might be found in [30]. Here only an outline of the major results is given. The general expression for the biharmonic function is presented in a form proposed for the first time by Schroder [47]:

Vd

p2-R2 r

cosmr

d

d

u ; vn = 2R

p

1

5T rf

p2 sinQ 59

„

/

dVd

, vfl =

(2.37) p sinS dp

where (p,cp,9) is a spherical coordinate system, centered in the droplet, *F d is the stream function and, u is a harmonic function (Aud = 0). The boundary conditions are completely analogous to (2.23) [30]. They are only slightly modified by replacing the flat interface (z - 0) with the droplet's surface hd{r)= R[ 1 - •N/7 - ( r / R ) ]. The final integral equation for the radial velocity vf [f,hd(r)j is presented in two alternative forms [30]:

Hydrodynamic Interactions and Stability of Emulsion Films

231

^ [^, /7- (F)] - spr^^ ^') - ^ ^ ^^f? "ff ^ ^ - ^^ ^ ) 1 - ^ - ^ + ^(s )j 2n

{ a o \r2 + ff -2r,r, cosy,)

=wwm

'

\

<238a)

and

J

z

0[r

+r,

-2rlr,cos(pl)

where y*[r,hd(r)]=

V

'f'h

^

. The term on the right-hand side (RHS) of the

last equation is the radial velocity on a tangentially free interface, i.e. when one has a bubble, instead of a droplet. At hd(r) = 0, Eqs.(2.38a-b) turn into the integral equations for the radial velocity U(r) on a flat liquid/gas interface (2.28) and (2.29), respectively. One peculiarity, besides the familiar coupling factor sjl (hydrodynamic factor fh as given by Eq. (2.10b)), is the additional term that appears only in Eq. (2.38). It is of the order of the geometrical factor s , related to the dimensions of the thin layer (~£>(e)). Thus, there is a slight difference in the obtained solutions, and in the respective drag-force expressions as compared to the flat interface case in subsection 2.4.1. Two asymptotic cases are of particular interest: (1) sjl < 1, with two possibilities which reflect the relative importance of the factors fh and s : (1.1) s j l « e «1 , i.e. j l « 1 and, (1.2) s « s p T « /, i.e. jl > 1. For both cases the solution of the integral equation is expressed as a power series in (ejl):

(2.39) where J',(f) might be found in the Appendix (Eq.(A.3)).

232

E. Mileva andB. Radoev

(2) sjl » 1. Insofar as one has always e « 1, this is possible only for high viscosity in the droplet. Thus, the tangential velocity is suitably presented as a power series in (sjl)"' or (Mo) :

vrrf0[r,^(r)] = -4[^(r)+^s)]+ 4f4:Tl \o [r. h " (F)] = MO[J'3 (F) + O{z )] + o\Mo)2 J

(2.40)

with J'3(r) given in the Appendix (Eqs.(A.4)). For the two major cases of low and high droplet viscosities the drag-force expression acquires the form of [30]: Fvhere=-^fVf^-{l

+ ^[Q2+^)]}

for

efi«7

(2.41)

ej!»7

(2.42)

and F^=^'V'^\I--L[R2+C?V^

Fsphere=-2izWVf^{l-Mo[R2+a{z)}}

or

for

The numerical quotients, g 2 and R2, are shown in the Appendix (Eqs.(A.5-6)). The force expressions (2.41-42) need some additional comments. The essential functional dependence on the parameters of the system is totally determined by the presence of (at least) one tangentially immobile boundary in the thin liquid layer. If all the rest conditions (namely Vf ,1, \if ,\xd,R) are the same, the drag force has a maximum value when the mobile interface is flat. The 'bending' of this interface into a droplet surface, results in a relative decrease of the shear stress on it. This effect (might be termed 'an opening of the gap' [30]) is characteristic of the lubrication-theory cases. It manifests itself regardless of whether the adjacent phase is gaseous, liquid or solid. The ratio of the factors outside the brackets in the drag force expressions is constant, irrespective of F • Fplam/Fsphere ~ 4 (see (2.35) and (2.41); (2.36) and (2.42)) [29,30,36,48].

Hydrodynamic Interactions and Stability of Emulsion Films

233

If one neglects this purely geometric effect and examine more closely the structure of the terms related to the viscosity in the contiguous phase, two effects are clearly outlined. The first effect is 'the pure influence of the viscosity in the droplet' and it is found in both asymptotic models (3.57 in (2.35) against Q2 in (2.41); 0.66 in (2.36) against R2 in (2.42)). The second effect (~ 6>(z)) appears only in the case of a spherical droplet (2.41-42). It is related to 'bending of the streamlines' inside the droplet, as compared to those in the semi-infinite liquid space. Insofar as this effect results in additional dissipation of energy, it leads to a relative (slight, however) increase in the drag force, as compared to the flat interface case. Thus, the main conclusion, to be drawn from the presented model investigations, is that at small gap widths between the particles (s «1), the effective flow zone inside the droplet shrinks uniformly around the symmetry axis and to the surface bordering the emulsion film. Therefore, the fluid flow interior the droplet could be regarded as a motion in a semi-infinite space (Fig.2a) [7,29-30]. This model situation constitutes the true specification of the 'primitive model of emulsion hydrodynamics', always when the flow in the emulsion film is of lubrication type. 3. ROLE OF SURFACTANTS The surfactants have a profound influence on the tangential mobility of the liquid interface [38]. The geometric anizodiametry of the thin layer between the droplets ( s « l ) results in a specific coupling of their mass transport and the hydrodynamics of the system [3]. One manifestation of this coupling is the relationship between the kinetic stability of the emulsion films and the preferential solubility of the surfactant species either in the film, or in the contiguous phase. As already mentioned, this effect is a firm experimental fact and is named a Bankroft rule [32]. An attempt to link it to hydrodynamics of emulsion films was first made by Lee and Hogdson [3], and was extensively studied for different regimes of the flow interior the droplet [11,12,25,31,33-35]. The essence of the mutual influence of fluid motion and surfactant mass transfer in the adjacent phases is incorporated in the boundary conditions (2.2). They are specified to include the Marangoni stress component that arise due to the interfacial tension gradient, and the surface viscosity effect [31,38,49]: (3.1) where y is the interfacial tension. In what follows, the issue of surface viscosity [Is [50,51] is neglected. The latter might routinely be included in our analysis. It

234

E. Mileva and B. Radoev

was shown in [49], that this effect results in a slight modification of the interface mobility expression. Upon the application of the potential theory approach an integrodifferential equation is obtained for the interface mobility, instead of the basic integral equation (3.27) (see Ref.[49] for details). As has been mentioned also in [31], for emulsion films, this surface viscosity effect is usually small compared to the other terms in (3.1). So, it won't be dealt here further on. Another important boundary condition is the surface mass balance of the surfactant species:

v t (rv/)-DX 2 r=y n

AT-) with jn= M^f)

(3.2)

-'' =-/„

(3.2')

Only the case of diffusion controlled fluxes towards the droplet's surface is presented. It was established that this possibility is of utmost importance for the interface mobility when thin liquid layer is formed in the gap between the fluid particles [21,22]. Depending on the preferential solubility of the surfactant in the respective cases, jn might acquire any of the forms in (3.2'). F is the surface concentration of the surfactant; Ds,Df,Dd are the surface and the bulk diffusion coefficients; cf ,cd are the bulk concentrations of the surfactant in the phases and they obey the steady mass transfer equations [38]: f

_ / / d dcf d2cf) f vf + v{ =D\ r + r dr dz \r dr dr dz~ J fdc

dr

f

fdc

dz

yr dr

dr

dz J

(3.3)

(3.4)

3.1. Scaling analysis The analysis, already presented in subsection 2.2, has to be modified to include additional scaling parameters related to the surfactant properties, as well

235

Hydrodynamic Interactions and Stability of Emulsion Films

Fig. 3. Mass transfer fluxes: (a) and (b) for a surfactant, soluble in the film; (c) for a surfactant, soluble in the droplet phase; c" - surfactant concentration in the meniscus;
options (see Fig. 3). The following new notations are introduced [34]: • Ac{., Acf - maximum value of the characteristic changes in the surfactant concentrations along the r, z -axes for the thin liquid layer; • Acd, Act - maximum value of the characteristic changes in the surfactant concentrations along the r, z -axes inside the droplets; • 5 f ,8 d - characteristic lengths normal to the interface in the film and in the droplet, within which the major concentration change occurs. The goal further on is to correlate these new characteristic changes with the length and dynamic scaling, already introduced in Section 2.1. The major result is the estimate of the tangential mobility of fluid interfaces, in the presence of a stabilizing surfactant in the film (the continuous phase) or/and in the droplets. 3.1.1. Mass transfer of a surfactant, soluble in the thin layer For a surfactant, soluble in the emulsion thin layer (continuous phase), the estimation of the diffusion flux in (3.2') is [34]:

ii~H~&^-

(3-5)

Additional assumptions are made: (i) The local value of the interfacial tension is related to the local concentration change as at equilibrium [3,34,49]: Vj~^=% ^ dr

dcf „ dr

-

^

^L

dcf „ Rf

(3 .6)

236

E. Mileva and B. Radoev

(ii) The scaling of the various terms in the surfactant balance equation (3.2) is as follows [34]:

V.-CrvJ-^-

and V , r — ^ c ' - ^ ^ -

(3.7)

All the quantities marked with a subscript | concern the respective equilibrium properties [3,34]. Following Levich [38], the surface concentration might be written as

r=r°+r'

(3.8)

where F " is the value in the absence of fluid motion, and F ' is the perturbation of F °, due to the flow in the system. Thus the maximal value of the changes in F isF". The equation of bulk mass transfer (3.3) is used to find the relation between the scaling parameters Ac/ and Ac/, namely [34]: Pef +(Rf/hf)\ f Ac = -,—• —Ac, ; Pe'+l A / y

UfRf Pe = — ; Df f J

s /

h' -

Rf , = yll + Pef

,.m (3.9)

Here Pef is the bulk Peclet number inside the thin liquid layer. Two asymptotic cases are particularly important: (1) Pef « 1. Due to the geometrical constraints in the film \Hf «Rf), one always has 8 f ~ Hf « Rf. Besides, Ac/ ~ c"' - cf and Ac/ ~ cf - c{ , where cm is the value of the bulk concentration in the meniscus far from the film; cf is the surfactant concentration in the film bulk; c/ is the concentration in the subsurface layer near the interface (see Fig. 3a). The mass balance equation is reduced to the simple Laplace-form (RHS of the Eq. (3.3)) and,

^ - f f r ) Ac/=s2Ac/

(3.10)

Thus, due to the anisodiametry of the thin liquid layer (e « / ) , the scale of the surfactant flux in a direction, normal to the film interface is considerably smaller than the scaling of the concentration change in the radial direction,

Hydrodynamic Interactions and Stability of Emulsion Films

237

Ac^ Ac^ namely: i{ ~ Df—z—~Dfz—r7-~ej{. Thus, conditions for the depletion of surfactant are created in the film bulk. In order to maintain the normal flux towards the interface new quantities of surfactant are required to enter the layer. They come from the meniscus region, thus moving along longer distance [3,34]. This effect is also familiar from foam films [20,21]. Rf (2) Pef » 1. In the case of Hf >bf—j^=, a diffusion boundary
(3.11)

The geometrical anisodiametry of the thin layer would not cause a difference in the scaling of the surfactant flux in the normal and radial direction inside the film bulk. So, the order of magnitude of the interface flux is the same as if there is no another interface in the vicinity [34]. This is a 'classical' diffusion boundary layer [38]. Based on the above-defined additional scaling parameters the following general estimate for the tangential mobility of the layer/droplet interface is obtained [34]: Uf

U<=

with

f

' + /*+//

™

V

f +f

» <

f

(3.12)

l + fh+ff

where fh is the same hydrodynamic factor as in (2.9) and, for the case of undeformable droplet interface, is given by (2.10); / / might be termed a concentration or Marangoni factor:

ff

T" c 5 / dj \ifDfZRfdc\ +

DfRf{Rf

It simplifies to:

)dcf

(Rf/5f)2+Pef l + Pe' n

1 + Pef

238

E. Mileva and B. Radoev

r" ay // =

fl

^

Pe7
for

(3.14)

and

r°

ay

E_ v

p*

+

ar s 7 7 7 D / / ac 0 V P ^ 7

;

Note that in the scaling relation (3.12), the effects of hydrodynamics and surfactant influence are delineated. Upon the insertion of (3.12) in the criterion for the validity of the lubrication approximation (2.12), one has

fo+//V('+/*+//)>*'• Alternatively, the interface velocity scale might be expressed via mobility factors: Moi=\

and

M/=

//

7

/„+//

=

M M

° °'\

(3.13')

Mo + Mofc

with Mo defined by (2.10a'). Therefore, UfMof

At/ 1 U = 7- with —- = l + Mof Uf 1 + Mof d f f and U =0, AU/U =1, if Mo =0; or , Ud =Uf, So, the validity criterion (2.12) acquires the form of Tjd

l

-^>z2 1 + Mof

(3.12) AU/Uf =0, if Mof ->oo. (2.12')

3.1.2. Mass transfer of a surfactant, soluble in the droplet Following similar procedure as in subsection 3.1.1, scaling relations are obtained for the case of a surfactant, soluble in the droplet phase [34]:

239

Hydrodynamic Interactions and Stability of Emulsion Films

Ud =

Uf

,

with

^ =

f

>+f"d

(3.16)

where the concentration factor is:

r" 5 ^ ^ (Rflbd)2+PeJ ff = V1***'*'* 7+ P ^ +

and

Ds (bd\dT Da'Rf[Fj'd?0

(Rflhd)2 + Ped 1 + Ped

P*J™-,

5 - = - ^ - - ^

(3.18)

Note that unlike the case of Pef < 1 when the maximum value of the concentration scaling normally to the film interface is 8f ~Hf, here 5 d ~ Rf{~ Rd ~ Hd) for Ped < 1 (compare Fig. 3a and Fig. 3c). Then:

r " ay d=_v/D?_dc\

Ds

F"

//=

for

dy

s

*'»'*'o'Jrt ;+

ped<J

dT

J

^

5 r

for

^

(3.20)

> 7

£

Alternatively, the interface velocity scaling (3.16) might be expressed via an interface mobility factor Mod: UfMod

TTd

U

.t, -T

with

d

.,

U MoMod

1

d

=

-r =

fh+ff

d

c

—r

Mo + Mo

d

c

and

1

— -= f

1 + Mo Mo

Af/

T

(3.16')

d

1 + Mo 1

Moc - —j.

f

(3.1V)

240

E. Mileva and B. Radoev

with Mo given by (2.10a'). Therefore, the criterion (2.12) is specified as

T^>z'

<2 12

- ">

Putting together the scaling relations ((3.19) and (3.14); (3.20) and (3.15); (3.19) and (3.20)), it is clearly outlined that a surfactant, soluble in the droplet would always has a less pronounced immobilizing effect on the interfacial mobility as compared to a surfactant, soluble in the emulsion layer. This influence is further diminished in thinner films, due to the factor s , irrespective of the particular mass-transfer-feeding pattern of the interface ((3.19) for Ped <1; (3.20) for Ped > 1). These scaling considerations show that the coupling of film flow conditions with the surfactant flux peculiarities might add the appropriate hydrodynamic background in understanding of the experimentally observed Bancroft rule. Only in the specific case of a diffusioncontrolled mass transfer of the surfactant, soluble in the emulsion film, it is possible to ensure the necessary conditions for the so-called 'controlled slip' [3] of the tangentially mobile interfaces that could efficiently immobilize these interfaces. In all the rest of the cases, the surfactant feed-up flux toward the film surfaces is very powerful and could not sustain low tangential mobility of the interfaces. The formation of the classical diffusion boundary layer alone, on either side of the mobile interface, is of no particular importance. It has none special effect on interfacial mobility and consequently, on the flow regime in the system (compare (3.15) and (3.20), which are of the same order of magnitude). The important issue for the surfactant, soluble in the fluid particles, is that the onset of a powerful flux from inside the droplet phase towards the film surface masks all the rest details. Thus, the film drains almost like in the surfactant-free case and even for a surfactant soluble in both phases, the effect would be exactly as by the preferential solubility inside the droplets (see Fig. 3c and find the details in Ref. [34]). 3.1.3. Interface concentration balance of the surfactant Let us focus on the surface concentration balance of the surfactant species, i.e. on the boundary condition (3.2). In itself, this balance is quite similar to the bulk mass-transfer equations (3.3-4). Therefore, it is possible to introduce additional dimensionless complexes like the bulk Peclet number, which account for the relative importance of the different factors included in (3.2). For details see Refs.[35-37,52], where the surface analogue of the bulk Peclet number, named surface Peclet number is defined as:

Hydrodynamic Interactions and Stability of Emulsion Films

241

This parameter is a measure of the relative importance of surface convection and surface diffusion of the surfactant species. Other authors [12,25] have also introduced similar complexes related to other characteristic interfacial changes due to the mobility of the film interfaces. Upon insertion of Eq. (3.21) in the boundary condition (3.2), the scaling of F ' is obtained in the form of [35]:

r ' =r ° f ^ - ^ l [Pe' + lJ

with

j=

4^7 DT°/(RfY

(3-22)

Here instead as in (3.7), the scaling of the surface diffusion flux is linked directly to the maximal possible change in F , i.e. DsV2zF ~DT" /(R1)2. Thus, the surface concentration change ( F ' ) is generally related to both the fluid flow and to the mass-transfer fluxes. The characteristic parameter (j) is the ratio of the scales of bulk and surface diffusion changes, in the case of soluble surfactant [35-37]. So, the boundary condition (3.2) might therefore, be written in a dimensionless form:

[( Pes+1

J

tT

(Pes+l)'

T

J [Pe'+lJ

J

'

where F = F / F °. For full analysis of (3.23) and the consequences from this particular formulation for the validity of the lubrication approximation, including the case of insoluble surfactants, see Refs.[35-37,52]. 3.2. Important example The explicit solution of the basic surfactant-free case, which imparts the essence of the close-distance emulsion hydrodynamics, has already been presented in section 2.2.1 (Fig. 2a). The aim here is to specify this asymptotic model to account for the presence of surfactants, soluble either in the thin liquid layer, or inside the semi-infinite liquid space [36,37]. The additional scaling parameters from subsection 3.1 are used for the mathematical formulation of the interaction problem. The basic equations and boundary conditions are: • in the thin layer - the lubrication theory equations (2.13); • inside the liquid semi-infinite space - the creeping flow equations (2.18);

242

E. Mileva and B. Radoev

• mass transfer of the surfactant in both phases is described by a reduced version of (3.3-4), namely V 2 c ; = 0, or V2cd =0 (Laplace equations); • on the flat fluid interface the conditions (3.1-2) are specified as follows: pf(r,0)=Prdz(r,0) + ~tdr

(3.24)

(
\-D'°f(r,z = 0) j,=\ f

(3.25)

dz

• additional relations are also designated, namely

^-[r,h'(r)]=0,

(')=(/)(');

r(r^)=r"

(3-26)

The Stokes stream function is presented as a biharmonic function (2.22). Therefore Eqs.(2.13) and (2.22), together with the boundary conditions (2.23,3.24-26) result in an integral equation for the tangential mobility on the plane interface, which is a modification of (2.27) [36,37]:

U(r)-,p±m

^

2cosq y dz ) _n

= U'(r)

4\iJ or

(3.27,

This constitutes the basic result for the model on Fig. 2a in the presence of a surfactant in the system. 3.2.1. Surfactant, soluble in the liquid semi-infinite space Several additional simplifications are introduced here. They do not change substantially the essence of the effect of the surfactant on the interfacial mobility of the fluid interface, but make the interaction problem solvable in terms of the potential theory. First, the perturbation in the surface concentration is small, i.e. F = F " + F ' , r ' « r °. Second, the surface diffusion flux is neglected. This effect is taken into account in [49] and it does not change drastically the results. The tangential mass balance (3.25) becomes: Flrd

rnVrU{r)=DJ—(r,z dz

= 0)

(3.28)

243

Hydrodynamic Interactions and Stability of Emulsion Films

Eq.(3.28) may be viewed as a normal derivative of cd(r,(p,z) on the plane fluid interface. Insofar as the surfactant concentration balance obeys the Laplace equation, and in combination with (3.28), its solution might be presented as a single layer potential [36,37]: c (r,ra,z)=

f

=r—

(3.29)

2xD+rj-2rr,co!fal-v)+z3]/2

Besides,

(dr)

dc'Xar)^

The second multiple on the RHS of (3.30) is the tangential derivative of the single layer potential (3.29) at the plane interface. The integral equation (3.27) might be written in two forms which are modifications of Eqs.(2.28) and (2.29): _h'(r) YjA,, [U {r, )]cosy ,r,dr,d, ^ l/7 + eu.— I IT— — — yjj 2n nh[r +r, -2rr,cosq,\

and W

sa-JJlL

2

7c[n-f2cos(
4ndc"aii'Dd

h'fa) J

-2l\

II

\_2

oo

[r, +r2 - 2r,r2cos{q2 -
-,

/

\V

/2

cosyfldrldql I I— 1

—?

J [r' + rf -

~,

. \

l/2

.. '

2rr,cosy,)

The respective solutions for the asymptotic cases of low and high viscosities of the liquid semi-infinite space are:

244

E. Mileva and B. Radoev

(i)6jr
(3.32)

with J,{r) and J2{r) presented in the Appendix (Eqs.(A.l),(A.7)), and F = -%\ifVf — \l + 3.57&ix+0.66s-^——-%

(3.33)

The mass transfer of the surfactant acts towards an effective decrease of the tangential interface mobility, and consequently, increases the respective drag force (compare (3.33) with (2.35)). (2) Ejl > 1

Ui^-Llj^

+ e-j-^-rpj 4n \iJD

8(j. |_

J4{r)\

(3-34)

dc 0

J3(r) and J4{r) could be found in the Appendix (Eqs.(A.2),(A.8)). F = 6n^fVf—

7-0.39— + 0.05-^—,T -%-

(3.35)

As judged against to the surfactant-free case (Eq.(2.36)), here only to a slight decrease of the interfacial mobility term, related to the finite \\.d-value is observed. 3.2.2 Surfactant, soluble in the emulsion layer The calculation procedure is completely analogous to that in the previous subsection. The respective results are [36,37]: (l)efT<7

£/(,)= U-if)- *!pJ,lf)2%

Zp-^r? fy JAr)

4% \iJDJ

dc1 „

F = -n]ifVf —[ 1 + 3.57z\i+ 0.66-^-— •%•

2

H

l{

*

vfDf dcf J

(336) (3.37)

Hydrodynamic Interactions and Stability of Emulsion Films

245

In the present case the impact of the surfactant species is most prominent, as compared to all the other options (Eqs. (3.32-35), and see further (3.39)). The difference between (3.37) on one hand, and (3.33) and (3.35) on the other, goes through the film geometric parameter e . While it is present in the Marangoni factor in the case of the surfactant, soluble in the adjacent phase, here for a surfactant soluble only in the emulsion layer, it is absent. This is just the same situation, as has already been shown in the scaling considerations (see again Eqs. (3.14) and (3.19)). (2)sp:>7

4% |V DJ dcJ „

sjj. |_ F = 6iz\xfVf—

1-0.39^ / [

+ 0.05 Ejl

T / d

. -% E\i Df dcf

(3.39) 0\

4. RUPTURE OF THIN EMULSION FILMS It is well known that when the gap widths between the dispersed particles get too small, the onset of additional specific interactions is observed. These are usually termed surface force interactions [53]. In turn, such surface forces per unit area represent the so-called disjoining pressure 77 [53,54]. A simple relation exists between this disjoining pressure and the Gibbs energy (per unit area) G(h) of a thin layer: n(h)

= -^on

(4.1)

where h is the thin layer thickness. If the layer is a film of constant thickness, then h is simply its thickness; if the layer thickness is variable then h must be considered as the local value dependent on the position in the gap between the two particles. The relation (4.1) permits to assign the respective component of the disjoining pressure FI } , to a respective component of the energy G7. The particular type of interaction that governs the film behavior depends on its thickness. At relatively large separations (10'8m
246

E. Mileva and B. Radoev

Other types of interactions (steric, hydrophobic, undulation, etc.), known as non-DLVO interactions [53], become important at thickness h ca. 10'8 m. Insofar as the problems discussed in the present chapter concern mainly the hydrodynamics of relatively thick films (h>J0'7m), when considering the problems of the stability of emulsion films we shall further deal predominantly with the DLVO-interactions. 4.1. Linear instability condition Two emulsion droplets approach at a small separation, so that a thin layer is formed between them (see also Section 2, Fig.l). A particular profile of the thin layer is established under the action of the hydrodynamic and capillary pressures. Two limiting cases are possible: (/) quasi-nondeformed, with spherical surfaces (at relatively high surface tensions) and (//) profiles comprising plane-parallel films in their thinnest sections, i.e. thin layers of quasi-constant thickness. The transition between these two states passes through the so-called dimple formation [58-60]. Further on we shall focus on the stability of plane-parallel films that are the simplest and the most comprehensively studied system. At the end of this Section (see below Eq.(4.2"')) we shall try to generalize the results for the cases of more complex profiles. As it is known, the instability of the thin layers is most often manifested by breaking or rupturing of thin liquid lamellae - a phenomenon which is encountered in both real and model systems. Let us consider a plane-parallel film formed between two emulsion droplets that approach against one another. Since the liquid surfaces are not even but corrugated by thermal fluctuations [61], the problem of the overall film stability is focused upon the stability of the individual fluctuation modes. The universal approach to describe the fluctuation modes is the Fourier analysis. Further on we shall employ the term Fourier component as equivalent to a fluctuation mode. Corrugation results in larger surface area and consequently, to an increase in the energy, proportional to the surface tension. The resistance to this increase is expressed in the appearance of capillary pressure. For this reason the thermal fluctuations on the surface of deep liquid are stable, since only capillary forces are operating in such a case. With thin films, the fluctuations of the surface shape are related not only to the increase in area but to changes in local film thickness, as well (see also Eq. (4.2)). As was mentioned above, the energy of a thin film is a function of its thickness. Hence, upon surface corrugation the interactions between the surfaces are also perturbed. Two cases can be observed with the decrease in thickness: (i) Gibbs interaction energy decreasing, which leads to progressing surface corrugation and (ii) increase in the Gibbs interaction energy, which is equivalent to damping of the surface corrugation. In the first case, unstable Fourier components are present, while in the second case all Fourier components are stable. Since the

Hydrodynamic Interactions and Stability of Emulsion Films

247

film has two interfaces, the correlation effect of their fluctuations has to be taken into account. If there are no specific force interactions in the layer, the fluctuations on the two surfaces are independent. When, however, disjoining pressure is present, correlation arises and is expressed in the frame of two collective modes, bending mode - also called undulation mode - and squeezing, or peristaltic mode [62,63]. The bending mode preserves the thickness of the film and does not contribute to its rupture. Important for rupture is the squeezing mode, since it leads to change in local thickness. It is the squeezing mode, which we shall have in mind as a model here. Therefore, it is very important to study the dynamics of the squeezing waves (see Fig. 4) through the pressure difference Ap, of the perturbed state and the initial state in the film [64]: -A/? = p T ( O + n ( / 2 + 2 C ) - n ( / 0

(4.2)

Here py is capillary pressure, C is deformation of a single film surface from its initial state; n (h) and n (h + 2^) are the disjoining pressures in the initial state and in the perturbed one, respectively. Note that the factor of 2 before
(4.2')

where yV% is the familiar expression for the capillary pressure at small perturbations, with y being surface tension. The linear relation between driving pressure Ap and perturbation C gives the name to the linear stability analysis.

Fig. 4. Squeezing mode in a parallel film (a sketch): h - film thickness; Q - deformation amplidude.

248

E. Mileva and B. Radoev

The comparison of the role of the two perturbation pressures in the interaction energy, i.e. the estimation of the relative weight of the two terms in (4.2') is commonly performed via Fourier transformation: ^p(q) = {-iq2 + 2dn/dh)i,

(4.2")

where q is the wave number of the respective Fourier component, and Ap,C, are Fourier images of Ap.t^ . As we shall show further on (see Eq. (4.7)), film stability follows directly from the sign of the factor in brackets in front of (, (-yq2 + dYl /dh). Before that, however, we shall pay attention to the important mode with wave number qc, for which Ap, or what is the same, Ap becomes zero: q2_2dYl/dh

(43)

Y

In the literature [18] the wave qc is called critical mode, or critical wave but, in fact, this is an equilibrium steady wave; its amplitude neither grows nor decreases. It follows from (4.3) that the spectrum of the short waves (q>qc; q=2n/X, X is wave length) is stable, since the capillary forces prevail over the van der Waals interactions. In contrast, the long waves (q 0. In the opposite case, i.e. at dYl /dh < 0, the root qc is imaginary, which physically corresponds to a stable film. Since the disjoining pressure FI (h), and consequently qQ(h), depend on the film thickness h, there exists such a equilibrium wave for every thickness value. This is an important circumstance for the thinning films (see below), where the spectrum of the unstable waves includes higher frequences. The theory of the linear stability presented here was for the first time proposed for foam films by Scheludko in 1962 [17,18], but it is utterly valid for emulsion films, as well. Emulsion specificity is accounted for though the respective values of dYl /dh and y. As it is well known, the model of a parallel film considered here is hardly the common form occurring at the mutual approach of two emulsion droplets (or two gas bubbles). At relatively large distances the surfaces are practically spherical and the thin film is adequately approximated by a parabolic gap (see Fig.2b, Section 2.4.2 and Refs.[30,52] for details). The other frequently

Hydrodynamic Interactions and Stability of Emulsion Films

249

observed case is a thin film with a dimple. Essential in both cases is that the thickness (the profile) of the layer becomes a function of the radial coordinate r, hence, the interactions in the layer vary, i. e. n=TI[h(r)]. Consequently, the perturbations in such a layer do not spread in a medium with constant disjoining pressure, a fact that may become decisive for the thin layer stability. The problem with the variable thickness cannot be figured out in a simple manner because of the complexity of the balance (4.2"): CO

2

.

Ap(q)=-yq ( + 2 \—[n (q - q,)]((q^q,

(4.2'")

-QO

where the second term on the RHS is no longer an ordinary product but a convolution. We shall not go deeper into analysis of the integral equation, but shall do with an illustration of a typical distribution of corrugations around a local minimum of the film profile. This minimum can be an element of the thin layer at the center between two quasi-nondeformed droplets, as well as a part of the barrier ring around the dimple at the film periphery. The idea of Fig. 5 is to show that the condition Ap = 0 does not correspond to a single wave (qc) but is realized through a corrugation of variable (increasing) length and (decreasing) amplitude towards the film periphery. Moreover, at a sufficiently steep profile the thin layer may turn being stable, even when the condition of local instability (4.3) is fulfilled in the thinnest portions (see, for instance [66]).

Fig. 5. Squeezing corrugation in a parabolic gap (a sketch). Due to the variable interactions (IT[h(x)]) the deformations are with decreasing amplitude toward the gap periphery.

250

E. Mileva and B. Radoev

4.2. Role of the film drainage on the thickness of rupture The spectral analysis of (4.2") is very instructive for understanding the reasons and conditions of thin film stability. However, it does not answer the very important, from experimental and practical viewpoint question of the film lifetime or, what is the same: of the thickness of rupture. Vrij [39,67] has proposed a solution to the problem. The essence of his model is that the film ruptures when its two surfaces touch locally. The identification of the moment of touching with rupture is not a general condition and requires additional specification with respect to the action of the non-DLVO forces (for example, the appearance of films of higher stability (see e.g. instance [64]). In the more general interpretation, Vrij's model is applicable to estimating the time of appearance of black spots [23,68]. Therefore, all the important characteristics (lifetime of the film, including the velocity of coalescence in emulsion, thickness of rupture, etc.) directly depend on the estimate of the time of reaching the state of local contact of the surfaces. In other words, this is the question of evolution, i.e. of the kinetics of growth of the unstable waves. These are surface waves, a particular case of the relatively well-studied waves on liquids [38,69]. Bearing in mind (see 4.3) that the equilibrium waves Xc are as a rule larger than the thickness h {Xc>h) [17,64], the hydrodynamics of the unstable waves is adequately approximated with the already considered lubrication approximation (see Subsection 2.1, Eq.(2.10'); see also [67]): ^ = -^T(l dt 12\iJ

+ Mo)V2Ap

(4.4)

where Ap is the driving pressure (4.2'), dC, Idt is the velocity of deformation and Mo is a coefficient of the type already introduced in the previous subsections, accounting for the mobility of the film interfaces. Analogous mobility coefficient is introduced also in [70,71], where miscellaneous effects influencing the surface mobility of thin films are commented. Here we shall focus on the effects that are typical for emulsion systems. As is already shown in previous sections of this chapter, the surface mobility is controlled by the viscosity in the droplets and by the Marangoni effect (see e.g. Eqs.(2.10) and (2.10a'); Eqs.(3.12') and (3.17'); Eq.(3.37)). Note that both, hydrodynamic and Marangoni effects, might lead to Mo=0, i.e. practically to immobilization of the surfaces. The evolution of the perturbation C,(t) follows directly from the solution of (4.4), which, as we have found (see (4.2")) is suitably presented through Fourier transformation. Combining (4.4) with (4.2') yields the equation (4.5):

Hydrodynamic Interactions and Stability of Emulsion Films

^--^{l

+ MoWU-l^-V

12\iJ

dt

y

251

(4.5)

dh J

The solution of (4.5) poses two problems for thinning films: the initial condition ^{q.t = 0) and the time dependence h(t). The specificity of the initial condition is related to the fluctuation (random) character of the perturbation waves. In [65] this problem is reduced to the solution of the respective FokkerPlanck equation of the kind (see also Ref. [72]):

P^fy-4—1^W^4 \

dt

q- dh ) 8C,

(4.6)

DC,

where XCO is a distribution function (probability density) of the process, normalized on f a s J/(£ ,t]dC, = 1. In (4.6) p = 12\if jh\l + Mo)q4 is the drag coefficient, kB is Boltzmann constant, T is temperature. Since the fluctuation mean value is zero, i.e. (£) = 0, the analysis should be carried out on its dispersion (C,2(k,t)). Applying the standard stochastic methods, from (4.6) one obtains for the dispersion of the surface wave amplitudes the following evolution equation [65]:

dt

y

q dh p

'

with zero initial condition (C 2(k,t = 0)) = 0. Note that in contrast to (4.4) where zero initial condition is equivalent to trivial (zero) solution, Eq. (4.6) has non-zero solution due to the inhomogeneous term kgT. Thus, for example, for a steady state film (/?=const), where all factors in (4.6) are constant, its solution takes the form:

Eq. (4.7) demonstrates explicitly the conclusion from (4.2") that the long wave spectrum {q
252

E. Mileva and B. Radoev

while the short wave spectrum (q>qc) is stable (£2{k,tJ) (decreasing with time). It is worth pointing out that initially, i.e. .at

\f - 2(dn /dh)/q2

t«l,

the

process for both spectral branches acquires the character of Brownian motion: ((,2{q,t^0)) =

k

-^t.

(4.7')

For the kinetics of the process, the resulting equation (4.7') means that the longest time is spent around the initial state, followed by an exponential growth (or recession). For emulsion systems, the important question is whether coalescence would occur at the collision of two droplets. Translated into the language of the thin-layer model this is equivalent to the problem of the wave evolution in a thinning film. As we already pointed out, the solution of (4.6) for these cases is complicated because of the dependence of /? and IJ on the film thickness h, which in its turn is a function of time, h{t). This problem finds an elegant solution within the frame of the linear perturbation approach, i.e. in the absence of coupling between the film dynamics and the perturbation waves. Indeed, in this case the film thickness h does not depend on the perturbation £"(0 and, as a unique function of time h(t), provides the correctness of the substitution: d/dt = Vfd/dh, where Vf --dh/dt is the velocity of film thinning (Eq. (2.3), Section 2.1) (see e.g. Ref.[64]). By means of this substitution (4.6) becomes:

dh

\

q an y

'

In Eq. (4.8) time is eliminated and the amplitude of perturbation is presented as a function of the film thickness (C2\(h), which is convenient for obtaining an explicit solution for the thickness of rupture h=hrupt. Following Vrij [39], the formal condition of rupture, reads: ^2)(h

= Km) = hnipt/2

(4.9)

Here, however, comes another problem related to the random nature of the surface waves. It can be seen that the direct inverse Fourier transform of

Hydrodynamic Interactions and Stability of Emulsion Films

253

(4.8.) is divergent and the reason for this is the formal infinite correlation of the surface perturbation £(r). The correct approach requires taking into account the finite spatial correlation when estimating the mean square amplitude (f,2) (t, (r)C, (r,)). Already Vrij refers to the presence of spatial correlations of the fluctuation waves [39]. More detailed analysis is presented in [73,74]. There the term 'correlated subdomain', i.e. the region in which ((^(r)^(r ; )):£0, is defined, as well as the important for the film quantity number of uncorrelated subdomains. The main result of the stochastic analysis in the above cited studies is that the larger (by area) films contain larger number of uncorrelated subdomains, which leads to shorter life-time and, respectively, to larger thickness of rupture hrupt. Besides, these results confirm the well-known fact that the larger (macro-) systems are less stable. The authors in [73] give a numerical solution for the particular case, which corresponds to the approximate relation: K,,P,~R'fiL

(4-10)

For the traditionally studied microscopic films (film diameter from 0.1 to 1.0 mm, see Refs. [65,75]) the already commented stochastic effects, i.e. the effects of correlation, are negligible with respect to the effect of the so-called most rapidly growing wave [39]. The reason is the sharp maximum of the factor (~ I2 v^2 ~ 2{d£l /dh)\) in Eq. (4.6), which determines the most rapidly growing wave in the entire Fourier spectrum causing the film rupture. The literature abounds of formulae for estimating the thickness of rupture, based on the above considerations, whose results do not differ significantly [64,67,76]. An example of an excellent agreement with the experiment is the result obtained in [65]: *.(*!/**«, yu7

(4

.n)

It is worth pointing out that the relation (4.11) does not require the use of a thinning-film model and experimental data can be directly applied instead. The latter is a great advantage of (4.11), as with larger films, the kinetics of thinning is rather complex and its simplification within the scheme of a given model increases the danger of erroneous hrup{ estimate [77]. The fact that fluctuation wave growth competes with film thinning, i.e. that the wave velocity DC, /8t is scaled with the film velocity Vf (see (4.8)), is of a decisive importance for the role of surfactants in the film stability. As we have pointed out above, the influence of the surfactant on the wave kinetics is

254

E. Mileva and B. Radoev

accounted for by the mobility factor Mo (see (4.4)), but the mobility of the film surfaces affects to the same extent the film velocity Vf, as well. Moreover, since Vf ~ 7/p ((3 is proportional to the factor inside the brackets on RHS of Eq. (3.39), Section 3.2.2), the product F7(3 in Eq.(4.8) is independent of Mo. An important consequence of this theoretical result is that hrupt must be independent on the surface mobility. From the experimental viewpoint, the rupture thickness independence on Mo is equivalent to its independence of the surfactant concentration. This, however, is not consistently supported by the experimental results [64,78]. Such a discrepancy between theory and experiment is still an open question. Most frequently, its solution is sought in non-linear effects [76,79], a hypothesis that has not found convincing evidence so far. Another possibility for such an influence of the surfactant on hrupl is the effect of macroscopic heterogeneities in the film thickness caused by hydrodynamic factors. At higher drainage rates Vf, these non-homogeneities are more pronounced [65,77]. Since lower surfactant concentrations correlate with higher Mo-coefficient or lower coefficient fc (see Eqs. (3.19),(3.33),(3.35), (3.37),(3.39)), i.e. with higher Vf, it is to be expected that generally, lower surfactant quantities will incite stronger (macro-) heterogeneity. On the other hand, more and larger non-homogeneities are registered as higher mean film thickness [65,77]. This may be the true reason for obtaining higher than the actual values oihrupl, and not the effect of decreasing surfactant content [80]. 5. CONCLUDUNG REMARKS One very important dynamic characteristic feature of the emulsion systems is the mobility of the interfaces of the droplets. The treatment in the present chapter shows that it is determined by two major factors: (1) the relative viscosity of the droplets and the continuous phase; (2) the presence of surfactants. The viscosity factor fT was first introduced by Rybczinski and Hadamard [38]. At close separations between the fluid particles, this effect is modified due to the hindered outflow in the thin liquid layer. In flow characteristics, it appears always in a combination with the geometry of the emulsion film in the form of the hydrodynamic factor fh. The impact of the surfactant mass-transfer on the hydrodynamics of emulsion systems is related to the onset of flow-driven surface-tension gradients that tend to immobilize the interfaces, and to the responding surfactant fluxes. At small separations, the coupling of the surfactant balance and the fluid motion is modified by the formation of thin emulsion layers, thus leading, in some particular cases, to an enhanced suppression of the tangential mobility of the

Hydrodynamic Interactions and Stability of Emulsion Films

255

surfaces. This effect might be accounted for as a specification of the interfacial mobility scaling by introducing the so-called Marangoni factor: fc. For the majority of the cases of correlation between the flow motion and the surfactant fluxes, both effects, fh and fc act in an autonomous additive manner. Their overall impact on the emulsion film hydrodynamics might be accounted for via a unified parameter termed mobility of the fluid surfaces (Eqs. (3.12'),(3.17'))As for the overall stability of the emulsion systems, it is determined both by the capillary forces and the surface force interactions, exactly as by the foam systems. The interfacial mobility of the droplets is important, insofar as it ensures sufficient slowdown of the emulsion film drainage. For higher interface mobility, the drainage velocity is also increased. This increase is related to the onset of well-defined non-homogeneities of the film thickness, and therefore, in a more rapid coalescence. As a rule, higher surface mobility is observed when surfactants are solvable in the droplets, or almost equally solvable in the contiguous phases.

REFERENCES [I] J. Sjoblom, Emulsion and Emulsion Stability, Marcel Dekker Inc., NY, 1996. [2] T. Gurkov and E. Basheva, in Encyclopedia of Surface and Colloid Chemistry, A. Hubbard (ed.), Marcel Dekker Inc., NY, 2002. [3] J. Lee and T. Hodgson, Chem. Eng. Sci., 23 (1968) 1375. [4] S. Haber, G. Hetsroni and A. Solan, Int.J.Multiphase Flow, 1 (1973) 57. [5] E. Rushton and G. Davis, Int.J.Multiphase Flow, 4 (1978) 357. [6] A. Sharma and E. Ruckenstein, Coll&Polym Sci., 266 (1988) 60. [7] E. Mileva and B. Radoev, Theor.&Appl. Mechanics, Bulg. Acad. Sci., 12 (1981) 49. [8] E. Mileva and B. Radoev, Coll&Polym.Sci., 263 (1985) 587. [9] E. Mileva and B, Radoev, Coll&Polym.Sci., 264 (1986) 823. [10] E. Mileva and B. Radoev, in Particulate Phenomena and Multiphase Transport, T. N. Veziroglu (ed.) Vol.4, Hemisphere Publ.Corp., (1988) 261. II1] E. Klaseboer, J. Ph. Chevaillier, C. Gourdon and O. Masbernat, J.Coll.Interface Sci., 229 (2000) 274. [12] L. Y. Yeo, O. K. Matar, E. S. P. de Ortiz and G. F. Hewitt, J.Coll.lnterface Sci., 257 (2003)93. [13] S. Jeelani and S. Hartland, J.Coll.lnterface Sci., 156 (1993) 467. [14] S. Jeelani and S. Hartland, J.Coll.lnterface Sci., 164 (1994) 296. [15] A. Chester and I. Bazhlekov, J.Coll.lnterface Sci., 230 (2000) 229. [16] A. Scheludko, G. Dessimirov and K. Nikolov, Ann.Sof.Univ., 49 (1954/55) 127. [17] A. Scheludko, Proc. Konikl. Ned. Akad. Wetenschap., B65 (1962) 87. [18] A. Scheludko, Adv.Coll&Interface Sci., 1 (1967) 391. [19] B. Radoev, E. Manev and I. Ivanov, Kolloid Z., 234 (1969) 1037. [20] I. Ivanov, D. Dimitrov and B. Radoev, Colloid Journal, 41 (1979) 36 (in Russian). [21] B. Radoev, D. Dimitrov and I. Ivanov, Coll&Polym.Sci., 252 (1974) 50.

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E. Mileva and B. Radoev

[22] I. Ivanov and D. Dimitrov, in Thin Liquid Films, I. Ivanov (ed.), Marcel Dekker Inc., NY (1988) 379. [23] D. Exerowa and P. Kruglyakov, Foams and Foam Films, Elesevier, Amsterdam, 1998. [24] R. Sedev and D. Exerowa, Adv.Coll&Interface Sci., 83 (1999) 111. [25] L. Y. Yeo, O. K. Matar, E. S. P. de Ortiz and G. F. Hewitt, J.ColI.Interface Sci., 241 (2001)233. [26] K. Jansons and J.Lester, Phys.Fluids, 31 (1988) 1321. [27] S. Yiantsios and R. Davis, J.ColI.Interface Sci., 144 (1991) 412. [28] R. Davis, J. Schonberg and J. Rallison, Phys.Fluids, Al (1989) 77. [29] E. Mileva and B. Radoev, Coll&Polym.Sci., 266 (1988) 368. [30] E. Mileva and B. Radoev, Coll&Polym.Sci., 266 (1988) 359. [31] T. Traykov and I. Ivanov, Int. J. Multiphase Flow, 2 (1976) 397. [32] J. Davies and E. Rideal, Interfacial Phenomena, Academic Press, NY, 1960. [33] T. Traykov, E. Manev and I. Ivanov, Int. J. Multiphase Flow, 3 (1977) 485. [34] E. Mileva and B. Radoev, Coll&Polym.Sci., 264 (1986) 965. [35] E. Mileva and B. Radoev, Coll&Surf., A74 (1993) 259. [36] E. Mileva and L. Nikolov, Coll&Surf., A74 (1993) 267. [37] E. Mileva and L. Nikolov, in Proc. First World Congress on Emulsions, Paris, France, Vol 2 (1993) 310.1-6. [38] V. Levich, Physicochemical Hydrodynamics, Prentice Hall, Engelwood Cliffs, 1962. [39] A. Vrij, Disc. Faraday Soc, 42 (1966) 23. [40] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics Prentice Hall, Engelwood Cliffs, 1965. [41] L. Sedov, Methods of Similarity and Dimensional Theory in Mechanics, Nauka, Moscow, 1977 (in Russian). [42] N. Gunter, Die Potentialtheorie und ihre Anwendungen auf Grundlagen der Mathematischen Physik, Leipzig, 1957. [43] V. Volterra, Theory of Functionals and of Integral Equations, Dover, NY, 1959. [44] F. Odquist, Math. Z., 32 (1930) 329. [45] O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, NY, 1969. [46] S. H. Lee and L. G. Leal, J.ColI.Interface Sci., 87 (1982) 81. [47] K. Schroder, Math. Z., 49 (1943) 110. [48] V. Beshkov, B. Radoev and I. Ivanov, Int.J.Multiphase Flow, 4 (1978) 563. [49] E. Mileva and D. Exerowa, in Proc. Second World Congress on Emulsion, Bordeaux, France, Vol.2 (1997) 2-2-235/1-10. [50] L. Scriven, Chem.Eng.Sci., 12 (1960) 98. [51] T. S0rensen, in Dynamics and Instability of Fluid Interfaces, T. S0rensen (ed.), SpringerVerlag, Berlin (1979) 1. [52] E. Mileva and B. Radoev, Commun.Dept.Chem., Bulg.Acad.Sci., 24, (1991) 513. [53] J. Israelashvili, Intermolecular and Surface Forces, Academic Press, London, 1992. [54] B. V. Derjaguin, Kolloid Z., 69 (1934) 155. [55] B. V. Derjaguin and L. Landau, Acta Physicochim. URSS, 14 (1041) 633. [56] E. J. W. Verwey and J. Th. Overbeek, Theory of Stability of Liophobic Colloids, Elsevier, Amsterdam, 1948. [57] B. V. Derjaguin, Theory of the Stability of Colloids and Thin Films, Consultants Bureau, New York, 1989. [58] R. S. Allan, G. E. Charles and S. G. Mason, J.ColI.Interface Sci., 16 (1961) 150. [59] D. Platikanov, J.Phys.Chem., 68 (1964) 3619.

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257

[60] [61] [62] [63]

K. A. Burrill and D.R. Woods, J.Coll.Interface Sci., 42 (1973) 15. L. Mandelstamm, Ann.Phys., 41 (1913) 609. A. Vrij, J.Coll.Interface Sci., 19 (1964) 1 J. G. H. Joosten in Thin Liquid Films, Surfactant Science Series, vol. 29, Marcel Dekker Inc., NY, 1988. [64] I. Ivanov, B. Radoev, A. Scheludko and E. Manev, Trans.Faraday Soc, 66 (1970) 1262. [65] B. Radoev, A. Scheludko and E. Manev, J.Coll.Interface Sci., 95 (1983) 254. [66] K. W. Stockelhuber, B. Radoev, A. Wenger and H. J. Schulze, Langmuir, 20 (2004) 164. [67] A. Vrij and J. Th. Overbeek, J.Am.Chem.Soc, 90 (1968) 3074. [68] D. Exerowa, A. Nikolov and M. Zacharieva, J.Coll.Interface Sci., 81 (1981) 419. [69] J. Lucassen, van den Terapel, A. Vrij and F. T. Hesselink, Proc. Konikl. Ned. Akad. Wetenschap., B73 (1970) 108. [70] R. Tsekov, H. J. Schulze, B. Radoev and Ph. Letocart, Coll&Surf, A142 (1998) 287. [71] R. Tsekov and B. Radoev, Int.J. Min. Processing, 56 (1999) 61. [72] E. W. Montroll and J. L. Lebowitz (eds.), Fluctuation Phenomena, North-Holland, Amsterdam, 1989. [73] R. Tsekov and B. Radoev, J.Chem.Soc.Faraday Trans., 88 (1992) 251. [74] R. Tsekov and B. Radoev, Adv.Coll&Interface Sci., 38 (1992) 353. [75] A. Scheludko and E. Manev, Trans Faraday Soc, 64 (1968) 1123. [76] D. Valkovska, K. Danov and I.B. Ivanov, Adv.Coll&Interface Sci., 96 (2002) 101. [77] E. Manev, R. Tsekov and B. Radoev, Disp.Sci.Technol., 18 (1997) 769. [78] E. Manev, A. Scheludko and D. Exerowa, Coll&Polym.Sci., 252 (1974) 586. [79] A. de Wit, D. Gallez and C. Christov, Phys.Fluids, 6 (1994) 3256. [80] J. Angarska and E.Manev, Coll&Surf., A190 (2001) 117.

APPENDIX

jfrh)f^U'%)V°SV^'t 2

2

oo \r +r,

-2rrlcos(pl)

cos

's&h-Yi^Nl, J',(r)= "fc L A

II



* **>**>v/2

^

\FdF)

(A.2) <»*<*'*>>

\2fr{r,)-h%)]\(r'+r?-2rrlCosvy2

(A3)

25 8

E. Mileva and B. Radoev

Q

(A 5)

^-'l\u*(-l%(-Y>^

/?,= f7

—

-

^J'(r)

(A.6)

. ,.>. _ Yfi - r, cosy, )Vr;U° (r, frdr.dy, J

2\r)-}}

rr2 oo

z~2 i n

^A-^

W3

[r +r, - JrrjCOSip,)

i (-\-TflTrfc ^''"JJlJJ no [no

~T'cosW-
/2

f,dr,cosyld<s?l \(-2 -2 -,— J \r + r, - 2rr,

V2 cosy,)

(A.8) —

Id

Vr=-f-r

(A.9)

r or

\ = Vr-^-4r or

r

(A- 10 )

Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 7

Structure and layering of fluids in thin films D. Henderson a , A. D. Trokhymchukab and D. T. Wasan0 a

Department of Chemistry and Biochemistry, Brigham Young University, Provo UT 84602, USA b

Institute for Condensed Matter Physics, National Academy of Sciences of the Ukraine, Lviv UA 79011, Ukraine c

Department of Chemical and Environmental Engineering, Illinois Institute of Technology, Chicago IL 60616, USA 1. INTRODUCTION During the past century there have been many investigations of the properties of electrolytes near electrodes and of aqueous and organic solvents near surfaces and confined by dispersed colloidal particles. Further, the last decade has seen an intensified interest in understanding the nature of those processes that are carried out in many real systems and nontraditional systems of technological importance that comprise colloidal particles of various sizes and are under various external fields and spatial confinements [1-3]. This is the case of the large number of chemicals and consumer products, such as polymeric latexes, paints, inks, coatings, emulsions, gels, foams, cosmetics and Pharmaceuticals, food and beverages. The properties of confined colloidal suspensions are essential for processes such as sedimentation, lubrication, paper making, soil remediation, detergency, wetting of solids by micellar solutions, etc. In general, we can talk about a complex system characterized from the point of statistical mechanics by four distinct length scales: a molecular scale (a few angstroms) of the primary suspending medium due to water, some organic solvents and electrolyte ions; a submicroscopic scale (1 nm to 100 nm) that encompasses nanoparticles or surfactant aggregates called micelles; a microscopic scale (1 ^m to 100 fim) that characterizes the size of liquid droplets or bubbles in emulsions or foam systems; and a macroscopic scale. The most popular and widely used statistical mechanical model for colloidal systems, which we can call the colloid primitive model, results when the

260

D. Henderson, A.D. TrokhymchukandD.T. Wasan

molecular nature of the suspending fluid is neglected. This model has a long history and is grounded in the fact that a small size ratio of suspending fluid species to colloidal particles (typically 10H and smaller) allows one to neglect the discrete nature of the molecular solvent under some circumstances. Due to this small size ratio, it is assumed that the solvent molecules and electrolyte ions and/or the smaller (e.g., submicroscopic) species are invisible on the length scale of the giant micro- and macrocolloids and may be considered to be a continuum. The theory of colloidal dispersions that treats the suspending medium as a continuum is that due to Derjaguin and Landau [4] and Verwey and Overbeek [5] (DLVO). The DLVO theory balances the repulsive electrostatic double layer and the attractive van der Waals forces between the particles and has long been the basis of an explanation of the structure and stability of colloidal dispersion and is a classic. However, the advent of new instrumentation during recent years, such as the surface force apparatus and the thin film capillary force balance as well as atomic force microscopy and total internal reflection microscopy, has resulted in a new area of research directed to an understanding the nature of the properties of thin fluid films formed from aqueous latex suspensions, surfactant micellar solutions, and microemulsions. A direct forcedistance measurement in fluids confined between two mica sheets in the surface force apparatus made by Horn and Israelachvili [6,7] clearly showed that the interaction between two surfaces becomes oscillatory as the macrosurfaces are brought together because of the finite size of the solvent molecules. Furthermore, these authors showed that the measured interaction energy is more repulsive than that given by the DLVO theory. In other sets of the surface force apparatus experiments by Kekicheff et. al [8-10] with confined supramolecular fluids, such as micellar solutions and microemulsions, the oscillatory force caused by the supramolecule species has been measured. Further experiments by Wasan and Nikolov [11] using differential and more common interference methods, low-angle transmitted light and Kossel diffraction techniques, digital optical imaging, and video microscopy have clearly established the presence of structural forces and depletion forces as have theoretical simulation studies of particle layering and in-layer particle ordering in colloidal suspensions confined between the film surfaces. These important experimental and theoretical findings revealed the role of excluded volume forces and it is becoming apparent that it is time to move beyond the colloidal primitive model and consequently move further beyond the classical DLVO theory. A great advantage of a statistical mechanical approach is that once an adequate model is formulated and an appropriate level of an approximate description is established, a broad set of properties (including thermodynamics, structure and dynamics) and their origin and interdependence may be examined

Structure and Layering of Fluids in Thin Films

261

simultaneously within the framework of the same (theoretical) "experiment". The recent development of theoretical approaches and their results have already exposed the inadequacy of the DLVO model. This has been discussed previously [12] for the dynamics of colloidal suspensions and illustrates that the continuum has a strong effect via Stokesian friction and hydrodynamic interactions, but it is still widely believed that the discrete nature of the suspending fluid has no influence on the static properties. However, the statistical mechanical approaches [13-15] for aqueous and organic solutions, with the solvent and solute particles taken into account explicitly, agree with the experimental observations. We hope that even such a brief discussion of the state of the art of confined colloidal suspensions and molecular solutions highlights sufficiently the applied and basic impact of the layering phenomenon and show the existing problems and the perspective for future development. The particular goal of this chapter is to highlight the important role that the non-traditional structural forces, seen originally in experiment, play in the confined colloidal suspensions and properties of electrochemical interfaces and show the link between statistical mechanical theory and experimental observations. We argue that an adequate understanding of confined colloidal suspensions cannot be based on a colloid primitive model that neglects the discrete nature of the suspending medium. To proceed with this we have applied the hard-core models for the suspending medium to electrochemical interfaces and confined colloidal suspensions and developed a formalism based on the Ornstein-Zernike relation that treats homogeneous bulk fluids, electrochemistry, and colloidal science in a unified manner. In particular, we report some statistical mechanical results for the local density, electrical potential, and disjoining pressure and interaction energy of a film formed from a colloidal suspension. The scope of this chapter is as follows. In the next section we introduce the general statistical mechanics model of the complex colloidal system and present the way in which the Ornstein-Zernike equation can be applied to the dispersion problem. Section 3 is dedicated to the non-charged systems while in Sec. 4 we discuss the application to charged systems. The layering phenomenon in the films formed from non-charged and charged colloidal suspensions is discussed in Sec. 5. A summary and outlook are collected in Sec. 6. 2. STATISTICAL MECHANICS OF LAYERING PHENOMENA The system we are interesting can be modeled in a general manner as a homogeneous (or bulk) M-component fluid mixture whose components have diameters dk and densities Pk - Nk IV ; where Nk is the number of particles of species k, V is the volume, k-\,...,M. There is also a giant (G) particle, whose diameter is D, that represents a confining surface or colloidal particle.

262

D. Henderson, A.D. TrokhymchukandD.T. Wasan

We assume that there are only one or two giant particles in the mixture. Thus, the giant particle is present in zero concentration, PQ = 0 . Further, we assume pGD = 0 ; otherwise, the giant particle would affect the entire fluid. In other words, the giant particles are part of the bulk fluid. The system as whole can be called a suspension with the giant particles referred to as the species belong to the suspended phase, while the M-component fluid mixture forms the suspending fluid. 2.1. Ornstein-Zernike equation The properties of a suspending fluid near one or two giant particles can be obtained by computer simulation, using either the Monte Carlo (MC) or molecular dynamics (MD) techniques, or by means of the integral equation theory (IET) based on the Ornstein-Zernike (OZ) equation [16]

hxfli.R\2) = c^i.R\2) + Y,Pv\hXvi.Rn)c^i.R%i)^,

(1)

v

where the Greek subscripts range over all the suspending fluid components 1, ..., M, as well as giant particle G, and r a is the position of particle « . Finally, The functions hAu(R)= h^x(R) = gA/l(R)-\ are called the total correlation functions (TCFs) for a pair of particles A, and M that are separated by the distance R. The functions 8xn (R)= 8M (R) are the pair or radial distribution functions (RDFs). These functions give the local probability of a pair of particles being separated by the distance R , and are normalized so that the RDF is unity when R is large while the TCF vanishes when R is large. The functions C A/Y (-^) = C/U (^) are the direct correlation functions (DCFs) that specify the direct correlations between two particles. The integral on the RHS of Eq. (1) (a convolution integral) is the indirect part of the total correlation function. The idea behind the OZ equation can be seen most easily by considering a fluid consisting of only one component. At low densities, there are only direct correlations. Thus, h(Rn) = c(Rn).

(2)

At higher densities, a third particle comes into play. As well as being directly correlated, particles 1 and 2 are indirectly correlated by particle 3 that is directly correlated with either of particles 1 or 2. It is convenient to write this indirect term by a convolution integral. Thus,

Structure and Layering of Fluids in Thin Films

h(Ri2) = c(Rl2) + pjc(Rl3)c(R32)dr3

.

263

(3)

Continuing when more extended clusters contribute, we obtain h{Rn) = c(Rx2) +

p\c{RX3)c(R32)dr3

7 r + p2 \c(R]3)c(R34)c(R42)dr3drA

(4) +...

A convolution integral in real space is an algebraic product in Fourier space. It is common to call a series such as that in Eq. (4) a chain sum. We define the Fourier transform (FT) of a function f(R) by

J(k) = - ^ \Rf(R) sin(kR)dR k o

(5)

and take the FT of Eq. (4), yielding

h(k) = ?(k) + p£2(k) + p2c\k)

+ ....

(6)

Summing

m=

Hk)

•

(7)

l-pc(k) Equation (7) can be written as )i(k) = c?(k) + ph(kMk),

(8)

which in real space is h(Rn) = c{R{2) + p\h{R,3)c{R32)dr3.

(9)

This is the OZ equation for a single-component fluid. Equation (1) is a straightforward generalization of Eq. (9) for a multi-component fluid. The OZ system of integral equations is only an identity that defines the DCFs. By itself this system of equations is not the basis of a theory. However, by coupling the OZ equations with an approximate closure, a potentially useful theory results.

264

D. Henderson, A.D. TrokhymchukandD.T. Wasan

In this chapter, three closures will be mentioned. First, there is the PercusYevick (PY) equation [17], h^{.R)-c^{R) = y^(R)-\,

(10)

with yAM(R) = exV[J3uZu(R)]gAM(R)

(11)

called the cavity function or background function. The function u^ (R) is the interaction potential between a pair of particles and ji = \l kBT, with kB and T being the Boltzmann constant and temperature, respectively. A second closure is the hypernetted chain (HNC) approximation K(R)-cx^R) = \nyx^R).

(12)

The H N C approximation is an offspring with many parents. For a full set of references, see Barker a n d Henderson [18]. T h e name comes from t h e fact that, in contrast to the simple chains of Eq. (4), more complex chain integrals with cross links have been included. Many model systems consist of particles with a hard core Too,

R < dAu

"*<*>-Ucn «>
The infinity, oo, for R

impenetrability of the pair of interacting particles due to the presence of a hard core while wx (R) is responsible for the interaction beyond the hard core. Of course, real particles do not have an infinitely hard core. However, the repulsive part of the pair potential is steep so that approximating this repulsive part by a hard core is not unreasonable. A third closure that is useful for hardcore systems is the mean spherical approximation (MSA) [19], h^(R) = -\,

R
(14)

R>dXM.

(15)

together with

%W = - M , ( 4

Structure and Layering of Fluids in Thin Films

265

Equation (14) is an exact statement of the fact that particles with a hard core cannot overlap. Equation (15) is an approximation and can be regarded as linearized form of the HNC approximation. The advantage of the MSA is that useful analytic solutions can be obtained for a variety of model systems, including pure hard spheres, charged hard spheres and hard spheres with a dipole. The PY approximation is identical to the MSA for hard-sphere fluid, where w^M (R) = 0. Perhaps this is why the PY approximation yields a useful reliable analytic result for hard spheres but is not particularly useful for most other fluids. The HNC approximation does not yield analytic solutions. It tends to be useful for many model systems with long range interactions but not for pure hard-core systems. For many model systems, the MSA approximation combines the advantages of the PY approximation and HNC. Because the MSA yields analytic results, our attention here will be directed to this approximation. For historical reasons, when applied to hard-sphere systems, the MSA will be called the PY approximation. 2.2. Interfacial applications of the Ornstein-Zernike equation Returning to the OZ equation, Eq. (1), and letting for the density pG of giant particles be vanishingly small, we obtain a system of three equations

htJ{Rn) =Cij{Rn)+Y,Pk

,

(i6)

+ Y.Pk lhGk(Ru)cki(R32)dr3,

(17)

K(*,3:M*32)^3

k

h-Mi)

= cM2)

k

and hGG(RX2)

= cGG{Rn)

+ Y^Pk lhGk(Rl3)ckG(R32)dr3,

(18)

k

where subscripts are range over the suspending fluid components only, i.e., i,j,k = 1,2,...,M. The first equation, (16), is just the OZ equation for the homogeneous M-component fluid mixture. Equations (17) and (18) involve the giant particles and, due to this, play an important role in the statistical mechanics of inhomogeneous fluid systems, i.e., systems involving mesoscopic objects. Namely, Eq. (17) describes the fluid inhomogeneity by means of the local density profiles,

266

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

of the suspending fluid species / near the giant particle or surface. In turn, Eq. (18) describes the correlations between two giant particles mediated by the suspending fluid, and yields information about the effective interaction W(R) between a pair of giant particles via the exact relation W(R) = -kBT\ngGG(R).

(20)

The function W{R) also is known as potential ofmean force. Equations (17) and (18) are applicable to any size D of the giant particle. If D » dj, for practical purposes it is more convenient to rewrite Eqs. (17) and (18) using as variables different definitions of particle separations for each equation. First we use the particle center-to-center separations, Rt , as variables. The volume element is dr3

=2*Y23dRudR23,

(21)

where Ry are the three sides of a triangle. Since diameter of giant particles D is large, it is better to change variables so that the distance from the surface of the giant particle is used as the variable. Thus in Eq. (17), we can introduce variable z = Rn-D/2,

(22)

that denotes the distance of the center of a fluid particle from the surface of a giant sphere, and s = Rl3-D/2.

(23)

Dropping the subscript G for an easier notation, Eq. (17) yields the equation due to Henderson, Abraham and Barker [20], 00

h-Xz) = c,(z) + In^pj j

00

\ hj{s)ds \tClJ{t)dt. -oo

(24)

\:-s\

Equation (24) is often called the HAB equation. To transform Eq. (18), we introduce variable x = Rn-D,

(25)

Structure and Layering of Fluids in Thin Films

267

that is the normal distance or gap width between the surfaces of the two giant spheres, and s = Ru-D/2

(26)

and t = R23-D/2.

(27)

Thus, Eq. (18) has the form 00

GO

h(x) = c(x) + xDYJPj \hj(s)ds \cj(t)dt, j

—00

(28)

X—S

a result obtained by Attard et al [21] and Henderson [22]. Again the subscript G has been dropped for a more convenient notation. The HAB equation, (24), in accordance with the definition (19), usually serves to yield the local density profiles /?, (z) of the suspending fluid species in the vicinity of a giant particle (or near a single wall). In addition, it yields the local density profiles pt (z, x) between the two surfaces separated by a normal distance or gap width x [sometimes instead of x we will use notation H that is more common to denote the thickness of a slit-like film]. Statistical mechanics provides a relation between the local density pt(z,H) and the force/unit area f(H) that the species of suspending fluid exert on the inner side of the slit-like film surfaces in the direction normal to the surfaces, f(H) = -kT^ i

)du'^H>)Pl{z,H')dZ.

(29)

o

where u^z,!!) is the "bare" interaction between fluid species / on the distance z from confining surfaces fixed at separation H. Very often the force / ( / / ) is referred to as the solvation force; f(H = <x>) corresponds to the bulk fluid pressure p , i.e., pressure of a homogeneous phase of suspending fluid that is in an equilibria with film fluid. The solvation force measured relative to the bulk pressure defines the so-called disjoining pressure, Tl(H) = f(H)-p.

(30)

268

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

In practice, the disjoining pressure can be measured by displacing one of the surfaces a distance dH and calculating the work per unit area to bring the surfaces from the separation H to infinite separation. Thus, the film energy per unit area, E(H), can be calculated as CO

E{H)=

\U{H')dH'. //

(31)

Alternatively to the HAB equation, Eq. (28) usually serves to obtain the interaction energy between a pair of giant spheres. In particular, applying the HNC closure (12) to describe the correlations between a pair of suspended giant spheres, Eq. (28) becomes OO

In g(x) + pu{x) = TTD^PJ j

CO

\hj{s)ds \Cj{t)dt, 0

(32)

x-s

where u(x) is the "bare" interaction energy of the two giant spheres at the gap width x. Taking into account relation (20), the effective interaction between the two giant spheres mediated by a suspending media, has form QO

00

fiW{x) = pu(x) + «£>£PJ \ hj (s)ds jCj (t)dt. j

(33)

X-S

-OO

It is useful to note that Eq. (28) and consequently Eqs. (32) and (33) can be easily differentiated to give d\h(x) - c(x)]

n *^ = ~~zDTPj

'

^

^

j

"r. , , ,

, ,

\hj (s)cj(*"

s ds

)

.. .. (34)

•

-00

Therefore, the force F(x) between the two giant spheres separated by gap distance x is given by „ „

Fix) =

dW(x)

du(x) n -^

%, . . ,

.,

- ^=

^ - -£>£pj \hj(s)cj(x - s)ds.

dx

dx

2

, J

._,.

(35)

J -00

Thus, the HAB equation, (24), in conjunction with Eqs. (29)-(31), provides a description of a slit-like geometry while Eq. (28) combined with Eqs. (32)-

Structure and Layering of Fluids in Thin Films

269

(35) provides a route to the sphere-sphere geometry. The connection between these two geometries can be obtained from the Derjaguin construction [23],

nD

nD

dx

which gives the energy per unit area of two flat surfaces in terms of the force per radius between two giant spheres. We recall that here x = R-D is the gap width between two spherical surfaces and H is the separation between two plane parallel surfaces. Another form of the Derjaguin approximation concerns the disjoining pressure between two flat walls can be expressed through the second derivative of the potential of mean force between two giant spheres

dH

nD

dx2

Equations (36) and (37) are obtained by regarding the giant spheres as being composed of a collection of planar integration elements and integrating. If we suppose that the same approximation has been used in the set of OZ equations (16)-(18), then Eqs. (36) and (37) can be used to validate the Derjaguin construction. Summarizing this section, we wish to make some general and useful comments. First of all, we note that the initial set of the three OZ equations (16)-(18) form a hierarchy. The homogeneous equation, (16), can be solved without reference to Eqs. (17) and (18) that involve the presence of the giant particles. Equation (17) requires as input the DCFs of the homogeneous fluid either from Eq. (16) or some other source, such as a computer simulation. Equation (18) requires as input the correlation functions (or local densities) for the suspending fluid near a giant particle, either from Eq. (17) or some other source. Both Eqs. (16) and (17) can be solved numerically using an iterative algorithms. In contrast, a numerical solution of Eq. (18) does not require an iterative algorithm. Secondly, in applications to a particular problem, it is not necessary to use the same closure for each equation of the set (16)-(18). For example, in electrochemical applications it is often convenient to use the MSA closure for Eq. (16) and the HNC closure for Eq. (17). This procedure is called the HNC/MSA theory. Finally, in Eq. (18) the correlation functions for giant particles appear only on the LHS of this equation. Thus, applying a particular closure for the giant particle correlations need not relate to the closures used for the correlation functions of the suspending fluid on the RHS of Eq. (18). For example, using

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D. Henderson, A.D. Trokhymchuk and D. T. Wasan

the PY closure (10) to describe the correlations between giant spheres, Eq. (18) becomes CO

y(x)-l

= nD^PJ j

GO

\hj(s)ds -oo

Jc,(t)dt.

(38)

x~s

By comparing Eq. (38) with Eqs. (32) and (33), we obtain /3W(x) = -\ngHNC(x)

= -hPY{x),

for x>0,

(39)

where use has been made of the fact that for hard spheres y(R) = g(R) outside the hard core. The result (39) is consistent with the energy of interaction W{x) being proportional to the diameter D of the giant spheres. Thus, the HNC approximation predicts correctly that W{x) is proportional to D whereas both the PY approximation and MSA lead to the incorrect prediction that W(x) is proportional to the logarithm of D. This is because both the PY approximation and MSA are linearized versions of the HNC approximation. If the PY approximation or the MSA are used for the giant particle correlations then the ansatz, /3W(x) = -h(x),

(40)

should be applied. This ansatz is consistent with Eq. (39) and is equivalent to using the HNC for the correlation between giant spheres. In this case, using the MSA or PY closures for Eqs. (16) and (17), yields the HNC/MSA/MSA or HNC/PY/PY approximations, respectively. 3. UNCHARGED FLUIDS NEAR SURFACES 3.1. Simple hard-sphere fluid The simplest system to which the equations of the previous section can be applied is giant hard spheres dispersed in a one-component (or simple) hardsphere fluid. Wertheim [24] and Thiele [25] have solved analytically the OZ equation (16) with the PY closure (10) for a homogeneous hard-sphere fluid and obtained the DCF and thermodynamic properties. Subsequently, Wertheim [26] obtained the Laplace transform (LT) of the RDF for this fluid. Baxter [27] has given an alternative, and in some ways simpler, solution in terms of FTs and contents himself with the DCF and thermodynamic functions. Barker and Henderson [18] used Baxter's method to obtain the RDF in real space for d < R < 2d (d is fluid particle diameter) analytically. This method can be

Structure and Layering of Fluids in Thin Films

271

extended to greater values of R as has been done numerically by Perram [28]. Alternatively, Smith and Henderson [29,30] have inverted analytically the Wertheim's LT of RDF for d < R < 6d using a zonal expansion technique. However, our main interest here is not a homogeneous hard-sphere fluid but the suspending hard-sphere fluid with the presence of giant particles. The starting point for such study is a binary mixture of hard spheres. Lebowitz [31] has extended Wertheim's analysis to obtain, within the PY approximation, the LTs of RDFs and thermodynamics of a mixture of hard spheres. Using Lebowitz's result, Henderson [14] has obtained the LTs of RDFs for the system composed of giant spheres diluted in a single-component (call it species 1, since M = 1) hard-sphere fluid. Together with Wertheim's result [26], these LTs are

and

£|#)1,'"|!'"l-"A'!';-M'ti|lO, -Lx(s)e~sd' +S(s) where

(43)

variable s is the LT variable and 77, = npx 16. The fluid particle

diameter is dx=d, and t]xd\ = is the volume fraction of the suspending fluid. The functions L\(s) and S(s) will be defined shortly. The inverse LTs (41)-(43) can be evaluated by means of a line integral in the complex plane, evaluated analytically usually by residues. Since the giant spheres are present in extreme dilution (pG = 0), the basic behavior of all correlations in the system must be that for a suspending fluid. This yields the result that the denominator in LTs (41)-(43) is the same. Henderson [14] exploited this fact and has related all three LTs to a single LT that he inverted analytically using a zonal expansion. These results are plotted in Fig. 1. Indeed, the profiles of all three curves are quite similar having an exponentially

272

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

decaying oscillatory shape with the same "periodicity" and decay length but with a different magnitude and phase. As is seen in Fig. lb, the fluid density near the giant sphere (or, for that matter, a surface) is stratified or layered just as it is in the vicinity of a fluid sphere in Fig. la. These results, first given in Ref. [20], were perhaps the earliest indication of layering near a surface. Later, this has been observed experimentally [7,11,32]. An alternative, easier but numerical, method to calculate the correlation functions in real space can be developed by converting the LTs into FTs that can then be inverted numerically by simple quadrature. With this method one can obtain numerical results for any value of R. An advantage of the inversion by means of residue theorem is that it provides one with simple analytical results for the correlation functions at short distances as well as in the asymptotic regime of large R. In particular, for the correlation function between a pair of giant spheres at their contact, x = 0, we obtain h(0)=^^.D,

(44)

2(1 -f d while at large distances x the same correlation function behaves as [15]

h(x) ~ 2 | yHS | cos{coHSx + arg{yHS})exp(-/cHSx),

(45)

where coHS and KHS are the real and imaginary parts of the pole of c[g{R)\ with the smallest imaginary part, and yHS is the residue of £.[g(R)] at the same pole. The latter result (45) is plotted in Fig. lc as well and we can see that the agreement between a leading asymptotic term (45) and a complete FTNC/PY/PY result (43) is rather good except for short distances of about one layer of suspending fluid, where an asymptotic formulae should not be expected to be applicable. The results of Eqs. (44) and (45) are valuable, since, in accordance with ansatz (40), both provide information on the effective interaction between the giant spheres mediated by the suspending fluid. Recently, Roth et al [33] exploited the fact that the asymptotic term (45) describes accurately the oscillatory structure of the total correlation function and have developed a simple parametrization for the interaction energy W(x) between two macrosurfaces suspended in an ordinary hard-sphere fluid. Their parametrization provides an accurate fit to the density functional theory (DFT) results as well as to the existing computer simulation data.

Structure and Layering of Fluids in Thin Films

273

Fig. 1. Correlation functions for a pair of giant spheres of diameter D suspended in a simple hard-sphere fluid. The suspending fluid consists of spheres of diameter d whose volume fraction is <j> = 0.35 . Part a gives the RDF of a bulk phase hard-sphere fluid. Part b gives the fluid-sphere RDF, which is the normalized density profile of the suspending fluid near a giant sphere or a flat surface. Part c gives the TCF of the pair of giant spheres which is related to effective interaction or mean force potential; the dashed line represents the calculations using a leading asymptotic term (45).

In Fig. 2 we plot the interaction energy, W{x), obtained from Eqs. (40) and (43) together with DFT-based results of Roth et al [33]. Note that the energy is oscillatory and exhibits stratification. As the two giant spheres are brought closer together, layers of suspending fluid species are ' squeezed' out. At very close separations, all of the suspending fluid species have been squeezed out. When the gap between the pair of giant spheres is depleted of the species of suspending fluid one may speak of a depletion force that is attractive. This effect was first noted by Asakura and Oosawa (AO) [35], who examined the depletion force at low densities of the suspending fluid and found W(0) = -3D(#/2J. We can see that in the low density limit, expression (44) reduces to the AO result [35]. This is not surprising since the PY theory is known to be correct at low densities.

274

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

Fig. 2. Interaction energy W between two giant spheres whose diameter is D, and disjoining pressure n , between two plane parallel surfaces suspended in a simple hard-sphere fluid. The suspending fluid consists of spheres of diameter d whose volume fraction is <j> = 0.314 . The dashed lines are the results of the HNC/PY/PY approximation while the solid lines represent the parametrization of Roth et al [33] for the interaction energy, and that of Trokhymchuk et al [34] for the disjoining pressure. The symbols for the disjoining pressure correspond to the MC data of Wertheim et al [38],

The agreement of the HNC/PY/PY result [ Eqs. (40) and (43) ] with the DFT-based result is remarkably good for all range of separations except for those separations near contact at x = 0 , where the agreement is only qualitative. The result of Eqs. (44) is larger in magnitude by roughly a factor (1 - £,y2. This is the penalty for using the PY approximation to treat the fluidfluid and fluid-giant sphere correlations. Beyond contact, as the gap width between macrosurfaces increases, the interaction energy calculated within HNC/PY/PY approximation approaches the DFT-based data. Particularly, as we can see from Fig. 2, beyond the depletion region, defined by the position of the main repulsive maxima in the energy profile, both results are practically indistinguishable. This is an expected result, since, as we already noted [36,37], both approaches have common roots. Trokhymchuk et al [34], using the asymptotic term (45) for the interaction energy W(x) and Derjaguin construction (36) and (37), have given simple analytical expressions for the film energy per unit area E(H) and disjoining pressure Tl(H) of a hard-sphere fluid between two flat surfaces, as a function of suspending fluid density. These expressions are parametrized to satisfy with some known exact relations for a confined hard-core fluid and are designed to be easily implemented in the calculations of film properties in the various

Structure and Layering of Fluids in Thin Films

275

applications of the type reported recently by Wasan and Nikolov [3]. Figure 2 shows the disjoining pressure Tl(H) exerted by a hard-sphere fluid film that is calculated from parametrization of Trokhymchuk et al [34] and compared with MC data for the same system studied by Wertheim et al [38]. 3.2. Binary and many-component hard-sphere fluid and size polydispersity It is interesting to generalize Eq. (39) to an arbitrary number of components in the suspending fluid. Here we present such a generalization that is intuitive but not unreasonable. The result [37] is 3^?]le-sd!-(3S2s2/2-3S]s

£fe(*)] = ^

sld+d)

^

i<j

+ 3S0) d

D,

(46)

i

where

hiJ=36TJiTiJ(di-dJ)2,

(48)

Li:(s) = 12f7,.[(1 + ^3 /2> 2 J,. + (1 + 2S3)s] ^ , » + £ [ 1 8 7 ^ / 4 ( 4 " dj)s2 + ^ ( 1 - sdtJ)]

(49)

j*i

and S(s) = \25Q(l + 2S3)s-\SS22s2

-6S2(\-52)s3

-(l-S3)2sA (50)

+ ^dhiJ[l-S(di+dj)] The definitions of L^s) and S(s) are based on the definitions of Lebowitz, changed slightly for an easier generalization to an arbitrary number of components, and differ somewhat from the definitions used by Wertheim. Because a large number of components leads to a large number of zones, Eq. (46) is not suitable for a zonal expansion. However, numerical inversion is

276

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

not difficult. The energy and force in a case of binary suspending fluid (M = 2) with a size ratio, small-to-large 1:10, are plotted in Fig. 3. Because of the large size asymmetry, the depletion interaction scenario observed for two giant spheres in a one-component suspending fluid composed of only large particles, is significantly affected when the fine species of suspending fluid are taken into account. The gap between two giant spheres, that is depleted of the large particles in a one-component fluid, becomes filled by the fine particles in the case of a two-component fluid. Now the small species of diameter dx = ds exhibit stratification since they are confined by giant spheres of diameter D and the large suspending fluid species of diameter d2= d. This changes qualitatively the shape of both the interaction energy and force profiles that now show fine oscillatory structure at separations near the contact of the giant spheres; the oscillations are governed by the diameter of small species. To a reasonable approximation, the energy and force between two giant spheres suspended in a bidisperse fluid can be viewed as a superposition of single-component results obtained using the diameters and, perhaps, effective densities of the both fluid components. Similar trends were seen previously for a bidisperse colloidal suspension between two flat parallel walls [39]. This, of course, is to be expected since energy per unit area, E(H), and disjoining pressure, Yl(H), for a pair of flat parallel surfaces are related simply, by means of Eqs. (36) and (37), to W(H) and its derivatives.

Fig. 3. Interaction energy, W, and corresponding force, F, between two giant spheres whose diameter is D, and dissolved at infinite dilution in a two-component hard-sphere fluid thatconsists of the small spheres of diameter d\ = ds and volume fraction ^ =0.15 and the large spheres of diameter d2 = d = \0ds and volume fraction ^ 2 = 0-20.

Structure and Layering of Fluids in Thin Films

277

To a reasonable approximation, the energy and force between two giant spheres suspended in a bidisperse fluid can be viewed as a superposition of single-component results obtained using the diameters and, perhaps, effective densities of the both fluid components. Similar trends were seen previously for a bidisperse colloidal suspension between two flat parallel walls [39]. This, of course, is to be expected since energy per unit area, E(H), and disjoining pressure, Yl(H), for a pair of flat parallel surfaces are related simply, by means of Eqs. (36) and (37), to W(H) and its derivatives. Similar results are plotted in Fig. 4 for a four- and ten-component suspending fluids. Each of these multi-component fluids include the fine particles as in the case of a binary fluid. Again, there is layering but the "period" of the layering reflects the sizes of all the constituent species of suspending fluid. In general, an increase of the number of components of the suspending medium shows a tendency to destroy the fluid layering. As the giant spheres are brought together, the interfacial region becomes depleted of the larger fluid particles but can still be filled with smaller particles. Thus, oscillations are present at a wide range of separations determined by the size of the smallest and the largest suspending fluid particles.

Fig. 4. Interaction energy, W, and corresponding force, F, between two giant spheres whose diameter is D, and dissolved at infinite dilution in a four and ten-component hard-sphere fluid. In each case suspending fluid consists of fine species of diameter d\ = ds and volume fraction \ = 0.15 . In the four-component case (solid line) the other species have diameters d2 = d = \0ds , and (\±0.5)d diameters are d =\0ds, (\±Q.2)d

whereas in the ten-component case (dashed line) the , (]±0A)d

, (1 ± 0.6)rf , and (l±0.8)J . In both cases,

the total volume fraction of the suspending species, other than fine component 1, is 0.2.

278

D. Henderson, A.D. TrokhymchukandD.T. Wasan

At this point we wish to show how our results for a many-component hardsphere suspending fluid can be extended to describe the polydisperse hardsphere suspending fluid with a continuous size distribution /(<x) that is normalized to be a probability density, i.e., \f{cr)d(j = 1. In a rigorous way the concept of polydispersity has been discussed by Salacuse and Stell [40] as well as others and applied to the polydisperse fluid of hard spheres by Blum and Stell [41]. Following their recipe, the introduction of probability density /(cr) projects a countable infinitude of the fluid components into a continuous distribution of sizes, replacing the species number densities pt by the average density //(cr,) where p is the fluid total number density. Then Eq. (46) has the form £[g(R)] =

5

, , , 3?7\f{x)e~sxdx - (\S2s2 - 3S]S + 3S0) 2*12 \\f{x)f{y)h{x,y)e-s{x+y)dxdyr,\f{x)L{x;s)e-sxdx

,

(51)

+ S{s)

where Sn=tjjf(x)xadx,

(52)

h(x,y) = 36(x-y)2,

(53)

L(x; s) = 12[(1 + —> 2 x + (1 + 2 S3 )s] 2

(54) 2

+ n J/O0[l W (x ~y)s + h{x, y){\ - s

*^)]dy,

S(s) = US0(\ + 2S3)s - 18£2 V - 652(\ - S3)s2 - (1 - 53)2s4 + \l2 \\f(x)f(y)h(x,y)(\ In these equations rj = npl6.

- s[x + y])dxdy

Structure and Layering of Fluids in Thin Films

279

Figure 5 shows the effect of polydispersity on the interaction energy between two giant spheres suspended in a fluid with the Shultz size distribution

( V+z

where (a) is the average diameter and the degree of polydispersity is controlled by parameter z. For the sake of comparison, the results of AO polydisperse model at a low density are shown as well. The AO results are calculated from oo

pWA0{x)=7^pd\f{G){(T-xfda,

(57)

x

where x = R-D. We can see that results of both, Eqs. (51) and (57), are quite similar showing that an increase of the polydispersity at low volume fraction of the suspending medium leads to an increase of the range of depletion attraction.

Fig. 5. Effective interaction energy, W, between two giant spheres whose diameter is D, and dissolved at infinite dilution in a hard-sphere suspending fluid with different degrees of size polydispersity. Part a shows results at a low volume fraction 0.01; the dashed lines represent the results of the AO polydisperse model. Part b shows results at a high volume fraction 0.3925.

280

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

As we already learned from the results of Fig. 2, a high volume fraction of monodisperse suspending medium promotes an oscillatory structural interaction between suspended species. The results presented on part b of Fig. 5 shows that introducing size polydispersity at a fixed volume fraction results in a decrease of both the attractive well at x - 0 and structural repulsive peak as the polydispersity goes up. Eventually, this leads to a smoothing the oscillatory features of the interaction energy profile. The effects of size polydispersity, similar to those of Fig. 5, have been observed by Walz and Sharma [42]. 3.3. Fluid of dumbbells, flexible linear chains and more complex aggregates The equations in the previous subsections are appropriate for a suspending fluid composed of spherical hard-core species with different diameters. An extension of the analysis to the suspending fluids comprising non-spherical particles can be provided by generalizing the set of OZ equations (16) - (18). One of the ways to do this is by using the ideas and models from the theory of site-site associating fluids. We will illustrate this by considering three models of a suspending fluid: a fluid formed from dimerizing hard spheres or dumbells, a fluid of flexible linear chains and a network-forming fluid. To formulate the fluid models we assume that the hard-sphere particles that compose a suspending fluid are of the same diameter d and each may have one, two or four attractive sites placed on the particle surface. The potential of interaction between two fluid particles for such model fluids becomes angulardependent and consists of a hard-core repulsion as in an ordinary hard-sphere fluid, plus a contribution describing a highly directional associative (AS) attraction between the sites of two different fluid particles,

fljn

^

\~£ab>

I 0,

x

ab

<x

c

r,-Q\

xab>xc

where 1 and 2 denote the position and orientations of the two fluid particles while a and b are the notations for sites on the particle 1 and 2, respectively. The variable xab is the distance between the sites a and b of the particles 1 and 2; an associative bond between two particles is formed if their two sites are within the distance xc. The parameter sah reflects the strength of an associative interaction; when sab - 0, the models converge to the ordinary hard-sphere fluid. Models with one and two sites describe the dimerizing and polymerizing fluids, respectively [43-45]. The four-site model mimics symmetrical and nonsymmetrical network-forming fluids that may consist of very complicated

Structure and Layering of Fluids in Thin Films

281

aggregates, e.g. linear branches, loops, crossing rings, etc [46]. Therefore, the four-site model is able to reproduce various kinds of macromolecules and a network of bonds by appropriate choices of the energy parameter sah . The properties of these model fluids in a bulk phase have been investigated intensively during the last decade [45-49] using a multidensity version of the OZ equation (16) that is usually called the Wertheim OZ (WOZ) equation [43,44]. For a simplest case of a dimerizing fluid, i.e., with only one site per fluid particle, WOZ reads

hf{\,2) = cf (1,2) + X \Kr {\,l)PrEcf (3,2)
(59)

ye

where the Greek superscripts a,/3 denote the bonding status (equal to 1 when site is bonded and 0 when site is non-bonded, i.e., free) of the sites located on the pair of particles 1 and 2, respectively. In a Wertheim multidensity formalism [43,44] the fluid particle total and direct correlation functions //,-,-(1,2) and c,7(l,2) become matrices with the elements, h"p{\,2) and c,^(l,2), called the partial total and partial direct correlation functions, respectively; the fluid density p also is replaced by the density matrix whose elements represent the number densities of the fluid particles with a bonded status of their sites

k"]=[^ *"].

(60)

where p° refers to the density of non-bonded particles. In contrast, none of the partial correlation functions has the physical meaning; as in the case of a simple hard-sphere fluid the physical meaning is attributed to the total correlation function that in this case is a linear combination of partial correlation functions

^(l,2) = ^ 0 (l,2) + 2[ P ° k°1(l,2) + f /7 ° j ^'(1,2).

(61)

Moreover, the partial correlation function /z°°(l,2) is a part of the selfconsistency relation [43,44] that splits the total fluid density on the density bonded and non-bonded particles

282

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

p = p° + (p°J {[/z°°(l,2) + ll/;.f (\,2)d2,

(62)

where ftfs (1,2) - exp[-u°f (1,2)/kT]-1, is the associative analogue of the Mayer function. Of course, if there are more than one site per particle, i.e., in the case of polymer chain (two sites per particle) and network (four sites per particle) fluids, Eqs. (59) - (62) are more complex [45,46]. However, using the associative generalization of the PY closure and within the two-particle density formalism for a dimerizing fluid [43,44], and the ideal chain [45] and ideal network [46] approximations to the description of bulk polymerizing and network fluids, respectively, leads to a significant simplification in the theory. In particular, the WOZ equation (59) may be solved analytically in the zerolimit for the range of associative interaction (58), i.e., xc —>0, by a method analogous to that used by Baxter [27] in the case of hard-sphere model. This sticky limit for the association is considered with the angle-averaged associative Mayer function being the singular form:

(ff (1,2)) =-*"*(*> [(e-M-^y

1] = K8(R - d),

(63)

where K is the parameter uniquely determined by the strength of associative interaction between sites a and b. The calculations are simplified further by the assumption of energetic equivalence of attractive sites, i.e., sab = s. To show the effect of the non-spherical nature of suspending fluid on stratification phenomena, in contrast to preceding subsections 4.1 and 4.2, here we explore the route utilizing Eq. (17), i.e., route based on the HAB equation. The route that is based on Eq. (18) has not been elaborated for the associating fluids so far. The associative version of hard-sphere HAB equation has been proposed by Holovko and Vakarin [47] and for dimerizing fluid reads

Ko(Rn) = c%(Ru) +1 K f (Rn)pPrcJ?(h2)dr,,

(64)

Pr where hjG°(R) and cfg(R) are the partial total and partial direct fluid speciesgiant sphere correlation functions, respectively. The superscript 0, that corresponds to giant sphere subscript G, indicates the absence of any bonded states at the surface of giant spheres if an associative interaction of the type of (58) between fluid particles and giant sphere is not considered.

Structure and Layering of Fluids in Thin Films

283

Recently, Duda et al [50] have applied the associative HAB equations of the type of Eq. (64) to evaluate the local density p(z,H) = phlG(z,H) for the suspending fluid composed of dumbells, flexible linear chains and network aggregates confined to a slit pore. The fluid-wall total correlation function hiG{z,H) was calculated as a linear composition of the partial fluid-wall functions hiG(z,H) and h]G{z,H) in the way similar to the bulk fluid case, Eq. (61). Some representative density profiles are plotted in Fig. 6 as a function of fluid particle distance from the confining surfaces. To mimic the reality where confining surface can be lyophilic or lyophobic, the fluid species-giant sphere interaction, ujG(z,H), was used as a (9,3) Lennard-Jones potential with and without attractive term, respectively. For the comparison, the local densities of an ordinary hard-sphere fluid are shown as well.

Fig. 6. Density profiles p(z,H)of a simple hard-sphere fluid (thin solid line), fluid consisting of dumbbell-shaped particles (dashed line), linear flexible chains (short dashed line) and network-forming fluid (thick solid line) in a slit-like film. The film thickness is H = 8d.

284

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

The quite tall and narrow peaks that are seen in Fig. 6 next to the confining surfaces reflect the well defined surface layers for a hard-sphere fluid. These peaks decrease for a fluid of dimers, decrease and widen for polymer chains, and bifurcate and almost disappear for fluid composed of network aggregates, especially in the case of the repulsive fluid-surface interaction. As the distance from the substrate increases, the heights of the peaks decrease and their widths increase. Nevertheless, we still can observe at least three well-defined layers even for flexible linear chains. At the same time, it is worth noting that the left shoulder of the double first peak of the network fluid density profile, that we observe near the attractive surfaces, totally disappears near the repulsive surface. It means that the fluid comprised of species of irregular shape tends to segregate from the confinement diminishing the effect of stratification. To understand the qualitative trends in the film properties initiated by the shape of suspending fluid species and different level of fluid stratification, in Figs. 8 and 9 we show the results for the force between film surfaces that is mediated by suspending fluid. Experimentally the force/radius between the two crossed cylinders of radius a-D/2 can be determined directly in the method of Israelachvili [7]. To obtain results that are comparable with measurements we have added to the fluid mediated force the long range van der Waals contribution [the so-called Hamaker force] that arises from the direct interaction between the surfaces. This interaction is obtained by integrating the - R~6 term in (say) the Lennard-Jones interaction (a four-fold integration) over the volumes of the two interacting bodies. The result is complex. However, if the bodies are giant, the result is quite simple and in the case of the two crossed cylinders equals - ADlYlx. Thus, the calculated force, which can be compared with the observed data, has the form [see, e.g., Eqs. (31) and (36)]

F

^

a

=

^

\2x2

+ n [U(H )dH ,

(65)

I

where A is the Hamaker constant and H is the gap width between confining surfaces. Figure 7 shows, by a dashed line, the force/radius calculated according to Eq. (65) between a pair of macrosurfaces that are suspended in a hard-sphere fluid of density pd3 =0.85. Taking a hard-sphere diameter d =0.9nm such fluid roughly simulates octamethylcyclotetrasiloxane (OMCTS), which is the frequently investigated experimental system [7]. The inset shows the measured force law between two cylindrically curved mica surfaces in OMCTS [6]. OMCTS is an inert silicone liquid whose non-polar molecules are quasispherical in a shape. To take into account the effect of non-sphericality, we

Structure and Layering of Fluids in Thin Films

285

assumed that the OMCTS molecule is slightly elongated in one direction and this shape can be represented by a dumbbell composed of two fused hard spheres, each of diameter dd. Choosing the diameter dd from the condition that the volume of the dumbbell-shaped molecule is the same as the volume of the spherical molecule of diameter d = 0.9nm, we performed calculation of the force in a dumbbell fluid with diameter dd =0.73nm, maintaining fluid volume fraction the same as in a monomer hard-sphere fluid. We can see that two calculated force profiles differ with the dumbbell model improving an agreement with experimental data.

Fig. 7. Force between two macrosurfaces suspended in a hard-sphere fluid of diameter d = 0.9 nm [dashed line] and in a dumbbell fluid of diameter dd =0.73nm [solid line]. The dotted line shows the van der Waals force. The fluid volume fraction in both cases is the same, pd

= 0.7. The inset shows the measured force law between two cylindrically curve

mica surfaces in OMCTS with average molecular diameter - 0 . 9 nm. The inset is adapted from Fig. 5 of Ref. [6].

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D. Henderson, A.D. Trokhymchuk and D. T. Wasan

Figure 8 shows the predicted results for the force between two macroscopic surfaces in a chain and network-forming fluids. These two results can be compared with measured force laws between mica surfaces observed by Christenson et al [51] across straight-chained liquid alkanes such as ntetradecane and n-hexadecane, and by Gee and Israelachvili [52] measured in a branched alkane [iso-parafin] 2-methyloctadecane. We see, that the force in linear chain fluid still exhibits oscillatory behavior, similar to the film of spherical molecules, but is shifted to the attraction region. The force mediated by a network fluid is completely different: it is repulsive until a thickness of about 1 nm [of order of two-to-three single bead diameters] and then becomes attractive and almost totally monotonic. The overall qualitative agreement between the calculated and experimental results is quite good.

Fig. 8. Force between two lyophilic macrosurfaces in a fluid of linear chain (thin solid line) with d - 0.4 nm, and in a network-forming fluid (thick solid line) with d = 0.45 nm. The density of both fluids is pd =0.85 .The dotted line shows the van der Waals force. The inset shows the measured force laws between mica surfaces in straight-chained liquid alkanes such as n-tetradecane and n-hexadecane (molecular width —0.4 nm), and in the branched alkane (iso-paraffin) 2-methyloctadecane. The inset is adapted from Fig. 13.5 of Ref. [7], page 272.

Structure and Layering of Fluids in Thin Films

287

The repulsive force between lyophilic surfaces at small separations originates from the first adsorbed layer of suspending fluid and extends on the separations of order of the thickness of this layer. The fluid layering near substrate can be identified from the oscillations of local density [see Fig. 6]. Two density maxima that we observe in a network-forming fluid next to the surface can be treated as the spliting of a rather thick first layer, which remains disordered or amorphous. The local density in such layer is always higher than the corresponding bulk density, and the force exerted by the fluid on the surfaces is repulsive. The fact that the chain structure of the suspending fluid can result in an attractive or repulsive contribution to the solvation force has been pointed out by Israelachvili [7] who calls this effect a positive or a negative solvation force. 4. CHARGED FLUIDS NEAR SURFACES In electrochemistry and colloidal science, the suspending medium is usually an electrolyte containing charged species - ions and polar solvent molecules. Further, the confining surfaces - electrode and/or the colloidal particles are charged. The MSA can be applied to such complex systems. 4.1. Bulk electrolytes The simplest model of an electrolyte is a system of charged hard spheres (the ions) in a continuum dielectric medium whose dielectric constant is s (the solvent). To keep the model simple we assume that the dielectric constant inside the charged hard spheres is the same dielectric constant as that of the electrolyte so that we need not worry about induced charges on their surface. This model is called the primitive model (PM) of an electrolyte. If the charged hard spheres all have the same diameter, this model is usually called the restricted primitive model (RPM). Waisman and Lebowitz [53] have solved the OZ/MSA equations analytically for the RPM. Their results have been simplified by Blum [54], who has also solved the OZ/MSA equations for the more general case of differing diameters, i.e., for the PM. In the case of the bulk RPM, the ion-ion RDFs are given by

J

s(\ + Yd)2

R

where d is the ion diameter. For the moment, all ions are considered to have the same diameter. The function gHS(R) is the hard-sphere RDF, f(R) is a

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D. Henderson, A.D. Trokhymchuk and D. T. Wasan

function that will be defined shortly, and F is a parameter that is related to the Debye screening length, K , defined by

-2 =4 fl>,V,

(67)

through the relation K = 2Y(l + Td).

(68)

Note that at low concentrations, when K is small, F = K/2 .

The LT of f(R) is given analytically, L[f(R)] = -7

,

•

(69)

Henderson and Smith [30] have inverted this LT and obtained f(R) as an infinite sum that involves spherical Bessel functions. From Eq. (69), we see that for small K (low concentrations) L[f(R)] = — -

(70)

S+K

So that f(R) = e~KR.

(71)

Thus, at low concentrations, the MSA result for the ion-ion RDFs becomes fin n

a-K(R-d)

If we make the approximation, valid at low concentrations,

{\ + Ydf

( d\2 K =[\ + =l + xd,

(73)

Structure and Layering of Fluids in Thin Films

289

neglect d in the argument of the exponential, and use gHS(R) = l,

(74)

the MSA result becomes the Debye-Huckel (DH) result [55]. We have obtained the MSA result from an integral equation whereas the DH approximation is usually obtained from a differential equation [often called the Poisson-Boltzmann (PB) equation]. The DH or PB approximation is the zero-diameter version of the MSA for charged hard spheres. In the DH/PB approach, Eq. (15) is assumed to be valid for c(R) for all distances. The MSA is a systematic generalization of the DH theory (strictly speaking in its linearized version). The factor 1 of the DH/PB approach becomes gHS(R), the exponential function becomes f(R), and (1 + KT/) becomes 2 (1 + Yd) . We will see that the factor (1 + Yd)2 overcomes the asymmetry that is a problem in the DH/PB theory. In the DH/PB approximation, only the central ion is given a size. In the MSA theory, all ions are treated consistently. The appearance of g (R) leads to oscillations that reflect the stratification phenomenon. If the PM is used, the density of the hard spheres in an electrolyte is low and these oscillations are correspondingly small. However, if an explicit molecular model of the solvent is used, the hard-sphere oscillations become important. The function f(R) also oscillates but this is not very important as the concentration of ions is small, whether or not an explicit model of the solvent is used. The oscillations in f(R) become more important with divalent ions. In a molten salt, the oscillations of f(R) would be of even greater importance. The renormalized screening parameter, 2F, is smaller than K SO that the system does not become overscreened. In the DH/PB theory \lK can become smaller than d. This is an unphysical situation because it would mean that the screening atmosphere is inside the central ion. We have commented that the MSA is a generalization of the linearized DH or LDH theory. The HNC is an appropriate generalization for the nonlinear DH theory [which does not yield an analytic solution]. Our attention here is restricted to the MSA and LDH approximation. The generalization of Eq. (66) to the case where the electrolyte ions have different diameters is difficult. However, considerable progress has been made by Blum [54]. As long as most of the ions have the diameter d and there are only a very small number of ions with different diameters, the ion-ion RDF is given by

290

D. Henderson, A.D. TrokhymchukandD.T. Wasan

where gjj (R) is the RDF of a hard-sphere mixture. The symmetric treatment of the ions [lacking in the DH/PB theory] is apparent in Eq. (75) and is of particular importance in applications to charged colloidal suspensions. To progress beyond the PM, we must include a molecular model of the solvent. A particularly simple model of the solvent is obtained using a system of hard spheres together with the dielectric background. This model has been called the solvent primitive model (SPM). At first glance, this would seem to be an unpromising model of a polar solvent. However, this model does recognize that the solvent molecules occupy space. Within the MSA, results obtained for a bulk electrolyte using the SPM are similar to the results obtained above with the PM. The solvent molecules are just another species of charged hard spheres that happen to have zero charge. Thus, Eq. (75) still applies but the density of hard spheres now includes the density of solvent and is large and g{fs(R) has pronounced oscillations. The second term in Eq. (75), that controls the electrical response of the system, is unaffected by SPM. Of course, this separation of the charged and non-charged terms is a feature of the linearized nature of the MSA. It would not be occur in a nonlinear theory, such as the HNC approximation where the effect of the SPM would propagate to the charged terms. A somewhat more sophisticated model of a solvent is a system of dipolar hard spheres. Now there is no dielectric background. The dielectric constant is the result of the non-zero dipole moment, /z, embedded into the solvent spheres. Wertheim [56] has used the MSA to obtain the properties of the dipolar hardsphere model solvent. He found \6s = A2(l + A)4

and

(76)

Structure and Layering of Fluids in Thin Films

291

where ps is the number density of the solvent. Note that Eq. (76) is a cubic equation in X1 that can be solved analytically. For s=\, X=\ and for £-78, X-2.65. Thus, X is a weak function of s. These equations give the relation between dipole moment /j and s. Although approximate, this relation is an improvement over the earlier Onsager formula. The value of /J , corresponding to the dielectric constant of water, is greater than the dipole moment of water vapor molecules. The usual view is that at high densities Eqs (76) and (77) give an effective dipole moment that accounts for such effects as polarization. Blum [54] has solved the MSA for a mixture of dipolar and charged hard spheres [loosely called the ion-dipole model] and has obtained results for the thermodynamic properties for this model electrolyte. The MSA results for this ion-dipole model will be discussed further in the next section. 4.2. Electrolyte near an electrode Equation (75) can be applied to an electrolyte near an electrode. Many electrodes are planar but some, for example a hanging drop mercury electrode, are non-planar. The planar electrode can be regarded as a giant sphere. The RDF for a pair consisting of a giant sphere of charge Q and radius a - D/2 and an ion of charge qt and diameter d, or in other words the normalized density profile gt(R) = pt(R)f p of the ions near an electrode is

M w-a w- n ?*L,/(*-f-f) a s{\ + Td)2Ta

v

™

2 2J

or gi(z)

=g f ( z ) - ^ . O o / ^ - 0

(79)

where z = R-D/2 and % =- ^

(80)

is the electrostatic potential at the giant sphere. Equation (68) has been used to obtain Eq. (79) which can be rewritten as g,(z) =g -(z)-^/z-*a £K \ 2

(81)

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D. Henderson, A.D. Trokhymchuk and D. T. Wasan

where E is electric field at the electrode. The electrostatic potential of the electrode, V, is <J>0 plus the potential difference, Ed I'2s, across the region of thickness d 12 between the electrode and the distance of closest approach of the ions. If the PM is appropriate, the PB theory is called the Gouy-Chapman (GC) theory [57], which actually predates the DH theory. If the GC/PB theory is applicable, Ed

ir

.

Ed

E

^ = ^ + ^o = — + —• 2s 2s SK

(82)

Using the MSA, gives

v

( 83 )

= 4^'

s2F and with Eq. (68),

1 1+IW 1 d — = = - + - + ... 2T

K

K

(84)

2

the GC/PB theory is recovered. The region near the macrosurface 0 < z d 12, this is described by the GC/PB or MSA theories, is often called the diffuse layer. The whole system is called a double layer because there is a layer of charge on the electrode and a compensating layer of charge in the electrolyte. In this picture, the potential difference across the double layer consists of the potential difference across the inner layer and the potential difference across the diffuse layer. The double layer acts similarly to two capacitors in series. Indeed, there is some seemingly convincing experimental evidence [58] supporting C"1 =C-t}+C~D\

(85)

where CD is the diffuse layer capacitance, as predicted by the GC/PB theory, and CH is the inner, or Helmholtz or Stern, layer capacitance that is independent of the concentration and other properties of the ions. As will be seen shortly, this picture is overly simplified. As seen in Eq. (84), two terms appear in the MSA expression for V only if the expansion is truncated at two terms.

Structure and Layering of Fluids in Thin Films

293

To bring Eqs. (82) and (85) into agreement with experiment, the properties of the inner layer must be adjusted. One popular scheme is to say that the dielectric constant, s *, in the inner layer is different from the bulk value. Thus, Eq. (82) becomes

2s*

SK

A value s * =3~6 seems appropriate. A reduced value for s * seems reasonable since the solvent is expected to be less polarizable near the electrode. However, this empirical approach is less than satisfying. Is it reasonable to confine the interfacial region of the solvent molecules to a region right at the electrode? Is a sharp dielectric boundary reasonable? Maxwell's equations require a polarization charge at a dielectric boundary. This effect has been satisfied only in part because no attempt has been made to ensure that the induced charge does not disturb the GC/PB ionic profiles. For that matter, induced charges have been ignored at the electrode surface also. However, this latter point may be reasonable there since 78 is well on the way to infinity [the value of the electrode dielectric constant, assuming the electrode to be metallic]. Since the value of s* is empirical, one might argue that the effect of the induced charges has been included in the value of s *. For this to be the case one would need to establish that the induced charge at the interface is independent of concentration [or at least only weakly dependent].

Fig. 9. Ion density profiles pt(z) calculated from the GC/PB theory (solid lines) compared with MC simulation data (symbols). Part a gives typical results for a monovalent salt. Part b gives typical results for a divalent salt.

294

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

As was the case for a bulk electrolyte, the PB theory is the point charge [or low concentration] limit of the MSA theory. For the double layer, the PB theory is called the GC theory. In contrast to the homogeneous version of the PB [the DH theory], the GC/PB equations can be solved analytically even when the electrode charge is large and the equations are nonlinear. The nonlinear version of the MSA theory, the HNC approximation, requires a numerical solution. Linear or nonlinear, the GC/PB results for concentration, charge, and potential profiles are monotonic and the term double layer is sensible. In contrast, in the MSA or HNC approximations these profiles can oscillate, again reflecting the presence of stratification phenomenon. As is seen in Fig. 9 where a comparison of the GC/PB ion density profiles with MC simulation data is shown, this is particularly apparent for divalent salts where the oscillations in the density profile are large enough that there is charge inversion where the density of co-ions can exceed the density of counter-ions. In this case the charge in the layer of counter-ions near the electrode exceeds the magnitude of the electrode charge and there is a further layer of co-ions. The total charge in the profiles is still equal and opposite to that of the electrode as overall charge neutrality requires. However, rather than a double layer, there is a triple layer of charges. The oscillations seen in Fig. 9 are relatively mild. Much more pronounced oscillations would be seen if a molecular model for solvent were used. One result of the GC theory is kTYpAd/2)

= kTp +

F2

,

(87)

where p = ^ pt is the total density of the ions. This is a special case of the exact, within the PM or SPM, result due to Henderson et al [59] kT^Pi(d/2) = p + ^ ,

(88)

where p is the bulk pressure (including both hard sphere and electrostatic contributions) of the suspending electrolyte. The term on the LHS in Eq. (88) is the momentum transfer to the electrode. This must equal the sum of the pressure and the second term on the RHS, which is the electrostatic (or Maxwell) stress. The GC/PB result is a special case of the exact result for the case where p = pkT (i.e., uncharged point particles with zero diameter). Neglecting terms that are higher order in E than linear, Eq. (88) becomes

Structure and Layering of Fluids in Thin Films

295

kTy£dPi(d/2) = p.

(89)

In contrast, the MSA and HNC theories give, instead of p, a geometric or arithmetic mean of the hard-sphere values of p and p/3(dp/dp), in Eqs. (89) and (88), respectively. At low concentrations and not too large values of the dimensionless charge parameter, flq^jlsd, the difference with p is small. However, if the concentration is significant, the valence of the salt is large or if the temperature, dielectric constant, or diameter are small, p can be large or, more usually, small, resulting in large or small contact values for the ion density profile. In contrast, the GC/PB, MSA and HNC results for the contact values of the ion density can only equal or exceed the bulk density. Thus far, only the PM and SPM have been considered. Carnie and Chan [60], Chan et al [61] and Blum and Henderson [62] have obtained an analytic solution of the MSA equation for the ion-dipole model of suspending fluid. For ions of diameter d and solvent dipolar hard spheres of diameter ds that all of the same size ds - d, the normalized ion density profiles have the form gl(z) = gHS(z) + Ahi(z)

(90)

and the normalized solvent density profile has the form gs(z,0) = gHS(z)+ l3Ahs(z)cos0, where gHS(x)

(91)

is the normalized density profile for neutral hard spheres (ions

plus solvent molecules) near a hard wall. Although analytic, the MSA expressions are not explicit. However, at low ionic concentrations an expansion may be made and explicit results can be obtained. For example, the contribution to the potential resulting from an integration of gt(z) is VI =E +Ed +..., (92) K 2 and the contribution to the potential resulting from an integration of gs (z, 0) is

K,=[^(l-I]l^ + ....

(93)

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D. Henderson, A.D. Trokhymchuk and D. T. Wasan

The total potential is

K . £ + flf, + £rl) + .... SK

2s V

(94)

1 J

For the more general case of differing diameters where ds •*• d, Carnie and Chan [60] have shown that Eq. (94) becomes

V = - + ^-[d + ^ d , ) SK

2s\

A

+

....

(95)

)

It is to be noted that there has been a cancellation between terms of order 1 and (1 - s) I s to produce a term of order 1 / s. This is no problem with analytic expressions but might be a problem in numerical calculations when s -80 as a small error in either term might result in an appreciable error in V.

Fig. 10. Values of Ahs(z) electrode.

from the MSA for a 0.1M monovalent electrolyte near an

The parameter Cs is a normalization constant that is not relevant to our

discussion. The solid curve is the MSA result, calculated using the ion-dipole model, whereas the dashed curve is a plot of exp[-xr(z - d 12)]. Adapted with permission from Schmickler and Henderson [63].

Structure and Layering of Fluids in Thin Films

297

When the higher order terms are neglected, Eqs. (94) and (95) are formally identical to Eq. (86). However, the interpretation is quite different. The potential consist of two terms only if the expansion is truncated at two terms. Further the "inner layer" term in Eqs. (94) and (95) results from an integration over all space. The interfacial region for the solvent molecules is as diffuse as that of the ions. As long as the screening of the ions is incomplete, there is an electric field and, as long as there is an electric field, the solvent molecules will be oriented. We do not have two capacitors in series. If anything, we have capacitors in parallel. Note in Eq. (95) that ds is weighted by the factor (s-\)/A. that for an aqueous solution is about 30. This means, that in agreement with experiment, the "inner" layer term is not affected appreciably by the nature of the ions in the electrolyte. The MSA function Ahs(z) is plotted in Fig. 10. Note that Ahs(z) oscillates and ultimately decays exponentially, which is consistent with our remark that the ion and solvent interfacial regions are of similar thickness.

Fig. 11. Potential profile for a 0.01M monovalent electrolyte near an electrode. The solid curve gives the MSA result, calculated using the ion-dipole model whereas the dashed curve gives the linearized GC result. For the region, 0 < z < d 12, the lower and upper curves give the GC result calculated using s*= 80.5 and 2.5, respectively. Adapted with permission from Carnie and Chan [60].

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D. Henderson, A.D. TrokhymchukandD.T. Wasan

The MSA potential profile is plotted in Fig. 11 and is also seen to be oscillatory. In contrast, the linearized GC/PB profile is monotonic and near the electrode contact, z = d 12 is quite different from the MSA result. We have commented that the GC/PB capacitance can be brought into agreement with experiment by empirical adjustment. Without empirical fitting the MSA iondipole result is in fairly good agreement with experiment. By taking into account [63] the polarization of the metallic electrons in the electrode, the MSA ion-dipole prediction is in very good agreement with experiment not only for a mercury electrode but for a wide variety of metal electrodes. It is not just that the lack of any need for an empirical fit that makes the MSA theory preferable to the GC/PB theory. We see from Fig. 12 that even if s* is adjusted so that the capacitance, a macroscopic quantity, agrees with experiment, the GC/PB values of the microscopic potential near the electrode are quite wrong. As a result, any description of electrochemical reactions that is based on the GC/PB theory, with or without empirical adjustment, will be misleading at best.

Fig. 12. Inverse capacitance of a monovalent electrolyte at the potential of zero charge as a function of the inverse diffuse layer capacitance obtained from the GC/PB theory. The points give the experimental results of Parsons and Zobel [58]. The light line gives the low concentration MSA results and the heavy line gives the full MSA results. Adapted with permission from Schmickler and Henderson [65].

Structure and Layering of Fluids in Thin Films

299

We have already pointed out Eq. (85) is thought to be in good agreement with experiment. This, after all, is the main justification for the division of the electrochemical interface into inner and diffuse layers. Figure 13 shows the full MSA result [65] for the ion-dipole model which is compared with experiment. We see that there is a small departure from linearity in the MSA results that is not predicted by the conventional theories. Note that the experimental results also show this departure from linearity, not just qualitatively but quantitatively. In summary, the conventional description of the electrochemical interface is misleading. There is no artificial "inner" layer. The interfacial region for the solvent molecules is as diffuse as the interfacial regions for the ions. The electrostatic potential close to the electrode is quite different from that given by the conventional picture. Finally, the density and potential profiles are not monotonic but oscillatory or stratified. 4.3. Charged colloidal suspensions Returning to Eq. (75), the TCF between a pair of charged giant spheres, i.e., colloids in this case, both of radius a=D/2, is

h(R-D) = hHS(R-D)

^ f(R-D) s{YD)2D

= h»s{x)--^f(x)

,

(96)

where x = R- D is the gap width between a pair of colloids. It is useful to note that we cannot obtain Eq. (96) from the DH/PB theory even with f(x) ~ e'** because of the asymmetric treatment of the spherical cores in the DH/PB theory, where only the central ion is given a nonzero size. Because the MSA is a linearized theory, we must use the ansatz (40) to obtain the interaction energy W{x) between colloid particles. Thus, the electrostatic and excluded volume contributions to W(x) are

J3W(X)

= -hHS (x) + ^

^ af(x).

(97)

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D. Henderson, A.D. Trokhymchuk and D. T. Wasan

In addition to these two contributions, in practice there is a short range dispersion interaction between the colloidal spheres, - Aa/\2x, where A is the Hamaker constant that can be measured experimentally. Thus,

™ = V ^ - - f + ^l /w . a

a

\2x

2

(98)

We have seen already [e.g., Eqs. (43) and (44)] that the hard-sphere correlation ETC"

function h between two giant spheres is proportional to the radius a. Hence, Wla is independent of a. Some years ago Derjaguin and Landau [4] and Verwey and Overbeek [5] proposed a theory (DLVO) for colloidal interactions. Their result is

^ a

=_

A+ ^ exp( _ ra) . \2x

m

2

Thus, the DLVO result is Eq. (98) in the limit of zero diameters for the ion species. The DLVO theory is another version of the PB theory. Of course, the derivation [4,5] differs from that given here. They solved the PB equation for an electrolyte between two plane parallel surfaces and then used the Derjaguin construction [15] to obtain the interaction between two giant spheres whereas we have obtained Eq. (98) from the OZ equation for two giant spheres using the HNC/MSA/MSA approximation. In the DLVO/PB theory, the interaction between colloidal particles is the superposition of an electrostatic repulsion that dominates at large separation and a dispersion attraction that dominates at small separations with a maximum at intermediate separations. This gives a pleasing account of colloidal stability. If the maximum is less than the thermal energy, the colloids flocculate whereas if the maximum is greater than the thermal energy, the colloids are stable. The assumption of zero ion diameters and the replacement of f(x) by an exponential that yields the electrostatic part of the DLVO theory are not too bad. However, neglecting the excluded volume contribution to the colloid-colloid interaction, -hHS(x), is a very poor assumption. The number concentration of the ions may be small but the density of the solvent molecule is not small. The presence of h (x) reflects the oscillatory structural force that results from the suspending fluid stratification in the gap between two colloids. In applications to colloids in an aqueous electrolytes an additional term, called as a dipole alignment contribution, -kThDW(x)la, should be added to Eq. (98) because of the reduced polarization of the water molecules near the

301

Structure and Layering of Fluids in Thin Films

colloidal particles. Henderson and Lozada-Cassou [4] have made a crude estimate of this term. Recently, Trokhymchuk et al [15] have made a more sophisticated derivation based on the MSA results for the ion-dipole fluid. Particularly, it has been obtained that similar to the excluded volume hardsphere contribution [see Eq. (45)], the decay of dipole contribution, h (x), has the same functional form

hDlp(x)~21

yDlp | c o s ^ x + arg{/D//>})exp(-/cZ)//>x),

where the coefficients coDIP, KDIP

and yDIP

(100)

are defined in the same way as

their hard-sphere counterparts a>HS, KHS and yHS but depend on both solvent density and dipole moment. The results of such calculation are presented in Fig. 13.

Fig. 13. The force between two giant charged spheres suspended in a 10

M aqueous

electrolyte. Part a shows the electrostatic contribution only: PB results [dotted line], HLC semiempirical approximation [long dashed line], MSA dipole alignment complete result [solid line] and asymptotic Eq. (100) [short dashed line]. Part b shows the total force (solid line) calculated according to Eq. (98) plus dipole alignment term, - kTh

I a.

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D. Henderson, A.D. Trokhymchuk and D. T. Wasan

Similarly to the excluded volume contribution, the dipole contribution has an oscillatory behavior and increases the total electrostatic repulsion at short separations, improving the quantitative agreement with experiment. The dipole alignment contribution extends until about 3-4 solvent layers near the surface and this region can be associated with an assumed region of lower dielectric constant [the inner layer] that has been discussed above to account for the observed capacitance of electrodes in electrochemical measurements. 5. FILMS FORMED FROM COLLOIDAL SUSPENSIONS Recently, considerable attention has been concentrated on the phenomenon of the layering of the nano-sized or submicroscopic colloids themselves when they are added to macrodispersions such as foams, emulsions, etc. The thin films formed from colloidal particles in such systems can be used as a tool to probe the stratification phenomenon within a colloidal substance [11]. It has been observed that thin colloidal films stratify, become thinner, in a regular step-wise manner. In this section we present some results obtained in the way when the OZ based formalism is combined with computer simulations to make a progress in studying a complex colloidal system. Colloidal films represent a complex system with the superposition of few (not just two as in the case of regular colloidal suspensions) distinct length scales. Such complex systems can be studied by an approach outlined in preceding sections using a many-component version of the OZ equations assuming that the submicroscopic colloids now are a part of a suspending medium. Proceeding in this way, we have found [66] that the hard-sphere colloidal suspension tends to be ordered in a monolayer structure next to a macrosurface. It has been argued that such enhanced layering or stratification is driven by the excluded volume forces that are entropic in origin and can be revealed only if the molecular nature of the primary suspending fluid, i.e., molecular solvent is taken into account. As we can see from Fig. 14, only 1% of the colloid particles dispersed in a hard-sphere solvent comprised of 15% of the fine (solvent-to-colloid size ratio is 1:10) species clearly indicates the formation of a surface-localized monolayer of colloid particles. This is a strong evidence of the prominent role that the fine particle medium plays in colloidal dispersions to enhance a structural forces both within the colloid particles and between the colloid particles and a macrosurface. We turn our attention to this fact, since no oscillations in correlation functions, i.e., no stratification, are expected when a colloid primitive model of a hard-core colloidal suspension at a low bulk volume fraction, such as 1% is used.

Structure and Layering of Fluids in Thin Films

303

Fig. 14. Normalized local density distribution of a suspension of hard-core colloids next to a film surface and its pictorial two-dimensional interpretation.

As the colloid volume fraction in a bulk phase increases, a well-defined second monolayer of the colloid particles next to a film surface also is observed; it is composed by the colloids that are adsorbed on a surface layer of the solvent species. In such an environment, the fine species prefer to be adsorbed on the surface of the colloids and fill the cavities made by the colloidal particles and the film surface, forming an effective film surface coverage. In practice, most colloidal suspensions are composed of charged colloidal particles or macroions. The explicit many-component modeling of such a system becomes progressively more complicated since besides the different length scales, similarly to non-charged colloidal suspensions, the total number of components present in suspension is increased: the molecular solvent is replaced by an electrolyte solution that additionally consists of cations and anions; plus there are counterions to maintain the electroneutrality of the system. Although the explicit many-component OZ approach still can be applied to such a system [15], here we wish to elaborate an approach that is based on the concept of mean or effective interaction [67-70]. This concept exploits the fact that essential forces experienced by colloidal objects in a suspending medium are those mediated by medium. Thus, the colloidal suspension can be viewed as a fluid of macroions interacting via the effective potential obtained by an averaging out the macroscopic degrees of freedom due

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to suspending medium. Usually the DLVO potential (99) is used as an effective potential between colloids. The effects of the solvent and electrolyte ions are incorporated in the DLVO potential via a continuum approximation, i.e., they are present in determining only the screening length. The main goal of the study discussed here is to take into account the excluded volume effects that result from both the finite size of the simple electrolyte ions and from the molecular nature of the solvent. The basic theory of interactions between spherical macroions suspended in an electrolyte solution has been discussed in the preceding section. Due to this, we use the one component fluid model with an effective interaction between two macroions that includes both electrostatic and excluded volume interactions in accordance with Eq. (97). Trokhymchuk et al [71] have made such a study by applying canonical Monte Carlo method that was combined with a simulation cell where the colloid film region and a colloid suspension bulk phase are connected as has been first suggested by Gao et al. [72].

Fig. 15. Monte Carlo data for the normalized local density distribution of the macroions in a film formed at 0.049 volume fraction. The thick vertical lines mark the film left and right boundaries. The film contains four layers and has the thickness that corresponds to seven and half hard-core diameters of macroions. The dashed line shows the Monte Carlo data obtained from the simulation with DLVO forces only. The macroion charge number is 30 in both cases.

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Figure 15 displays MC data for the normalized local density distribution of charged macroions (presumably micelles) across the film region of thickness H. To reveal the role that the excluded volume forces due to suspending fluid are playing, the dashed line shows MC data for the macroion density distribution in a film formed under the same bulk conditions but assuming that the colloids interact only via the DLVO potential. Both models show that the colloid particles are forming the film that has a thickness between seven-to-eight hardcore colloid diameters and are organized into four well-defined layers. In both cases there is an evident difference between the layers next to the film surfaces and those in the middle of the film. Each surface layer is rather narrow with the higher density of the particles condensed directly on the film surface that steeply decreases going to the layer boundary. In contrast to the surface layers, the middle-film layers are thick and more diffusive, i.e., less organized. The thickness of the each of two middle-film layer in Fig. 16 is around two colloid particle hard-core diameters while the surface layer thickness only slightly exceeds the one particle diameter

Fig. 16. Snapshots of MC generated configurations of macroions at a bulk volume fraction 0.049 that corresponds to a high surfactant concentration, 0.10 mole/1. Number of charges carried by macroions is 30. The film with three layers has a thickness of five hard-core diameters of the macroions while film with two surface layers has a thickness of three and half hard-core diameters.

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The main differences between the model with the suspending fluid contribution and the DLVO-like model are found in the surface layers. In the case of a model with a suspending fluid taken into account, the surface layers themselves show structuring with respect to the film surfaces. This results in each surface layer consisting of a well-defined mono-layer in the immediate vicinity of the film surface and one or two similar sub-layers that are less pronounced and are separated by an "effective" layer of a suspending solution. The shape of the density profiles of the surface sub-layers in this case has a 5 - like form indicating that surface sub-layers are the quasi-two-dimensional monolayers. The surface layers formed within the DLVO model, although thinner than the middle-film layers, are still far from being monolayers. As a result, the segregation of the middle-film particle layers from the surface layers is not observed in this case. As for the middle-film layers for both models only some quantitative differences in the particle local density distribution are observed. When the separation between surfaces decreases, we find that the next two films that provide the local minima of the film energy (per film particle) have a thickness around five and three colloid hard-core diameters, respectively. In particular, one of the films shown in Fig. 16 has thickness H - 5D and contains three layers of colloid particles, i.e. has one layer less than in the film discussed in Fig. 15. Again we observe that the middle-film layer is almost completely separated from the surface layers. The thickness of the middle-film layer decreases slightly when the separation between film surfaces decreases. In contrast, the surface layers do not change notably. This quasi-stability of the surface layers becomes even more evident by analyzing the local density distribution in the film that has thickness around three times the micelle hardcore diameter, H = 3.5D, and contains two surface layers only (Fig. 16). We conclude that the surface layers for three considered films remain largely unaffected and are almost identical. It follows that during the film thinning process the film changes its thickness by squeezing out one middle-film layer of particles (the so-called "squeezing layer" mechanism ). Then the height of the step-wise layer-by-layer thinning will depend on the effective thickness of the squeezed layer. To verify this assumption, the effective thickness of the squeezed layers have been calculated as the difference between the metastable thickness of the films containing four and three, three and two particle layers for both the low and high surfactant concentrations and compared with observation for the heights of the thinning steps. The film containing two surface layers is mostly stable and its thickness corresponds to the final film thickness in the film thinning process at the surfactant concentrations considered in the present study. These also agrees qualitatively well with the force measurements of Richetti and Kekicheff [8].

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6. SUMMARY A statistical mechanical approach to study the layering or stratification of the species of suspending fluid that occurs in a vicinity of suspended macrosurfaces has been presented and discussed. This approach is based on the solution of the system of Ornstein-Zernike (OZ) equations that relate the direct and total correlation functions of the giant solutes and the species comprising the suspending medium. In a self-consistent way this approach provides with the bulk phase properties of a discrete suspending medium, its inhomogeneous properties, i.e., local density distributions in an external field of the suspended species, and with the interaction energy between a pair of giant species that is induced by a discrete suspending medium. There are two main driving forces for suspending fluid stratification excluded volume, or entropy, and electrostatics. Excluded volume forces rely on the volume fraction occupied by the species comprising suspending medium, i.e., on their sizes while electrostatic forces contribute by the number of charges carried by suspending species. Within the mean spherical approximation (MSA) excluded volume and entropic contributions are additive. Using the MSA solution we have shown the way in which phenomena of the suspending fluid stratification are related and affect the properties of electrochemical interfaces such as double layer and electrostatic potential, as well as the stability of colloidal suspensions. In a transparent manner we have obtained that the conventional theories of suspending electrolytes such as due to Debye and Hiickel, Gouy and Chapman, and Poisson-Boltzman and DLVO theories all are zero-diameter limits of the suspending fluid species within the MSA theory. Although the assumption of zero ion diameters seems to be reasonable for electrolyte ions, the density of the solvent molecule is not small. Consequently, all these traditional theories neglect the excluded volume contribution to the stratification that results in a series of drawbacks related with their application. In particular, the conventional description of the electrochemical interface is misleading. There is no artificial "inner" layer. The interfacial region for the solvent molecules is as diffuse as the interfacial regions for the ions; the electrostatic potential close to the electrode is quite different from that given by the conventional picture. The excluded volume contribution to the effective interaction between two giant spheres in the case of a simple suspending fluid is characterized by (i) a monotonic depletion attraction for separations between confining surfaces that roughly are smaller than half of the fluid particle diameter, (ii) a repulsive maximum, located at about three quarters of the fluid particle diameter, (iii) a secondary minimum just after a separation of one diameter of the suspending fluid species, and (iv) has a shape of an oscillatory structural repulsion for separations larger than one suspending fluid particle diameter. In the limit of

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low density of the suspending fluid this results are reduced to the AsakuraOosawa depletion interaction. We have shown that using a one-component, two-component and manycomponent modeling of a suspending medium there is a possibility to mimic the whole spectrum of colloidal dispersions from the submicroscopic to macroscopic scales. The real objects to which such approach can be applied include complex colloidal dispersions composed of solid surfaces, emulsion droplets, etc., all dissolved in aqueous or non-aqueous suspensions of colloidal particles, surfactant micelles, i.e. fluid systems where the constituents with a few competing length scales are involved. Numerical calculations have been performed for one-, two-, four- and ten-component model systems. A onecomponent model of suspending medium is essential and represents a crucial necessary step that allows one to move beyond the primitive modeling of colloidal suspensions, fn particular, it provides with an effective interaction energy between two mesoscopic surfaces in a molecular solvent. Proceeding in this way we have shown the importance of taking into account the molecular nature of suspending fluid that reveals the primary molecular solvent stratification in the vicinity of a single surface and in the film confinement formed by a pair of macrosurfaces. This theoretical picture is supported by the surface force measurements conducted by Israelachvili and his colleagues [7]. An extension of the simple suspending fluid to a bidisperse solution comprising species with highly asymmetric sizes, i.e., both nanosized colloids and species of molecular solvent, has revealed some new qualitative features for which the stratification phenomenon is responsible. The most notable is that the effective interaction between a pair of giant spheres in a two-component suspending fluid of colloid and fine species is no longer monotonic in the gap region depleted by colloid particles but shows an oscillatory repulsive behavior, reflecting the filling of this region by the fine species. This modeling prediction agree well with observed layering of water molecules known as a repulsive hydration forces [7]. Taking into account the presence of the molecular solvent component also affects the layering of colloid particles near the surface of giant sphere. It is interesting to note, that due to the large asymmetry in the particle size in a bidisperse suspending fluid, the contribution of each of the component to the interaction energy is split on the length scale. Due to this, it seems that to some extent the effect of each component with a competing length scales associated with particle diameters can be treated as a superposition of the results for two monodisperse fluids with corresponding particle size and, probably, with an appropriate effective density. Another interesting finding is that the contribution to the effective interaction between a pair of colloids due to the fine particles of bidisperse fluid medium seems to acquire characteristics that are similar to an adhesive sticky potential when compared to the

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contribution of the larger particles. This suggests that it may be possible to mimic the presence of the fine species in a simple manner by means of a sticky potential. Colloidal suspensions invariably contain particles with size polydispersity ranging from several percent in the very carefully controlled polymerization process to hundreds of percent in a typical emulsification process. An increase in the number of species that represent the colloid particles with different but similar diameters (four- and ten-component fluid models), i.e., a primitive attempt to explore the role of colloid particle size polydispersity, shows a tendency to diminish the effects observed when the colloid particle component is monodisperse; however, this does not affect the features introduced by the taking into account a monodisperse fine ("solvent") component. Additionally, we have presented the way in which continuous size polydispersity can be treated. Finally, we have combined the OZ/MSA approach with computer simulations techniques to study the complex colloidal systems involving more than two distinct length scales that is a case of emulsions and foam systems. In particular, we have employed an effective interaction energy between two giant spheres in a bidisperse suspending fluid, i.e., in a colloidal suspension in a MC study of the ionic micelle stratification. We have shown that the forces operating at the submicroscopic scale between colloidal particles are governing the stability of both micro- and macrodispersions. ACKNOWLEDGEMENTS This work was supported in part by the National Science Foundation under Grant No CTS 01-00854. We are grateful to Alex Nikolov and Yurko Duda in collaboration with whom some of the results reported here have been obtained.

REFERENCES [I] [2] [3] [4] [5]

J. Texter and M. Tirrell, AIChE J., 47 (2001) 1706. H. A. Stone and S. Kim, AIChE J., 47 (2001) 1250. D. T. Wasan and A. Nikolov, Nature, 423 (2003) 156. B. V. Derjaguin and L. Landau, Acta Physicochim., 14 (1941) 633. E. J. W. Verwey and J. Th. G. Overbeek, The Theory of Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [6] R. G. Horn and J. N. Israelachvili, J. Chem. Phys., 75 (1981) 1400. [7] J. N. Israelachvili, Intermolecular and Surfaces Forces, 2nd ed., Academic Press, London, 1992. [8] P. Richetti and P. Kekicheff, Phys. Rev. Letter, 68 (1992) 1951. [9] J. L. Parker, P. Richetti, P. Kekicheff, and S. Sarman, Phys. Rev. Letter, 68 (1992) 1955. [10] P. Kekicheff, and P. Richetti, Prog. In Colloid and Polym. Sci., 88 (1992) 8. II1] A. D. Nikolov and D. T. Wasan, J. Colloid Interface Sci., 133 (1989) 1.

310

D. Henderson, A.D. Trokhymchuk and D. T. Wasan

[12] T. Biben, J.-P. Hansen, H. Loven, in Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, (S.-H. Chen, J. S. Huang, P. Tartaglia, Eds.) Kluwer Academic Publishers, Dordrecht, Netherlands, 1992, 23. [13] D. Henderson and M. Lozada-Cassou, J. Colloid Interface Sci., 114 (1986) 180. [14] D. Henderson, J. Colloid Interface Sci., 121 (1988) 486. [15] A. Trokhymchuk, D. Henderson, and D. T. Wasan, J. Colloid Interface Sci., 210 (1999) 320. [16] L. S. Ornstein and F. Zernike, Proc. Acad. Sci. Amsterdam, 17 (1914) 793. [17] J. K. Percus and G. L. Yevick, Phys. Rev., 110 (1958) 1. [18] J. A. Barker and D. Henderson, Rev. Mod. Phys., 48 (1976) 587. [19] J. L. Lebowitz and J. K. Percus, Phys. Rev., 144 (1966) 251. [20] D. Henderson, F. F. Abraham, and J. A, Barker, Mol. Phys., 31 (1976) 1291. Reprinted as Mol. Phys., 100 (2002) 129. [21] P. Attard, D. Bernard, C. Ursenbach, and G. N. Patey, Phys. Rev. A., 44 (1991) 8224. [22] D. Henderson, 11th Symposium on Thermophysical Properties, Boulder CO, June 23-27, 1991; Fluid Phase EquiL, 76 (1992) 1; J. Chem. Phys., 97 (1992) 1266. [23] B. V. Derjaguin, Kolloid Z., 69 (1934) 155. [24] M. S. Wertheim, Phys. Rev. Letters, 10 (1963) 321. [25] E. Thiele, J. Chem. Phys., 39 (1963) 474. [26] M. S. Wertheim, J. Math. Phys., 5 (1964) 643. [27] R. J. Baxter, Aust. J. Phys., 21 (1968) 563. [28] J. Perram, Mol. Phys., 30 (1975) 1505. [29] W. R. Smith and D. Henderson, Mol. Phys., 19 (1970) 411. [30] D. Henderson and W. R. Smith, J. Stat. Phys., 19 (1978) 191. [31] J. L. Lebowitz, Phys. Rev., 133 (1964) A895. [32] M. Toney, J. N. Howard, J. Richer, G. L. Borges, J. G. Gordon, O. Melroy, D. G. Wiesler, D. Yee, and L. B. Sorensen, Surface Sci., 335 (1995) 326. [33] R. Roth, R. Evans and S. Dietrich, Phys. Rev. E, 62 (2000) 5362. [34] A. Trokhymchuk, D. Henderson, A. Nikolov and D. T. Wasan, Langmuir, 17 (2001) 4940. [35] S. Asakura and F. Oosawa, J. Chem. Phys., 22 (1954) 1255. [36] D. Henderson, D. T. Wasan, and A. Trokhymchuk, J. Chem. Phys., 119 (2003) 11989. [37] D. Henderson, D. T.Wasan and A. Trokhymchuk, Condens. Matter Phys., 4 (2001) 779. [38] M. S. Wertheim, L. Blum and D. Bratko, in Micellar Solutions and Microemulsions (S.H. Chen, R. Rajagopalan, Eds.) Springer: New York, 1990; Chapter 6, p.99. [39] A. Trokhymchuk, D. Henderson, A. Nikolov and D. T. Wasan, J. Colloid Interface Sci., 243(2001) 116. [40] J. J. Salacuse and G. Stell, J. Chem. Phys., 77 (1982) 3714. [41] L. Blum and G. Stell, J. Chem. Phys., 71 (1979) 1300. [42] J. Y. Walz and A. Sharma, J. Colloid Interface Sci., 168 (1994) 4851. [43] M. S. Wertheim, J. Stat. Phys., 42 (1986) 459; 42 (986) 477. [44] M. S. Wertheim, J. Chem. Phys., 85 (1986) 2929; 87 (1987) 7323. [45] J. Chang and S. Sandier, J. Chem. Phys., 102 (1995) 437; 103 (1995) 3196. [46] E. Vakarin, Yu. Duda and M. F. Holovko, Mol. Phys., 90 (1997) 611. [47] M. F. Holovko and E. V. Vakarin, Mol. Phys., 84 (1995) 1057. [48] M. F. Holovko and E. V. Vakarin, Mol. Phys., 87 (1996) 1375. [49] E. Vakarin, M. F. Holovko and Yu. Duda, Mol. Phys., 91 (1997) 203. [50] D. Duda, D. Henderson, A. Trokhymchuk and D. T. Wasan, J. Phys. Chem. B, 103 (1999)7495.

Structure and Layering of Fluids in Thin Films

311

[51] H. K. Christenson, D. W. R. Gruen, R. G. Horn and J. N. Israelachvili, J. Chem. Phys., 87 (1987) 1834. [52] M. L. Gee and J. N. Israelachvili, J. Chem. Soc, Faraday Trans., 86 (1990) 4049. [53] E. Waisman and J. L. Lebowitz, J. Chem. Phys., 52 (1970) 430; 56 (1972) 3086, 3093. [54] L. Blum in Theoretical Chemistry: Advances and Perspectives, 5 (H. Eyring and D. Henderson, Eds) Academic Press, New York, 1980, 1. [55] P. Debye and E. Httckel, PhysikZ., 24 (1923) 185. [56] M. Wertheim, J. Chem. Phys., 55 (1971) 4281. [57] G. Gouy, J. de Physique, 9 (1910) 457; D. L. Chapman, Phil. Mag., 25 (1913) 475. [58] R. Parsons and F. G. R. Zobel, J. Electroanal. Chem., 9 (1965) 333. [59] D. Henderson, L. Blum, and J. L. Lebowitz, J. Electroanal. Chem., 102 (1979) 315. [60] S. L. Carnie and D. Y. C. Chan, J. Chem. Phys., 73 (1980) 2949. [61] D. Y. C. Chan, D. J. Mitchell, B. W. Ninham and B. A. Pailthrope, J. Chem. Phys., 69 (1978)691. [62] L. Blum and D. Henderson, J. Chem. Phys., 74 (1981) 1902. [63] W. Schmickler and D. Henderson, J. Chem. Phys., 80 (1984) 3381. [64] L. Blum, D. Henderson, and R. Parsons, J. Electroanal. Chem., 101 (1984) 389. [65] W. Schmickler and D. Henderson, Progr. Surface Sci, 22 (1986) 323. [66] A. Trokhymchuk, D. Henderson, A. Nikolov, and D. T. Wasan, Phys. Rev. E, 64 (2001) 012401. [67] W. G. McMillan and J. E. Mayer, J. Chem. Phys., 13 (1945) 276. [68] L. Beloni, J. Phys.: Condens. Matter, 12 (2000) R549. [69] A. A. Louis, Philos. Trans. R. Soc. London, Ser. A359 (2001) 939. [70] C. N. Likos, Phys. Rep., 348 (2001) 267. [71] A. Trokhymchuk, D. Henderson, A. Nikolov, and D. T. Wasan, J. Phys. Chem. B, 107 (2003) 3927. [72] J. Gao, W. D. Luedtke, and U. Landman, J. Phys. Chem. B, 101 (1997) 4013.

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Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 8

Theory of emulsion flocculation D. N. Petsev Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131 1. INTRODUCTION Emulsions are colloidal dispersions of liquid in liquid, for example, a mixture of oil and water. As a result of the mixing, one may obtain oil droplets in water (O/W) or water droplets in oil (W/O). The sizes of the droplets could be in the micrometer and even submicrometer range. A problem of both fundamental and practical importance is that of the emulsion stability against flocculation. Colloid stability is often analyzed in the framework of the Derjaguin-LandauVerwey-Overbeek (DLVO) theory [1-4]. DLVO theory suggests that the stability against aggregation in a colloidal dispersion (e.g., emulsion) depends on the balance between van der Waals attraction and electrostatic repulsion. However, these two forces do not represent a full account of the whole variety of interactions that may occur in colloidal systems. These include steric repulsion [5,6], depletion attraction [7-12], hydration and hydrophobic interactions, oscillatory surface forces, etc. [13]. In the case of emulsions, the colloidal particles are fluid. The droplet fluidity and interfacial mobility may have a strong impact on the emulsion behavior and particularly on their stability against flocculation [14-26]. Emulsion stability also always requires the presence of surface-active molecules or fine solid particles that adsorb at the droplet interface and prevent the droplets from rapid coalescence [4]. Even in the presence of a stabilizing additive, emulsions are thermodynamically unstable. The contribution of the interfacial free energy is proportional to the total area of contact between the two phases and is usually positive. Destroying the droplets and separating the phases macroscopically allows for considerable reduction of this unfavorable term, and therefore of the overall free energy. The time scales on which such event occurs may vary from seconds to years. Hence, emulsions are usually kinetically stable. The analysis of emulsion stability provides not only fundamental challenges of great scientific interest, but is also very important for various

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practical aspects. In some cases (as in food industry, paint production), the main concern is to obtain stable emulsions while in others (e.g. oil recovery) destabilization and oil separation is desired. Knowledge of the basic principles and mechanisms governing emulsion stability presents both academic and applied interest. Stability loss in emulsions may occur in four different ways [27,28]: • Creaming (or sedimentation) is due to density difference between the two immiscible liquids, the lighter phase will tend to go up while the heavier will move downwards. In the case of an O/W emulsion the oil droplets will accumulate at the top, forming a cream layer. • Flocculation occurs when the droplets stick to each other and form aggregates due to the presence of a minimum in the interaction energy. The aggregates could be compact or may form an expanded gel-like structure. However, the individual droplets remain separated by a layer of the continuous phase - thin liquid film (see Fig. 1). Flocculation may be weak and reversible or strong and irreversible. Flocculation enhances creaming since it forms large floes. It is also often a prerequisite step to coalescence.

Fig. 1. Sketch of two deformed droplets in the presence of long range repulsion (e.g., electrostatic). If no long range forces are present, the droplets deform upon contact of their surfaces. Otherwise a liquid film with thickness h is formed.

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• Coalescence is the process of the fusion of two or more droplets to form a larger droplet. In this case the droplets are no longer separated by the continuous phase and have become a single entity. The thin liquid film that may have separated them before in the flocculation stage has ruptured under the action of the attractive forces, or due to hydrodynamic instabilities [20,29-31]. • Ostwald ripening is a process of molecular diffusion transfer of oil from the smaller droplet to the larger droplet and is driven by chemical potential differences. The present chapter discusses only emulsion flocculation. Coalescence is outlined in Chapter 9 of this book. As mentioned above flocculation is a process during which the droplets aggregate, but remain separated by thin films of the continuous phase as shown schematically in Fig. 1. The formation of such configuration is due to an attractive interaction but the stability of the film between the interfaces proves the presence of repulsive force at shorter distances. Attractive and repulsive energy contributions have different dependence on the separation of the droplets and their superposition may exhibit a complex distance behavior.

Fig. 2. A sketch of DLVO interaction energy as a function of the separation (liquid film thickness) between two infinite flat surfaces. This picture is more relevant to solid surfaces. The droplets shown are to indicate qualitatively the effect of the attractive forces on the film formation (more detailed discussion is given in the text).

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In DLVO interaction, it is described by the combination of van der Waals attraction and electrostatic repulsion [1-4]. Van der Waals energy diverges as \lh at small separations, h, and decays as \/h6 at large distances. The electrostatic repulsion is finite at small distances and decays exponentially. Because of this, the overall interaction energy is dominated at small and large separations by attraction, while the intermediate region could be repulsive because of the electrostatic contribution, see Fig. 2. Fig. 2 suggests the presence of a strong and short-ranged repulsion also at very small separations. This repulsion is not included into the original version of DLVO theory [1-4]. Its possible origin is discussed briefly further in this chapter. The particular shape of the energy curve in Fig. 2, however implies that flocculation may take place in the far (secondary) minimum II or in the near (primary) minimum I. The attraction in the secondary minimum is much weaker than that in the primary and so is the flocculation. Droplets, flocculated in the secondary minimum would separate more easily than those in the primary. Flocculation in the primary minimum could have been preceded by such in the secondary minimum. Emulsions with an energy barrier for coalescence could still be amenable to flocculation. The kinetics of flocculation, as well as that for transition between secondary to primary minimum and film rupture depends on both direct (e.g., vane der Waals, electrostatic, etc.) and hydrodynamic interactions [15,20,26,29,31-35]. In the present chapter we discuss the droplet interaction energy and its relation to the kinetics of flocculation. A brief overview of the typical interactions that might be encountered in emulsion systems is given in Section 2. Section 3 outlines the mechanism and kinetics of droplet flocculation and film thickness transitions. Section 4 presents briefly some of the experimental methods for studying droplet interactions and flocculation and Section 5 contains the conclusions.

2. INTERACTION ENERGY BETWEEN TWO EMULSION DROPLETS 2.1. Energy density per unit area Let us consider the interaction between two infinite plane-parallel surfaces (representing the interacting droplets) separated by a layer of the different medium (representing the disperse phase, e.g., water for aqueous suspensions). This is a one-dimensional problem and its theoretical treatment is relatively simple. It also provides a foundation for further more elaborate treatment relevant to more realistic particle shapes (see below). The possible interactions between emulsion droplets may have a different physical origin [1-5,11-13]. Some of them are briefly outlined below.

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2.1.1. Electrical double layer (electrostatic) interactions Electrostatic interactions are due to the charge that many colloidal particles (or emulsion droplets) acquire when immersed in water. The reason for this charge is dissociation of surface ionic groups or adsorption of ions from the solution. In the particular case of emulsions, the surfaces charges are usually due to the use of ionic surfactants as a stabilizing additive. Electrostatic interactions are repulsive in general and play an important role in explaining colloid stability in DLVO theory [1-4]. In some special cases like interaction between surfaces with different charge (or potential) attraction at very short distances could be observed [1,2], see also [14]. Even the simple case of two plane parallel surfaces has no exact and general solution for the electrostatic interactions. There are, however, two reasonable approximations, briefly outlined below. Low surface potentials (weakly charged droplets). This approximation refers to the case (e$!olkT}<\ where e is the charge of the electron, AT is the thermal energy and ^Fo is the dimensional surface potential. For room temperature (T = 298 K) weakly charged droplets are those with ^Fo ^ 25 mV. The electrostatic energy density per unit area as a function of the separation, h, in this case is [1-4,12]

/J(/i) = 4 7 r e o £ « ^ l - t a n h ^ j .

(1)

The superscript *F means that the surface potential remains constant as the separation between the infinite plates varies. This assumption leads to significant mathematical simplifications but does not necessarily correspond to the real situation. It is a result of the boundary condition imposed at the droplet surface. An alternative is to assume that the potential may vary but the surface charge, a remains constant as the separation between the droplet surfaces, h, changes. Then [1-4,12]

/ ; lh) =

ZL_ i _ cothf—1 .

(2)

The parameter, K, takes into account the charge screening due to the presence of electrolyte (inverse Debye length), and for symmetric (z:z) electrolyte is

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2 2

1/2

(3)

«= ^ - 7 f Q

where Ce/ is the electrolyte number concentration and z is the charge number of the salt ions, s is the relative dielectric permittivity of the solvent and SQ is the dielectric constant of vacuum. Weakly overlapping double layers. In this case no constraints on the magnitude of the surface potential (or charge) are enforced. There is still an assumption, however, that the two interacting double layers are weakly overlapping, or nh > 1. This is a widely used approximation, known as the nonlinear superposition [12] and reads (see also[l-3,36]) fel{h)-

64CelkTK-1 tanh2 ^

exp(-^).

(4)

There is a number of other and more elaborate expressions for the electrostatic interactions available [1-3,36], and there are always the alternatives of numerical solutions, but the expressions shown above cover a very wide range of reasonable experimental conditions relevant to emulsions and offer rather accurate results. The surface charge (or potential), of emulsion droplets, could be conveniently controlled by using a mixture of ionic and nonionic surfactants [37]. 2.1.2. Van der Waals (dispersion) interactions Van der Waals interactions are due to the forces acting between the individual molecules in the macroscopic colloidal particles (e.g., droplets). These forces are between dipoles and induced dipoles and are typically short ranged. Integration of all these molecular interactions over the volumes of the interacting macroscopic bodies gives an overall energy that is considerably longer ranged. This was first done by de Boer [38] and Hamaker [39], assuming pairwise additivity of the interactions between the individual molecules in the colloidal particles. Such integration, sometimes referred to as Hamaker approach, could often be performed analytically for a number of particle shapes and geometries. Van der Waals interactions are always present between particles immersed in a liquid of a different refractive index and dielectric constant. Their magnitude may be negligible, especially when compared to other interactions that are present. However, van der Waals interactions diverge at contact as \lh and for sufficiently small particle separations they may easily become the dominant force.

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The van der Waals energy of interaction per unit area for two infinite parallel slabs is [1-3,12,13,36]

AH is the Hamaker constant and depends on the material properties of the droplets and the surrounding fluid. It was shown later [40,41] that due to phase shift between the interacting dipoles, AH might actually decay with the separation and hence, is not truly a constant. This effect is called electromagnetic retardation and may become substantial for h> C/LJ, where c is the speed of light and co is the frequency of dipole oscillation. Further developments of the theory of dispersion interactions [42-44] (see also [12]) lead to elaborate approaches for calculation of the Hamaker constant. These approaches usually require a substantial numerical effort but useful approximate analytical results are available [12] for the non-retarded

and fully retarded

AH =±kT 4

£L^2+iV e, - e 2

47T«2 nf + n{

«±ZA:'l

(7)

h

cases, where S\ and si are the dielectric permittivities, while n\ and n2 are the refractive indices for the droplets and the surrounding medium respectively. hp is Planck's constant, a> is the fluctuation frequency of the interacting molecular dipoles and c is the speed of light. Obviously, retardation affects only the second (frequency dependent) term of Hamaker constant [see Eqs. (6) and (7)]. The important difference between the two expressions above is that the nonretarded AH is a true constant with respect to the separation, h. The second term in the right hand side of the expression (7) for the fully retarded AH, however, decays as \lh [compare to (6)]. Hence, retardation leads to a faster decay of the van der Waals energy. A very useful expression that interpolates between (6) and (7) has been suggested {see [12]}

320

D.N. Petsev

where H = «2 njf + rq\ v '

— c

and

.

(8)

i~2/ 3

3 2

F(H) = ±H /•"C + ^)°iK-^) < f c M 1+ M '

Equation (8) covers the whole range from nonretarded to fully retarded van der Waals interactions. The presence of electrolytes in the solution containing the interacting colloids decreases the value of the Hamaker constant [45]. The first term in (6), (7) and (8) is the only one affected by electrolyte screening and decays exponentially with the separation h. While the electrolyte effect is substantial for high concentrations (e.g., about 1 M NaCl), it has virtually no effect when the electrolyte amount is lower [12]. Typical numerical values of Hamaker constants for oil/water/oil or water/oil/water systems (like most emulsions) are between 3 and 4xlO"21 J (see e.g., [13]). Dispersion interactions may also induce repulsion along with the attraction. It is due to the interactions between atoms and molecules at very small separations. Analysis of this repulsive contribution for colloidal particles has been done recently [46] and showed that the respective free energy of repulsion has the form

fZP{h) = Const

x^.

h

(9)

2.1.3. Steric interactions Steric interactions occur when there is an overlap of the adsorbed polymer layers that may cover the colloidal particles. Emulsion droplets covered by a nonionic surfactant are stable because of the steric repulsion between the hydrophilic headgroups of the amphiphile. Since the adsorbed polymers induce steric interactions, their interactions with the solvent are extremely important [5,12]. For good [47,48] and ©-solvents [49], the interaction is usually repulsive, see also [50], but for poor solvents it could be attractive [12,51]. The first two (repulsive) cases could be modeled theoretically relatively simply, the last one allows only for numerical treatment. Nonionic surfactants usually have

321

Theory of Emulsion Flocculation

polyoxyethylene headgroups, which are known to become more hydrophobic with increasing temperature. Hence, emulsion stabilized by such a surfactant may switch from a stable to an unstable (flocculated) state by varying the temperature. The free energy of interaction per unit area due to polymer steric repulsion in good solvent is approximately given by [47,48]

fsl(h) = 2kTI3/2Lg

h-5l4+HlIA-^

for h<2Lg

(10)

where Lg is the polymer layer thickness in a good solvent defined by Lg= N\Tls\

, hg=hl2Lg

and Y denote the polymer adsorption. For ©-

solvents the interaction is given by [49]

fAh) = FkT^-[^-ln

*l£\

for h<^3L0 J

fsl(h) = 4rkTexp

.

(11)

for h>^f3LQ 2 Lo

L0=ly/N is the mean-square end-to-end distance of the polymer molecule dissolved in the solvent layer separating the surfaces; / and N are the length of a fragment unit and their number in the polymer chain respectively. There are much more elaborate theories for the interaction between layers of adsorbed polymers (or nonionic surfactants) available based on single molecule mean field [52] or self-consistent field [53] approaches. None of them are analytical and they require numerical analysis 2.1.4. Depletion interactions Emulsion droplets often exist in the presence of much smaller species like polymer molecules, surfactant micelles, microemulsions droplets, etc. These species may not adsorb at the droplet interface but still affect the interactions and hence the stability of the emulsion. This is due to the depletion attraction [7,12]. It leads to destabilization and depletion flocculation [8,9,54]. According to the simple model of Asakura and Oosawa [7], the depletion attraction per unit area between two infinite and parallel surfaces is

322

UP(h)

D.N. Petsev

= -^(d-h)

fdep{h) = 0

for

for

h
(i

^

h>d

where d is the diameter of the small colloids and is their volume fraction. The physical origin of depletion interaction is in the osmotic under pressure in the gap between the surfaces when it becomes inaccessible to the small species. In some cases, small colloids could be trapped in the gap between the surfaces. Then the energy may have a complex shape with several minima and maxima corresponding to the number of particle layers [55,56]. A rigorous analysis of this case was suggested by Henderson and Lozada-Cassou [57]. 2.1.5 Hydrophilic and hydrophobia interactions Hydrophilic and hydrophobic interactions between colloids in general are related to the interactions between the particles (emulsion droplets) and the surrounding solvent (e.g., water). If the droplet surfaces "like" water, the molecules in the gap are "reluctant" to be expelled and oppose particle approach. Hydrophobic surfaces, on the other hand, do not "like" water and prefer to be in direct contact with each other, thus minimizing the unfavorable interaction with water molecules. Hence, hydrophilic interactions lead to repulsive (hydration) force, while hydrophobic interactions are attractive. There is a great number of theoretical [57-63] and experimental [13,6467] studies on these forces but nevertheless a clear quantitative explanation is still missing. Empirical analysis of experimental results suggest an exponential decay of the hydration and hydrophobic interactions with distance [13,67] //^=/oexP--

(13)

where X is the decay length and/ 0 is the surface free energy at contact. Both are determined by fitting experimental data. In the case of hydration repulsion / 0 « 3 - ^ 30 mJ/m and A £»0.6-^ 1.1 nm, while for hydrophobic attraction / o = — 27 with 7 « 1 0 — 50 mJ/m2 being the interfacial tension for a single particle/water interface) and the decay length is A « 1 — 2 nm [11,13].

Theory of Emulsion Flocculation

2.2.

323

Interaction energy between curved fluid interfaces

2.2.1. Interfacial shape of interacting emulsion droplets The shape of a fluid interface is given by the Laplace equation of capillarity [68], see also Ref. [69] and the excellent review by Princen [70]. In the case of two interacting fluid surfaces (as between two emulsion droplets) the Laplace equation should be modified to account for the interaction. For not very small droplets, this could be accomplished by introducing a disjoining pressure [1,2] term in the Laplace equation [71-74], see also [75]. If the system has rotational symmetry, it obtains the form (see Fig. 3 and Ref. [18])

d\j(x)sma(x)} dx

7(x)sina(x) x

,N

where a(x) is the running slope angle of the interface, Pc is the capillary pressure and n(x) = II[//(JC)] is the disjoining pressure between the two droplet surfaces at point x.

Fig. 3. A qualitative illustration of the smooth shape of the transition region between the film and the surrounding curved droplet surfaces. H(x) is actually the shape of the droplet surfaces, a(x) is the running contact angle, ho is the distance in the middle of the film, Rc and a c are the effective film thickness and contact angle respectively, as is the radius of the spherical part of the droplet centered at yo. x\, is a computational parameter (see the text and Ref. [35]) Reproduced by permission of Academic Press, N.D. Denkov, D.N. Petsev, K.D. Danov, J. Colloid and Interface Sci. 176 (1995) 189.

324

D.N. Petsev

H(x) is the local film thickness and the y(x) is the local interfacial tension. To solve Eq. (14), one needs to know these two functions. The equations for them are dH(x) — = 2tancv(x) and

(15)

dx - ^

= -n(x)sina(x).

(16)

The boundary conditions for solving Eqs. (14-16) are (see Fig. 3 and Ref. [18]) l{*B) = l + \f{HB),

(17)

HB=H{xB) = 2 ^ - ^a] - x\) and

(18)

tan«(x g ) =

%B

(19)

^as - xB

The disjoining pressure, U(x) and surface free energy fix) are related by the expression

f{h)= [°°n{H)dh

(20)

*J h

and y is the interfacial tension of undeformed droplet [18]. Since the fluid in the droplet is incompressible, the volume, V, does not change upon deformation, or V = ^a3=\fH\x2{H)dH

+ \{3as~p),

p{as) = as + ^

- x\

(21)

where a is the radius of the undeformed droplet, as is the radius of the spherical part of a deformed droplet in a doublet (see Fig. 3) and x{H) is the function of the droplet shape in the region ^h$
Theory of Emulsion Flocculation

325

F = 2TrJX"ll(x)xdx = 0.

(22)

The dependence of the disjoining pressure Tl(x) on the radial coordinate x is shown in Fig. 4. The repulsion between the droplet surfaces in the thin film region is balanced by the attraction that is due to the surrounding spherical parts. The shape of the droplets in an equilibrium doublet is calculated using the following procedure [18]: equations (14-16) together with boundary conditions (17-19) are solved numerically to obtain the first approximation of the shape in the region 0 < x < xB. The remaining part of the droplet is assumed to be perfectly spherical with radius as. Then condition (22) is checked and if it is not fulfilled, a new starting value for y0 is chosen, which leads to a new value for as calculated by means of Eq. (21). This is repeated until y0 and as satisfy both (21) and (22) simultaneously. Extending the droplet interfacial area gives rise to the following contribution to the pair interaction energy Us = 4TT7 JX" x^l + tan2a(x)dx + asp(as)-

Fig. 4.

2a2 .

Disjoining pressure against the film radial coordinate (see Fig. 3).

parameters are droplet radius as = 2 p.m, Hamaker constant AH = l x l 0 ~

(23)

The 20

J,

interfacial tension y = 1 mN/m, surface potential \&0 = 1 0 0 m V , and monovalent electrolyte concentration Cei = 0.1 M. The vertical dotted line defines the thermodynamic film radius. Reproduced by permission of Academic Press, N.D. Denkov, D.N. Petsev, K.D. Danov, J. Colloid and Interface Sci. 176 (1995) 189.

326

D.N. Petsev

The value of as quantifies the extent of the droplet deformation. Note that this definition of the interfacial tension contribution to the overall droplet interaction does not account for the interaction between the surfaces which is discussed separately below. More details are given in Ref. [18]. 2.2.2. Derjaguin's approximation Calculating the energy of interaction between finite size colloidal particles (e.g., emulsion droplets) is not as simple as for the case of infinite parallel plates discussed above. This difficulty follows from the curvature of the interacting surfaces. This was partially resolved by Derjaguin [76] {see also [1,2,12,13,36]} who suggested that for small separations between the particle surfaces, hi a «; 1, the interaction energy could be given by the general formula

U(h) = 2nf™f{H{x)]xdx.

(24)

h is the shortest distance between the deformed droplet surfaces (see Fig. 3) and it may change as the droplets approach or separate. If the free energy of interaction per unit area is known (see Section 2.1 above) then calculation of its contribution to the pair energy can carried out using Eq. (24). For example, introducing Eq. (5) into (24) we obtain the van der Waals contribution to the droplet pair energy of interaction

Uvw = 27rj

xVl + tan 2 «(x) -

J 0

I

H[

J

\2ITH

>[ dx

(25)

[X)

while Eq. (4) provides the respective electrostatic term Uel = 2TTJX" x^l + tan2 a(x)

64CelkTn~] tanh2 ^ - exp[-K#(x)]Lt. (26)

A considerable simplification could be attained if the deformed droplet shape is represented by that of a truncated sphere, like in Fig. 1. This approximation may give very similar results to those that are obtained by the approach presented above [18]. For DLVO type of interactions (van der Waals, electrostatics and surface extension - see below) and wide range of parameters, the differences were s 7%. In this truncated sphere approximation, we distinguish a well-defined plane-parallel film surrounded by the spherical droplet surfaces (see Fig. 1). Then [14] {see also [15,17,18,21,25]}

Theory of Emulsion Flocculation

H(x) = h

forx
H(x)^h + -(x2-R2)

forx>/?"

327

h is again the closest distance between the surfaces and according to Fig. 1 it is the actual thin film thickness, R is the film radius and x is the running radial coordinate. Combining (24) and (27) one obtains for small deformations,

(rlaf <1 U(h,R) = 7rR2f(h) + nafC°dHf(H)

(28)

Jh

where f(h) is the energy density per unit area as defined in Section 2.1. The first term in (28) accounts for the interaction energy across the plane-parallel film while the second is due to the surrounding curved portions (see Fig. 1). If there is no deformation, r = 0, the first term in (28) disappears and Derjaguin's expression for interacting spheres is obtained [1,2,12,13]. Hence, Eq. (28) allows for an easy and accurate way to calculate the different terms in the total pair energy between deformed droplets which is discussed below in this chapter. An alternative method for calculating the interaction energy of curved interfaces is based on a perturbation approach [75]. Using Eq. (28) and assuming a model truncated sphere shape for the interacting droplets allows for obtaining very simple expressions for the different types of interactions that could be present - electrostatic, van der Waals, steric, depletion, salvation, etc., see Section 2.1. The van der Waals interaction, however, could be calculated explicitly for the model shape which is demonstrated below. 2.2.3. Van der Waals (dispersion) interactions between droplets Van der Waals interactions could be calculated exactly for some simple shapes of the interacting colloidal particles. This approach was developed by Hamaker [39], who calculated the dispersion energy between bodies of different shape including two spheres of different size. This result is particularly relevant to emulsions where the spherical shape of the droplets is governed by the interfacial tension. In some cases (low interfacial tension), the droplets deform and their shapes could be approximated with that of truncated spheres separated by a plane-parallel film. In this case (see Fig. 1), the van der Waals energy can be calculated exactly following the integration procedure of Hamaker and was

328

D.N. Petsev

done independently in Refs. [14] and [77]. The general result for two deformed spheres with equal radii is [14]

>K

12 \{L + h)2

' +

h(2L + h)

(L

+

hf

^ . _ _ ^ 2i?2 2 h h(L + h) (L + h)[2(L-a) + h] (h2+4R2)yh2+4R2-h)

2R2a(2L2+Lh + 2ah)

2h[2(L-a) + h]

h(L + h)2[2(L-a) + h\

+

where L = a + va 2 — R2 . Eq. (29) could be significantly simplified if one assumes that the deformations between the droplets is small, or (Vla) < 1. In this case, the van der Waals energy contribution becomes

rr /, N Uvw(h,r) =

A

4 2 H <3 4a2 , hUa + ti) !L\ J + TT, TT + 2 1 n ~ T 12 (2a + hf h{4a + h) (2a + hf LV

J

.

(30)

5 2

l28a R 2

h (2a + hf (4a + hf \ The first three terms on the right hand side of (30) are equivalent to the Hamaker result for spheres, while the last term is due to the formation of plane-parallel film between the droplets. 2.2.4. Surface extension and bending energies Emulsion droplets normally consist of incompressible fluid. Therefore any deformation is accompanied by extension of the droplet interface from the optimal (spherical) to the new shape (e.g., truncated sphere) [16]. The energy contribution related to this (again for small deformations, (r la) < 1) is [14]

E/,(r) = ™2 ^ 4 + — 4 V

^

2 a4

64

8 a

(31)

Theory of Emulsion Flocculation

329

where y0 is the interfacial tension of the spherical droplet in absence of

deformations

and

EG = (d-y/d\nS)s=s

= (dj/d\nr2)

is the Gibbs'

elasticity of the surfactant monolayer at the droplet interface. The second term on the right hand side of (31) is often negligible and only one proportional to the interfacial tension, y0, remains. For oil/water/surfactant systems with ultra low interfacial tension [78], the Gibbs' elasticity term may become important. Film formation between droplets also leads to energy contributions due to the bending of the interface [21], viz.

Ub = -2,r2B0H 1 - - £ - = *nke t f -^ 1 - f . 2HQ

[a) a0

(32)

2a

Bo is the bending moment, kc is the bending elasticity constant [79], H = — 1 / a and Ho = — 1 / a0 are the interfacial and spontaneous curvatures respectively (a and a0 are the droplet and the interfacial spontaneous curvature radii). It is important to emphasize that the surface extension and bending energy contributions depend only on the extent of deformation expressed by the film radius R and not on the separation (film thickness) h. 2.3. Structure of droplet aggregates 2.3.1. Doublet of two interacting droplets Colloid particles [ 1,2] as well as emulsion droplets may flocculate in the distant (secondary) or in the closer (primary) energy minimum - see Fig. 2. The overall energy of interaction between droplets is determined by the superposition of different energy contributions like those described above and depends on both the interdroplet distance and film radius (if any deformation is present)

U{h,r) = YJUi{Kr)

(33)

i

where the subscript / stands for the respective energy term (electrostatic, van der Waals, steric, depletion, etc.). If the droplets have low interfacial energy, they are deformable and the contributions of the interfacial deformation (surface extension and/or bending) have to be taken into account. Whether the droplets forming a doublet will deform or not is determined by a variety of factors including the interfacial tension, droplet size, charge, electrolyte concentration and magnitude of the attractive energy [14-18,21,23-25]. Hence, it is not

330

D.N. Petsev

possible to define a simple criterion for the occurrence of deformation. Usually, the total interaction energy has to be minimized with respect to the variables h and r (the film thickness and film radius) [17]. In cases when the parameters do not favor film formation the minimization predicts r = 0. As rules of thumb we find that, see also Ref. [80]: > Attractive interactions favor the deformation (film formation). Hence, strong van der Waals forces or depletion interactions, induced by smaller colloids, may lead to droplet deformation. > Repulsive interactions suppress film formation. A typical example is electrostatic repulsion: high surface potentials and/or low electrolyte concentrations do not favor the formation of plane a parallel film between the droplets in a doublet. Addition of electrolyte decreases the repulsion and therefore facilitates deformation. > The lower the interfacial tension the higher the droplet deformability. Interfacial tension resists any extensions of the droplet interface and thus the formation of film in the doublet. Similar is the role of high bending energy which increases the interfacial rigidity. ^ Larger droplets deform more readily than small ones if all the remaining parameters are the same. This could be attributed to the higher capillary pressure of small droplets that make them behave more like rigid spheres.

Fig. 5. Countour diagram of the interaction energy of two charge stabilized droplets. a= 1 urn, AH = l x l O " 2 0 J , y = l mN/m, tf0 = 100 mV, Ce, = 0.1 M.

Theory of Emulsion Flocculation

331

This is illustrated in Fig. 5, which presents a contour diagram of the interaction energy between two droplets as a function of the film thickness, h, and film radius, r. The droplet size chosen for this calculation is 1 \im, the electrolyte concentration corresponds to 0.1 M monovalent salt and the Hamaker constant is AH = l x l ( T 2 0 J, ^ 0 = 5 0 mV and y = 1 mN/m. A minimum in the energy surface is formed at r/a = 0.055 and h/a = 0.0069. The attraction energy is -29.1 kT. If the deformability is ignored, the energy vs. distance dependence runs along the ordinate axis and the energy minimum is -27.3 kT at h/a = 0.0058. From the ratio r/a (at the energy minimum), one may calculate the equilibrium contact angle, a using [18] a = sin—-. a

(34)

Increasing the electrolyte concentration or lowering the droplet surface potential leads to greater deformation. More details on the structure (values of the film radius and thickness) are given elsewhere [14,17,18,21,24,25]. These also include the effect of other types of colloidal forces like steric, depletion and structural. While steric repulsion acts similarly to the electrostatics, depletion may turn into oscillatory structural force at a high volume fraction of the small colloids [21,25]. These forces lead to metastable states of macroscopic emulsion films containing one or more layers of trapped particles [55,56,81,82] and their role for miniemulsion droplets (submicrometer sized and below) is usually destabilizing, leading to a deep depletion minimum without any small particles in it [21,25]. 3. KINETICS OF DROPLET FLOCCULATION The kinetics of flocculation of solid colloidal dispersions was first analyzed by Smoluchowski [83-85]. The theoretical model he developed was based on the notion that the time determining step is the approach of two colloidal particles due to diffusion. It has subsequently undergone numerous refinements and generalizations [2,86-96]. These include more realistic direct and hydrodynamic interactions and account for higher order particle correlations but mostly for solid colloidal particles. Still, the major differences between solid and fluid particles are the fluidity and flexibility of the droplet interfaces. 3.1. Droplet motion and hydrodynamic interactions The motion of a viscous droplet in unbounded viscous fluid without surfactants presents a relatively simple hydrodynamic problem and was solved

332

D.N. Petsev

by Rybczynski [97] and Hadamard [98]. However, in most real cases there are always surface-active substances present and they change the droplet movement considerably. Analyses of single droplet motion in the presence of surfactants were given by Levich [99]. The relative motion of two droplets presents a more difficult problem. The case of absence of surfactant and interfacial tension gradients has been considered by Davis et ah, who suggested numerical solutions for the mutual mobilities of non-deformable droplets [100] as well as accounting for the deformation in lubrication approximation [101,102] or using a boundary integral algorithm [103]. More detailed analysis of the hydrodynamic interactions between droplets is also given in Chapters 10 and 11 of this book. Emulsion droplets in absence of surfactants are usually unstable against coalescence and are unable to form doublets or higher order aggregates of individual droplets separated by thin films of the continuous phase. The presence of surfactant stabilizes the emulsion droplets and strongly affects the hydrodynamic interactions. When two droplets are close to each other, the main contribution to the hydrodynamic resistance (energy dissipation) comes from the thin gap between them, or the thin liquid film. The hydrodynamics of displacing solvent from the film during the approach of two surfactant stabilized emulsion droplets has been studied extensively [31,33,34,104-106]. These studies considered millimeter and sub millimeter sized droplets where the main force, bringing the two droplets together, is usually buoyancy. Knowing the magnitude of the buoyancy force and the hydrodynamic resistance, one may calculate the velocity, V{h,R) and hence the time of film thinning, r, viz. r{hm,hf,R) = ^ ^ - ^

V(h,R) = FBC(h,R)

(35)

where FB is the buoyancy force and <^h,R) is the hydrodynamic resistance that depends on the film thickness, h, and radius, R. For small (micrometer and submicrometer) sizes, the Brownian motion is bringing the two droplets together [15]. The kinetics of flocculation of Brownian droplets can be described following the Smoluchowski method modified to account for the droplet deformation if such occurs. Note that the hydrodynamic force between the droplets may induce deformation and it usually occurs at larger separations [15], than those that correspond to a minimum in the interaction energy. The relative diffusion approach of two droplets at small separations is related to the hydrodynamic drag force. The thin film formed between the droplets is characterized by two variables thickness, h, and radius, R and in Brownian systems they both fluctuate when subjected to the random

Theory of Emulsion Flocculation

333

Fig. 6. Random force acting on a thin liquid film between two droplets and the respective trajectory in the 2-dimensional space of the film thickness h and radius R. This is a qualitative illustration of film thickness transition due to Brownian fluctuating forces Fh and FR.

force, F, with normal, and radial components Fh and FR respectively, see Fig. 6. Hence,

\FR}=

^RR ^Rh\[VR

{Fh}

QhR ChhAVh '

(26)

For linear hydrodynamic flows one may apply the Onsager theorem [107] which implies that (Rh — QhR. The components of the friction tensor, £, are given by [26], see also [15,31] 6TTT]R2

h{\-£s) _ ShR—^Rh—

_W 2h

3nVR3 J2—

^'>

, , R2 , R4 ' ah

l l ah

where rj is the solvent viscosity and ss is a parameter that takes into account the interfacial mobility and droplet bulk viscosity - see [31,104]. For tangentially immobile droplet interfaces, ss = 1. This is the case when the interfaces are

334

D.N. Petsev

Fig. 7. Components of the hydrodynamic resistance tensor as defined by Eq. (37). These are valid for small separations between the droplets.

Theory of Emulsion Flocculation

335

completely saturated with adsorbed surfactants that are soluble in the continuous phase only. If there are no surfactants, or if they are soluble only in the droplet phase then ss decreases and may become about 0.001 [31,104,105]. In absence of deformation (R = 0) and tangentially immobile droplet surfaces, ChR =CRR = 0 a n d (hh =3-7rrya2/2/z, which corresponds to the result for two solid spheres of radius a [12]. Another limiting case is a very thin film with a large radius, Rlh^>\, when the contribution of the curved surfaces around the film could be ignored. In this case the Reynolds formula for the approach of two solid parallel disks [108], (hh = 3nrjR 12}?, is applicable. The component QhR increases as R3 while (RR becomes infinite for the case of tangentially immobile droplet surfaces and finite radius since es — 1, see Eq. (37). Fig. 7 represents the dependence of the different tensor components on the film thickness, h, and radius, R. 3.2. Brownian Flocculation 3.2.1. Effect of the droplet interactions on the flocculation kinetics If the direct and hydrodynamic interactions are known one may calculate the steady flux of droplets toward a given central droplet, J, and hence - the rate constant of pair flocculation

j

n^

WF

where nx is the droplet number concentration at infinite distance from the central droplet, Do is the Stokes-Einstein diffusion coefficient [109] and WF is the Fuchs factor defined by [2,12,86]

WF=4D0af dr Jo

[

\\ D{r)

2 2 r

J

-

(39)

The integral in (39) is taken over the distance between the flocculating droplets. U(r) is the interaction energy and D{r) = kTI(,(r) is the diffusion coefficient at small separations [15,26], also see below. The rate constant, kf, defined by (38), can be introduced in the kinetic equation for the formation rate of doublets of droplets [15]

336

D.N. Petsev

£ =*/-*,

CO)

«2 is the number of doublets formed and nx is the single droplet number at infinity. Detailed analysis of the above equation for the particular cases of droplet flocculation and coalescence is performed in Ref. [110]. The flocculation and coalescence kinetics of Brownian emulsions is not affected by deformability for diluted systems, in the presence of only attractive interactions [15]. In such systems the time determining step for a flocculation or coalescence of two droplets is the approach from large distances. On the other hand, the presence of strong repulsive interactions, e.g. high surface potentials, may prevent the droplet deformation for low electrolyte concentrations ^ 0.1 M of monovalent electrolyte. Therefore, the kinetics of flocculation and coalescence of very diluted or charged emulsions with low concentration of electrolyte present will be identical to solid dispersions with the same particle size, Hamaker constant, surface potential and other interaction determining parameters [15], see also [111,112]. In the case of charge-stabilized emulsions at high ionic strength (above 0.1 M monovalent electrolyte), the situation could be different. For such high salt concentrations droplet flocculation often occurs and is accompanied by droplet deformation [32,113-116]. The droplets may flocculate in the secondary minimum of the DLVO energy [117] and eventually jump into the primary similarly to macroscopic foam and emulsion films [29,118]. The primary doublets may further coalesce. This case was recently analyzed [26] and was shown that the mean time for transition between the secondary into the primary energy minimum, r depends on the droplet deformation and film formation. This mean transition time could be obtained by solving the following equation [26] (see also Fig. 6) —

^

V\Peq (R,h)D(R,h).Vr(R,h)\ = - 1 , where

611

Peg (/?,/,) = e x p - ^ P

X

'

(41)

and D = kT^(R,hf

Since the inverse friction matrix is [26] >•-! _

1 ( (>RR ~ C M | _

1

|

(M

the components of the diffusion tensor, D, become

~ChR\

,,2\

337

Theory of Emulsion Flocculation

kT

Chh

D K R =

CRRCM

kT

=

~ChR

+

,

ChR CRRCHH )

n

~kT<:hR -

- n -

U

U

hR — Rh — ~—";

~kT

T2~ —

r<m 2

"

'

'

SRRShh ~ ChR

CwChh j _ ChR ChR CRRChh kT > i ChR

kTQm = CRRChh ~ChR

CRRCM-I

The dependence of the diffusion tensor components on the film thickness, h, and radius, 7?, is given in Fig. 8. Note that DRR is infinite at R = 0 and becomes zero for es — 1. The transition (or coalescence) time equals twice the mean time necessary for a system (doublet of droplets) to move from the secondary minimum (Rmin,hmm) to the saddle point (RsaddAadd) (see Fig. 5). Hence, the boundary conditions for Eq. (41) are T R

{ min. Kin ) =T>

T R

(44)

{ sadd, Kadd ) = °

The multiplication by two is needed to account for the fact that a system at the saddle point can either cross it or return with equal probability [119,120]. In the case of tangentially immobile droplet surfaces DhR = DRh = DRR = 0 , Eq. (41) becomes [26]

"^^.^.ijiJlML-l. Peq{R,h)dh

H

,45)

dh

Equation (45) can be integrated directly and c Km,

dz

r°°

^ • ^ ^ ) = J L f ( > M ^ M J , dyPeq{R,y).

(46)

If hmin is at the secondary minimum and hsadd corresponds to the top of the barrier that has to be overtaken, Eq. (46) gives the mean transition time between the secondary and the saddle point (Fig. 6) [26]. It is interesting to compare (46)

338

D.N. Petsev

Fig. 8. Components of the diffusion tensor as defined by Eq. (43). These are valid for small separations between the droplets.

Theory of Emulsion Flocculation

339

to (35). The latter applies to film thinning and thickness transitions due to the action of a well-defined deterministic force (e.g., buoyancy), while the former is valid for processes that are driven by random Brownian forces. That is why the mean times given by (46) and (41) are statistical averages and depend on the probability distribution in the configuration space defined by Peq(i?,/?) = exp —U(R,h)I kT , as well as on the diffusion coefficient Dhh(R,h) = kT/(hh(R,h) and/or the diffusion tensor D(R,h) = kT[C)(R,h)\'1. Analysis of the dynamics of secondary-primary film transitions, based on (44) demonstrated that the deformation of the droplets may increase the mean transition time with orders of magnitudes when compared to referent nondeformable particles, for more details and examples see Ref. [26]. Eq. (46) and the underlying model do not take into account wave disturbances of the droplet (and more importantly, film) surfaces. Such disturbances are known to trigger film thickness transitions and breakups in macroscopic films (e.g. large millimeter sized droplets) [29-31]. According to experimental evidence [22,118,121] and some theoretical estimates [122], film surface corrugations and wavelike deformations do not occur for small films, like those that form between Brownian micrometer and submicrometer droplets. Still the analysis of some model systems based on Eq. (44) showed that the effect of the droplet deformability could be very substantial at high electrolyte concentrations [26]. 3.2.2. Flocculation and coalescence kinetics of dilute emulsions Calculating the flocculation and/or coalescence rate constant for diluted emulsions is relatively simple if only the Dhh(R,h) = kT /(,hh(R,h) term is taken into account and the other tensor terms are assumed to be zero, DhR = DRh = DRR = 0 [15]. In this case the problem could be solved by calculating the steady flux of droplets toward a single one as suggested by Smoluchowski for solid colloidal particles [83-85]. The important difference in case of droplets is the possibility for interfacial deformation due to hydrodynamic and direct forces. The droplet deformation in this case may start at distances that are substantially larger that those corresponding to the distant secondary minimum of the direct droplet interactions. The reason is the addition of an effective Brownian force to the action of the surface colloidal forces. The droplet flux is

J =4 ^

D W

* G +£ & * 4 2 L coml [ dr

kT

(47)

dr

where r is the distance from the central droplet, nx is the droplet number concentration at infinity, D(r) is the diffusion coefficient, P(r) is the probability

340

D.N. Petsev

to find a droplet at a distance r from the central droplet, and U(r) is the pair interaction energy. Eq. (47) can be solved for the probability function using the boundary conditions P -> 1, U -> 0 for r -»oo P^O, forr^O

(48)

and the result is [15,123]

pooexp[^(r)/^r] , . P(r) w = exp

U[r)\ —-1 itr

*r

D(r)r2 ^yi— . r ooexp U(r)lkT\ I ^—df Jo D(r)r 2

(49)

To determine the total force (Brownian and direct), it is convenient to rewrite Eq. (47) in the form [15] J<;(r) =kTd\nP(r) 2

47rr « 00

|

dU(r)

^

(5Q)

dr

where C,{r) — kTID[r) is the hydrodynamic resistance to the mutual droplet approach. The mean droplet velocity of motion toward the central droplet is [15] V r

() =

/

, ,-

(51)

4irr2nooP(r) This velocity is related to the mean total force, Fj{r), by the relationship FT(r) = t{r)V{r)

(52)

or [15]

FT(r) = tr*¥& dr

+

«W. dr

(53)

Theory of Emulsion Flocculation

341

The first term corresponds to the mean Brownian force, while the second is due to the direct interactions (e.g., van der Waals, electrostatic, etc.). Hence, the total force is

47rr2WnoP(r) For the special case of droplet approach with simultaneous deformation due to the force bringing them together, one may compute the hydrodynamic resistance coefficient using the fact that the radial, VR, and normal, Vh, film velocities are related by [15,26] Vr> R(\-e.\ -*- = - i *-!-. Vh 2h

(55)

Then Eq. (52) becomes Fr=Chh+ChR^^Vh

(56)

where Qhh + C,hRR{\ — es)/2h is the effective friction coefficient. Knowing the total force, one could easily obtain the deformation separation distance h, (where the surface curvature inverses its sign, and therefore the subscript /) from the relationship [31,34], see also [15] h -

FT

J

^rd)

p - p l

r

\

(
where rd = hf + 2a is the droplet mass center-to-center distance (see Fig. 9). The probability at the point of deformation is

Eq. (57) together with (56) gives [15]

342

D.N. Petsev

IrT

k=-Z^L7

(59)

where pr, rd ((r) Jo

l

r C,(rd)

\U(r)-U(rd)] kT

This integral has to be evaluated over the three regions Ac, ,4/and Ad (see Fig. 9).

Fig. 9. A qualitative illustration of the pair probability distribution function P(r) for deformable interacting droplets. Aa represents the region of droplet approach with no deformation, Aj correspond to the interdroplet distances where the droplets deform, A/ is the region of film thinning and Aa is the final region where coalescence may occur. rd is the distance between the droplet mass centers at which the deformation starts while rj is the value where the film starts to thin and rcr is the critical distance for film rupture. It is often convenient to work with film thicknesses instead of mass center distances. Thus ht is the film thickness where the deformation starts or the surface curvature inverts its sign (hence the subscript /). Sometimes it is assumed (for simplicity) that while the films forms and obtains radius rj the thickness remains constant and equal to h, (region Ad). Reproduced by permission of the American Chemical Society, K.D. Danov , N.D. Denkov, D.N. Petsev, I.B. Ivanov, R. Borwankar, Langmuir 9 (1993) 1731.

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343

If the emulsion is diluted, it can be assumed that the contribution from region Ac is much smaller than the contributions from the other two regions [15]. This assumption can be justified by the fact that the film rupture and droplet coalescence is usually accompanied by a sharp decrease in the interfacial energy and the exponential in (60) is small. Also, the rate of coalescence is much greater than the rate of approach and film thinning, which indicates that the ratio £(r)/£(r r f )
*=-«-, 4ir-yaf0

/0 =2/'<4^M*2L 1 Jra r C,[rd)

(61)

kT

Eq. (61) allows for determining ht if C,(f), U(r) and rcr {hcr = rcr — 2a), where hcr is the critical thickness of film rupture [31,124-126] 2

2

^,=0.243/*,.-^-,

1/7

.

(62)

2-7T7 hj

Eq. (61) is the last piece of information needed to obtain the Fuchs factor [see Eq. (39)], [15] and hence, the droplet flocculation kinetic coefficient [Eq. (38)]. Eq. (62) was derived for attractive van der Waals forces without any long range repulsion. Therefore its application to charged droplets is an approximation. Following this approach, one may also calculate the Wuchs factor and the kinetic constant, see equations (38) and (39) respectively. If the film between the droplets remains intact at the final stage, then such a calculation refers to the kinetics of flocculation. If, however, the film breaks (at distance hcr), it refers to the kinetics of coalescence. 4. EXPERIMENTAL STUDIES OF EMULSION FLOCCULATION AND INTERACTION BETWEEN FLUID INTERFACES This section presents a brief outline of some recently developed techniques for studying the interaction and flocculation between emulsion droplets and measuring the forces between fluid interfaces. 4.1. Direct measurement of the forces between emulsion droplets Performing well-defined model experiments with emulsions became possible due to the method for obtaining monodisperse sample developed by

344

D.N. Petsev

Bibette [54]. This method avoids all the ambiguities stemming to their polydispersity and allows for better comparison with theoretical models that are readily available for droplets of equal size. A method for directly studying droplet-droplet interaction was suggested recently, which is based on monitoring the force-distance relationship law between equally-sized magnetic emulsion droplets [127]. A monodisperse sample of octane ferrofluid (octane containing 10% Fe2C>3 grains) was prepared following [54]. Applying a magnetic field to the emulsion gives rise to a force pressing the droplets against each other, while monitoring the Bragg diffraction of visible light from the sample provides the necessary information about the distance between them. This powerful technique has a great potential in studying different types of interactions that might be present in emulsions [127,128]. The main inconvenience of this method is that Fe2O3 grains must always be present in the dispersed phase to provide the response to the externally applied magnetic field. 4.2. Liquid surface force apparatus Liquid surface force apparatus is a device for measuring the forcedistance profile between a droplet (e.g. oil) formed at the tip of a flexible micropipette and a macroscopic oil-water interface [129-132]. The force exerted on the droplet while pressed towards the interface is determined by the deflection of the pipette shaft. The disjoining pressure in the film between the droplet and the interface could be obtained by knowing the pressure inside the droplet. The latter is controlled throughout the experiment. This method allows for direct investigation of the important case of a droplet interacting with the macroscopic phase of the same material and extensive experimental studies have proven that it is very informative and reliable [129-131]. Its relevance to the interaction between two small droplets in an emulsion is a little less straightforward. 4.3. Experimental cells for studying emulsion films The formation and evolution of thin liquid films could be conveniently traced in specially designed experimental cells [29,133]. The cell is usually a capillary (with a diameter of the order of a few millimeters) in which the oil phase is separated in two parts by a thin layer of water to simulate an oil in water emulsion. The capillary is connected to a pressure controller and the film thickness could be changed as well as monitored by interferometry. The experimental setup allows for obtaining the forces acting across the thin water film between the oil phases. The main disadvantages of such a cell are [118]: (i) the typical film diameters formed are above 100 urn (much larger) and (ii) the capillary pressures are much lower than those in real emulsion systems. A step forward in the design of cells for thin film studies was the construction of a

Theory of Emulsion Flocculation

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miniaturized version [118]. Using laser micro machinery a cell diameter of about 280 um was produced that allowed for the formation of films with radii in the range 20-220 urn. Although a significant improvement, these radii are still too large when compared to miniemulsions, where the film's radii could be in the nanometer range. 4.4. Video-enhanced microscopy This method was developed recently [134-136] and is based on direct observation of the kinetic processes in emulsions. The kinetics of flocculation (including reversible) [134,135] and flocculation combined with coalescence [136] were studied. The method is well designed for kinetic investigations of emulsions but it is also applicable to solid dispersions. 5. CONCLUDING REMARKS This chapter presents an overview of the droplet interactions in emulsions and their effect on the stability against flocculation. Surfactant stabilized emulsions are often treated theoretically in the same way as suspensions of solid particles. This is justified for small droplets (below one micrometer) with high interfacial tension because they are virtually undeformable. Large droplets (between tens of microns to millimeters in diameter) represent the opposite limiting case where all the important interactions are across the plane parallel liquid film that forms between them at small separations. There are many other situations, however, where the droplets are deformed and the overall interaction is due not only to the plane parallel liquid film, but also to the contribution from the curved surfaces surrounding the film. Examples for such emulsion systems are small droplets (a few hundreds of nanometers in diameter) with low interfacial tension. A special case are microemulsions where the interfacial tension could be extremely low and even droplets smaller that 100 nm in diameter are amenable to deformation. Such droplets exhibit Brownian motion and deform under the action of the hydrodynamic and surface forces present. Hence, all theories for the interactions and kinetics of flocculation have to be modified to account for the effect of the deformation. This chapter summarizes some of the work that we have done in this direction. We suggest expressions for the most common types colloidal interactions (electrostatic, van der Waals, etc.) that take into account the droplet deformation along with the contribution of the droplet surface extension. The theoretical analysis of the flocculation kinetics requires knowledge of the hydrodynamic interactions between the droplets. We have derived expressions for the friction tensor and developed a procedure for calculating the mean transition time over the energy barrier that might be present between two approaching droplets. More details and accurate numerical analysis of these interactions is presented in Chapter 10 of this book. For more

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information on coalescence and droplet breakup the reader may refer to Chapters 9 and 11. Complete understanding of the details of emulsion flocculation and coalescence is important because it helps in developing more accurate models describing emulsion systems. Such models would have better predicting power and can be tested with the constantly improving experimental methods for studying emulsions. REFERENCES [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [II]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

B.V. Derjaguin, N.V. Churaev and V.M. Muller, Surface Forces (Plenum, New York, 1987). B.V. Derjaguin, Theory of Stability of Colloids and Thin Films (Plenum, New York, 1989). E.J.W. Verwey and J.T.G. Overbeek, Theory and Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). J.T.G. Overbeek, Stability of Hydrophobic Colloids and Emulsions, in: H.R. Kruyt (ed.), Colloid Science, Vol. 1 (Elsevier, Amsterdam, 1952) 302-341. D.H. Napper, Polymeric Stabilization of Colloidal Dispersions (Academic Press, New York, 1983). G. Hadziioannou, S. Patel, S. Cranick and M. Tirrell, J. Am. Chem. Soc, 108 (1986) 2869. S. Asakura and F. Oosawa, J. Chem. Phys., 22 (1954) 1255. M. Aronson, Langmuir, 5 (1989) 494. P. Richetti and P. Kekicheff, Phys. Rev. Lett., 68 (1992) 1951. J.L. Parker, P. Richetti, P. Kekicheff and S. Sarman, Phys. Rev. Lett, 68 (1992) 1955. P. Somasundaran, B. Markovic, S. Krishnakumar and X. Yu, Colloid Systems and Interfaces - Stability of Dispersions Through Polymer and Surfactant Adsorption, in: K. Birdi (ed.), Handbook of Surface and Colloid Science (CRC Press, New York, 1997). W.B. Russel, D.A. Saville and W.R. Schowalter, Colloidal Dispersions (Cambridge University Press, Cambridge, 1989). J.N. Israelachvili, Intermolecular and Surface Forces (Academic, New York, 1991). K.D. Danov, D.N. Petsev and N.D. Denkov, J. Chem. Phys., 99 (1993) 7179. K.D. Danov, N.D. Denkov, D.N. Petsev, I.B. Ivanov and R. Borwankar, Langmuir, 9 (1993) 1731. N.D. Denkov, P.A. Kralchevsky, C. Vassilieff and I.B. Ivanov, J. Colloid and Interface Sci., 143(1991)157. N.D. Denkov, D.N. Petsev and K.D. Danov, Phys. Rev. Lett., 71 (1993) 3226. N.D. Denkov, D.N. Petsev and K.D. Danov, J. Colloid and Interface Sci., 176 (1995) 189. J.A.M.H. Hofman and H.N. Stein, J. Colloid and Interface Sci., 147 (1991) 508. I.B. Ivanov and P.A. Kralchevsky, Colloids Surf, 128 (1997) 155-175. D.N. Petsev, N.D. Denkov and P.A. Kralchevsky, J. Colloid and Interface Sci., 176 (1995)201. D.N. Petsev and J. Bibette, Langmuir, 11 (1995) 1075. D.N. Petsev and P. Linse, Phys. Rev . E., 55 (1997) 586. D.N. Petsev, Physica A, 250 (1997) 115.

Theory of Emulsion Flocculation

[25]

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

[36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60]

347

D.N. Petsev, Interactions and Macroscopic Properties of Emulsions and Microemulsions, in: B.P. Binks (ed.), Modern Aspects of Emulsion Science (RSC, London, 1998). D.N. Petsev, Langmuir, 16 (2000) 2093. B.P. Binks, Emulsions — Recent Advances in Understanding, in: B.P. Binks (ed.), Modern Aspects of Emulsion Science (Royal Society of Chemistry, London, 1998). B.P. Binks, Annu. Rep. Prog. Chem., Sect. C, 92 (1996) 97. Scheludko, Adv. Colloid Interface Sci., 1 (1967) 391. C. Maldarelli, R.K. Jain, I.B. Ivanov and E. Ruckenstein, J. Colloid Interface Sci., 78 (1980) 118. I.B. Ivanov and D.S. Dimitrov, Thin Film Drainage, Chap. 7, in: I.B. Ivanov (ed.), Thin Liquid Films (Dekker, New York, 1988). J. Bibette, Langmuir, 8 (1992) 3178. I.B. Ivanov, Pure Appl. Chem., 44 (1980) 1241. I.B. Ivanov, D.S. Dimitrov, P. Somasundaran and R.K. Jain, Chem. Eng. Sci., 40 (1985) 137. P.A. Kralchevsky, K.D. Danov and N.D. Denkov, Chemical Physics of Surface and Colloid Science, in: K. Birdi (ed.), Handbook of Surface and Colloid Science (CRC Press, New York, 1997). J.T.G. Overbeek, The interaction between colloidal particles, in: H.R. Kruyt (ed.), Colloid Science, Vol. 1 (Elsevier, Amsterdam, 1952)245-277. R. Aveyard, B.P. Binks, J. Esquena and P.D.I. Fletcher, Langmuir, 15 (1999) 970. J.H.d. Boer, Trans. Faraday Soc, 32 (1936) 10. H.C. Hamaker, Physica, 4 (1937) 1058-1072. H.B.G. Casimir and D. Polder, Nature, 158 (1946) 787. H.B.G. Casimir and D. Polder, Phys. Rev, 73 (1948) 360. E.M. Lifshitz, Soviet Physics JETP, 2 (1956) 73. I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Adv. Phys., 10 (1961) 165. N.G.v. Kampen, B.R.A. Nijboer and K. Schram, Phys. Lett, 26A (1968) 307. J. Mahanty and B.W. Ninham, Dispersion Forces (Academic Press, New York, 1976). D. Henderson, D.-M. Duh, X. Chu and D.T. Wasan, J. Colloid Interface Sci., 185 (1997)265. S.J. Alexander, Physique, 38 (1977) 983. P.G.d. Gennes, Adv. Colloid Interface Sci., 27 (1987) 189. A.K. Dolan and S.F. Edwards, Proc. R. Soc. London Ser. A, 337 (1974) 509. T.F. Tadros, Steric Interactions in Thin Liquid Films, in: I.B. Ivanov (ed.), Thin Liquid Films, Vol. 29 (Marcel Dekker, New York, 1988). H.J. Ploehn and W.B. Russel, Adv. Chem. Eng., 15 (1990) 137. I. Szleifer and M.A. Carignano, Tethered polymer layers, in: I.P.a.S.A. Rice (ed.), Advances in Chemical Physics, Vol. 94 (Wiley, New York, 1996). J.M.H.M. Scheutjens and G.J. Fleer, Macromolecules, 18 (1985) 1882. J. Bibette, J. Colloid and Interface Sci., 147 (1991) 474. X.L. Chu, A.D. Nikolov and D.T. Wasan, J. Chem. Phys., 103 (1995) 6653. D.T. Wasan, A.D. Nikolov, P.A. Kralchevsky and I.B. Ivanov, Colloids Surfaces, 67 (1992) 139. D. Henderson and M. Lozada-Cassou, J. Colloid Interface Sci., 114 (1986) 180. S. Basu and M.M. Sharma, J. Colloid Interface Sci., 165 (1994) 355. N.A.M. Besseling, Langmuir, 13 (1997) 2113. D. Henderson and M. Lozada-Cassou, J. Colloid Interface Sci., 162 (1994) 508.

348

[61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98]

D.N. Petsev

S. Marcelja and N. Radic, Chem. Phys. Lett., 42 (1976) 129. J. Forsman and C.E. Woodward, Langmuir, 13(1997) 5459. V.N. Paunov and B.P. Binks, Langmuir, 15 (1996) 2015. J.N. Israelachvili and H. Wennerstrom, Nature, 379 (1996) 219. J. Petkov, J. Senechal, F. Guimberteau and F.L. Calderon, Langmuir, 14 (1998) 4011. R.M. Pashley, J. Colloid Interface Sci., 80 (1981) 153. R.M. Pashley, Adv. Colloid Interface Sci., 16 (1982) 57. P.S. Laplace, Traite de Mecanique Celeste; Supplements au Livre X, Vol. 4 (GauthierVillars, Paris, 1806). R. Finn, Equilibrium Capillary Surfaces (Springer-Verlag, New York, 1986). H.M. Princen, Shape of Interfaces, Drops, and Bubbles, in: E. Matijevic (ed.), Surface and Colloid Science, Vol. 2 (J. Wiley, New York, 1969) 1. P.A. Kralchevsky and I.B. Ivanov, Chem. Phys. Lett, 121 (1985) 111. P.A. Kralchevsky and I.B. Ivanov, Chem. Phys. Lett, 121 (1985) 116. A.D. Nikolov, P.A. Kralchevsky, I.B. Ivanov and A.S. Dimitrov, AIChE Symp. Ser., 252(1986)82. I.B. Ivanov and P.A. Kralchevsky, in: I.B. Ivanov (ed.), Thin Liquid Films (Marcel Dekker, New York, 1988) Chapter 2. S.J. Miklavcic, Phys Rev. E, 57 (1998) 561. B.V. Derjaguin, Kolloid Z., 69 (1934) 155. J.K. Klahn, W.G.M. Agterof, F.v.V. Vader, R.D. Groot and F. Groenweg, Colloids Surf., 65(1992) 151. R. Aveyard, B.P. Binks, S. Clark and J. Mead, J. Chem. Soc. Faraday Trans. 1, 82 (1986) 125. W. Helfrich, Z. Naturforsch., 28c (1973) 693. D.N. Petsev, Mechanisms of Emulsion Flocculation, in: A. Hubbard (ed.), Encylcopedia of Surface and Colloid Science (Marcell Dekker, New York, 2002) 3192. K.G. Marinova, T.D. Gurkov, T.D. Dimitrova, R.G. Alargova and D. Smith, Langmuir, 14(1998)2011. A.D. Nikolov, D.T. Wasan, P.A. Kralchevsky and I.B. Ivanov, J. Colloid and Interface Sci., 133(1989) 1. M.v. Smoluchowski, Phys. Z., 17 (1916) 557. M.v. Smoluchowski, Phys. Z, 17(1916) 585. M.v. Smoluchowski, Z. Phys. Chem., 92 (1917) 129. N.A. Fuchs, Z. Phys., 89 (1934) 736. P.M. Debye, Trans. Electrochem. Soc, 82 (1942) 265. F.C. Collins and G.E. Kimbal, J. Colloid Sci., 4 (1949) 452. T.R. Waite, Phys. Rev., 107 (1957) 463. T.R. Waite, J. Chem. Phys., 28 (1958) 103. G. Wilemski and M. Fixman, J. Chem. Phys., 58 (1973) 4009. J.M. Deutch and B.U. Felderhof, J. Chem. Phys., 59(1973) 1669. D.F. Calef and J.M. Deutch, Annu. Rev. Phys. Chem., 34 (1983) 493. G.H. Weiss, J. Stat. Phys., 42 (1986) 3. J. Keizer, Chem. Rev., 87 (1987) 167. A.A. Ovchinnikov, S.F. Timasheff and A.A. Belyy, Kinetics of Diffusion Controlled Chemical Reactions (Nova Scientific, New York, 1989). W. Rybczynski, Bull. Intern. Acad. Sci. Cracovie Ser. A Sciences Mathematiques, 1 (1911)40. J.S. Hadamard, Compt. Rend. Acad. Sci. (Paris), 152 (1911) 1735.

Theory of Emulsion Flocculation

[99]

349

B.G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, 1962). [100] X. Zhang and R.H. Davis, J. Fluid Mech., 230 (1991) 479. [101] R.H. Davis, J.A. Schonberg and J.M. Rallison, Phys. Fluids A, 1 (1989) 77. [102] S.G. Yiantsios and R.H. Davis, J. Colloid Interface Sci., 144 (1991) 412. [103] A.Z. Zinchenko, M.A. Rother and R.H. Davis, Phys. Fluids A, 9 (1997) 1493. [104] T.T. Traykov and I.B. Ivanov, Int. J. Multiphase Flow, 3 (1977) 471. [105] T.T. Traykov, E.D. Manev and I.B. Ivanov, Int.J. Multiphase Flow, 3 (1977) 485-494. [106] K.D. Danov, D.S. Vlahovska and I.B. Ivanov, Journal of Colloid and Interface Science, 211 (1999)291-303. [107] L. Onsager, Phys. Rev., 37 (1931) 405. [108] O. Reynolds, Phil. Trans. Roy. Soc. London A, 177 (1886) 157. [109] A. Einstein, Ann. Phys., 17 (1905) 549. [110] K.D. Danov, I.B. Ivanov, T.D. Gurkov and R.P. Borwankar, J. Colloid Interface Sci., 167(1994)8. [ I l l ] S. Dukhin and J. Sjoblom, Kinetics of Brownian and Gravitational Coalescence in Dilute Emulsions, in: J. Sjoblom (ed.), Emulsions and Emulsion Stability (Marcel Dekker, New York, 1996). [112] N.A. Mischuk, S.V. Verbich, S.S. Dukhin, O. Holt and J. Sjoblom, J. Disp. Sci & Technol., 18(1997)517. [113] M.P. Aronson and H. Princen, Nature, 286 (1980) 370. [114] M.P. Aronson and H. Princen, Colloids Surf., 4 (1982) 173. [115] J. Bibette, T.G. Mason, H. Gang and D.A. Weitz, Phys. Rev. Lett., 69 (1992) 981. [116] J. Bibette, Langmuir, 9 (1993) 3352. [117] T.G.M.V.d. Ven and S.G. Mason, J. Colloid Interface Sci., 57 (1976) 517. [118] O.D. Velev, G.N. Constantinides, D.G. Avraam, A.C. Payatakes and R.P. Borwankar, J. Colloid Interface Sci., 175 (1995) 68. [119] C.W. Gardiner, Handbook of Stochastic Methods (Springer, New York, 1985). [120] N.G.v. Kampen, Stochastic Methods in Physics and Chemistry (Elsevier, New York, 1992). [121] B. Radoev, A. Scheludko and E. Manev, J. Colloid Interface Sci., 95 (1983) 254. [122] R. Tsekov, Colloids Surfaces, 141 (1998) 161. [123] S.G. Yiantsios and R.H. Davis, J. Fluid Mech., 217 (1990) 547. [124] S.K. Chakarova, M. Dupeyrat, E. Nakache, C D . Dushkin and I.B. Ivanov, J. Surf. Sci. Technol., 6 (1990) 17. [125] C. Maldarelli and R.K. Jain, in: I.B. Ivanov (ed.), Thin Liquid Films (1988) 497. [126] A. Vrij, F. Hesselink, J. Lucassen and M.v.d. Tempel, Proc. K. Ned. Acad. Wet. B, 73 (1970) 124. [127] F.L. Calderon, T. Stora, O.M. Monval, P. Poulin and J. Bibette, Phys. Rev. Lett., 72 (1994)2959. [128] O.M. Monval, F.L. Calderon, J. Philip and J. Bibette, Phys. Rev. Lett., 75 (1995) 3364. [129] R. Aveyard, B.P. Binks, W.-G. Cho, L.R. Fisher, P.D.I. Fletcher and F. Klinkhammer, Langmuir, 12(1996)6561. [130] B.P. Binks, W.-G. Cho and P.D.I. Fletcher, Langmuir, 13 (1997) 7180. [131] W.-G. Cho and P.D.I. Fletcher, J. Chem. Soc. Faraday Trans., 93 (1997) 1389. [132] P.D.I. Fletcher, Interactions of Emulsion Drops, in: D. Mobius and R. Miller (eds.), Drops and Bubbles in Interfacial Research (Elsevier, New York, 1998). [133] B.V. Derjaguin, A.S. Titievskaya, I.I. Abrikosova and A.D. Malkina, Faraday Discuss. Chem. Soc, 18(1954)27.

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[134] O. Holt, O. Saether, J. Sjoblom, S.S. Dukhin and N.A. Mischuk, Colloids Surf., 123124(1997) 195. [135] O. Holt, O. Saether, J. Sjoblom, S.S. Dukhin and N.A. Mischuk, Colloids Surf, 141 (1998)269. [136] O. Saether, J. Sjoblom, S.V. Verbich, N.A. Mishchuk and S.S. Dukhin, Colloids Surfaces, 142(1998) 189.

Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 9

Coalescence kinetics of Brownian emulsions N.O.Mishchuk Institute of Colloid Chemistry and Chemistry of Water, National Academy of Sciences of Ukraine, Vernadskogo avenue, 42, Kyiv 03142, Ukraine 1. INTRODUCTION Investigation of the reasons and regularities of emulsion destabilization, including droplet coalescence, is of considerable scientific and practical importance [1-3]. Despite the progress in theoretical [3-7] and experimental [4, 8-11] investigation of properties of thin films, there is not much information about coalescence rate in real emulsions. It is quite difficult to study the process of coalescence in emulsions directly and therefore coalescence between two droplets or between a droplet and a layer of liquid [12-14] or the general destabilization process are investigated. Although calculation of the total number of droplets [8] or use of dielectric spectroscopy [9] to establish coalescence rate are quite informative, these methods do not allow to follow the details of existent processes and, hence, to distinguish the rate of rupture of thin films between droplets from the rate of the other processes accounting for the approach and emergence of the films. Absence of industrial equipment for automatic measurement of the number of singlets, doublets and multiplets is one of the reasons for such unfavorable scientific and technological situation. Therefore, the characteristic time of coalescence r c in emulsions was established experimentally for the first time by quite laborious analysis of emulsion photos made at certain time intervals [15-20]. It should be also noted that, when developing a method of coalescence analysis, not only the technical support of investigations matters but the general methodological approach and availability of theoretic models ensuring the adequate consideration of the multiple factors influencing the investigated processes are of great importance as well. In particular, there are still no reasonably accurate theoretical models and reliable experimental verification of Van der Waals forces retardation and their screening with electrolyte [21, 22]. Practically, no theoretical models taking into account the variety of factors determining the stability of emulsions with medium and high concentration exist

352

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[23, 24]. Not enough attention is paid to investigation of coagulation reversibility. Emulsion destabilization is quite often considered as a combination of irreversible aggregation and coalescence (see, for instance, [8, 25]), although the aggregation reversibility may be an important factor influencing the rate of coalescence and destabilization regularities in whole [26- 29]. The role of aggregation reversibility diminishes or increases depending not only on the interaction energy of specific droplets in a doublet but also on the correlation of rates of the aggregate disintegration, emergence of a new aggregate and coalescence. Thus the aggregation irreversibility may be quite relevant for certain emulsions that are used for practical purposes. Yet, in many other cases, the aggregate reversibility in the secondary energy minimum - and sometimes in the primary minimum as well - may turn out to be the main factor determining emulsion properties. Hence, to develop general fundamental ideas of emulsion properties, it is imperative to select such conditions of experiment that would allow separating all the processes that are of importance for understanding of emulsion stability, including the aggregation reversibility. At the same time, it is necessary to ensure high accuracy of measurements and the possibility of univocal interpretation of obtained results, i.e., to conduct the experiment under conditions ensuring a theoretical description of the system with a minimum number of constants and arbitrary assumptions. For instance, the use of monodisperse emulsions or emulsions with inconsiderable scattering of droplet sizes allows not only to weaken the role of orthokinetic (gravitational) coagulation but also to avoid the difference in the value of energy of interaction between different droplets, which affects both aggregation kinetics and the coalescence process. Formation of multiplets is also one of the complicating factors. The energy of interaction between droplets in a multiplet and, therefore, coagulation reversibility and coalescence rate in these aggregates depend on the quantity of interacting particles and the character of their packing. Moreover, when multiplets are formed, initially monodisperse emulsion quickly changes into polydisperse and that leads to transition from purely Brownian coagulation to Brownian-gravitational one. Therefore Dukhin and Sjoblom recommended to use singlet-doublet emulsions, i.e., those in which forming of multiplets is unlikely, as a standard method for determining the characteristic time of an elementary coalescence act [26]. Such emulsions may be obtained at a certain combination of the time of rapid coagulation and the effective time of aggregate disintegration. Articles [28, 29] provide a thorough theoretical analysis of the possibility of obtaining singlet-doublet emulsions by selecting a droplet surface charge, electrolyte concentration and droplet volume fraction. The review [28] also analyzes the influence of various factors (retardation and screening of Van der Waals forces with electrolyte [30-32], presence of surfactants [33], etc.) on the energy of

Coalescence Kinetics ofBrownian Emulsions

353

interparticle interaction and, hence, the probability of aggregation in the primary and/or secondary minimums and, correspondingly, the probability of coalescence. The present chapter is based methodologically on the approach suggested in [26]. However, unlike [26], it explores not only the significance of coagulation reversibility but also investigates the role of transitions from the secondary minimum to the primary one. The theoretical analysis is limited to diluted monodisperse singlet-doublet Brownian emulsions that satisfy the aforementioned requirements. To provide a clearer view of the processes taking place and their influence on the kinetic dependencies that are measured, the investigation starts with simpler systems. Hence the first part of the chapter deals with stable thin films. This allows not only to carry out a qualitative and quantitative analysis of the processes taking place but also to dwell on certain moments of aggregation description that are of interest both for emulsions and dispersions. High emphasis is placed on discussion of the question of correct interpretation of obtained experimental data. 2.THEORY 2.1. Definition of Brownian emulsion and the main limitations on the investigated system The main property ofBrownian diffusion is thermal motion of droplets. If droplets are sufficiently small (with radius of micron order or smaller), it is the Brownian motion and not sedimentation that defines the coagulation behavior of emulsion. Emulsions with larger droplets may be referred to Brownian ones only in case that dispersion medium has almost the same density as emulsion droplets, which is almost impossible in real systems. Even if we suppose that there is ideal monodispersity and complete stabilization of the system, when large droplets with density p^ other than that of the medium pm are used, there will be complications with creaming of emulsion. At sedimentation or floating-up of droplets, they will concentrate in the lower or upper part of the cuvette. In the opposite part of the cuvette, volume fraction of droplets will decrease, which may be erroneously taken for coagulation. The investigated picture is also modified by the counter flow of dispersion medium, which compensates the sedimentation transfer of droplets from one part of the cuvette to another and distorts their Brownian motion. If coagulation takes place in the monodisperse system, a deviation from the regularities of Brownian coagulation may also emerge due to different velocity of sedimentation or floating-up of singlets and doublets (or enlarged droplets when they coalesce). The bigger is the difference between the density of droplets and the density of medium and the larger are the droplets, the more

354

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important is the gravitation effect. Significance of the effect of this process on Brownian coagulation may be defined using the model [34, 35], developed on the basis of the main ideas of gravitation coagulation [36]. Without dwelling on the details of these works, it should be noted that, when the superposition of Brownian and gravitation factors is assumed (which is admissible only in case of small effect of sedimentation or floating-up of droplets on Brownian coagulation), a change of the number of singlets could be described using rather simple formulas. In particular, the following expression is obtained for monodispersion [35]:

*.«).*, (.{•"^o[ 7 ^-'" ( ' + 2 ' /r - ) ]] {

(It™

(1)

2

8(l + l/r OT ) JJ

where N\{t) = N Q l(\. +111 Sm) is the change of the total number of particles at rapid purely Brownian coagulation [37], t is the time counted from the beginning of the process, tgm - n rj aQ / a kT is the time of rapid coagulation calculated according to Smoluchowski, Pe0-47taQAp/3kT is the Peclet number characterizing the relative role of gravitation and diffusion factor in coagulation and corresponding to the initial state of system with the droplet radius a0, TJ is the dynamical viscosity of the dispersion medium, a is the volume fraction of droplets, kT is the energy of Brownian motion, k is Bolzmann constant, T is the absolute temperature, Ap = \p^ - pm is the difference between the densities of droplets pj and medium pm, and g is gravity acceleration. According to Eq. (1), deviation of the number of singlets from the value calculated on the basis of Smoluchowski formula, increases in time. Let us assume that the accuracy of investigation may be considered acceptable when the error of determining the number of singlets caused by gravitation does not exceed 10%, i.e., N(t)/N{(t) = 0.9. The typical time t* that is necessary for appearance of such inaccuracy and which is calculated according to Eq. (1), is presented in Fig. 1. As the figure shows, the larger is the value Ap, the smaller is, at fixed radius a 0 , the value of t * /rSm, which leads to considerable deviation from the Brownian nature of coagulation. Dependence of t*lt^m on the radius of droplets is even sharper. It is not surprising taking into account that the coefficient of diffusion of Brownian particles according to Stokes-Einstein formula is inversely proportional to their radius D o -kT/(mr]a§, and the velocity of sedimentation of droplets is proportional to squared radius

Coalescence Kinetics of Brownian Emulsions

355

Used - 2aQ&pgl9ri [38]. The larger the value t*ltSm, the later, in comparison to Smoluchowski time, a deviation from the regularities of purely Brownian coagulation appears. Using the above figure, it is possible to determine the size of droplets for which, at the given Ap, the gravitation effect may be ignored. If the coagulation process within the Smoluchowski time is to be investigated, the intersection of the dotted line with the curves for relevant Ap (see fig.l) corresponds to the maximum acceptable droplet size. For instance, at Ap = 0.5, the radius of droplets should be less than 0.5 micron. Only in this case emulsion (or suspension) may be considered as Brownian. If emulsion is to be investigated within a longer period of time, the condition becomes even more rigid. Dependence of the critical droplets size on the difference between the densities of the medium and droplets is shown in Fig.2. As is seen from the figure, the gravitation effect may be ignored only at very small radiuses OQ , or only at very small differences of densities Ap. Similar calculations can be performed for non-monodisperse emulsion, showing that, at small scattering of droplets radiuses (for instance, when the volumes of the smallest and largest droplets differ twice), a considerable

Fig.l. Dependence of typical time /* , normalized to Smoluchowski time Tgm, on the radius of droplets. Numbers near curves refer to the difference of densities Ap. The dotted line shows equality t* = t^m. Fig.2. Critical radius of droplets fln^at gravitational coagulation as a function of the difference of densities Ap for investigations during / < T$m (curve 1) and / <\Qzgm (curve 2).

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N.O.Mishchuk

distinction from the monodisperse system exists only in the beginning of the process, before multiplets are formed in the initial monodisperse emulsion, i.e., when the latter has not begun to behave as non-monodisperse emulsion. At A/? < 0.1, results of numerical calculations for such non-monodisperse system differ from the monodisperse one not more than for several percents. Thus, the above estimation of critical particle sizes is suitable also for moderate nonmonodispersity. Unfortunately, there is another restriction on droplet size as well. Droplets cannot be much smaller than a micron because, due to Ostwald ripening [39], nanosize droplets disappear quite quickly and therefore Brownian aggregation and coalescence are complicated by this process. Another important restriction is concerned with the fact that existing theoretical models calculate the time of rapid coagulation, assuming free Brownian motion of droplets, i.e. independence of particle motion from the neighboring particles. Such approach is correct when surface forces exist only at short distances, in other words, when droplets move independently of each other within the time interval between their collisions. This is true for aqueous dispersion mediums, where the radius of surface forces action is usually considerably smaller than droplets size and, correspondingly, considerably smaller than the distance between droplets even in rather concentrated emulsions. However, this restriction can prove to be important at investigation of "water-in-oil" emulsions, where, due to low ion concentration in non-aqueous medium, the extension of double electrical layer K~ can be not only commensurable to but even distinctly larger than droplet radius CQ, i.e., satisfying condition KOQ «1. Indeed, the volume fraction of droplets in emulsion a could be expressed through the diameter of droplets 2ag and the distance between their centers 2Z>Q as a ~ (OQ Ibo) . Let us suppose that droplet motion may be considered sufficiently free when a droplet gets in the field of neighboring droplets (roughly speaking, when double layers intersect) at the distance not exceeding 5% of the average distance between them b§ - OQ, i.e., 2K~ «(b§ - ao)l2§. Taking into account the expression for a, a restriction on the thickness of the double electrical layer is obtained: rcag »40\fa/\l-^faj. It follows from this that, at a = 0.001, 0.01 and 0.1, it is necessary that AT^Q were larger than 4, 10 and 35, correspondingly. Hence, the thickness of the double layer should not exceed 1/4, 1/10 and 1/35 of the droplet radius. These conditions may be fulfilled in real non-aqueous mediums only for large droplets OQ » l/^m • However, in this case even at small Ap gravitation plays an important part, i.e., the emulsion is not Brownian but Brownian-gravitational.

Coalescence Kinetics of Brownian Emulsions

357

2.2. Basic notions of reversible and irreversible coagulations in Brownian emulsion Kinetic behavior of miniemulsions is a manifestation of competition between surface forces and Brownian motion [2, 15, 16, 40]. According to DLVO theory, in the general case, energy of pair interaction of charged particles is characterized by the existence of two minimums with a barrier between them [2], which result from different amplitudes and characteristic lengths of action of electrostatic and Van der Waals forces (Fig.3). Energy of interaction at the presence of other forces (hydration, hydrophobic, steric etc.) has the same qualitative appearance. The scheme of theoretic and experimental investigation of coalescence kinetics presented in this chapter is based on the main ideas of Brownian emulsions. All processes taking place in emulsions, except for coalescence, are caused by Brownian motion of droplets. In dilute emulsions, droplets diffuse freely until they get into the area where surface forces of other droplets act.

Fig.3. Energy of interaction of two particles U as a function of distance between their centers r and scheme of fluxes in a system (see Section 2.3): J\ is the diffusion flux from infinity to droplet, J 2 t n e fux from secondary minimum to infinity, JT, the flux from secondary minimum to primary one, J4 the flux from primary minimum to secondary one, J5 the flux from primary minimum to coalesced state. R = 2OQ is the closest distance between the centers of particles with radiuses QQ . Barrier, secondary and primary energy pits are shown using letters B, S, P.

358

N.O.Mishchuk

Depending on the energy of interaction between droplets, the following scenarios of further developments may be distinguished [41, 42]. When getting into energy minimum (pit), droplets are attracted and, at sufficiently big depth of the minimum, they maintain this stable position, i.e. form a doublet. Since there are two minimums (a deeper, primary, minimum and a shallower, secondary, one), there may be two types of doublets, which, following the established terminology [41, 29], are called here as primary (PD) and secondary (SD). When droplets approach, they first find themselves in the secondary minimum (S) and hence form a secondary doublet (SD). If the minimum is very shallow, i.e., attraction energy is small, the doublet is quickly disintegrated. This is reversible aggregation (or fiocculation). When the barrier is not very high (B), transition from the secondary (S) to the primary (P) minimum is possible. Since the latter is usually quite deep, the aggregate that is formed (PD) is stable and thus irreversible coagulation takes place. At the distances between the surfaces of particles having order of several nanometers, due to surface hydrophylicity additional repulsion may also appear, hindering further approach in the primary energy minimum. Adsorption layers, creating a steric barrier between droplets, act in a similar way. Sometimes the length of the layer of adsorbed macromolecules or macroions may be big enough to overlap the primary minimum, either in part or in full, and thereby to prevent irreversible coagulation. An opposite situation is possible when droplets are weakly charged and the barrier is negligible or inexistent. In that case two minimums become confluent. Not only the possibility of formation of primary and secondary aggregates but also the probability of their disintegration considerably depends on the change of interaction energy at the approach or moving away of droplets. For each pair of approached droplets, the probability and, consequently, the time of change of their aggregate state (aggregate disintegration or transition from one minimum to another one) depends on the thermal motion of surrounding molecules and the motion of the Brownian droplets themselves. For the same period of time, some doublets may disaggregate; some may transfer from one minimum to another, and others may retain their state. However, in general, transition from one state to another in the system has certain statistic character. Therefore it is convenient to consider transition processes as droplet fluxs Jj from one state to another (Fig.3) and, by analogy with multi-stage chemical reactions, to introduce characteristic times corresponding to each stage of such "reaction" (Fig.4). Unlike disperse particles, deformation is of importance for droplets. Since, in the absence of other forces except surface ones, deformation may take place only at attraction between droplets, it is possible in the energy minimum only. Apparently, in most cases, it may be accomplished only in the primary minimum, where interaction energy is sufficient for noticeable deformation.

Coalescence Kinetics of Brownian Emulsions

359

This means that no characteristic time, except for the period of disintegration of the primary doublet r pcj S(j, depends on droplet deformation. Since surface flattening intensifies interaction between droplets, their deformation leads to deepening of the primary minimum and increase of coagulation irreversibility. Thus, the process of coalescence depends on droplet deformation for two reasons: due to increase of stability of the primary doublet and because both the form of film and its thickness and area change as the gap flattens. Taking into consideration that the process of deformation depends not only on the interaction energy of approached droplets (which decreases as the droplets become smaller, according to the DLFO theory) but also on Laplace pressure [43] (which increases as droplet size decreases), deformation is more peculiar to quite large droplets, with the radius exceeding a micron. Moreover, as the study [25] shows, for micron droplets, change of interaction energy due to droplet deformation is not too significant and therefore can be ignored. Since the chapter is focused on rather small droplets, the role of deformation in aggregation will not be analyzed here in detail. Furthermore, to simplify theoretic models, we will assume that coagulation is irreversible in the first minimum, i.e., r p j sc[ -> oo even without droplet deformation. As regards coalescence, even though it depends on droplet deformation, the chapter is aimed at determining the time of coalescence irrespectively of the details of this process and therefore the mentioned issue will not be considered here. Investigation of small droplets also allows to consider them as solid particles, i.e., to neglect the effect of the surface and internal mobility of droplets on their sedimentation and on interparticle interaction. Since usually droplets of emulsion are covered with a layer of adsorbed maeromolecules, their mobility, especially for micron-sized droplets, diminishes to zero.

Fig.4. Scheme of aggregation, disintegration and coalescence. Characteristic times of different stages: TS SC{- collision of two singlets (formation of a secondary doublet), TSC]^Sdisintegration of a secondary doublet, rsd^pd ~ transformation of a secondary doublet into a primary one, rpd sd~ transformation of a primary doublet into a secondary one, TC coalescence time.

360

N.O.Mishchuk

2.3. Method of calculation of characteristic times The most consecutive model of aggregation and disaggregation in dilute suspensions, including calculation of characteristic times of subprocesses, is the model of Muller [41, 42]. This model could be directly used for description of non-coalescing droplets and expanded with account of coalescence [26-29]. If emulsion is rather dilute, multiplets practically do not exist and therefore it is enough to take into account the fluxes of single particles (singlets) Jj directed to or from a certain arbitrarily chosen particle (fig.3). Let us suppose that a system has reached a quasi-equilibrium, in other words, the aggregation and disaggregation are mutually equilibrated and the concentrations of singlets «1, primary rip and secondary ns doublets are constant. Naturally, at such quasi-stationary regime, diffusion fluxes are constant and can be described by the equation [41, 42]

L dr

kT dr J

y }

where v(r) is the concentration of singlets at the distance r from a chosen particle, U{r) is the energy of interaction of two particles, D = 2DQ is the coefficient of reciprocal diffusion of two singlets, defined through the diffusion coefficient of single sphere Do = kT 167rr/a0. At short distances D(r) = 2DQ /p(r), where p(r) is the factor taking into account hydrodynamic resistance between the particles approaching each other (for example,

p{r) = l +

ao{r-2ao)[44]).

To define fluxes J\ -J4, presented in Fig.3, Muller proposed to solve Eq.(2) with the following boundary conditions. The flux Jx leads to appearance of secondary doublets and satisfies the boundary conditions of Smoluchowski [45] - the absence of particles in the secondary pit and constant concentration of particles at the infinity v(r e S) - 0

and v(r = <x>) = NQ = const

(3)

The flux J2 leads to disintegration of a secondary doublet and satisfies the conditions |V(r)4;zr2dr = 1

and v(r = <x>) = 0

(4)

i.e., at the moment of doublet disintegration there is only one particle in the

Coalescence Kinetics of Brownian Emulsions

361

secondary pit and, during the process of disintegration, the particle that moves away interacts only with the second particle from the doublet. The flux J 3 defines the probability of transformation of secondary doublets into primary ones and satisfies the condition v(r e P) = 0

(5)

and the first condition in (4), that is, the absence of a particle in the primary pit and the presence of a single particle in the secondary pit. Finally, the flux J 4 defines the probability of transformation of primary doublets into secondary ones and satisfies the conditions

^v(r)4nr2dr = \

H

v(r€S) = 0

(6)

(rsP)

i.e., at the moment of transition there is a single particle in the primary pit and no particle in the secondary pit. According to Eq.(2) and conditions (3-6), the expressions for fluxes can be presented in the following way [42]:

R2TS

R2WTS

R2WTP

where 1^,1^,1^, are the "powers" of primary and secondary minimums, Wis the retardation factor [46, 47] which can be presented as

^ = \r2

exp(- U{r)) \P}Q txV{U(r'))dr' dr « r^

(8)

(r'Y

W=

f (reB)

P(r

'hxp{O(r'))dr'

fcxp(-O(r))r2dr

Fs= (reS)

(9)

(r'Y >

v

I> =

Jexp(-£/(r)/ kT)r2dr (reP)

(10)

362

N.O.Mishchuk

Here the distance r is normalized on diameter of particle 2a 0 U(r) = U(r)/kT. Taking into account the achievement of a local equilibrium J] n\ = 2 J2 %

H J^fip-J^ng

and

(11)

and existence only of singlets and doublets, the characteristic times of subprocesses have been expressed [42] as

r

s sd

'

-

l

1

-

J,

sd s

T

J3

-J_-^l

> ~J2~2D0'

R2WT^

1 l

sd,pd

~

SnD0RN0'

—

„ _. ^-^0

i '

l

pd,sd~

j

R2WTP

~~ ~ r. *M ^^O T

m\

(

v1J/

Although Eqs.(7-13) are received for a quasi-equilibrium state, which can be reached only at t » Ts sd'^sd,s'^sd pd>^pd,sd> they c a n t>e a l s o u s e c l when analyzing transition to an equilibrium state [42]. It is interesting to note that the time of transition from a secondary to a primary pit depends not only on the barrier height but also on the depth of the secondary pit, and the characteristic time for inverse transition depends on the depth of the primary pit. The larger the depth of the primary and secondary pits and, correspondingly, the larger the values of FS,TS and Tp, the less "readily" the particle leaves its place and the larger is the time of transition from the secondary pit into unconstrained state TS(J}S or into primary pit fsdnd • As a consequence, the flux J\ is divided into two fluxes J2 and J 3 , the intensities of which, according to Eq.(7), correlate as J2 UT, = W^s l^s ^W . This means that, due to the barrier, the probability of formation of a primary doublet is W times smaller than the probability of disintegration of an aggregate. It should be stressed that, according to the model presented above, even if a secondary pit is very shallow, i.e. the characteristic times Tsj s and/or ^sd,pdare small the transition from two singlets to a primary doublet formally has two stages: transition from singlets to a secondary doublet and transition through the barrier from a secondary doublet to a primary one. Although coalescence was not investigated in [42], formally it is possible to introduce the notion of the flux J 5 and the characteristic time of rupture of

Coalescence Kinetics ofBrownian Emulsions

363

thin film between two droplets (or the time of coalescence) Tc = 1/ J 5 , as it was done, inter alia, in [19, 26]. Some problems of correct calculation of interaction energy and characteristic times, necessary to ensure the proper description and understanding of existent processes, were discussed in the review [29]. 2.4. System of differential equations for description of coagulation and coalescence According to papers [42] and [19, 26], the change of concentrations of singlets (n s ), primary ( n p d ) and secondary ( n s d ) doublets can be described by interconnected differential equations

dns^_nj T

d?

+2nsd

+npd

r

s,sd

(i4)

f

sd,s

c

^ = -3L-»J^- + - U+ -^L 2r

d?

s,sd

dn

pd

{Fsd,.!

nsd

=

npd

r

&

r

pd,sd

npd

r

sd,pd

sd,pd)

(15)

r

pd,sd

r

c

or, with introduction of dimensionless values, by the equations djh^_n2+2^L dt

T

+^ L

(1?)

T

sd,s

c

d

-^=i-J—+-^1

dt dn

pd dt

2

T

y sd,s nsd

= T

sd,pd

as)

T

sd,pd)

npd T

c

where the concentrations of singlets and doublets are normalized on the initial droplet concentration NQ: ns=nsl NQ, npd =rtpdl NQ and nsd =nsd INQ and all characteristic times are normalized on the time of formation of secondary doublets Tj = T; I Ts sd .

364

N.O.Mishchuk

Since the size of the chapter is limited, the possibility of transformation of a primary doublet into a secondary one and correspondent terms npd I Tpd scj at transition from Eqs. (14-16) to Eqs. (17-19) are neglected. Eqs.(17-19) take into account the process of formation of secondary doublets, their possible disintegration into two singlets or transformation into primary doublets. Coefficient 2 in Eq. (17) shows that disintegration of a doublet results in two singlets and coefficient 1/2 in Eq.(18) shows that two droplets form a doublet. Eqs. (17-19) assume that formation of triplets is negligible. This is possible only in the case when the number of doublets in the system is not too high. The condition necessary for this will be discussed in Section 2.5. The terms npci I rc describe coalescence. The number of coalesced droplets is directly proportional to the number of formed thin films, i.e., the number of primary doublets, and inversely proportional to the characteristic time of their rupture r c . Since existent experimental methods do not allow to measure relatively small changes of droplet size, the coalesced droplets are included into Eqs. (17) and (19) as initial singlets. Taking into account that it is practically impossible to create an absolutely monodisperse emulsion, the 26% difference between the radiuses of initial (<3Q) a n d coalesced (a\ — 42CIQ) droplets cannot strongly reduce its non-monodispersity. Eqs.(17-19) should be solved together with initial conditions ns(t = 0) = l

nsd(t = 0) = 0

npd(t = 0) = 0

(20)

To better understand the role of coalescence and the way of its experimental definition, first, the process of coagulation will be investigated for the case of stable thin layer between droplets. Such analysis is important both for emulsions with thick adsorbed layers of macroions or macromolecules preventing coalescence and for suspensions, whose aggregation in primary and secondary energy minimums is underinvestigated. In this particular case, conditions (20) should be supplemented with the additional condition of the constant total number of droplets n

s + 2nsd + 2nPd =!

(21)

2.5. Single energy minimum, non-coalescing droplets The simplest case to be considered is the existence of a single energy minimum combined with high stability of thin film. Such situation takes place when the energy barrier is very high or the first minimum is inaccessible due to steric repulsion and, as a result, the primary doublets cannot form. Another case is when the surface charge is very low and, thus, a barrier is absent and only one

Coalescence Kinetics ofBrownian Emulsions

365

minimum exists, and thin film is stabilized with a dense layer of macromolecules. In this case Eqs.(17-19) can be reduced to two differential equations ^ .

=

_

n

2

dt

dnsd

+ 2 ^ T

_ sd,s

_ ns

dt

2

(22)

nsd T

sd,s

where the indexes corresponding to the secondary minimum are preserved, although, as was written above, the nature of a minimum could vary. With account of two first conditions (20) and condition (21), the solution of this system of equations can be presented as „ _ q O ~ c 2 ) - c 2 (l - q)exp(- (q - c2)t)m 0 " c2) ~ (1 - c, )exp(- (c, - c 2 ) 0

_^~ns 2

where c

i = ^

2r

;

C

2=^-^

M^

( 25 )

lT

sd,s

At ? -> oo, the quasi-equilibrium concentrations of singlets and doublets are established 1-Ci

ns (t -> co) = d

« ^ (/ -> oo) = - ^ - i -

(26)

Solution (24) is obtained at an arbitrary ratio of characteristic times. However, since the focus is made here on singlet-doublet emulsions, the following limitation for the characteristic times can be obtained from Eq.(25) nsd «1

or

^sd,s « l

(27)

With account of these conditions, the expressions (24) can be simplified: ns » l - ( l - q X l - e x p ( - ( q -c2)t)) , nsd * ^f>(l

-exp(-(q -c2)t))

(28)

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N.O.Mishchuk

The accuracy of the obtained approximation depends on the correlation between time t and the multiplier in the exponent (q -c-i). Comparison of (28) and (24) shows that at t > (q -c 2 )~ the deviation of (28) from the exact solution is about 0.1%. At t<(c\-C2)~ and strict observance of condition (27), the deviation is less than 0.25%. When condition (27) is not fulfilled, the deviation of approximation (25) from exact solution increases (at TS(J S ~ 1, it is less than 2% and, at Tsds ~ 5, it reaches 15%). Analysis of numerical and analytical solutions of differential equations shows that a quasi-stationary state is reached during the time tst that is a few times larger than tsci s. For example, if it is assumed that doublets have reached the quasi-stationary state when their concentration is equal to 90% from possible maximum value, the critical time of stationarity can be obtained from (28) as _ ln(10) _, 2.3 r ^ , S t

~~

/1

C]-c2

A

*•

'

Jl + 4TsdjS

With account of (27), quasi-stationary concentrations of singlets and doublets can be evaluated as ns (t -> oo) = 1 - rsdtS + 2x\s

;

nsd (;->«,) = ^

~^sd's

(30)

The larger the time of doublet existence, the higher is their concentration and the lower is the concentration of singlets. Under condition (27), nscj «1/8. At such small number of doublets, the appearance of triplets is very unlikely, i.e., this is really singlet-doublet emulsion. Examples of numerical calculations according to Eqs.(24) are shown in fig. 5. Since the quasi-stationary state could be reached very quickly, the best way to investigate experimentally the aggregation in the secondary minimum with account of possible disintegration of doublets is to measure the stationary concentration of doublets or the total concentration of "particles" ns + n^ using light absorption. The values should be stable during a very long time. Modern equipment allows to measure light absorption with very high accuracy during very short time t « TSCJ S , i.e., long before the quasi-stationary state is reached. However, the interpretation of such results could be incorrect. Indeed, as could be seen in Fig. 5, the beginnings of curves at considerably different values of TSCJ s are quite close to each another. Therefore accidental

Coalescence Kinetics of Brownian Emulsions

367

Fig.5. Dependence of concentrations of singlets ns (curves 1-6) and doublets nsci (curve 1'6') on time t. Disintegration timeTscj^s equals to : 1000 (curve 0) , 1 (1,1',1"); 0.8 (2,2',2"); 0.6 (3,3',3"); 0.4 (4,4',4"); 0.2 (5,5',5"); 0.1 (6,6',6"). Curves 0*-6* - sum of singlets and doublets ns + nsci, presented in Figs. 5c and 5d in different scales.

deviations of concentrations can lead to inaccuracy of measurement comparable to the distances between the curves. It is also necessary to stress that the decrease of slope of beginnings of the curves representing reversible coagulation in comparison to the slope of the curve of rapid coagulation (see fig. 5) can be

368

N.O.Mishchuk

wrongfully considered as manifestation of slow coagulation related with the overcoming of energy barrier between particles or droplets. However, in reality, as can be seen from curves in fig.5, at high energy barrier that prevents transformation of secondary doublets into primary ones, the slope of curves changes owing to aggregation reversibility. In contrast, at large depth of a secondary minimum (curves 0, 0*), aggregation looks as rapid coagulation in a primary minimum, although droplets do not overcome the energy barrier. 2.6. Secondary minimum of limited depth and infinitely deep primary minimum. Non-coalescing droplets The problem with account of transition into a primary minimum without coalescence can be described by three differential equations

* ' = - „ ? +2 "** dt

(31)

T

sd,s

dn

n

sd

S „ f

dt

2

dn

Pl dt

=

i

l

l

m

x

T

\Jsd,s

sd,pd )

^cL_ T sd,pd

(33)

Differentiation of Eq.(32) with account of Eqs. (31), (33) and (21) leads to the differential equation of the second order

d\£L 2

dt

= _2rhn^_dn^(2ns T

sd,pd

dt

{

+

J_

+

T

sd,s

_L_ > |

(34)

T

sd,pd)

the solution of which could be received according to the scheme proposed in [19, 26], where the linearization of a similar equation was performed by replacing function ns (t) with its initial value ns (t = 0)« 1. The exactness of solution at t > tst (29) could be higher when ns is replaced with its quasiequilibrium value ns « ns (t -> oo) - c\ (26), which was obtained for simpler emulsion without transition into a primary minimum. After presentation of Eq.(34) as

^V^^L dt2

dt

\

]-2c,-^+ T- ^ + ^— T T sd,s

sd,pd)

sd,pd

(35)

369

Coalescence Kinetics ofBrownian Emulsions

its solution is expressed very simply: a f

e-

_e~a2'

\

2(af2-a,) where

1 1 2ci ^,=20,+-!-+—!-;?,=—*T

sd,s

T

(38)

T

sd,pd

sd,pd

Concentrations of primary doublets and singlets are obtained from Eqs.(33, 36, 21) as

a2(l-e-^)~a,(l-e-a^) "/«/= o ^"7°

H;

. . . 2 «5=l- n^-2«pd

..„ (39)

Emulsion will remain singlet-doublet if, in addition to condition (27), it satisfies the condition of slow transition into the primary minimum, which is possible if hd,Pd»hd,s

and ^d,Pd>1

( 4 °)

In the opposite case, the number of primary doublets increases considerably, i.e., the total number of doublets rises and the possibility of triplet formation appears. Under conditions (40) q\ <1, p\ » 1 and a\ « 1 a2 » 1 and, therefore, at t »1 /«2 solutions (36, 39) can be simplified:

2«2

2*sd,pda\a2

{ a2

T

sd,pd^\«2)

The deviation of these simple expressions from exact numerical solutions during the period tst
370

N.O.Mishchuk

The rate of transformation of secondary doublets into primary ones can be found by differentiation of obtained functions (41): dn

dnscLa_a}_e-aV^ dt 2a2

vd ._, e~a* dt 2Tsd^pda2

'

(42)

These expressions become more obvious after expansion of parameters a, (37) in series dn

sd dt

T

., 2r

2 sd,s

sd,pd

T

c

sd,s Tsd^pd

T

dnpd dt

Tsd^

^

sd,s Tsd^pd

2t

sd,pd

The rate of formation of primary doublets is not only inversely proportional to the time of overcoming of barrier Tsdtpd but also proportional to the time of existence of secondary doublets rsd^s. The longer is the existence of secondary doublets, the larger is the probability that disintegration of secondary doublets will be replaced with their transformation into primary ones. The decrease of the number of secondary doublets caused by their transformation into primary ones is also inversely proportional to the characteristic time rsd pd. However, the obtained expression contains, instead of the primary degree of disintegration time, the secondary one r , . This fact, SCI jS

strange at the first glance, is actually quite correct. On the one hand, the rate of decrease of the number of secondary doublets is proportional to the probability of their transformation into primary doublets rsd s I rsd pd; on the other hand, it is proportional to the number of secondary doublets nsd(see Eq.(30)) which grows as the time of existence of secondary doublets r ^ 5 increases. Taking into account that in the investigated case of short disintegration time Tsds «1 (27), the rate of change of the number of secondary doublets dnsd I dt is lower than the rate of formation of primary ones dnpdl dt (see Eqs.(43)), the total concentration of doublets nd - nsd +npd grows. It is reasonable that under condition (40) the intense transition from a secondary to a primary pit takes place when the number of secondary doublets is sufficiently high. Joint analysis of the numerical solution of Eqs.(31-33) and the analytical solution (24) of the problem of aggregation without transition into a primary minimum showed that during the time / < tst (26) they practically

Coalescence Kinetics of Brownian Emulsions

371

coincide. Thus, at the short time of investigation ttst, by expressions (36, 39) or (41). The position of the maximum on the kinetic curve representing the evolution of number of secondary doublets can be found by differentiation of Eq.(38) and the use of the condition of extremum dnsd Idt = 0 as

, m= ]5C«iZ«a) a

2

<44)

~a\

Obtained value of rmax is very close to the time when the kinetic curves for a single minimum reach their quasi-stationary values (Fig.5,6). However, at ? m a x , bends of the curves representing the number of singlets and total number of doublets appear. The values of the maximum concentration of secondary doublets and concentration of singlets in the point of the bend can be found by substitution of Eq.(44) into Eqs.(36) and (39). Comparing the theoretical and experimental maximums of doublets concentrations and bends of curves for singlets can give a lot of information as that allows to determine the region of transition to quasi-stationary state and the value of TSCJS . The slope of curves after the transition region allows to define the value of rsd,pd • Let us compare the rates of processes at t < tst (29) and at t > tmax(44). In both cases the rate of investigated processes is defined by exponential function of time. In the first case, according to Eq.(28), the characteristic time is l/(cj -C2)- ?sd,s I ^+ ^Tsd,sx

T

sds~^T

d

while in the second case,

according to Eq.(41), it equals \l ct\ « rsc( ^ ( 3 + l/TS(jfS). On the basis of these expressions one can conclude that at t > tmax the change of singlets concentration caused by transformation of secondary doublets into primary ones is a i / ( q -c2)={r2sds-2T3sdJ/(xsdpd(\ + 3zsd^)) time slower than the change at t
372

N.O.Mishchuk

Fig.6. Dependence of concentrations of singlets ns (curves 1-6), secondary nsj(\'-6') and primary « p j (4"-6") doublets on time t. tsd,pd =1000 and Tsd^= '000 (curves 0,0') ; 0.03 (1,1'); 0.1 (2,2'); 0.3 (3,3'); r ^ ^ ^ = 1 0 and r^ ;iS =0.03 (curves 4,4',4"); 0.1 (5,5',5"); 0.3 (6,6',6"). Curves 0*, 4*-6* represent the sum of singlets and all doublets ns +npd +nsci; curves 4**-6** show the sums of primary and secondary doublets npd+nsdzero and the concentrations of singlets and secondary doublets coincide with the concentrations presented in Fig.5. A decrease of transition time Tsd^pd creates conditions for transformation of secondary doublets into primary ones (see

Coalescence Kinetics ofBrownian Emulsions

373

curves 4"-6"). As a result, the concentrations both of singlets (curves 4-6) and secondary doublets (curves 4'-6') decrease. However, since characteristic times for all the processes strongly differ (rsej>pci » TS^SCJ » TSCI^ ), at first, the faster processes equilibrate and that leads to coincidence of the beginnings of curves for singlets (1-3 and 4-6) and doublets (l'-3' and 4'-6'). A difference between the curves at the same value of TS(Jp(j but different values of TSCI s is clearly seen in Fig.6. Therefore, both the barrier that should be overcome by droplets and their location and interaction energy are of importance. The lower the interaction energy in the secondary minimum and, correspondingly, the smaller the time of secondary doublet disintegration, the lower is the probability of transition into the primary minimum at the same height of the barrier. Since secondary doublets transform into primary ones slower than they disintegrate, the concentration of primary doublets in the beginning of the process is low and does not affect the concentration of singlets and secondary doublets. However, as the number of primary doublets increases, the number of droplets participating in aggregation and disaggregation in the secondary minimum decreases and that leads to a decrease of singlet concentration. As a result, maximums of the curves 4'-6' and bends in the curves 4-6, 4*-6* appear demonstrating the qualitative change of the investigated process. Similarly to aggregation in the secondary minimum (Section 2.5), in the present case, not only the time of transition to a quasi-stationary state but also the values of corresponding concentrations depend on the characteristic times of each subprocess. Both in case of insurmountable and surmountable barriers, the concentration of secondary doublets grows and the concentration of singlets decreases as the time of doublet disintegration TS(JS increases (see transition from curves 1, 1' to 3, 3' and from curves 4, 4',4" to 6, 6', 6"). At very low TSCJs, the probability of secondary doublets disintegration is to such extent higher than the probability of their transformation into primary doublets that the concentration of secondary doublets becomes independent of the latter process. This conclusion is proved by small difference between curves 1' and 4', which coincide at the given scale of the figure. It is useful to compare the kinetic curves at fixed TSJS and different T

sd pd ( s e e Fig-7)- The decrease of the barrier leads to acceleration of

transitions into the primary minimum and affects the number of secondary doublets and residuary singlets. However, even at the transition times that are commensurable with the time of rapid coagulation (TSJ nd=^> curves 5, 5',5") or even smaller ( r 5 j n ( i < l, see curves 6,6",6"), the general rate of coagulation is considerably lower than the rate of rapid coagulation. This means that at a very

374

N.O.Mishchuk

Fig.7. Dependence of concentrations of singlets ns (curves 1-6), secondary nscj (V6') and primary rip^ (l'-6') doublets and the sum of doublets and singlets (l*-6*) on time / . The calculations are performed at TSCJ s =0.1 and

TSCJ n ^ = 1 0

(curves 1,1',1",1*); 5

(2,2',2",2*); 3 (3,3',3",3*); 2 (4,4',4",4*); 1 (5,5',5",5*) and 0.3 (6,6',6",6*), correspondingly. Curve 0* is calculated at zsc[ s =1000 and TSCJ pj =1000. Figs.c and d present the same results on different scales.

shallow secondary minimum and, consequently, low xs£s, droplets scatter sooner than they receive an opportunity to overcome the barrier. The analyzed peculiarities of primary doublet formation could be used for

Coalescence Kinetics of Brownian Emulsions

375

theoretical explanation of the well-known experimental fact - at reduction of electrolyte concentration, the deceleration of coagulation in comparison with rapid coagulation is smoother than according to the calculation of the retardation factor (see, for example, [46, 47]). Let us compare Figs.7d and 5d. Curve 1 * in Fig.7d coincides with curve 6* in Fig. 5d and the fan of curves 2*-6* in Fig.7d is the branching of the same curve 1 * that appears owing to the change of transition time TSCJpcj. Since in Fig.7 all values of x^

pcj

are larger than TscjfS , at short time t < TSCJs curves

l*-6* practically coincide. This means that if the measurement is performed within the time interval 0 < t < TS^^S , there will be a difference between curve 0* for rapid coagulation and curves l*-6*, caused not by transition into the primary minimum but by disintegration of secondary doublets. In other words, this will be "slow" coagulation, although not in its classical meaning. It will be slow due to reversibility of aggregation in the secondary minimum. The qualitative illustration of two different types of "retardation" is shown in Fig.8, where the factor of retardation at transition into the primary minimum without disintegration of doublets (curve 1) and the factor of the "retardation" caused by the change of the slope of kinetic curves related with the reversibility of aggregation in the secondary minimum (curves 2,2',2") are presented.

Fig.8. Factor of retardation W as a function of electrolyte concentration. Curve 1- traditional factor of retardation (see Eq.(9)), curves 2,2',2' - "retardation" factor caused by reversibility —21 of aggregation in the secondary minimum. Hamaker constant is ^ = 2 1 0 J, surface potential y/ = 25mV, radius OQ =0.5/jm. Curve 2" is calculated at the linear change of surface potential (// from 25mV at C=\molll to 50mV at C = 10~ moll I. Volume fractions of droplets are 1% (curve 2) and 10 % (curves 2',2").

376

N.O.Mishchuk

The numerical calculations were performed for arbitrary chosen parameters with account of retardation and screening of van der Waals interaction according to the models [30, 31] that particularly were used in [28, 29]. It was also assumed that experimental measurements of the number of droplets were performed at t - 0.2, which corresponds to the practically linear section of the kinetic curves in Fig.7d. As seen from the curves in Fig.8, at reversible aggregation, theoretical "retardation" of the process at reduction of electrolyte concentration could be even slower than that obtained experimentally, for example, in [48, 49]. The degree of "retardation" depends both on the change of volume fraction of droplets (2, 2') and on possible change of surface forces related with the change of surface potential (curves 2', 2"). The dependence of the position of curve 2 on volume fraction could be used to prove the "non-traditional" mechanism of coagulation retardation and to separate it from other mechanisms, for example, those proposed in [48] and other papers. Choice of the time of measurement is also important. For example, in Fig. 7d, the kinetic curves at /<0.2 look linear, i.e., seemingly it is enough to do the measurements at t - 0.2 and use them to define the retardation factor. However, curve 6* in Fig.5d (the same time interval but different scaling) shows the evident deviation from linearity. Thus, measurement of droplet concentration at smaller t will produce another value. That is why it is very important to study aggregation not only in the beginning of the process but within the large time interval giving more information about the investigated process. It is necessary to stress that, when investigating "water in oil" emulsion, one should take into account the hydrophobicity of droplet surface and, correspondingly, the possibility of hydrophobic attraction between droplets, which can considerably decrease the height of energy barrier and accelerate both aggregation in the primary minimum and rupture of thin films between droplets [50]. The dissolution of air in dispersion medium could also affect the rate of aggregation [49] since it changes the van der Waals interaction [51]. Essential influence on the rate of aggregation can be made by the mobility of droplet surface [52] and presence and behavior of adsorbed layers [3, 6]. Since the above examination of aggregation was carried out without account of coalescence, all obtained results are also valid for solid particles. The performed analysis of the role of TSJ>S in formation of primary doublets is especially important for "water in oil" emulsion, where the thickness of the double layer is very large and, consequently, TSCJ^ diminishes to zero. 2.7. Single energy minimum. Coalescing droplets The simplest version of coalescence takes place after aggregation directly in the primary minimum, i.e., without complicative transition from the

311

Coalescence Kinetics ofBrownian Emulsions

secondary minimum to the primary one. This is possible at low surface charge when two minimums merge in one. The system of three differential equations (17-19) in this case is reduced to two: ^

= -,,2 + 2 - ^ + ^

dt

T

(45)

x

pd,s

c

dn

pd n2s f 1 l) -jr = ^-npd\ +— dt 2 \rpd,s Tc)

where zpds

( 46 )

has the same meaning as rsds

in Section 2.5. The system of

equations can be solved using the scheme of linearization presented in Section 2.6. The solution for the number of doublets is e-ft'-e-/ht nPd rnd=—-. 2{/32-px)

(47)

where

A , 2 = " f [-1± P f ]; P2=2cXp + ^ ^ ; 9 2 = ^ 2

1

i

T d s

p \ \

P>

Tc

Tc

(48)

and quasi-equilibrium concentration of doublets c\p is expressed by Eq.(25) with the replace of ts^^s by rp^s. Since the total number of droplets at coalescence is not preserved, the concentration of singlets can be found from Eqs.(45, 46) as

(ns + 2npd)=-

y

or

ns=\-2npd

-nc

where

is the decrease of the total number of droplets caused by coalescence.

(49)

378

N.O.Mishchuk

The location of the maximum for doublet concentration can be found similarly to Eq.(44) as

_ln(/V/?2) 'max

KJ1J

n n

P2 ~ P\

However, contrary to the previous case, where the similar expressions defined the location of maximum number of secondary doublets and the bend of kinetic curve for the sum of primary and secondary doublets, in the given case, this is the location of the maximum concentration of one type of doublets. In Fig.9 presented below, calculations are performed specially for shallow minimums and, correspondingly, quick disintegration of doublets, in order to make the role of doublets disintegration more illustrative. As seen from the figures, the rate of coalescence depends on aggregation reversibility. The larger the doublet lifetime r ^ , the higher is the probability of coalescence. The increase of

TSC[S

(compare Figs. 9a,b and 9 c,d) leads not only to

stronger decrease of ns and ns +np(^ but also to more pronounced maximums of npci that demonstrate faster coalescence. It should be underlined that the intensification of coalescence is caused not by the decrease of the time of film rupture r c , but by the increase of the number of droplets that are in the energy minimum and wait for the rupture of the film. Attention should be also paid to the fact that although the total number of droplets decreases, according to the curves in Fig.9, the emulsion remains to be singlet-doublet, since npci lns < 0.15 when the time of investigation is not too long. To show the role of doublet disintegration for a wider range of parameters, the decrease of the total number of droplets is calculated for a few sets of characteristic times of disaggregation and coalescence (see Fig. 10). The influence of rp^s depends on its correlation with the time of coalescence r c . If these values are of the same order (see curves 1, 1' or 2, 2'), the number of coalesced droplets changes weakly. However, when they differ to the great extent (compare curves 1 and 1" or 2 and 2"), the number of coalesced droplets decreases considerably. It is interesting to note that, excluding the beginning of the curves, i.e. the time interval t
Coalescence Kinetics of Brownian Emulsions

379

exponent is a slow function of time and even at t » 1 can be approximated using two first terms of expansion into series as e~^]t ~ 1 - f5\t, reflecting the linear dependence of nc on t and, correspondingly, the linear dependence of the total number of droplets on time.

Fig.9. Dependence of concentrations of singlets ns (curves 1-4), doublets tip^ (l'-4') and their sum ns + npd (l'-4') on time t. tpd,s

=

^ ' (Figs, a, b), Tpd>s = 0.3 (Figs, c, d) and

TC = 1000 (curves 1,1', 1"); 10 (2, 2', 2"); 3 (3, 3', 3"); 1 (4, 4',4"). Curve 0* is calculated for T

pd

5

= 1000 and rc = 1000.

380

N.O.Mishchuk

Fig. 10. Dependence of the decrease of total number of droplets nc on time at tp^s = 100 (1 3); Tpd,s=™ (1'- 2') and

tpdjS=\

(l"-2") for r c =30 (1,1',1"); r c =100 (2,2',2");

r c =1000(3).

Taking into account that, at numerical calculations for Fig. 10, rather large values of coalescence time r c were used, the coefficient fi\ for this limiting case can be written in a simpler view:

A=^ Tc

!

(52)

2c]p+l/TpdfS

Thus, it is inversely proportional to the coalescence time. This is not surprising since it is coalescence that is the slowest process limiting the rate of emulsion destabilization in the given system. 2.8. Two energy minimums. Coalescing droplets Let us analyze a more general case of coalescence with the possibility of transition from the secondary minimum to the primary one that can be described by three equations (17-19). Since the solution of these equations is more complicated than in the previous cases, fist of all the numerical solution will be analyzed (see Figs. 11 and 12). The comparison of Figs. 11 and 12 clearly points to the influence of characteristic time isd,pd o n m e formation of primary doublets and, consequently, on the rate of coalescence at a fixed time of rupture of thin films rc. The lower energy barrier leads to quicker appearance of primary doublets

381

Coalescence Kinetics ofBrownian Emulsions

and rupture of thin layers and that, in its turn, causes quicker decrease of numbers of singlets and secondary doublets. Similarly to the case of noncoalescing droplets, at the fixed value of rsd S , the slope of curves in the

Fig.ll. Dependence of concentrations of singlets ns (curves 1-5), secondary « 5 ^(l'-5') and primary tip^ (2"-5") doublets, sum of doublets n^ (l*-5*) and sum of singlets and doublets n +n

s d

Tsd,pd

(curves ]**-5**)

On

time t.

Curves 0,0** are received at r ^ ^ =1000,

=1000, r c = 1 0 0 0 ; curves l,l',l", 1*,1** at 7 ^ = 0 . 3 ,

TC =1000; the others at r ^ ^ =0.3, rsd,pd

=1°

and 7

r^^=1000

and

c =1000 (2, 2', 2", 2*, 2**); 10(3,

3', 3", 3*, 3**); 3 (4, 4', 4", 4*, 4**); 0.1 (5, 5', 5", 5*, 5**), correspondingly.

N.O.Mishchuk

382

beginning of aggregation is independent of TSC[p j , which was taken considerably larger than r ^ s. The initial slope of curves is also independent of rc,

the value of which is comparable with TSCJS and is considerably smaller

than TSC{pc[. The latter result is caused by the need to overcome a barrier before coalescence starts. This means that instead of comparing TSCI s and r c , it is necessary to compare the values of TS(jtS and T

T

T

sd sp + c >> sd s'

me

T

sd,sp

+T

c • Since

initial slopes of curves are defined by a faster

process- the disintegration of secondary doublets.

Fig. 12. The same as Fig. 11, but rsd^s = 0.3 is replaced with tgd^pd =

Coalescence Kinetics of Brownian Emulsions

383

As seen from numerical analysis, slow rupture of thin film in case of quite high barrier (large TSCJpj) leads to a slow change of the total number of droplets. This means that the system should be examined for a very long time and therefore the exactness of the investigation can be low owing to different accompanying processes, the influence of which increases with time. The decrease of the total number of droplets calculated for a few sets of parameters is shown in Figs. 13 a,b. The curves in Fig. 13a are similar to curve 2" in Fig. 10, which is also shown in Fig. 13a to make comparison easier. Indeed, the calculations for curve 2" were performed at xc =100, r ^ =1 (since in Fig.10 a single minimum was investigated, rpci s is the analogue of ts^^s), therefore the curves in Fig. 13a can be considered as the complication of the process shown in Fig. 10. The droplets represented by curve 2" participate in three processes (appearance of a doublet, its disintegration or coalescence). In the case shown in Fig. 13a, the additional process appears, that is the overcoming of a barrier. It might seem that, when a barrier exists, coalescence should be slower. However this is true only for a short period of time when the number of primary doublets is very small (see beginning of curves 2,3 and 2" in Fig. 13a). Later, at a low barrier (curves 1-3), coalescence occurs quicker than in a single minimum case (curve 2"). This non-trivial result has a very simple explanation. The disintegration of a doublet from a single shallow minimum occurs quicker than its possible coalescence. In case of two minimums, after formation of secondary doublets, some of them disintegrate into singlets, while others transform into primary doublets, which wait until thin films rupture. For a single energy pit, the number of coalesced droplets is proportional to the ratio of times iS(j s lxc . For two pits, the TSC[ S/TSCJ sp part of secondary doublets transforms into primary ones, and the l / r c part of transformed doublets coalesces. Thus, the rate of coalescence is proportional to Tsd,s/[rsd,spTc)- This very rough qualitative evaluation of the role of two energy pits is reflected by kinetic curves for the number of coalesced droplets. At a high barrier (TSCJ^P = 10, curve 4), when the probability of formation of primary doublets is low, at fixed tc, the rate of coalescence for two minimums is lower than for a single minimum. In contrast, at low barrier (and small TSCJsp, see curves 1,2), the probability of formation of primary doublets is high and therefore the rate of coalescence for two minimums is higher than for a single one. It is also worthwhile to analyze results of numerical calculations at fixed r sd,pd' a s presented in Fig. 13b, in which curve 3 coincides with the same curve in Fig.l3a and curve 2" from Fig. 10 is also shown repeatedly. The shorter the

384

N.O.Mishchuk

Fig.13. Dependence of the decrease of the total number of droplets nc on time a) TC = 100, TSCJ^S = 1 and Tsd,pd = °-' (curve 1), 1 (curve 2), 3 (curve 3), 10 (curve 4); 6) TSCI pd =0.1 and r c = 1 0 0 (curves 1-3) and r c = 1 0 (curves l'-3'); xscjs

equals O.I

(curves 1,1'), 0.3 (curves 2,2') and 1 (curves 3,3'). Curve 2" is copied from Fig. 10.

time TSCI s, the lower is the number of coalesced droplets. Droplets from secondary doublets, instead of transformation into primary doublets, "prefer" to leave aggregates and therefore the probability of coalescence decreases. The decrease of r c , when other parameters are fixed, leads to quicker coalescence, as shown by curves l'-3' inFig.l3b. Unfortunately, the analytical solution of differential equations for a low barrier and slow rupture of thin layer is rather difficult. Therefore we will analyze analytically only the opposite limiting case of a relatively high barrier and quick rupture of thin films. This case is more interesting from the experimental point of view than the quick rupture of film and a low barrier, since the latter process would look as rapid coagulation and on its background it would be difficult to define the rate of coalescence. When condition T

r

sd,pd »

(53)

c

is fulfilled, it may be assumed that each primary doublet merges into a single droplet immediately after it is formed, i.e., as may be described by the condition dn

pd dt

nsd

=

T

sd,pd

_npd=Q T

c

( 5 4 )

385

Coalescence Kinetics ofBrownian Emulsions

In the framework of the above scheme, the solution of differential equations (17-19) with account of (54) takes the following view: e-nt

n

_e-r2l

e-r]t_e-nt

T

sd =

"pd = —

(55)

T

n-n

sd,Pd

n-n

and d /

n

\

—\ns+2nsd) at

pd

=

or

n

=1 -2nsd

TC

- In

-nc

(56)

ft=-S_

(58)

d F

where

*Tsd,Pd

nnTsd,pd

ri,2=^f-l± P f 1 P3=2c 1+ -U^- ; 2

I

\|

P3 I

T

sd,s

T

sd,pd

T

sd,pd

The maximum of the kinetic curves for secondary doublets is reached after the time _ Hn 172)

(59)

\jy;

' m a x ~~

72-71 In the limiting case when xc I tsdnd ~^> 0, the same time corresponds to the maximum of primary doublets (curve 5"in Fig. 12b) and, consequently, to the maximum of the total number of doublets (curve 5* in Fig.l2d). However, when ratio ?clTsd,pd'v& n o t t 0 ° l ° w ' condition (53) is violated and the maximum of curves for primary doublets and the total number of doublets shift to larger times (see, for instance, curves 3", 4" in Fig.12b and curves 3*, 4* in Fig.l2d). 3. DESIGN OF EXPERIMENTAL INVESTIGATIONS Comparison of kinetic curves for total concentrations of singlets and doublets (see Figs. 7c, 9c, l i e and 12c) shows that trends of curves seem to be very similar at considerably different subproccesses of emulsion destabilization.

386

N.O.Mishchuk

This means that there may be different interpretation of the experimental data obtained by absorption or scattering of light. In particular, absorption of light does not allow discrimination between two aggregated droplets and two droplets that have already coalesced. Moreover, taking into account that absorption of light depends on the length of the used wave, droplet size, size and shape of the aggregate and total concentration of droplets, the applicability of the used method and equipment for the specific investigated system should be thoroughly controlled. It is clear that for this aim it is necessary to develop a special technique that allows to count not only the general number of droplets and aggregates but also singlets, doublets and multiplets separately. Such apparatus based on the combination of streaming ultramicroscopy and single particle light scattering is used for a long time to investigate aggregation in suspensions [53, 54]. As distinct from streaming ultramicroscopy that owing to hydrodynamic flow could affect the stability of secondary doublets, transition into primary minimum and the process of coalescence, videomicroscopy [15-20] is a method that, in principle, does not influence the state of emulsion. Moreover, videomicroscopy allows not only to count the number of doublets and multiplets but also to observe the behavior of chosen doublets (see, for instance, Fig.3 in paper [17]) and follow general changes in emulsion. Therefore it seems to be the only reliable way to define the rate of rupture of thin films against a background of all subprocesses that lead to drop coalescence . Owing to the fact that the analysis of the videopictures [15-19] was not automated, the information about the rupture or coalescence of doublets was not sufficiently complete. Development of special computer programs for videoimage processing could allow to analyze large data files. Only in this case, careful study would allow separation of primary (irreversible) and secondary (reversible) doublets and direct definition of the mechanism of slow coagulation. Beside the development of special videotechnique and corresponding software tools, details of experiment arrangement also play an important part. As it was repeatedly written above, at a given ratio between the density of droplets and medium, droplet size should be limited to avoid the influence of gravitation. The scattering of emulsion size should be minimal, as that could allow not only to weaken the gravitational component of coagulation but also to avoid the change of interaction energy and, correspondingly, characteristic times for different pairs of droplets. One can easily see that the strict monodispersity is less important for rapid coagulation and more important for all other subprocesses of aggregation. Indeed, the most important factor for rapid coagulation is the efficiency of collision between droplets. The analysis of collision efficiencies [55] for identical 4DQQQ and different (Z)o + Z)jX^o +a\) droplets with radiuses QQ a n ^

Coalescence Kinetics of Brownian Emulsions

387

a\ shows that, owing to inverse proportion between diffusion coefficients and radiuses Dq\~\l a§\, t n e efficiencies differ weaker than the radiuses. For example, the efficiencies of collision between initial (a 0 ) droplets only and between initial ( CIQ ) and coalesced (a\ = iflciQ = 1.26ao) droplets differ for 1.3%. When two types of quite different droplets (ag ar>d a\ - 2«o) a r e present, the efficiencies differ for 12.5% only. Emulsion with such scattering of droplet sizes, which does not considerably affect Smoluchowski time, can be easily obtained using even very old methods (see, for instance, [56]). Unfortunately, all other characteristic times depend on radiuses considerably stronger and therefore the aforementioned radius scattering is too large to provide a small difference between the values of TS^JS, tsci,pd a n d Tc for different pairs of droplets. This is caused both by the dependence of characteristic times r,- on droplet radiuses f,- ~ R l2D§~a^ (12-13) and by their exponential relation (see Eqs. (12, 13) and (8-10)) with the energy of interaction U{r), which is a function of particle radiuses 2a o «i /(ao + a\) [2]Indeed, calculation of the characteristic time of doublet disintegration [28, 29] showed that a relatively small change of droplets radius leads to a sharp change of its value. It is clear that the larger the scattering of radiuses and of characteristic times, the more inaccurate is the definition of the time of thin film rupture. Thus, the design of the experiment should provide for the preparation of emulsion with maximum monodispersity. The experiment also requires to select such electrolyte concentration and volume fraction of droplets that would allow forming of singlet-doublet emulsion with prevalence of singlets, relatively low number of doublets and almost total lack of triplets and other multiplets. It is this type of emulsion that makes it possible to observe kinetics of singlet and doublet concentrations, define the characteristic time of disaggregation and calculate the time of thin film rupture. For simplification of theoretical analysis all times were normalized on the time of formation of secondary doublets Ts>scj. However the choice of the optimal value of this time is very important. It should be about several minutes; otherwise bends in kinetic curves, which are of significance for interpretation of investigated processes, could be lost in the time interval between preparation and investigation of emulsion. Since, at the given difference of droplets and medium densities, the choice of droplet size is limited because of gravitation influence, it is necessary to change both the size and volume fraction of droplets. Thus, for example, for one-percent emulsion (a = 0.01) and radius a0 = 0.5jam, Ts sci is approximately equal to 10 sec, but for OQ =\/MI it is already 1.5 min.

388

N.O.Mishchuk

Decreasing the volume fraction in 10 times results increases tgm to 15min. With account of condition (26), the dimensionless disintegration time should be T sd s K< 0-25, therefore the bend in kinetic curves should appear within a few seconds, a few dozens of seconds or a few minutes, correspondingly. It is also necessary to draw attention to the fact that while at rapid coagulation of emulsion the main developments take place in the bulk of emulsion, at slow coagulation the key role could be played by wall effects. First of all, this refers to the abovementioned emulsion concentration or dilution near the upper or bottom wall of the cuvette caused by gravitation. This means that behavior of droplets should be investigated not near the wall but in the bulk of emulsion. For this aim the cuvette should satisfy a few conditions. It should be vertical with the height about 5-10 cm so that the state of emulsion in the central part of cuvette was independent of creaming. With account of the droplet sedimentation velocity Usecj =2aoApg/9r/,

for example at Ap =

O.Olg/cm , such height of the cuvette allows to investigate the properties of emulsion with one micron droplets during a week, with 3-micron droplets during two days and so on. The thickness of the cuvette should be about several hundreds of microns to avoid natural convection. It is also necessary to stress that investigations may not be carried out just near the vertical wall of the cuvette since, owing to hydrodynamic resistance between a droplet and the wall, the latter hampers diffusion [57] and, correspondingly, slows down the rate of droplet collision. The requirement of studying emulsion at a certain distance from the wall does not cause any considerable complications, since investigations should be performed at strongly diluted emulsions, which are sufficiently transparent to allow video filming in the depth of the cuvette. Another important fact to be considered is the attraction of droplets to walls, in result of which the droplets are either attached to the walls or spread on their surface. Therefore the walls should have a strong charge of the same sign as the droplets to create a high energy barrier between them. They can also be covered with non-charged macromolecules, overflowing primary pits in the gap between walls and droplets and in this way preventing aggregation in the primary minimum. For example, in [28] it was proposed to use for this aim agarose gel, which immobilizes electric double layer hydrodynamically [58] and, hence, is supposed to overflow the primary minimum. In addition, the adsorption of macromolecules should be irreversible, since even their small admixture in emulsion changes its behavior. In conclusion, it should be said that the theoretical investigation presented in [26-29] and, for a more general case, in the given chapter shows the role of different factors in destabilization of emulsions and observable kinetic of coalescence. The comprehensive experimental study and theoretical analysis of

Coalescence Kinetics ofBrownian Emulsions

389

kinetic behavior of singlets and doublets allow to discriminate the role of reversibility of aggregation, transformation of secondary doublets into primary ones and coalescence in the general process of emulsion destabilization. The characteristic time of rupture of thin film between two droplets regardless of the nature of this rupture could be determined. Although the experience of application of videomicroscopy in such research is not quite wide, the accomplished experimental investigations [15-20] confirm the possibility of designing a device to measure coalescence and disintegration time of doublets, which, in combination with emulsion dynamics modeling, could be a useful tool in analyzing emulsion properties. Development of theoretical modeling and experimental investigation that takes into account overcoming of energy barrier and provides a thorough analysis of all existing subprocesses would refine the insight on emulsion properties and allow to optimize emulsion technologies with respect to stabilization and destabilization. REFERENCES [I] J.A. Kitchener and P.R. Musselwhite, in: "Emulsion Science", I.P. Sherman (Ed.), Academic Press, New York, 1968. [2] B.V. Derjaguin, Theory of Stability of Colloid and Thin Films, Plenum, New York, 1989. [3] I.B. Ivanov and P.A. Kralchewski, Colloids Surf. A, 128 (1997) 155. [4] I.B.Ivanov. (ed.), Thin Liquid Films. Marcel Dekker, New York, 1988. [5] K.D. Danov, I.B.Ivanov, T.D.Gurkov and R. Borwankar, J. Colloid Interface Sci., 167,

(1994)8. [6] [7] [8] [9]

I.B.Ivanov, K.D. Danov and P.A. Kralchewski, Colloids Surf. A, 152 (1999) 161. D.N.Petsev, Langmuir 2000, 16, 2093-2100. R.G.P. Borwankar, L.A. Lobo and D.T. Wasan, Colloids Surf. A, 69 (1992) 135. J. Sjoblom, in: J. Sjoblom (ed.), Emulsion and Emulsion Stability, Marcel Decker, 1996, p. 393. [10] V.Mishra, S.M.Kresta and J.Masliyah, J. Colloid Interface Sci., 197, (1998) 57. [II] K. Khristov, S.E. Taylor, J. Czarnecki and J. Masliyah, Colloids Surf. A, 174 (2000) 183. [12] T.Gilespie and E.K.Rideal, Trans.Faraday Soc, 52 (1956) 173. [13] E.G.Cockbain and T.S.McRoberts J. Colloid Sci. 8 (1953) 440 [14] H.Sonntag and H.Klare. Z.Phys. Chem. 223 (1963) 8. [15] S.S. Dukhin, O. Saether and J. Sjoblom, in: K.L. Mittal and P. Kumar (eds.), Emulsions, Foams, and Thin Films, N.Y., Basel, 2000. [16] S.S. Dukhin, O. Saether and J. Sjoblom, in: J. Sjoblom (ed.), Encyclopedic Book of Emulsion Technology, Marcel Decker, 2001, 71. [17] O. Holt, O. Saether, J. Sjoblom, S.S. Dukhin and N.A. Mishchuk, Colloids Surf. A, 123124(1997) 195. [18] O. Saether, J. Sjoblom, S.V. Verbich, N.A. Mishchuk and S.S. Dukhin, Colloids Surf. A, 142(1998) 189. [19] O. Holt, O. Saether, J. Sjoblom, S.S. Dukhin and N.A. Mishchuk, Colloids Surf. A, 141 (1998)269. [20] O. Saether, J. SjQblom, S.V.Verbich and S.S.Dukhin, J. Disp. Sci. Technol., 20 (1999) 295.

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[21] W.B.Russel, D.A.Saville and W.R.Schowalter, Colloidal Dispersions, Cambridge University Press, New York, 1989. [22] J.N. Israelachvili, Intermolecular and Surface Forces, 2 nd Edition, Academic Press, London, 1991. [23] D.N.Petsev, V.M.Starov and I.B.Ivanov, Colloids and Surf. A, 1 (1993) 65-81. [24] W.Albers and J.Th.G. Overbeek, J. Colloid Interface Sci., 14 (1959) 518. [25] K.D. Danov, N.D. Denkov, D.N. Petsev and R. Borwankar, Langmuir, 9 (1993) 1731. [26] S.S. Dukhin and J. Sjoblom, J. Dispers. Sci. Technol., 19 (1998) 311. [27] S.S. Dukhin, J. Sjoblom, D.T. Wasan and O. Saether, Colloids Surf. A, 180 (2001) 223. [28] N.A.Mishchuk, R.Miller, A.Steinchen and A.Sanfeld, J.Colloid Interface Sci., 256 (2002) 435. [29] S.S.Dukhin, N.A.Mishchuk, G.Loglio, L.Liggieri and Miller R., Adv. Colloid Interface Sci., 100-102(2003)47. [30] Ya.I. Rabinovich and N.V. Churaev, Kolloid Zh., 52 (1990) 309. [31] V.N. Gorelkin and V.P. Smilga, Kolloid Zh., 34 (1972) 685. [32] N.A. Mishchuk, J. Sjoblom and S.S. Dukhin, Kolloid Zh., 57 (1995) 785. [33] E.D. Shchukin, E.A. Amelina and A.M. Parfenova, Colloids Surf. A, 176 (2001) 35. [34] N.A. Mishchuk, S.V. Verbich, S.S. Dukhin, O. Holt and J. Sjoblom, J. Disp. Sci. Technol., 18(5) (1997) 517. [35] A.S.Dukhin, Kolloid Zh., 49 (1987) 630. [36] V.M.Voloschuk and Y.S. Sedunov, Coagulation Processes in Disperse Systems. Hydrometeoizdat: Leningrad, 1975 (in Russian) [37] M.Smoluchowsi, Z.Phys.Chem. 92 (1918) 129. [38] V.G.Levich, Physico-Chemical Hydrodynamics, Prentice Hall, New York, 1962. [39] A.S.Kabalnov, A.V.Pertsov, Yu.D. Aprosin and E.D.Shchukin, Kolloidn.Zh. 47 (1985) 1048. [40] S.S. Dukhin and J. Sjoblom, in: Emulsion and Emulsion Stability, J. Sjoblom (Ed.), Marcel Decker, 1996,41. [41] V.M. Muller, Kolloid. Zh., 40 (1978) 885. [42] V.M. Muller, in: 'Surface Forces in Thin Films", B.V. Derjaguin (Ed.), Nauka, Moscow, 1979,30. [43] S.Ljunggren, J.C.Eriksson and P.A.Kralchevsky, J.Colloid Interface Sci. 191 (1997) 424. [44] V.M. Muller, Colloid J., 58 (1996) 598. [45] M.Smoluchowski, Phys.Z., 17 (1916) 557, 585. [46] N.A. Fucks, Z.Phys., 89 (1934) 736. [47] B.V. Derjagin and V.M. Muller, Doklady Acad./Nauk SSSR, 176 (1967) 869. [48] H.Kihira, N.Ryde and E. Matijevic, J.Chem.Soc.Faraday Trans., 88 (16) (1992) 2379. [49] D.Snoswell, J.Duan, D.Fornasiero and J.Ralston, J.Phys.Chem., B, 107 (2003) 2986. [50] RJ.Pugh, Adv.Colloid Interface Sci., 64(1996)67. [51] N.Mishchuk, D.Fornasiero and J.Ralston, J. Phys. Chem., A. 106 (2002) 689-696. [52] S.A.K. Jeelani and S.Hartland, J.Colloid Interface Sci., 206 (1998) 83. [53] N.Buske, H.Hedan, H.Lichtenfeld, W.Katz and H.Sonntag. CoI.Polym.Sci. 258 (1980) 1303.

[54] [55] [56] [57] [58]

H.Sonntag, V.Shilov, H.Gedan, H.Lichtenfeld and C.Durr. Colloids Surf., 20 (1986) 303. N.A.Fuchs, The Mechanics of Aerosols, Pergamon, Oxford, 1964. M.A. Nawab and S.G. Mason, J.Colloid Sci.13 , 179, 1958. J.Goldman, R.G. Cox and H.Brennet, Chem.Eng. Sci., 22 (1967) 637. C.J. van Oss, R.M. Fike, R.J. Good and J.M. Reinig, Analyt. Biochem., 60 (1974) 242.

Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 10

Hydrodynamical interaction of deformable drops A.Z. Zinchenko and R.H. Davis Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309-0424 1. INTRODUCTION For small emulsion drops with diameters in the range of 1-100 /im, the surface deformation is typically small, and much theoretical work has been done to date to investigate drop interactions with negligible shape distortion. Another relevant assumption for small emulsion drops is the neglect of fluid inertia on the microscale, making it possible to use simplified (Stokes) equations for the fluid motion. Exact semi-analytical solutions have been constructed to describe hydrodynamical interaction of two surfactant-free spherical drops at arbitrary surface separations and orientations, both in a quiescent liquid and shear flows [1-8]. These results allowed us to study the effects of drop interactions on the collision efficiency of Brownian and non-Brownian drops in gravity-induced and shear-induced motion, and on the non-Newtonian rheology of semi-dilute emulsions [4,8,911]. For 'well-mixed' concentrated emulsions of spherical drops, numerical multipole techniques were used to evaluate the effective properties (viscosity, sedimentation rate) [12]. These studies have been complemented by asymptotic analyses for two-drop interactions at small surface separations [4-5,13-15]. In particular, it was found that the lubrication resistance of the liquid film between two non-deformed drops with arbitrary viscosities is inversely proportional to the square root of the gap, and therefore this resistance does not preclude drops from coming into contact (in a finite time) even without van der Waals attraction. This finding is in stark contrast to solid-particle interaction; it is well known (e.g., [16]) for the latter case that the lubrication resistance of the thin film is inversely proportional to the gap, and so collisions of small solid particles are not possible

392

A.Z. Zinchenko and R.H. Davis

without attractive molecular forces, non-continuum effects, or microscale surface roughness. The present review focuses on the effects of surface deformation on drop interactions. While deformation is relatively unimportant for describing the behaviour of solitary and well-separated drops of small size (1-100 lira) in dilute emulsions, it may have a crucial effect, as it turns out, on the efficiency of drop coalescence and on the collective behaviour of highlyconcentrated emulsions. The reason is that the deformation of the thin film between two surfaces approaching each other is known to preclude drops from coming into contact, unless van der Waals forces become important [17-18], and so the coalescence efficiency of slightly deformable drops is a result of subtle interplay between the resistance of the thin film and short-range molecular attractions. The degree of deformation on the drop lengthscale is measured by the capillary number Ca ~ /ief//cr

,

where /j,e is the dynamical fluid viscosity outside the drops, U is the characteristic velocity of the flow relative to the drops, and a is the constant surface tension (the effects of surface tension gradients due to surfactants or temperature variations are not included here). When Ca
Hydrodynamical Interaction of Deformable Drops

393

useful for two or more drops with interactions. A relatively new element is multipole acceleration (Sec. 3.2), which, in particular, extends the capabilities of dynamical boundary-integral simulations to very large systems of drops (on the order of a thousand drops) with large deformations [23-25]. In Sec. 4, multipole-accelerated, boundary-integral calculations are used to obtain exact results for two drops in very close approach at Ch -C 1, and thereby to verify the asymptotic theory of drop coalescence of Sec. 2 and find its range of validity. In Sec. 5, we review some results of the boundaryintegral analysis [26] for two drops or bubbles with large deformations in buoyancy/gravity-driven motion. It is demonstrated how interactions give rise to cusped shapes, drop breakup and capture. The capture phenomenon is of particular interest; it was first discovered experimentally by Manga and Stone [27-28] for two bubbles in a highly-viscous corn syrup. In Sec. 6, multidrop simulations in concentrated emulsions with large interface deformations are discussed. Of particular interest here is the phenomenon of sedimentational instability due to deformations [23,25], which, we believe, is akin to Koch-Shaqfeh type of instability predicted previously for a dilute suspension of solid, rod-like particles [29]. Multidrop rheological simulations in a shear flow [24] are also discussed in Sec. 6; even a small amount of deformation is shown to make an emulsion non-Newtonian (at high drop volume fractions). As another application of the multipole-accelerated boundary-integral method, the problem of a large bubble or drop motion through a concentrated emulsion of deformable freely-suspended drops [25] is discussed in Sec. 6. It is shown how to calculate the steady-state settling velocity of the bubble/drop from first principles, by rigorous multidrop simulations, and thus to avoid difficulties with the constitutive equation for the emulsion. In Sec. 7, we discuss some unresolved issues and open areas for future research. 2. HYDRODYNAMICAL THEORY OF COALESCENCE OF SLIGHTLY DEFORMABLE DROPS Consider two Newtonian drops of non-deformed radii a\ and a,2, viscosity fjf (assumed to be the same for both drops, just for simplicity) and density p' moving in an unbounded Newtonian liquid of viscosity /ie and density pe. The Reynolds number, Re = peUai//j,e, for the flow relative to the drops is assumed to be small (which is the most practical case for emulsion drops) and so the Stokes equations for the fluid motion inside and outside the drops apply [30]: ^V 2 u = Vp ,

V •u = 0

,

(1)

394

A.Z. Zinchenko and R.H. Davis

where u and p are the fluid velocity and pressure, respectively. Two cases are of specific interest: (i) relative drop motion in a quiescent liquid under gravity/buoyancy (Ap = p' — pe ^ 0) and (ii) shear-induced motion of two freely-suspended drops (p' — pe), where the unperturbed flow around the drops has the form u^x) = (—7x2, 0, 0), with 7 > 0 being a shear rate. The capillary number is precisely defined as

for shear-induced motion, and a=^\vr-VT\

2(A+l)a,V~P.)g

p)

for gravity-induced motion. In Eq. (3), V?° is the settling velocity of an isolated spherical drop calculated by Hadamard-Rybchinski's formula [30], g is the gravity acceleration, and A = n'/ne

(4)

is the viscosity ratio. For Gi < 1, the problem of drop coalescence is best handled by matched asymptotic expansions. On the lengthscale of the drop size, the drops are considered as non-deformed fluid spheres, and they can reach contact in a finite time because the hydrodynamic resistance coefficient for mutual approach has an integrable singularity ~ /i~1//2 [4,13,15], as the surface clearance h -» 0. On the deformation lengthscale, however, there is a small non-uniform gap between the drops that evolves under the action of hydrodynamic and short-range, singular molecular forces; inclusion of molecular forces such as van der Waals attractions is imperative to make two deformable surfaces come into contact and, thereby, coalesce. It was shown in Ref. [20] that, at least, at not too high viscosity ratios A, relative tangential motion of drops in close approach is practically unaffected by small deformation, which allows us to calculate the time-dependent 'contact' force acting along the line-of-centres, from the 'outer' solution for spherical drops in apparent contact. This 'contact' force is equal to the lubrication force, which closes the local integro-differential equations for the thin film in the gap. Solving these thin film equations (see below) allows us to determine whether the physical contact of the deformed surfaces (which is identified with the onset of coalescence in the present approach) is reached for given initial conditions, or if the drops instead move past each other and separate without coalescing.

Hydrodynamical Interaction of Deformable Drops

395

To demonstrate this methodology, consider first gravity/buoyancydriven motion. Due to linearity of the Stokes equations (1), hydrodynamic forces acting on spherical drops 1 and 2 in apparent contact can be written as

F i = - 6 7 ^ 1 [A[2V + Ttn{Vl - V 2 ) x + T{2V^\ + F F2 = - 6 7 ^ 0 3 [A22V + Tt2l{V2 - V^

+ Tl2Vi] - F

,

(5)

where Vi and V2 are the velocities of geometric centres of the spheres, V is the common velocity of the drops along the line of centres, and F and — F are contact forces acting along the line of centres; _l_ denotes the vector components perpendicular to the line of centres. The non-dimensional hydrodynamic resistance coefficients, A\2 and A22, are related to the forces acting on drops in their motion as an aggregate along the line of centres; these coefficients depend on size ratio k = a\/a2 and viscosity ratio A, and are known from the exact solution of Ref. [31]. Likewise, the nondimensional coefficients T^Ak, A) are known from the exact solution [3] for the two-drop motion perpendicular to the line of centres in the limit of touching. Projecting the force balances Fi + |?raf Apg = 0 on the line of centres and excluding V yields the expression for the contact force: 4 (fc + 1)3

,

A.U-k2AL

where /3 is the angle between the centreline and the vertical (Fig. 1), and R = a\a2/(ai + a2) is the so-called 'reduced radius.' Similarly, projecting the force balances on the plane orthogonal to the centreline gives the equation for the relative tangential motion in apparent contact: M-K1

sm fJ ,

- - ^

+ a 2

^ ,

-TtiTt2

+ TtiTt2

•

W

Integrating Eq. (7) yields the contact force F(/3(t)) as an explicit function of time, given the initial conditions for (3{t) (see below). The values of the non-dimensional coefficients a(k, A) and Tl(k, A) are given in Table 1. For two freely-suspended drops in a shear flow uoo(x) = (—7x2,0,0), the centre-to-centre unit director p = {pi,p2,Ps) from sphere 2 to sphere 1 is characterized by two angles /3 and 9 (Fig. 2): Pi = sin 9 cos (3 , p2 = sin 9 sin (i , Pz = cos 9 .

(8)

Similar algebra gives the contact force: F(/3,9) = 37r/ueai(oi + a2)D*j sin2 9 sin/? cos /3

,

(9)

396

A.Z. Zinchenko and R.H. Davis

Fig. 1. Gravity-induced motion of two drops in apparent contact. The close-up shows the inner region with the deformed thin film.

Table 1

The values of a and T ( for gravity-induced motion k 0.10

A = 0.1 a = 72.870 T* = 0.713 0.15 a = 50.291 Tl = 0.728 0.25 a = 31.323 Tl = 0.733 0.35 a = 22.246 T* = 0.709 0.50 a = 14.162 T ( = 0.623 0.75 a = 5.792 T ( = 0.371 0.90 a = 2.118 Tl = 0.162

1 2 0.5 0.25 69.241 64.300 57.158 48.460 0.616 0.509 0.390 0.282 47.989 44.861 40.348 34.874 0.316 0.635 0.533 0.419 30.098 28.434 26.039 23.148 0.357 0.648 0.555 0.451 21.490 20.463 18.984 17.201 0.632 0.547 0.453 0.369 13.758 13.207 12.412 11.453 0.560 0.490 0.412 0.343 5.653 5.463 5.188 4.856 0.212 0.335 0.295 0.251 2.069 2.002 1.905 1.788 0.146 0.129 0.110 0.093

where g . = A + | D{K\2 + D\K\2 A+ 1

A22 + &A'i2

4 39.590 0.201 29.323 0.237 20.239 0.285 15.415 0.304 10.495 0.291 4.524 0.183 1.671 0.081

10 29.720 0.133 23.188 0.168 17.059 0.220 13.477 0.247 9.462 0.245 4.167 0.158 1.546 0.070

(10)

Hydrodynamical Interaction of Deformable Drops

397

Fig. 2. Shear-induced motion of two drops in apparent contact.

and the hydrodynamic coefficients D\ and D\ are obtained from the exact solutions [5,8] in the limit of touching drops. The dynamics of contact motion for two touching spherical drops in simple shear flow is described by

— = - 7 ( 1 - £ f )sin0cos0sin/3cos/3 at

,

where B* is the value of hydro dynamic relative mobility function in the zero-gap limit, found from the solutions [5,8]- Again, solving Eq. (11) with given initial conditions (see below) gives the contact force F({3(t),6(t)) as a function of time. The values of the non-dimensional coefficients D*(k, A) and Bt(k, A) are given in Table 2. Even though relative drop motion is three-dimensional for both gravitational and shear flows (Figs. 1,2), it was proved in Ref. 20 that the near-contact deformation (shown as a close-up in Fig. 1) is mainly axisymmetrical for Ca
398

A.Z. Zinchenko and R.H. Davis

Table 2

The values of D* and Bl 0.5 A = 0.1 0.25 k 0.538 0.545 0.547 0.1 0.057 0.127 0.213 0.15 0.734 0.749 0.762 0.047 0.103 0.174 1.050 1.088 0.25 1.015 0.032 0.072 0.122 1.237 1.299 0.35 1.185 0.024 0.053 0.090 1.368 1.456 0.50 1.299 0.037 0.064 0.016 1.280 0.75 1.358 1.459 0.011 0.026 0.046 1.227 1.321 1.154 1.00 0.010 0.024 0.042

1 0.539 0.325 0.767 0.267 1.129 0.189 1.376 0.141 1.570 0.101 1.596 0.074 1.450 0.069

Normal stress balance A (I 1\ p

-6^

=a

+

{^

2 0.515 0.445 0.756 0.369 1.160 0.265 1.450 0.200 1.694 0.146 1.750 0.109 1.596 0.101

4 0.476 0.553 0.728 0.466 1.174 0.341 1.509 0.262 1.803 0.194 1.894 0.147 1.733 0.137

10 0.421 0.664 0.682 0.571 1.172 0.432 1.555 0.339 1.906 0.256 2.035 0.197 1.870 0.184

a d ( dh\

,1rt.

a-2)-Yrdr- V Yr)

(12)

'

Momentum balance k f-~2dr~

}

9P

(13)

{6)

'

Local boundary integral u(r,t) = -, /o°° (r',r)f(tS)dr'

,

Mass continuity dh 1 d r , 1 —+ - — 77m=0

h2

dt r dr L J Integral force balance

, '

r(p-fi^i)2'rr*=F

u=u

•

(14) dp -£-

12/xe dr

,

(15)

y

'

(i6)

In Eqs. (12) and (16), A is the Hamaker constant, and — A/(6irh3) represents the disjoining pressure term for unretarded van der Waals attraction (other colloidal forces, such as electrostatic, are not considered herein). The analysis can be easily extended to retarded van der Waals force expressions, if needed. The boundary integral (14) can be written in an equivalent form:

f(t, r) = V f (r>, r ) [ ^ - ^ - | , ] «(t, r')dr' ,

(17)

Hydrodynamical Interaction of Deformable Drops

where ,, , s 1 ^(r',r =

r'

/-Trr r« / 1

2rr'cos6)i-i/2 n n = 5cos9d9

399

. . 18

is the elliptic-type Green function. Relation (15) accounts for the parabolic velocity profile in the film and generalizes the well-known lubrication equation of Hocking [32] for mobile films (u ^ 0); u(t,r) is the average fluid velocity across the film. Infinite limits of integration in Eqs. (14) and (16) are justified, since the film radius becomes infinitesimally small, as Ca —> 0. Most importantly, the contact force (6) or (9) from the outer solution enters the integral balance (16) for the film as the lubrication force. The thin-film system of equations (12)-(17) (or its simplifications for immobile or non-deformable films, constant driving force F, or no van der Waals attractions) have been derived and used by a number of authors [33,15,17,18,34] to study head-on collisions. In Refs. [20,21], film drainage equations (12)-(13), (15)-(17) were combined with the outer solution equations (6)-(7) (or (9), (11)) to describe three-dimensional coalescence in gravity- or shear-induced motions, but without the pressure-gradient term in Eq. (15), which has limited the theory in Refs. [20,21] to moderate viscosity ratios A
•

exp /

^—

'- dr\

,

(19)

where r is the centre-to-centre distance, and the mobility functions L and M, for instantaneous relative motion along and normal to the line of centres [4, 35], respectively, can be taken from exact two-sphere solutions [4]. For a trajectory analysis in shear-induced motion, see Refs. [8, 9, 21].

400

A.Z. Zinchenko and R.H. Davis

Given (3C (or (3C and 9C), the simplest matching strategy [20], suitable for moderate viscosity ratios A, is based on calculating the 'collision' time required for a spherical drop trajectory to reach contact from an initial configuration where the drops are slightly separated. The hydrodynamical lubrication force between two non-deformed drops in close contact has the form [4, 13]

F « - | f f V2V#3/2%^S

,

(20)

where /i min is the minimum gap between the drop surfaces; for nondeformed drops the minimum separation is located along their line of centres. On the other hand, this force is approximately equal to the contact force (6) or (9), and (3 K, (3C (Q J=S QC) during the mutual approach of spherical drops from a nearly touching configuration. These arguments give the collision time for spherical drops by integrating Eq. (20): tcoll

= I 7T3V2 n'R^ (h°mmf2/F((3c)

(21)

(for shear-induced motion, F(f3c) is replaced with F(/3C,8C)), where h°min is the initial gap at t = 0 between the spherical drops. In all cases of interest, short-range van der Waals attraction has no appreciable effect on matching (see Ref. [20]) and it was neglected in Eq. (21). In the spirit of the asymptotic method, h°min is subject to the limitations h°min cuCa. The second condition guarantees that the film deformation is initially unimportant, and allows us to take an initial film profile as h(t,r) = h°min + ^

,

t =0

,

(22)

which corresponds to local parabolic approximation of nondeformed shapes. Accordingly, the thin-film equations (12)-(17) are solved first from t — 0 to t — tcoii with the initial condition (22) and a constant driving force F((3C) (or F((3C,8C)), and then, for t > tcou, the time-dependence of the contact force through Eq. (7) or (11) is taken into account. The exact choice of the initial separation h°min does not affect the overall solution in the range of interest t > tcou (since, for most of the time during t < tcou, deformation is unimportant). This simplest matching strategy based on Eq. (21) is asymptotically correct for Ca -> 0 (except for the special case of sliding collisions, when F(fic) m 0 or F((3C,8C) & 0, which have a negligible probability in random emulsions), but it is limited to moderate viscosity ratios A
Hydrodynamical Interaction of Deformable Drops

401

Refs. [20, 21]. A more general matching condition [19] not subject to this restriction takes into account that, during an initial approach of highly viscous drops, the angles (3 and 9 and the contact force may have substantial variations due to slow film drainage. In addition, a more general lubrication form [14-15], suitable for A > 1, is used instead of Eq. (20): w -67r/i e - — $(p) — — , p = AJ ——, (23) h-min at N where $ ( p ) has an exact expression [14]:

H

^ ^ f ^ n + i ^ i + i ^ + e]

(24)

'

The lubrication form (20) is the limiting case of (24) for p -C 1. Equating (23) to the contact force (6) or (9), and integrating backwards from Kiin = 0 to hmin = h°min, yields

<25>

*""*'=£ IT* • where

^

*(p)

-

n(n + 1)

F

(2n + l)Po]

and po = X(2h^nin/R)1^2\ the contact force F and d^/di in Eq. (25) are taken from equations (6)-(7) or (9)-(ll). Equation (25) relates the collision angles (3C (or f3c and 0c) to the initial angles (30 (or (30 and 90) at t = 0. In particular, for gravity-induced motion,

sin/30 = s i n / 3 c e x p [ - ^ ^ l [ aApgR J

.

(27)

For relative centre-to-centre trajectories in the plane 9 = n/2 in shearinduced motion, Eq. (25) yields ! - ( ! - £ ' ) cos 2/?e = 8fcI(l-g') 1 - (1-Bt)co82/30 (l + k)3D*

, , '

{

(for general trajectories, 9 ^ TT/2, it is also possible to relate 9O, (30 to 9C, (3C through the second equation (11), but the algebra becomes more involved). Knowing /3O (or (30,90) and the initial film profile (22) allows us to start integrating the thin-film equations together with the contact-motion equations. Again, the choice of initial separation h°min has no effect on the solution in the range of substantial film deformations (for /i/a8- < O(Ca)).

402

A.Z. Zinchenko and R.H. Davis

For AGz1/2 < 1, the matching strategy (25) and that of Refs. [20, 21] yield practically identical results, and both do not contain any adjustable parameters; for \Call2 > 0(1), however, the matching rule (25) is the one to use. The numerical solution of thin-film equations (12)-(17) is greatly complicated by numerical stiffness, so that the simplest, explicit algorithms require extremely small time steps (especially at the initial stage of film deformation), and are very impractical for systematic coalescence efficiency calculations (when these equations need to be solved many thousand times). Far more efficient, semi-implicit, absolutely stable algorithms have been developed recently; see the original works [19, 20] for details. To demonstrate how the theory is used to calculate the coalescence efficiency, consider the case of two drops in gravity-induced motion with size ratio k — a\/a,2 = 0.5, viscosity ratio A = 1, Ca = 2.2 x 10~3, and the non-dimensional Hamaker parameter 5 = 1.1 x 10~3, where

Fig. 3. Dynamics of the minimum surface separation for two trajectories close to critical.

Fig. 3 presents the minimum surface separation, hmin/a2, scaled with the larger radius 02, versus the angle (3, for two different collision angles /3C = 1.035 rad (solid line) and 1.040 rad (dashed line). The minimum separation is located along the line of centres, or at the rim when a dimple forms. When (3C — 1.035 rad, the van der Waals attraction has enough time to pull the surfaces together, leading to coalescence. In contrast, for a slightly

Hydrodynamical Interaction of Deformable Drops

403

higher value of j3c = 1.040 rad, the strength of the van der Waals attraction is insufficient for coalescence, the drops reach minimum separation of hmin/o-2 — 7.6 x 10~6 at (3 « TT/2 and eventually separate. The critical value (3fu in the range 1.035 - 1.040 is slightly different from f3?it m 0.99 found in Ref. [20] for the same conditions, simply because we used in the present calculations much larger cutoff radii r max in the numerical solution of thin-film equations (12)-(16) for full numerical convergence. Fig. 4 shows the film profile evolution for a slightly supercritical value of pc = 1.040. The profiles are plotted in non-dimensional variables [20]: h = hR/b2

,

r = r/b

,

(30)

where b is defined by irb2a/R = ApgR3 and is a measure of the film radial extent. At /3 — 1.101, the film just starts to dimple. Subsequently, the dimple radius decreases with increasing (3. At (3 = 1.712, when the surface clearance hmin has reached its last local minimum (Fig. 3), the dimple disappears altogether, causing very rapid drop separation thereafter.

Fig. 4. Dynamics of the film profile for f3c = 1.04 rad.

Once the critical collision angle f3^u is found by trial-and-error (which typically requires extensive calculations and an efficient thin-film algorithm), Eq. (19) may be used to determine the corresponding critical offset parameter d^\ separating coalescence (c?oo < d^11) and noncoalescence (doo > d^f). The coalescence efficiency is defined as / dcrit •^12 =

\2 •

(<Jl)

404

A.Z. Zinchenko and R.H. Davis

Most importantly, knowing E\i allows us to solve the population dynamics equation (e.g. [36-38]) and predict the evolution of the drop-size distribution in dilute emulsions due to coalescence. It was found [20] that there is a narrow 'second coalescence zone' for collision angles j3c close to n/2, but its contribution to the coalescence efficiency £12 is typically very small, so Eq. (31), assuming a single coalescence zone d^ < d^lt, is quite adequate for practical purposes. Figure 5 shows the coalescence efficiency for slightly deformable (solid curves) and spherical (dashed lines) drops with Ca — 0.001 and A = 1 as a function of the non-dimensional Hamaker parameter S. Figure 6 presents E12 as a function of the capillary number at k = 0.5 and 8 = 4 x 10~4. It can be predicted from Fig. 6, for example, that deformation decreases the coalescence efficiency of two drops with a\ = 262 /xm, a-i = 525 /im, a = 10 dyn cm"1, A = 0.5, Ap = 0.1 g cm"3, and A = 10~14 erg (5 = 4 x 10"4, Ca — 5.8 x 10~3) by about five fold. The sharp dependence of S on the drop size (see Eq. (29)) is the reason for steep behaviour of Eyi in the transition range from spherical to deformed drops. Similar calculations can be performed for shear flow [21], although they are technically more involved, since the upstream interception area for deformable drops is no longer a circle, and two collision angles, j3^%t and #£"*, have to be found by trial-and-error. The collision efficiency, E\%, is defined as En = Jn/J°12

,

(32)

Fig. 5. Coalescence efficiency for gravity-induced motion at Ca = 0.001 and A = 1. Reproduced by permission from Ref. [20].

Hydrodynamical Interaction of Deformable Drops

405

Fig. 6. Coalescence efficiency for gravity-induced motion at k = 0.5 and S = 4 x 10 4 . Reproduced by permission from Ref. [20].

where J\2 is the flux of pairs through the upstream interception area, and Jf2 is the corresponding Smoluchowski's value [40] assuming the neglect of drop interactions, shape distortions and molecular attraction. Figure 10 from Ref. [21] shows, for example, that the deformation effect on the coalescence efficiency of two drops of ethyl salicylate in diethylene glycol for shear flow with k = 0.5, 7 = 1 s"1, a = 1.9 dyn cm"1, A = 0.1, /j,e = 0.35 g cm"1 s"1, A = 5 x 10~14 erg becomes significant when the average radius a = (ai + 02)/2 exceeds 300 /zm. At a = 300 /xm, the capillary number (2) is 0.005, which is expected to be within the range of validity of the asymptotic theory, even for this small A = 0.1 (see Sec. 4). For A = 0(1), it is also possible to perform scaling for the range of drop sizes where deformations have a significant effect on shear-induced coalescence. For two drops with A = 0(1), coming into 'apparent contact' without van der Waals attraction in a general 3D collision, the minimum surface separation h along the trajectory scales like h ~ RCa4^ (Sec. 4). Roughly, if the molecular force Fw ~ AR/h2 becomes competitive with hydrodynamic forces FH ~ HeiR2 at this separation, the drops will coalesce. Substituting the definition (2) of the capillary number Ca into the criterion Fw ~ FH yields r A5rr8 1 1/17

*~&]

(33)

In deriving this scaling, we neglected all non-dimensional numerical factors that depend only on k and A.

406

A.Z. Zinchenko and R.H. Davis

3. COMPUTATIONAL METHODS FOR INTERACTING DROPS WITH ARBITRARY DEFORMATIONS 3.1 Boundary-integral Method For drops with finite deformations (Ca = 0(1)), the asymptotic method of Sec. 2 is obviously not valid, and so theoretical analyses in this case rely more heavily upon numerical calculations. If the Reynolds number is still small, typically requiring a highly viscous matrix fluid, the boundaryintegral method is most appropriate. This method was pioneered by Rallison and Acrivos [22] for one-drop calculations, and since then it has received considerable development, including binary interactions [19, 26-28, 41-42], and multiple drops in a periodic box [23-25, 43-44] to simulate locally homogeneous emulsions. The essence of this method is to reduce the Stokes equations (1) in the fluid domains inside and outside the drops to a system of integral equations for the fluid velocity u on drop surfaces only, using the reciprocal theorem for Stokes flows [30] and the Green function (which corresponds to the Stokes flow generated in an unbounded fluid by a point force). In particular, for two drops moving in an unbounded liquid, this system of boundary-integral equations takes the form u(y) = 2K £ / ti(a:) • T(x - y) • n(x)dSx + F(y) 0=1

,

(34)

•/6<3

where K — (A — 1)/(A + 1), Sp (/? = 1, 2) is a drop surface,

T(r) = £ - ^

(r = x - y)

(35)

is the stresslet third-rank tensor corresponding to the free-space Green second-rank tensor (with unit viscosity):

and n(x) is the outward unit normal at x G Sp. The inhomogeneous term F can be written as Oil

ltl\

F{y) = ^ff 1+ A

9

+ -jf-VS

2

r

£ /, /(»)»(*) • G(x - y)dSx

, (37)

He{± + A) p=iJSl3

where u^iy) is the unperturbed flow (which would exist at point y in the absence of drops), f{x) - 2ak(x) + (Pe - p')gz

,

(38)

Hydrodynamical Interaction of Deformable Drops

407

k(x) = \{k\ + k,2) is the mean surface curvature at x (ki, ki being the two principal curvatures of the surface), g is the gravity acceleration, and z is the Cartesian coordinate in the direction of gravity. The first term on the RHS of Eq. (38) is related to the stress jump across the interface, since 2ak is nothing but the Laplace pressure caused by the surface tension. Knowing the fluid velocity u on drop surfaces from the solution of Eq. (34) at each time step allows us to advance the drop shapes (see below). For matching viscosities A = 1, Eq. (34) yields an explicit expression u(y) = F(y), which greatly facilitates calculation; otherwise, Eq. (34) must be solved by iterations. The most popular way is that of 'successive substitutions' of the previous iterate u^ into the right-hand side of (34) to get a new approximation «("+1) until such iterations converge. However, K = ±1 are known to be the spectral values for Eq. (34) (see, e.g., [45-46]), so that the homogeneous system (34) (i.e., with F = 0) has non-zero solutions u at K = ±1. As a result, for extreme viscosity ratios A < 1 or A > 1, when \K\ « 1, the system (34) is ill-conditioned and successive substitutions are very poorly convergent. A procedure called 'Wielandt's deflation' (e.g., [4546]) eliminates K = ±1 from the spectral values at the expense of slightly modifying the boundary-integral equation (34), which greatly accelerates iterations for well-separated drops with A < 1 or A > 1. For drops in close approach, however, the spectrum of K-values for Eq. (34) becomes nearly continuous, and so elimination of just marginal values n = ±1 from the spectrum by deflation does not alleviate all difficulties for extreme A; it is better to use in this case more sophisticated, biconjugate-gradient iterations of Lanczos [42], after deflation. For numerical solution of Eq. (34) (or its deflated version), the drop surfaces must be discretized. There are two general approaches to surface discretization: (i) global parametrization (e.g., [46-47]) and (ii) unstructured triangulation. The first approach (with a clear analogy to globe parametrization by a system of longitude and latitude lines) is losing popularity and is now considered generally inferior to the second method, primarily due to the presence of coordinate singularities (poles) in the global parametrization approach, with a negative effect on numerical stability. In dynamical simulations, drops often start from spherical shapes. Unstructured, almost uniform triangulations of a sphere can be constructed by simple techniques. In the first scheme [45], each face of a regular icosaedron inscribed into the sphere is divided into four triangles, the new vertices are projected radially on the sphere, and the process is repeated as many times as necessary, giving discretizations with N& = 20 x Ak (k = 0,1,2,...) triangles;

408

A.Z. Zinchenko and R.H. Davis

Fig. 7 shows an example for N& = 1280. The second scheme [42] differs only in that we start from a regular dodecaedron, project each pentagon face centre radially on the circumscribed sphere, connect the projection with the face vertices, and then proceed as in the first method; this scheme produces triangulations with N& — 60 x 4k [k = 0,1,2,...). Subdivision of each triangle face into m2 smaller triangles [43] may also be practiced; when the latter procedure for small m < 5 is combined with the first two schemes, it gives additional possibilities NA = 720, 1500, 2160, 2880, 6000, 6480, etc., but still with highly uniform unstructured triangulations (the maximum-to-minimum mesh edge ratio being within 1.19-1.22).

Fig. 7. An example of surface triangulation into 1280 elements starting from a regular icosaedron.

When drops move (and deform), one can advect the mesh nodes either with the interfacial fluid velocity u or with the normal velocity (u • n)n (both found from the solution of Eq. (34)), to update the drop shapes. A familiar difficulty with both strategies, making them unsuitable, is that the internode distances become highly irregular, invalidating the mesh after a short simulation time. One remedy [43] is to add an artificial tangential velocity to the node motion (which has no effect on shape evolution, for sufficiently small time steps); this additional velocity is constructed by some local rules to prevent mesh degradation for a long simulation time, based on the idea of internode 'springs'. Unfortunately, in the original form, this idea of 'grid tension' leads to numerical stiffness, with tight stability limitations on the time step. A more advanced, adaptive restructuring method [48] combines a dynamical spring-like mesh relaxation (performed iteratively) with topological mesh transformations (node addition/subtraction/reconnection). The latter approach is very flexible and was used, in particular, in drop pinch-off simulations [48]. With isotropic mesh restructuring into compact elements, however, it is not easy to reach convergence (i.e., independence of the global results from triangulation) for elongated shapes, because adequate azimuthal resolution requires a

Hydrodynamical Interaction of Deformable Drops

409

very large total number of boundary elements as a drop stretches. A very different approach to mesh control called 'passive mesh stabilization' [42, 26, 23-25] uses fixed topology triangulations (i.e., with a fixed number of nodes and fixed connections) and seeks to prevent mesh degradation by constructing an additional global tangential field on each surface Sa separately from the solution of a variational problem. These additional node velocities are required to minimize, in some sense, the 'kinetic energy' of disordered mesh motion (as opposed to 'potential energy' in the simplest grid tension method), thus avoiding excessive numerical stiffness. Suitable forms of the kinetic energy function were found both for compact [42] and highly stretched [26] shapes. This method provides less control over the mesh than does adaptive restructuring [48]. However, it was found to be surprisingly flexible in a variety of problems, including dimpled shapes and highly stretched drops closely approaching breakup [26]; most importantly, it was possible to systematically demonstrate, with passive mesh stabilization, the independence of the results (global shapes, dynamics of thin neck thinning, etc.) from the degree of triangulation even for a modest number of boundary elements [26]. The results for finite deformations discussed below are all obtained using passive mesh stabilization. Local surface fitting by a paraboloid has become a popular method to calculate the curvatures k(x) and normals n(x) in the nodes of an unstructured method. The first parabolic fitting algorithm is, probably, due to Rallison [49]; a much simpler, more general and efficient version was subsequently developed [42]. Numerical discretization of the singular integrals (34) and (37) by a trapezoidal rule [49] (see [42] for details) is preceded by singularity and 'near-singularity' subtractions [43] (with u(x*) and f(x*) subtracted from u(x) and /(as), where a;* is the mesh node on Sp nearest to y); the latter greatly improves the quality of the numerical solution. For gas bubbles or low-viscosity drops (A
410

A.Z. Zinchenko and R.H. Davis

A remedy allowing us to continue the calculation beyond the singularity formation is the 'curvatureless' form [26] of the boundary integral (37):

fSfik{xMx)-G{x-y)dSx = ±fSfl{l-*^^}dSx

. (39)

The right-hand side of Eq. (39) (understood as a principal-value integral) contains only normal vectors n(x) much less sensitive to discretization errors than is k{x), and eliminates the difficulty with the concentrated capillary force. However, the curvatureless form (39) is more singular at r —>• 0, and requires special desingularization techniques [26]. Using the curvatureless form is also crucial for successful drop breakup simulations, when a fixed-topology mesh is used (allowing boundary elements to stretch); examples are given in Sec. 5. Besides binary interactions in an unbounded medium (relevant to dilute emulsions), as described by Eq. (34), it is of great interest to simulate multidrop interactions with periodic boundaries. Namely, a random system of TV ^> 1 drops with centres in a box V is assumed to be continued triply-periodically into the whole space. The idea of periodic continuation is borrowed from statistical physics (e.g. [51]) and allows us to accurately simulate the effective properties of locally-homogeneous concentrated emulsions away from the boundaries with a limited number of drops N. For emulsion sedimentation (Sec. 6.2), the periodic box V is stationary, and can be taken as a cube (Fig. 8). For emulsion shear flow (Sec. 6.1), the box V is initially a cube, but it is then skewed by the flow, until the whole system repeats itself in a cyclic manner [43] (Fig. 9). The main modification to the boundary-integral equation (34) is in using Hasimoto's [52] periodic Green function. The details are involved, however, and the reader is referred to the original papers [23-25]. 3.2 Multipole Acceleration of Boundary-integral Calculations Direct point-to-point calculation of all boundary integrals (34) and (37) has 0(N2N^) computational cost per time step, which severely limits the number of drops N and/or the number of triangular elements N& per drop possible in dynamical simulations, especially for contrast viscosities (A
Hydrodynamical Interaction of Deformable Drops

411

Fig. 8. A schematic for multidrop sedimentation with periodic boundaries (a 2D sketch, not to scale). Surfaces S'p and S'p are periodic images. Reproduced by permission from Ref. [23].

Fig. 9. A schematic for multidrop simulation in shear flow with periodic boundaries. The non-dimensional periodic box V (contoured bold) is based on the lattice vectors ei, e^, ez\ initially V is a unit cube based on ei, e\, e%. When the strain 7 = 7* reaches 1/2, the periodic boxes are restructured for the first time, to avoid excessive skew. The third dimension is not shown. Reproduced by permission from Ref. [24].

412

A.Z. Zinchenko andR.H. Davis

and resolution N& ~ 103. In this scheme, elongated drops are first sliced into compact blocks (if a drop is compact, it coincides with its only block), and minimal spherical shells (V) are constructed around each block (Fig. 10).

Fig. 10. Multipole-accelerated calculation of near-field interactions. The contributions of block By (surrounded by shell Z>7) to the boundary integrals for y, y' and y" are evaluated, respectively, by (i) reexpansion of Lamb's singular series from x° to x°s, (ii) pointwise calculation of Lamb's singular series, and (iii) direct point-to-point summation. Reproduced by permission from Ref. [23].

The periodic Green function is partitioned into the free-space and 'farfield' part. Each block contribution to the free-space part of the boundary integrals is expanded as Lamb's singular series [30] (which may be viewed as an expansion in inverse powers of the distance from the block centre). Interactions between well-separated blocks (e.g., B1 and Bs, with shells P 7 and V$, respectively) are handled by Lamb's series reexpansions from singular to a regular form (i.e., in positive powers of the distance from the centre x°6). If the shells V^ and V1 overlap, but mesh node y' of block B% is 'well outside' shell X>7, Lamb's singular series for block B1 is used directly to calculate this block's contribution to the boundary integrals for node y' (Fig. 10). Only if node y" of block B( is inside shell £>7 (or is outside, but

Hydrodynamical Interaction of Deformable Drops

413

too close to X>7), should we use direct summations to calculate the contribution of block B7 to the boundary integrals for node y" (Fig. 10). The far-field contributions are handled by Taylor expansions about block centres. Thus, the essence of the algorithm is to use efficient multipole/Taylor expansions as much as possible, and employ direct summations only as the last resort. Technical details are quite involved, though, and the interested reader is referred to the original papers [23-25]. For N — O(102 — 103) and NA ~ 103, the computational gain of this algorithm over the standard summation is 2-3 orders of magnitude at each time step, which has made such large long-time simulations feasible [23-25]. Perhaps contrary to expectations, 3D boundary-integral simulations for only two drops in close approach is a very difficult problem, when the capillary number is small and the drops are nearly non-deformed. One reason is a very tight stability limitation on the time step stemming from the Courant condition. Another difficulty is localization of stress, requiring very high resolution in the narrow gap between two drops. Less obviously, the outer region (away from the gap) also needs unlimited resolution as Ca —> 0; the reason is that, in the non-dimensional form of (37), the curvature deviation (of order O(Ca)) from a uniform value is divided by Ca, to produce an 0(1) effect. As a result, for Ca
414

A.Z. Zinchenko and R.H. Davis

Fig. 11. Node partition into non-overlapping 'patches' in multipole-accelerated calculations for two drops with high resolution.

4. VALIDATION OF THE SMALL-DEFORMATION THEORY OF COALESCENCE The asymptotic theory of coalescence (Sec. 2), which is based on matching the thin-film solution in the gap with the outer solution for spherical drops, avoids formidable numerical difficulties inherent in boundaryintegral simulations for Ca
Hydrodynamical Interaction of Deformable Drops

415

an asymptotic film thinning to arbitrarily small separations. It is therefore of great interest to study if the small-deformation theory of Sec. 2 is indeed a correct approach to non-axisymmetrical interactions and to find its range of validity; in particular, it is of interest to see if (and when) the neglect of the pumping flow is an accurate assumption. Multipole-accelerated 3D boundary-integral calculations for two drops with very high resolution [19], albeit costly, are able to provide exact results for comparisons. Examples below are for two equal drops of radius a freely suspended in a shear flow (gravity-induced motion of unequal drops is additionally discussed in Ref. [19]). For simplicity, van der Waals attraction is neglected, and only centre-to-centre trajectories in the plane 6 = n/2 (Sec. 2) are considered.

Fig. 12. Shear-induced motion of two drops at A = 1 and Ca = 0.01. Close-ups at t = 5.5 and 6.5 show the gaps of 3.2 x 10~3a and 7.3 x 10~4a, respectively.

Fig. 12 presents snapshots of relative motion for A = 1, Ca = /j,eja/a = 0.01, and initial centre-to-centre offsets Ay0 = 0.7a (across the flow) and Ax 0 = 5a (along the flow), using a gap-adaptive 'projective' mesh [19] with N& = 34560 elements per drop; time is scaled with j 1. Spherical drops

416

A.Z. Zinchenko and R.H. Davis

would collide at (3C « 0.55 rad. In contrast, there remains a very small, but non-zero gap (reaching 7 X 10"4a) during the apparent 'contact' motion, and the drops eventually separate. This simulation was repeated for varying capillary numbers, to compare the surface clearance hmin{(3) with the asymptotic theories. In Fig. 13a,b, for Ca = 0.04 and 0.01, respectively,

Fig. 13a,b. Comparison of exact and asymptotic results for the surface clearance at A = 1.

the solid lines are from fully-convergent boundary-integral calculations (for Ca = 0.01, we had to use N& — 138240). Short-dashed lines are from the asymptotic theory [20] (which neglects the pressure gradient term in Eq. (15), assuming A = 0{Call2\) • 0 (Fig. 14), so that (3sep remains appreciably higher than 7r/2 for all attainable capillary numbers. The behaviour of j3sep at Ca —>• 0 may be found by rescaling [19] the thin-film equations in the vicinity of /3 = TT/2. Let to De the time moment corresponding to (3 — TT/2. The driving force in Eq. (16) can be linearized at t RS to: F = F'(to)(t — to). Non-dimensional variables are introduced:

Hydrodynamical Interaction of Deformable Drops

All

Fig. 14. Comparison between the exact (squares) and asymptotic (crosses, A = 0 theory) values of the separation angle for A = 1.

„_ r

~ _ hR

r

~n

'

p=—p

- _ Qa(t - t0)

TJ2 ' *~

2i?y

, u = —f— , f = -^—

,

40

where

Then, the thin-film equations (12)-(16) (without van der Waals attraction and the pressure gradient term in (15)) can be written as P

r dr {dr)

'

J

u(i,r) = f§°(r', f)f(t,

r')df'

dh

r°°

1 d , " .

2 dr

,

-

/^ \

^ - + - -W2 (rhu) = 0 , / prdr = -t . 42 Equations (42) are universal and do not contain any parameters. The separation time isep must be therefore 0(1). Using Eqs. (9), (11) to evaluate F'(t0), together with Eqs. (40)-(41) and (2), one can find that isep = 0(1) corresponds to f3sep — TT/2 = O(QJ 1 / / 3 ), which is in agreement with the asymptotic curve in Fig. 14 and shows, indeed, an extremely slow approach

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Psep —>• vr/2, as Ca —> 0. The same universality arguments can be used to find the scaling [19] 0{a Ca4^) for the minimum surface separation along the trajectory, which was used in Sec. 2. While, for A = 1, the A = 0 theory is preferred, the situation is reversed for highly-viscous drops. Figure 15a,b presents the non-dimensional

Fig. 15a,b. Comparison of exact and asymptotic results for the surface clearance at A = 4.

surface separation hmin/a for A = 4, Ay0 = 0.7a, Ax0 = 5a, and two capillary numbers Ca = 0.02 and 0.01. Fully convergent boundary-integral calculations (solid lines) are much closer to the predictions of the A ^ 0 theory (long-dashed lines) than to the results by the A = 0 theory (shortdashed lines). Both theories, however, give similar errors for (3sep (but of opposite signs), which slowly disappear, as Ca —> 0; for Ca = 0.005, the exact result (3sep = 2.06 rad is only slightly different from the prediction (3sep = 2.03 rad by the A / 0 theory. Similar calculations (Fig. 16a,b) for A = 10, Ax 0 = 5a, and Ay0 — 0.5a show that, in this case, the A ^ 0 theory becomes accurate for hmin{f5) at Ca < 0.01, and gives only a modest error for the separation angle (this error in (3sep further decreases to 1.2% at Ca = 0.005); the A = 0 theory was totally unsuccessful for A = 10, for obvious reasons. Interestingly, the behaviour of the separation angle at A = 10 is even farther from the naive expectation (3sep m ?r/2 than for A = 1: (3sep varies only from 2.40 to 2.16, as the capillary number is reduced from 0.04 to 0.005; to observe f3sep —>• TT/2, much smaller Ca would be required. Loewenberg and Hindi [41] predicted, by qualitative arguments without a detailed solution, that the minimum separation (min hmin) that two highly-viscous drops can reach along a trajectory scales like O{a Ca), as Ca -» 0 (unlike O(a Ca4/3) for A = 0(1), see above). Our results for A = 10 in Fig. 16 closely follow this predicted scaling.

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Fig. 16a,b. Comparison of exact and asymptotic results for the surface clearance at A = 10.

Overall, the above comparisons with exact boundary-integral results demonstrate the validity of the asymptotic method of Sec. 2 for generic, non-axisymmetrical collisions with offsets Ayo ~ a, for a wide range of Ca O(l). The method very accurately describes the dynamics of small surface clearance in the entire range of near-contact motion, except in the vicinity of separation. A modest discrepancy in the separation angle (3sep (which slowly disappears, as Ca —> 0) is most likely due to 'pumping flow' which drives the fluid away from the gap (during the tensile stage), thus delaying separation compared to the asymptotic result. Typical times of nonaxisymmetric interactions, however, are too short for the pumping flow to have a strong effect. Pumping flow will be obviously more important for collisions close to head-on, and there are current efforts [54] to incorporate this effect in the thin-film equations. For A < 1 , however, the asymptotic method of Sec. 2 is less successful, even for general, nonaxisymmetric collisions. Figure 17 presents hmin((3) for A = 0.25, Ay0 = 0.7a, Ax0 = 5a at a moderately small Ca = 0.04. The boundary-integral results for two adaptive triangulations are in close agreement (solid and short-dashed lines), but both differ considerably from the prediction of the A = 0 asymptotic theory (long-dashed line). For the same Ca = 0.04 and matching viscosities, A = 1, the asymptotic theory is much more accurate (cf. Fig. 13a). At A = 0.25, the primary reason for the discrepancy in the separation zone is a vigorous pumping flow. At the initial stage of approach, however, the discrepancy is mostly due to neglect of other high-order terms in thin-film equations (it is known [20] that, even apart from pumping-flow effects, the leading-order asymptotic theory is limited to A » O(Ca1/2\ lnCb|)). For A < 1, the range of validity of the existing theory is simply shifted to much smaller capillary numbers (which would be too difficult, however, to demonstrate numerically, due to low film resistance and particularly tight time-step stability limitations for

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Fig. 17. Comparison of boundary-integral and asymptotic results for the surface clearance at A = 0.25 and Ca = 0.04.

A • oo, A A; —> 0), we have a seeming indeterminacy oo • 0 for the normal stress jump 2ak in Eq. (38). The methods for 'spherical' drops, like those used in Sec. 2 for the outer solution, admit such an indeterminacy and simply bypass it by an alternative solution technique, namely, assuming spherical shapes as a first approximation and thereby not considering the boundary condition for the normal stress. Once such a solution is obtained, the normal stress jump could be used, if desired, to calculate this indeterminacy 2ak and the corresponding perturbation of drop shapes. This discussion clarifies that the theory for 'spherical' drops used in the outer region (where shape expansions in Ca are regular) does not ignore the 'global deformation', but simply includes the main, O(l)-effect of it; the non-dimensional form of the boundary integral (37) shows this effect as stemming from AA; = O(Ca) divided by Ca. To proceed beyond this level of approximation, one would have to consider O(Ca2) corrections to the spherical shapes, which is very impractical and also unnecessary for coalescence theories at Ca
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5. BUOYANCY OR GRAVITY-INDUCED MOTION OF TWO BUBBLES/DROPS WITH LARGE DEFORMATIONS Interaction of two strongly deformable drops or bubbles in the gravitational field offers a surprising variety of different types of behaviour. In the examples below, calculated by the curvatureless algorithm [26], the characteristic velocity V and time for nondimensionalization are \/\p\ga\/{&irjie) and CI2/V, respectively, where ai is the largest of the two non-deformed radii. Drops/bubbles start from spherical shapes. Each relative trajectory is determined by A, k = a\jai < 1, initial non-dimensional horizontal A i o = Axo/a2 and vertical Azo = Azo/a2 offsets of the drop centres, and the Bond number B — \Ap\gal/a (which is used in place of Ca to characterize deformation). Figure 18 shows relative motion of two bubbles for A = 10~3, k = 0.9, B = 14.3, Ax0 = 4.75, Ay0 = 0 and Azo = 15. Even though the initial shapes were smooth, interaction induces apparent surface singularities. Most interestingly, the side view in the (y, z)-plane (Fig. 18d) reveals that the larger, bottom bubble tends to develop a point-singularity, while the singularity formed on the smaller bubble by t = 8 is scallop-like, which is locally a 2D cusp. The occurrence of such sharp edges was also observed in related experiments [27,28] and is not a numerical artifact. If untreated, this singularity causes the calculation to stall. A special smoothing technique was developed [26], allowing for this and similar calculations to continue beyond the development of apparent singularities. The key idea is to add an artificial normal velocity to that obtained from the boundaryintegral solution at each time step. This additional velocity is on the order of e • 0, simultaneously increasing the number N& of triangular elements per bubble/drop, to produce convergent results for global motion, with increasingly sharp singularities. For realistic triangulations and supercritical Bond numbers (like that in Fig. 18), when the singularity in the exact solution is quite sharp on the drop lengthscale, it is not reasonable to expect convergent results with this procedure in the cusp region, but this local convergence is not necessary if only global motion is the goal. When only point singularities form, a combination of this cusp-smoothing

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Fig. 18. Relative buoyancy-driven motion of two bubbles with a\/a2 = 0.9, A = 10~3, B = 14.3 and 3840 triangular elements on each surface; no smoothing: (a-c) the bubble shapes in the (x, z)-plane; (d) in the (y, z)-plane. The calculation could not be continued due to the 2D cusp formation on the upper bubble at t w 8, without smoothing. Reproduced by permission from Ref. [26].

procedure with the use of conventional boundary integral (37) is generally successful and allows us to continue a simulation well beyond the singularity formation. When line singularities (2D cusps) form, however, it is crucial to use the curvatureless form (39) instead, in addition to smoothing, in order to eliminate a numerically unresolvable 0(1) cusp contribution. The above arguments, showing that detailed cusp resolution (which would be numerically impossible due to molecular scale of the cusp [50]) can always be avoided in the calculation of global motion when the curvatureless form is used, also indicate that additional physical mechanisms should not have a global effect, if they invalidate the classical model in

Hydrodynamical Interaction of Deformable Drops

423

the small cusp region only. For example, in practice, tip streaming often occurs for A
< q >a 11 - exp L ( < ^ i >

- l) I

,

(43)

where ea -C 1 is the smoothing parameter for surface Sa,

K={{k\ + kl)/2\1'2

(44)

is a measure of the local curvature (note K = k for a sphere), and < q >a and < K >a are the average values of |g| and K over Sa. In addition, an appropriate small constant is added to (43) to make zero the total flux through Sa (to conserve mass). On the main, smooth part of the drop or bubble, the modification to the normal velocity u, • rii is very small, of order ea < q >a (or even smaller, if the surface is nearly spherical, as is often the case in bubble cusping). In the cusp region, however, where K may attain high values, (43) gives a stronger negative correction to the normal velocity, and the exponential dependence keeps the cusp curvature under control. The principal curvatures k\ and ki for Eq. (44) do not have to be accurate, and so the method remains effectively curvatureless. Repeating the simulation in Fig. 18 with a small cusp-smoothing (ei = 62 = e = 6.67 x 10~3) allows us to proceed much farther and predict the further motion (Fig. 19). Namely, a larger bubble is being sucked into the dimple formed on a smaller one, resulting in bubble capture. This interesting phenomenon of deformation-induced capture was first discovered experimentally by Manga and Stone [27-28] for two bubbles in corn syrup, although adequate computer simulations were not available at that time. Manga and Stone [27-28], however, offered a far-field asymptotic analysis for well-separated drops with small deformations by the 'method

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Fig. 19. The same simulation as in Fig. 18 in the (x, z)-plane, repeated with smoothing (e = 6.67 x 10~3) and leading to bubble capture. Reproduced by permission from Ref. [26].

of reflections', to qualitatively explain the onset of bubble capture. Most strikingly, simulations in Figs. 18-19 show that the critical offset Ax% for bubble capture can be much larger than the geometrical Smoluchowski's value 1 + k, which is in accord with experiments [27-28]. In practice, this capture phenomenon is followed by bubble coalescence when the van der Waals attraction drains the thin film, but the actual coalescence time (which may be quite large [27-28]) depends on the details of near-contact physics. Additional calculations [26] confirm global numerical convergence of the results in Fig. 19 in the limit N& —> oo, e —»• 0, and their independence of the relation between e and N& in this limit, thus making the choice of smoothing (43) purely technical. The cusp-smoothing, in combination with the curvatureless algorithm, is therefore a correct method to predict the global shape evolution both with point and line 'singularities' (limitations are discussed in Ref. [26]). The local structure of cusps,

Hydrodynamical Interaction of Deformable Drops

425

however interesting, is irrelevant to predicting global dynamics of cusped shapes. A different mode of bubble capture is shown in Fig. 20. The parameters k = 0.7, A = 10"3, -6 = 7, Axo = 2.3 and Az0 - 10 are the same as in Fig. 9 of Ref. [26], but we used here much finer triangulations (N& = 15360 triangles per bubble, instead of N& = 3840 in Ref. [26]), and, accordingly, a four times smaller smoothing parameter ei = 8.2 x 10~4 fa — 0). Only a point singularity forms on the smaller bubble, while the other bubble remains smooth. Unlike in Fig. 19, the smaller bubble is swept around the larger one and eventually becomes sucked into the dimple formed at the rear of the larger bubble. This later mode of bubble capture is known experimentally [27-28] as 'entrainment.' The present results are close to those in Fig. 9 of Ref. [26] and show just a slightly faster capture; again, a good global accuracy is achieved without fully resolving the cusp. Calculations similar to those in Figs. 19-20 (although, for more limited resolutions) were used systematically [26] to find the critical offsets (Axo)cr for capture by trial-and-error (Fig. 21). The critical offset increases with B due to increased deformation-induced alignment but becomes only a weak function of B at large Bond numbers. Also, the critical offset is very sensitive to the size ratio and is the largest for bubbles of nearly the same size (although the mutual approach is slow in this case). Additionally, a comparison of the dark squares and the crosses in Fig. 21 at a\/a2 = 0.7 shows that the capture efficiency is slightly reduced when the bubbles (A = 10~3, which is representative of A < 0(1O~2)) are replaced with drops having A = 0.1. Further increase in A to 0(1) values leads to interaction-induced breakup, instead of capture. For ai/02 = 0.7, A = 1, B = 5.31, Ax0 = 1, and Azo = 5.09 (Fig. 22), the smaller drop is swept around the larger one, stretches due to hydrodynamical interaction, and starts necking. The reason for stretching is that the hydrodynamical field in the wake of the larger drop is mainly an extensional flow. The simulation in Fig. 22 was repeated for four different numbers of triangles iVAl = 2160, 3840, 6000 and 8640 on the smaller drop [26], to assess accuracy. After t = 6, the drops continue to separate, and the smaller drop experiences neck pinchoff, while the other drop remains compact. In Fig. 23, only the evolution of the breaking drop is shown. Despite some local mesh imperfections inherent in our fixed-mesh topology breakup simulations, the global convergence is quite good (Fig. 24), especially in the top bulbous area; in particular, the radius 03 = 0.920ai of the main (top) fragment after breakup is accurately predicted [26]. Moreover, these simulations accurately describe the temporal

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A.Z. Zinchenko and R.H. Davis

Fig. 20. Bubble capture through 'entrapment' at ai/a2 = 0.7, A = 10~"3, B = 7, NA = 15360, ei = 8.2 x 10" 4 .

Fig. 21. The non-dimensional critical capture offset far upstream for bubbles and low-viscosity drops. Dark squares are for A = 10~3, crosses are for A = 0.1 (at a.i/a.2 = 0.7 only). Reproduced by permission from Ref. [26].

Hydrodynamical Interaction of Deformable Drops

427

Fig. 22. Relative buoyancy-driven motion of two drops with 0,1/0,2 = 0.7, A = 1, B = 5.31; 3840 elements on the smaller and 1280 elements on the larger drop. Reproduced by permission from Ref. [26].

dynamics of the neck thinning. As breakup is approached, the local neck shape becomes axisymmetrical (even though the whole problem is 3D), thus allowing us to introduce an equivalent neck radius r = (<STOm/7r)1/'2, where Smin is the minimum area of cross-sections orthogonal to the line of maximum elongation. This observation supports, incidentally, the body of local axisymmetrical studies on viscous pinchoff (e.g., [59-62]). In particular, for A = 1, a local self-similar axisymmetrical solution [62] predicts a linear dependence r(t) at pinchoff, with the dimensional slope of —0.034cr/^e- For our simulations in Fig. 24, the dynamics of the equivalent neck radius is given in Fig. 25. The small lack of accuracy at the bottom of the drop at large times (Fig. 24) does not seem to appreciably affect the dynamics of the neck thinning, and excellent numerical convergence with respect to triangulations is observed in Fig. 25; this agreement may be due to the universality of self-similar thinning at pinchoff. Our dimensional slope dr/dt at the end of the simulation is about —0.029cr//xe, which is close to the theoretical result [62] —0.034a//ie; a plausible reason for small discrepancy is that the global drop shape at the end of our simulation

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A.Z. Zinchenko and R.H. Davis

Fig. 23. The stretching and breakup of the smaller drop. The parameters are the same as in Fig. 22, except that 8640 triangular elements are used for the smaller drop.

Fig. 24. Comparison of the absolute positions and shapes of the smaller drop in the (x, z)-plane for the simulation of Fig. 23 using JVAl = 3840 (dashed lines), 6000 (solid lines), and 8640 (dotted lines). Reproduced by permission from Ref. [26].

Hydrodynamical Interaction of Deformable Drops

429

Fig. 25. The non-dimensional neck radius vs. time for the simulations in Figs. 23-24 with different triangulations of the smaller drop. The results for N^i = 8640 and 6000 are practically indistinguishable. Reproduced by permission from Ref. [26].

is not quite axisymmetrical, causing a small error in the direction of the minimal cross-section. Another possible reason is that the ultimate slope dr/dt is approached only for very thin necks, which was seen previously in axisymmetrical simulations [62-63]. In any case, using dr/dt between —0.029cr//ie and — 0.034a/fie to extrapolate the neck radius in Fig. 25 to zero yields tight bounds on the non-dimensional breakup time: t^ = 7.95 — 8.02. Breakup simulations similar to those in Figs. 22-24 were also made for contrast viscosities (A ^ 1), and critical horizontal offsets (Axo)^. for breakup were also calculated [26]. An additional mode of 3D two-drop interaction in gravity-induced motion predicted theoretically [26] is 'combined capture and breakup.' Namely, after the smaller drop is swept around the large one, it becomes sucked into the dimple formed on the larger drop, like in the pure capture phenomenon (Fig. 20), but simultaneously undergoes considerable elongation and starts necking. This behaviour strongly suggests that the smaller drop will break without being released from the dimple, as has also been observed in axisymmetric simulations [63]. Another, and very interesting phenomenon, discovered experimentally [64] for moderate A and moderate-to-large B is 'pass-through', in which the leading drop forms a torus and the smaller drop passes through its centre, sometimes in multiple cycles reminiscent of leap-frogging. It can be noted that a gravity-induced breakup simulation for a set of parameters close to ours in Figs. 22-23 was also performed by Cristini

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et al. [65] using isotropic mesh restructuring [48] (with a growing number of boundary elements as a drop stretches). Their mesh algorithm is very versatile, with an excellent local control over the mesh, and even allows the calculations to proceed after primary breakup by using mesh splicing [48]. With isotropic restructuring, however, it is often more difficult to obtain convergent results (in particular, an accurate temporal dynamics of drop elongation in primary breakup), and a very large total number of elements may be needed for this purpose, compared to 'passive' mesh stabilization [26]6. MULTIDROP SIMULATIONS 6.1 Shear Flow of Concentrated Emulsions One of the useful applications [24] of multipole-accelerated, boundaryintegral simulations in a periodic box is to study non-Newtonian rheology of a concentrated emulsion of deformable drops in a steady shear (7x2, 0, 0) (where 7 > 0, without a loss of generality). For simplicity, monodispersity is assumed. A large number N of spherical drops of radius a with centres in a periodic box V (initially, a cube) starts from a random non-overlapping configuration (prepared by a standard Monte-Carlo method, e.g. [66]), and the system is subject to shear, which causes the drops to deform and the periodic cell V to skew (excessive cell skew is simply avoided [43, 24] by restructuring periodic cells V at half-integer strains jt). At every time moment t > 0, the quantity of interest is the space-average stress tensor Eij, which can be expressed in terms of surface integrals (e.g., Ref. [67]) over interfaces only. For N ^> 1, the only essential components are [68] the shear stress £12, the first £11 — £22 and the second £22 — £33 normal stress differences. It is convenient to introduce non-dimensional quantities M* = E12 /(/vy), M = (En - £22)/(/ie|7l) and N2 = (£ 22 - E M ) / ^ ! ) Fig. 26 presents a typical snapshot [24] of the simulations, at drop volume fraction c = 0.5, N = 200 drops in a periodic cell, N& = 1280 triangular elements per drop, and capillary number Ca — fieja/a = 0.1; the whole simulation was continued to strains jt ss 28 with about 3000 time steps. For the number of drops N attainable in present-day dynamical simulations (roughly, N = O(102 — 103) with multipole-accelerated codes [23-25]), a single configuration is not representative, and time averaging of the stress Ey is essential. Fig. 27(a-c) presents the trajectories of the dimensionless effective viscosity //* and normal stress differences JVi, N2 vs. strain 7^ at c = 0.5, A = 1, N = 100 and capillary numbers Ca = 0.025, 0.1 and 0.2. These results were produced by the algorithm

Hydrodynamical Interaction of Deformable Drops

431

Fig. 26. A snapshot of an emulsion shear flow simulation at c = 0.5, A = 1, Ca = 0.1, and strain -yt = 4.45. The centres of 200 independent drops have been mapped into the unit cube, an initial periodic cell. Reproduced by permission from Ref. [24].

[24], but we used here much higher resolution N& = 8640 at Ca = 0.025 essential for Ca • 0 is approached, however, averaging becomes a formidable task and requires strains jt of order 30-40, or more, especially for the normal stress differences. Additionally, time-step stability limitations and required surface resolution are much tighter, making, again, Ca -C 1 a far more difficult case. One physical reason why large (but still subcritical) deformations are easier to study is that such drops tend to orient and stretch along the shear flow, thus greatly reducing drop interactions and statistical fluctuations. Although small systems (TV ~ 10) can be simulated by the much simpler, direct boundary-integral method (without multipole acceleration), it is quite important to use large systems N > 0(1O2) in emulsion-rheology simulations at high drop volume fractions. First [24], the statistical fluctuations for N ~ 10 were found to be much larger, necessitating much larger strain intervals for averaging; however, dynamical calculations for N ~ 10 do not proceed much faster than for N ~ 102 when multipole acceleration is used [24]. Secondly, and most importantly, when statistical errors for N ~ 10 are eliminated by adequate time averaging, the systematic errors can still be quite large, especially for the normal stress differences. For example, at Ca = 0.05, A = 1, and c=0.5, the N = 10 approximation

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A.Z. Zinchenko andR.H. Davis

Fig. 27. The trajectories of the effective viscosity (a), and first (b) and second (c) normal stress differences for c = 0.5, A = 1, NA = 1280 - 8640 and N = 100, with Ca = 0.025 (thick lines), 0.1 (dashed lines) and 0.2 (thin solid lines). The initial configuration for Ca = 0.1 calculations (shown in a,b only) was taken from a steady state with Ca = 0.2

was shown [24] to overestimate the average of N\ almost 1.5-fold (note that the first normal stress difference in this case is essential, comparable to the shear stress). In contrast, the average results for iV = 100 and 200 showed very good convergence [24]. The time-averaged results for the effective shear viscosity (< fi* >) and normal stress differences (< Ni >, < iV2 >) at A = 1, N = 100 and different concentrations c (from 0.3 to 0.5) are shown in Fig. 28(a-c); at

Hydrodynamical Interaction of Deformable Drops

433

c = 0.55, the behaviour is complicated by "phase transition" [24] and will not be discussed here. For moderate capillary numbers {Ca > 0.1), the results are identical to those in Ref. [24], but for Ca = 0.025 and 0.05, we used much higher resolutions (N& = 8640 and 6000, respectively) in the present calculations to improve the accuracy. Several trends in Fig. 28(a-c) are interesting to discuss. The first is a sharp dependence of the emulsion viscosity on Ca at high concentrations (Fig. 28a), so that most of the shear-thinning occurs for drops with only small deformation. This shear-thinning occurs because the drops slide more easily past each other when the shear rate and, hence, capillary number are increased, due to drop deformation.

Fig. 28. Time-averaged (a) effective viscosity and (b,c) normal stress differences at A = 1.

Unlike for the viscosity < / / >, there are no general mechanical principles to predict the signs of < N\ > and < N2 >, but our calculations always yield positive < N\ > and negative < N2 > (which is in accord with

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more limited simulations [43-44J for small systems (N = 12) at 30% concentration, and with single-drop calculations [69]). However, our results for c = 0.3, 0.4 and 0.45 indicate that the first normal stress difference, < N\ >, extrapolates to small but negative values at Ca = 0 (Fig. 28b). The second normal stress difference, < N2 >, although inevitably subject to some statistical errors, is seen to remain negative as Ca —>• 0 (Fig. 28c). Negative signs of both < N\ > and < N2 > at Ca = 0 are also in agreement with the asymptotic rheological calculations [9] for semi-dilute emulsions of spherical drops on the level of pairwise interactions. The physical reason for predicted sign change of < N\ > at small Ca (Fig. 28b) is that, for moderate-to-large Ca, drop deformation has the primary effect on N\ while, at Ca and < A^ > at Ca —> 0 indicate that, in physical units, the normal stress differences En — E22 and E22 — £33 are linear, not quadratic functions of the shear rate |-y|, as 7 —> 0. On the other hand, in the phenomenological theory of simple nonNewtonian liquids (e.g. [68]), assuming certain 'smoothness hypotheses', the constitutive equation for slow flows is shown to be of Rivlin-Ericksen [70] type. This general form is, indeed, confirmed by a small-deformation analysis of a single drop in a linear flow [71-72]; in particular, the normal stress differences are expandable in even powers of the shear rate 7, as 7 -> 0. Our rigorous simulations (Fig. 28) reveal that the Rivlin-Ericksen phenomenological theory becomes inapplicable to concentrated emulsions due to drop interactions. It should be noted, however, that, for Ca to the capillary number in the range Ca -C 1 at high concentrations, so that most of the shear thinning is observed again for nearly non-deformed drops.

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Fig. 29. A snapshot of the emulsion shear flow simulation for c = 0.55, A = 3, Ca = 0.1 and jt = 6.38. The centres of 100 independent drops have been mapped into the unit cube, an initial periodic cell. Reproduced by permission from Ref. [24].

Fig. 30. The trajectories of the effective viscosity for c = 0.55, A = 3, and N = 100. Reproduced by permission from Ref. [24].

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A.Z. Zinchenko andR.H. Davis

6.2 Sedimentation of Concentrated Emulsions of Deformable Drops In this section, we discuss sedimentation (or, equivalently, creaming under buoyancy) of concentrated emulsions of deformable drops starting from a homogeneous initial state with no deformation. Although this problem did not attract attention in the literature until recently [23, 25], sedimentation of interacting drops with moderate-to-large deformations may have interesting applications in phase separations. Under normal gravity, the Bond number for emulsion drops is typically small, so the sedimentation would occur only with small drop deformation. If such an emulsion, however, is placed in a centrifuge (thereby increasing the acceleration due to gravity by an order of magnitude, or more), the Bond number may become 0(1), causing strong deformation (and drastically changing the sedimentation regime!). Another example is emulsion sedimentation in the miscibility gap, with low surface tension. The behaviour of interacting deformable drops settling under gravity is strikingly different from that for freely suspended drops in shear flow. Figure 31 presents snapshots of the simulation at Bond number B = Apga2/a = 1.75, drop volume fraction c = 0.35, viscosity ratio A = 1, N = 600 drops in a periodic cell and N& = 960 triangular elements per drop, for different non-dimensional times t (scaled with fj,e/(Apga)). The initial 'well-mixed' state of non-overlapping spherical drops of radius a (not shown) was prepared by the standard Monte-Carlo method. The main quantity of interest is the sedimentation rate, i.e., the volume-averaged fluid velocity over the drop phase, in the reference frame where the emulsion, on average, is at rest (only the vertical component, U, is essential). The non-dimensional rate U/Uo (where Uo is the settling velocity of an isolated spherical drop) for this simulation is presented in Fig. 32a by the thick line. An apparent non-existence of the steady state is due to drop clustering, and is akin to the instability of a dilute suspension of sedimenting solid rod-like particles predicted analytically by Koch and Shaqfeh [29] by far-field analysis of the pair distribution function (for more recent studies on rod-like particles, see Ref. [73-74]). Indeed, most drops in our simulation (Fig. 31) acquire prolate shapes and become partly oriented along the vertical, and there is some qualitative analogy between the two systems. The main difficulty of the emulsion sedimentation study lies in a strong dependence of the instability growth on the initial random configuration; thin lines in Fig. 32a are for nine other random initial conditions showing considerable dispersion, as time proceeds. Probably, to make one realization representative in a wide time range, systems with N > O(105)

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Fig. 31. Snapshots of the emulsion sedimentation simulation from a homogeneous initial state of spherical drops for B = 1.75, c = 0.35, A = 1 and TV = 600. The drops sediment downward. Reproduced by permission from Ref. [25].

Fig. 32. The dependence of the sedimentation rate on random initial configurations at B = 1.75, c = 0.35, A = 1 and NA = 960: (a) 10 realizations with N = 600; (b) 10 realizations with N = 300. Reproduced with permission from Ref. [25].

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would need to be considered (since statistical fluctuations decay slowly, ~ N~ll2), which is far beyond the present-day capabilities; ensemble averaging over many realizations must be performed instead at each time moment, with assumptions made about the probability density of the initial states. In what follows, we assume [25] that all initial configurations of non-overlapping spheres have equal probability; a standard Monte Carlo method for 'hard spheres' (e.g. [66]) is known to generate realizations satisfying this condition. To study the effect of N, similar calculations were done for 25 uncorrelated initial realizations with N = 300; the first 10 are given in Fig. 32b showing even larger dispersion. Remarkably, however, ensemble-averaged results for N = 300 and 600 are in excellent agreement over a wide time range (Fig. 33) and are thus representative of the macroscopic behaviour; in contrast, a smaller system N = 100 greatly underestimates the average sedimentation rate (Fig. 33), except for very small times. The physical reason why very large systems are imperative in this problem is due to clustering; as time grows, so does the cluster size, and more drops are required to make the results box-size independent. In addition to the effect of N, the numerical effect of drop triangulation on the results in Fig. 33 was also studied and found to be small in a wide time range [25]. Calculations were also performed at the same A and B to study the effect of drop volume fraction (from 0.15 to 0.4) on the ensemble-average

Fig. 33. The ensemble-averaged sedimentation rate for B = 1.75, c = 0.35, A = 1 and N& = 960, with 14, 25 and 20 realizations for TV = 600, 300 and 100, respectively. Vertical bars show statistical errors (with 67% confidence level) for N = 100 and 600; statistical errors for N = 300 are similar to those for N = 600. Reproduced by permission from Ref. [25].

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sedimentation rate (Fig. 34); at c = 0.15, quite large systems N — 1200 were used (Fig. 35). At c — 0.4, the instability and clustering grow slowly (Fig. 34). This slower growth is likely due to geometrical constraints at high concentrations hampering drop deformation. The instability growth rate is higher at c = 0.35. Emulsions with 25% drop volume fraction show the strongest instability, with the sedimentation rate increasing 2.15 times from its minimum by t = 150 (by this time, a single drop would have fallen a distance equivalent to 40 radii). At c = 0.15, the initial sedimentation rate is, of course, higher than at c = 0.25 (due to less backfiow and hindrance at lower concentration), but it grows somewhat more slowly as time proceeds. The average drop length and deformation at c = 0.15 are also somewhat smaller than at c = 0.25, thus helping to explain why the emulsion instability at c = 0.15 is not as strong as at c = 0.25. The existence of the optimum volume fraction for emulsion instability and clustering (about 0.15—0.25 for A = 1 and B = 1.75) simply follows from the fact that, as c —>• 0, there are no interactions, and, hence, no deformation of sedimenting drops - a primary cause of clustering and instability. It remains an open question, however, if the graphs of the average sedimentation rate for different drop volume fractions can intersect (so that a more concentrated system would start sedimenting faster than a less concentrated one). We

Fig. 34. The ensemble-averaged sedimentation rate for B = 1.75, A = 1, N& = 960 and different drop volume fractions. Twelve realizations with TV = 400, 14 realizations with N = 600, 14 realizations with N = 800 and 10 realizations with N = 1200 were used for c = 0.4, 0.35, 0.25, and 0.15, respectively. Reproduced by permission from Ref. [25].

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A.Z. Zinchenko andR.H. Davis

Fig. 35. A typical snapshot of a clustered configuration with B = 1.75, c = 0.15, A = 1, N = 1200 and N& = 960. Drops sediment downward, t = 140. Reproduced by permission from Ref. [25].

cannot rule out that further evolution would lead to massive, statistically significant drop break-up changing the trends in Fig. 34. Surprisingly, however, in none of the calculations for Fig. 34 have we seen individual drop break-up; the time-scales where drop break-up might have a significant effect on the sedimentation rate are much larger than those in Fig. 34. The phenomenon of instability of sedimenting emulsions is strongly sensitive to the Bond number: When B is decreased, the instability growth slows down dramatically, due to less deformation and clustering, which is demonstrated in Fig. 36 for two random realizations. It is expected, however, in the spirit of the Koch-Shaqfeh [29] theory for a dilute suspension of rod-like particles, that there is no critical Bond number for the instability. Similar calculations for A ^ 1 are much more expensive and could be done only for individual realizations (e.g., Fig. 37), without ensemble averaging. Even under this limitation, it was possible to study the effect of A on the sedimentation rate starting from the same random initial configuration (Fig. 38). The most interesting observation from Fig. 38 is that the instability growth accelerates dramatically, when A is decreased from 1 to 0.25, and the results for A = 0.1 show the same trend; in the latter case, however, we could not proceed to large times due to numerical difficulties.

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Fig. 36. The effect of the Bond number on the instability growth for two random realizations (1 and 2) with c = 0.35, A = 1, N = 300 and JVA = 960. Solid lines, B = 1.75; dashed lines, B = 0.9. Reproduced by permission from Ref. [25].

Fig. 37. Snapshots of the emulsion sedimentation simulation from a homogeneous initial state of spherical drops for B = 2.5, c = 0.4, A = 0.25, N = 125 and 7VA = 1500. Drops sediment downward. Reproduced by permission from Ref. [25].

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Fig. 38. The effect of the viscosity ratio A on the instability growth for an individual realization with B = 2.5, c = 0.4, N = 125. Reproduced by permission from Ref. [25].

6.3 Flow of a concentrated emulsion past a deformable drop or bubble Another problem [25] which can be studied by large-scale numerical simulations is the steady gravity-induced motion of a deformable bubble (or drop still called a 'bubble' in what follows) through a concentrated emulsion at low Reynolds numbers. When the bubble is much larger than the emulsion drops, the emulsion can be treated, in principle, as an effective medium with the macroscopic boundary conditions (no flux, continuity of velocity and stress) on the bubble surface. For a Newtonian form of the effective stress tensor and a spherical bubble, the classical solution of Hadamard and Rybchinski [75] shows that no deformation will occur, irrespective of the nonzero surface tension value; surface tension, however, is required for the stability of the drop. Unfortunately, a Newtonian form of the effective stress tensor is not an accurate approximation for concentrated emulsions of deformable drops (Sec. 6.1), and a more complicated constitutive equation valid for arbitrary kinematics would be required, which is problematic and could be done, at present, only in an ad hoc manner. For a bubble comparable in size with the emulsion drops, the problem can be studied instead by rigorous, first-principle numerical simulations, without a constitutive equation. A large bubble of non-deformed radius a& and surface tension G\, is placed in a cubic box, together with N 2> 1 drops of non-deformed radius a^ forming a random emulsion. The whole system is continued triply periodically into the whole space. The emulsion drops are deformable (with surface tension aa) and neutrally buoyant, so that the

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motion in the system is entirely due to density difference pb — pe between the bubble and the surrounding medium. The bubble and the emulsion drops are assumed to start with spherical shapes. The main quantity of interest is the settling bubble velocity (in the reference frame where the whole system is at rest) as a function of emulsion concentration c, bubble Bond number Bb = \pb- Pe\alg/ab

,

(46)

bubble-to-emulsion-drop size ratio ab/ad and surface tension ratio o^/orf, bubble-to-medium-viscosity ratio A& = w,//i e , and emulsion-drop viscosity ratio Xd = p-d/p-e- The limit N —> oo must be taken to approach the solution for a single bubble in an unbounded medium. The solid line in Fig. 39a shows the non-dimensional settling bubble velocity Ub/U0 (where

Fig. 39. The non-dimensional settling velocity of a bubble through a concentrated emulsion at c = 0.45, ab/ad = 2, \b = 0.05, ab/ad = 2, and \d = 1, with (a) N = 800, (b) N = 200. The bubble and each emulsion drop are discretized by 2160 and 1280 elements, respectively. The horizontal lines are the stationary levels for Bb = 4. Reproduced by permission from Ref. [25].

Uo is the isolated bubble velocity, in the absence of emulsion drops) in a 45% concentrated emulsion for N = 800 and Bb — 4; time is scaled with Me/(|/°6 — Pe\ga>b)- The initial decline in Ub/U0 is due to emulsion-drop interactions increasing the effective viscosity around the bubble; a statistical steady state is reached at t ~ 50. Despite a relatively large Bb = 4, the bubble deformation was found to be small (although an emulsion at c = 0.45 and Ad = 1 is noticeably non-Newtonian, see Sec. 6.1). In contrast, the emulsion drops experience large deformations in the vicinity of the bubble, with typical oblate and prolate shapes upstream and downstream, respectively (Fig. 40); the other drops (away from the bubble) are only slightly

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Fig. 40. A typical snapshot of emulsion drops in the vicinity of the bubble, from the simulation in Fig. 39 with N = 800 and -B& = 4 at t = 31. The bubble rises upward. Reproduced by permission from Ref. [25].

deformed, but they create a necessary background for the simulation. A similar simulation with N = 200 (solid curve in Fig. 39b) shows strong sensitivity of the initial velocity Ub/Uo (at t = 0) to N (0.68 for iV = 200 vs. 0.80 for N = 800; the limit must be 1 at N —>• oo). This dependence on N merely reflects strong artifact interactions between single bubbles in different periodic cells at t = 0, since strictly spherical emulsion drops with \d — 1 have no effect at all. Fortunately, at finite 5&, the necessary steady-state results for Ub/U0 are less sensitive to N (0.578 at N = 200 vs. 0.619 at N — 800), and thus the limit N —>• oo can be practically achieved in this problem. Similar calculations at Bb = 2 (dashed lines in Fig. 39a,b) give slightly worse convergence of the steady-state results (0.566 for N = 800 vs. 0.521 for N = 200). It appears that the screening effect of emulsion-drop deformation weakens as Bb is decreased, and convergent steady-state simulations for smaller Bond number may require N > O(103); the same is obviously true for larger size ratios ab/ad- Nevertheless, our simulations demonstrate the possibility of solving the problem without an ad hoc constitutive equation for an emulsion.

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7. SOME PROSPECTS FOR FUTURE RESEARCH In this chapter, we have demonstrated some contemporary progress made in the area of deformable drop interactions and coalescence, and concentrated emulsion flows by rigorous theories and first-principle numerical simulations. Still, there remains a large number of issues to be resolved, and a large room for further development. The hydrodynamical theory of coalescence (Sec. 2), based on matching the thin-film 'inner' solution with the 'outer' solution was shown to be a correct approach for arbitrary 3D interaction of slightly deformable drops. Applications to coalescence efficiency calculations, however, have been made so far [20, 21] only for classic, unretarded van der Waals attractions. To make this theory more realistic, additional account for electromagnetic retardation and other colloidal forces may be necessary. Besides, real emulsions are rarely uncontaminated, and it would be practically important to include surfactant effects on interaction and coalescence, which opens a wide area of research; some initial studies have already been made [76]. For multidrop interactions, it appears possible to further improve multipole-accelerated boundary-integral algorithms and make them suitable for accurate manythousand drop simulations with large deformations, with more powerful computer resources available in the near future. For solid particle simulations, the lattice-Boltzmann (LB) method was found to be a very promising tool [77-79]. It would be useful to explore if LB can become a competitive method for deformable drops, in terms of accuracy and speed. When fluid inertia cannot be neglected (which is often the case for large drops), the problems become much more involved; a front-tracking method [80] appears to be a suitable tool for deformable drops, although much remains to be done to improve the accuracy of this method. In the area of computational rheology of concentrated emulsions (Sec. 6.1), generalization for time-dependent shear flows would be straightforward; a major challenge, however, is to develop a simulation method suitable for an arbitrary history of macroscopic deformation (different from shear flow).

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REFERENCES [1] E. Rushton and G.A. Davis, Appl. Sci. Res., 28 (1973) 37. [2] S. Haber, G. Hetsroni and A. Solan, Int. J. Multiphase Flow, 1 (1973) 57. [3] A.Z. Zinchenko, Prikl. Mat. Mekhan., 44 (1981) 30. [4] A.Z. Zinchenko, Prikl. Mat. Mekhan., 46 (1983) 58. [5] A.Z. Zinchenko, Prik. Mat. Mekhan., 47 (1984) 37. [6] Y.O. Fuentes, S. Kim and D.J. Jeffrey, Phys. Fluids, 31 (1988) 2445. [7] Y.O. Fuentes, S. Kim and D.J. Jeffrey, Phys. Fluids A, 1 (1989) 61. [8] H. Wang, A.Z. Zinchenko and R.H. Davis, J. Fluid Mech., 265 (1994) 161. [9] A.Z. Zinchenko, Prikl. Mat. Mekhan., 48 (1984) 198. [10] A.Z. Zinchenko and R.H. Davis, J. Fluid Mech., 280 (1994) 119. [11] A.Z. Zinchenko and R.H. Davis, Phys. Fluids, 7 (1995) 2310. [12] G. Mo and A.S. Sangani, Phys. Fluids, 6 (1994), 1637. [13] A.Z. Zinchenko, Prikl. Mat. Mekhan., 42 (1979) 1046. [14] A.Z. Zinchenko, Prikl. Mat. Mekhan., 45 (1982) 564. [15] R.H. Davis, J.A. Schonberg and J.M. Rallison, Phys. Fluids A, 1 (1989) 77. [16] M.D. Cooley and M.E. O'Neill, Mathematika, 16 (1969) 37. [17] S.G. Yiantsios and R.H. Davis, J. Fluid Mech., 217 (1990) 547. [18] S.G. Yiantsios and R.H. Davis, J. Colloid Interface Sci., 144 (1991) 412. [19] A.Z. Zinchenko and R.H. Davis, J. Comput. Phys. (2004, submitted). [20] M.A. Rother, A.Z. Zinchenko and R.H. Davis, J. Fluid Mech., 346 (1997) 117. [21] M.A. Rother and R.H. Davis, Phys. Fluids, 13 (2001) 1178. [22] J.M. Rallison and A. Acrivos, J. Fluid Mech., 89 (1978) 191. [23] A.Z. Zinchenko and R.H. Davis, J. Comput. Phys., 157 (2000) 539. [24] A.Z. Zinchenko and R.H. Davis, J. Fluid Mech., 455 (2002) 21. [25] A.Z. Zinchenko and R.H. Davis, Phil. Trans R. Soc. Lond. A, 361 (2003) 813. [26] A.Z. Zinchenko, M.A. Rother and R.H. Davis, J. Fluid Mech., 391 (1999) 249. [27] M. Manga and H.A. Stone, J. Fluid Mech., 256 (1993) 647. [28] M. Manga and H.A. Stone, J. Fluid Mech., 300 (1995) 231. [29] D.L. Koch and E.S.G. Shaqfeh, J. Fluid Mech., 209 (1989) 521. [30] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Nijhoff, Dordrecht, 1973. [31] L.D. Reed and F.A. Morrison, Int. J. Multiphase Flow, 1 (1974) 573. [32] L.M. Hocking, J. Engng Maths, 7 (1973) 207. [33] A.F. Jones and S.D.R. Wilson, J. Fluid Mech., 87 (1978) 263. [34] I. B. Bazhlekov, A. K. Chesters and F. N. van de Vosse, Int. J. Multiphase Flow, 26 (2000) 445. [35] X. Zhang and R.H. Davis, J. Fluid Mech., 230 (1991) 479. [36] S.R. Reddy, D.H. Melik and H.S. Fogler, J. Colloid Interface Sci., 82(1981) 116. [37] J.R. Rogers and R.H. Davis, Metall. Trans., 21A (1990) 59. [38] H. Wang and R.H. Davis, J. Fluid Mech., 295 (1995) 247. [39] K. Barton and R. Subramanian, J. Colloid Interface Sci., 133 (1989) 211. [40] V. Smoluchowski, Z. Phys. Chem. (Leipzig, 92 (1917) 129. [41] M. Loewenberg and E.J. Hinch, J. Fluid Mech., 338 (1997) 299. [42] A.Z. Zinchenko, M.A. Rother and R.H. Davis, Phys. Fluids, 9 (1997) 1493. [43] M. Loewenberg and E.J. Hinch, J. Fluid Mech., 321 (1996) 395.

Hydrodynamical Interaction of Deformable Drops

447

[44] M. Loewenberg, Trans. ASME: J. Fluids Engng, 120 (1998) 824. [45] S. Kim and S. Karilla, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann, 1991. [46] C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous flow, Cambridge University Press, 1992. [47] G.P. Muldowney and J.J.L. Higdon, J. Fluid Mech., 298 (1995) 167. [48] V. Cristini, Blawzdziewicz and M. Loewenberg, J. Comput. Phys., 168 (2001) 445. [49] J.M. Rallison, J. Fluid Mech., 109 (1981) 465. [50] J.-T. Jeong and H.K. Moffatt, J. Fluid Mech., 241 (1992) 1. [51] J.P. Hansen and I.R. McDonald, Theory of Simple Liquids, Academic Press, 1976. [52] H. Hasimoto, J. Fluid Mech., 5 (1959) 317. [53] H. Yang, C.C. Park, Y.T. Hu and L.G. Leal, Phys. Fluids, 13 (2001) 1087. [54] M. B. Nemer, X.Chen, D. H. Papadopoulos, J. Blawzdziewicz and M. Loewenberg, Phys. Rev. Lett. 92 (2004) 114501. [55] R.A. de Bruijn, Chem. Engng Sci., 48 (1993) 277. [56] H.A. Stone, Ann. Rev. Fluid Mech, 26 (1994) 65. [57] W.J. Milliken, H.A. Stone and L.G. Leal, Phys. Fluids A, 5 (1993) 69. [58] J.D. Sherwood, J. Fluid Mech, 144 (1984) 281. [59] J. Eggers, Phys. Rev. Lett, 71 (1993) 3458. [60] J. Eggers, Rev. Mod. Phys, 69 (1997), 865. [61] M.P. Brenner, J.R. Lister and H.A. Stone, Phys. Fluids, 8 (1996) 2827. [62] J.R. Lister and H.A. Stone, Phys. Fluids, 10 (1998) 2758. [63] R. H. Davis, Phys. Fluids, 11 (1999) 1016. [64] J. Kushner IV, M.A. Rother and R.H. Davis, J. Fluid Mech, 446 (2001) 253. [65] V. Cristini, J. Blawzdziewicz and M. Loewenberg, Phys. Fluids, 10 1781. [66] P.K. MacKeown, Stochastic Simulation in Physics, Springer, 1997. [67] C. Pozrikidis, J. Comput. Phys, 169 (2001) 250. [68] G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGrawHill, 1974. [69] M.R. Kennedy, C. Pozrikidis and R. Skalak, Comput. Fluids, 23 (1994) 251. [70] R.S. Rivlin and J.L. Ericksen, Arch. Rat. Mech. Anal, 4 (1955) 323. [71] N. A. Frankel and A. Acrivos, J. Fluid Mech, 44 (1970) 65. [72] D. Barthes-Biesel and A. Acrivos, Int. J. Multiphase Flow, 1 (1973) 1. [73] M.B. Mackaplow and E.S.G. Shaqfeh, J. Fluid Mech, 376 (1998) 1493 [74] J.E. Butler and E.S.G. Shaqfeh, J. Fluid Mech, 468 (2002) 205. [75] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967. [76] A. K. Chesters and I. B. Bazhlekov, J. Colloid Interface Sci, 230 (2000) 229. [77] A.J.C. Ladd, J. Fluid Mech, 271 (1994) 285. [78] A.J.C. Ladd, J. Fluid Mech, 271 (1994) 311. [79] A.J.C. Ladd, Phys. Fluids, 9 (1997) 491. [80] S. O. Unverdi and G. Tryggvason, J. Comput. Phys, 100 (1992) 25.

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Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 11

The role of inertial effects and conical flows in breakup of liquid threads Vakhtang Putkaradze Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131-1141, USA email: [email protected]

We study the appearance and relevance of conical flows in the problem of droplet breakup. First, we give the theory of break-up of a slender jet of fluid in air [1]. We then proceed to the two-fluid break-up when a thread of one fluid is rupturing up while being surrounded by another fluid. The two-fluid break-up shows the appearance of conical shapes [2]. Inspired by these experiments, we develop a theory of exact solutions of Navier-Stokes equations which describe the flow of two fluids fluid separated by an interface in the shape of a single or double cone. We also describe the extension of these solutions to two dimensions. 1. INTRODUCTION When observing a dripping faucet in the kitchen, one can notice that the process of forming an individual droplet can be separated into two parts. The first part is the slow accumulation of water in the end of the faucet so that the gravitational force pulling the droplet down can overcome the surface tension holding the droplet inside the faucet. Once enough mass is accumulated, the droplet falls from the faucet with ever-increasing speed. The "slow" phase finishes when substantial volume of the droplet appears from the faucet; from that time on, the surface tension's role becomes destructive, snapping the liquid thread extending from the faucet to the droplet.

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The process of snapping the liquid thread and forming the droplet is very fast, so inertial effects are important for understanding this process. Formation of droplets and the role of inertial effects when air is the surrounding substance is understood quite well. Less well understood is the role of inertial effects when a thread of one fluid snaps while being surrounded by another fluid. The major difficulty of describing the fluid motion at the very moment of droplet formation lies in the fact that at this point in time and space Navier-Stokes equations exhibit a singularity. This singularity is physical, as at the moment when the droplet is born, the thread becomes infinitely thin in finite time. An amazingly beautiful and complex dynamic during droplet break-up has been revealed in great detail in [3]. We shall start by giving the reader an introduction into the theory of droplet formation in air, where a rigorous theory balancing surface tension, viscosity and inertia can be derived. 2. 2.1.

DROPLET FORMATION IN AIR Derivation of equations of motion

In this section, we follow the derivation of equations for fluid break-up in [1] without getting into too many technical details. The analysis of this problem has been mainly completed, and we will concentrate on giving the reader enough information to see the difference between this case and the thread break-up in case the ambient fluid is present. The reader is advised to refer to the original article in order to fill in the gaps in the derivation. Let us start with the derivation of the mass conservation at the point of breakup. We assume that the fluid is incompressible and the motion has radial symmetry with the axis of symmetry being z. By assumption, the velocity has only two components: vr, at the direction of the polar radius r (perpendicular to the thread's axis) and vz at the direction of z (along the thread's axis). To make further progress, let us identify the small parameter e inherent for this problem as the ratio of the typical scales in r and z direction. Thus, we re-scale r as r —> er, whereas the z coordinate remains unchanged: z —> z. Let us therefore assume the following expansion for the z component of the velocity: vz = v0(z,t)

+ e2r2v2{z,t)

+ ...

(1)

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

451

The incompressibility requires that WV

Id ~ rdr ^Vr'

+

dvz ~dz ~

'

^

which in turn yields an expansion for the r component of the velocity: Id

loot?

Suppose the equation for the thread is F(r, z, t) = h(z, t) — r = 0. The kinematic condition on the free surface requires that molecules which are on the surface remain on the surface, or DF/Dt = 0, where D/Dt is a full (material) derivative. This leads to

dh dt

dh dz

^7 + vz-

vr \z=h = 0

(4)

Integrating (2) with respect to rdr from r — 0 to r = h(z, t) and using (4) gives the equation of mass conservation -7T- + w (voh2) = 0.

(5)

Equation (5) is easy to understand, as the area of the radially symmetric thread at point z is given by S(z,t) = nh(z,t)2, so (5) describes simply the advection of volume of a radially-symmetric fluid element. The velocity of fluid VQ is still unknown. To complete the system, we must consider the z-component of Navier-Stokes equation, which connects VQ and V2- The full Navier-Stokes for three-dimensional velocity v in vector form reads: dv 1 — + (v • V)v = — V p + i/Av,

(6)

with p being pressure and v being kinematic viscosity. Let us consider the z-component of this equation and do appropriate rescaling of spatial coordinates (r —> er , z —> z), and velocities (1,3). Keeping only the terms of the lowest order in e we obtain dvo_ dt

dvo__ _ldp dz pdz

(. \

dhjo\ dz2 J

where p is pressure and v is kinematic viscosity. To connect pressure with the velocity components, we utilize the boundary conditions at the free

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surface. The normal dynamic boundary condition (vanishing of the stress normal to the surface) gives p = 7/c, where 7 is the surface tension coefficient and K is local curvature of the surface. Finally, the tangential dynamic boundary condition (vanishing of the tangential stress) yields the connection between VQ and v
3 dv0 dh

.

2

4 dz 2h dz dz Combining all these equations in one system, we obtain dv0 at

dv0 dz

7 d , , . ~v d fdv0 pdz h2dz \oz

f+!(**')=0.

2\

J

do.

For a thin slender thread the curvature is approximated as K = 1/h. However, it was demonstrated that because of numerical stability and other reasons, it is advantageous to keep higher order terms in curvature [4, 5]. The full expansion for curvature is 1 K = -

1 .

hy/l + e2h2z

2.2.

hzz

2

—g —.

7 1 + 62^

(11)

Analysis of t h e equations

We shall refer to (9,10) as Cosserat equations, which were first obtained by Green in 1976, with an inviscid version given by Lee in 1974, see [1] for references and history of these equations. (Technically speaking, equations (9,10) are only leading-order approximations to the Cosserat equations. However, we shall call them Cosserat equations for simplicity). The inviscid equations can approximate the solution quite well for some time and have interesting mathematical properties, such as linearization under Hodograph transformation and infinite hierarchy of conservation laws, see [6, 7] and references therein. To make quantitative analysis of the Cosserat equations, let us first introduce a length scale, constructed out of the physical parameters of the problem lv = ^ 2 p/7, and time scale tv = uzp2/j2. The ratio of a typical length scale L to the viscous length scale /„, F = L/lv, determines how important the viscosity is on different regimes. Numerical simulations of equations (9,10) reveal that there is a singularity forming in finite time. The exact timing of singularity, to, depends

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

453

on the initial conditions, but the structure of the solution on approach to the singularity has a universal self-similar pattern which we will briefly outline here [1]. The behavior of solutions observed in simulations and experiments is quite different for small and large values of F, and we will describe the difference below. Of course, as we have mentioned earlier, the viscosity is important no matter how small it is, but the effect of viscosity for a very large F shows only very close to the singularity. For large viscosities, F < 1 or even F ~ 1, the equations show excellent agreement with the experiment. Not only do the equations capture the results of experiments qualitatively, but excellent agreement in quantitative details is also achieved. Equations (9,10) correctly predict the length of the thin necking region and the shape and size of the drop [4]. These equations also show that the break-up of liquid threads close to the singularity is well approximated by a universal self-similar solution. To see how the self-similar scenario comes about mathematically, we suggest that, close to the singularity, all dimensional parameters of the problem may enter the solution only through the time and length scales tv and lu. Thus, we assume the following self-similar ansatz for radius and velocity describing the evolution of the thread close to the point of singularity z$ and timing of singularity to (here we use t' = (to — t)/tv and z! = (z-zo)/lv):

h{z,t) = t'{ri) vo(z,t) = (tT1/2Hv) v

(12)

= (t')-V2z'

We shall only consider the solution for t < to (before breakup). The selfsimilar functions 4>(r]), ip{rj) satisfy

I»

+

#') + W = |

+

3 < ^

i ( * 2 + ^(*2)') + ( « 2 ) ' = o

(13) (I")

To specify the boundary conditions on the solutions of (13,14), let us notice that far away from the singularity in non-rescaled variables, both height and velocity should be finite. Non-rescaled variables z',t', which are close to singularity, correspond to very large values of 77. This is referred to as matching of inner solution (12) and outer solution, describing the evolution of fluid far from the singularity. The inner and outer solution must match,

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Vakhtang Putkaradze

in particular, both the height and velocity of the inner solution should remain finite. Technically speaking, it is an assumption that both height and velocity should tend to a constant value (as this constant value can be zero). However, this assumption is well-justified by extensive comparison between the theory, numerical simulation and experiment. Thus, we posit (v) -> A±V2,

77->±oo,

n

rx

The boundary conditions (47) close the system and allow one to find a solution of (13,14). Since the conditions on the solution are asymptotical, the solution is not unique and additional solutions satisfying (13,14) and boundary conditions (15) were found [8]. The universal law of scaling was later studied in more details in [9]. Notice that the inner solution satisfying (15) grows quadratically away from the singularity. Let us now investigate how the inner solution described by the self-similar ansatz merges with the outer solution. Close to the singularity, when the length scale of the self-similar solution becomes small, we can see that when 77 —> ±00, | ^ | ^ . /

W

±

(

z

_

2

o

)

,

(16)

so the outer solution must be quadratic close to the singularity. More precisely, Viter-^-^o)2-

(17)

Thus, the outer solution for the neck's width h(z, t) consists of two parabolas with (very) different curvatures, joined smoothly at the singularity. It is interesting that in this particular case the microscopic processes driving the inner solution specify the asymptotics of the outer solution (17), whereas in most standard cases, it is the large-scale processes at the outer solution which are driving the asymptotic of the inner solution. The process we have described is universal and is observed in any fluid rupturing in air, provided that the process of rupture does not develop an instability. However, for very small viscosities, this universal self-similar process of rupture can be hard to observe, as the viscous length scale lv can become extremely small. On the scales much larger that lu, the outer solution forms conical shapes. To describe the flow of fluid in conical geometry, we shall later introduce a self-similar ansatz separating radial and

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

455

angular variables. While that description lacks the power of Cosserat equations describing the dynamics of rupture, it will introduce the contribution of nonlinear terms in a consistent way. However, before we do that, we will describe another set of experiments showing rupture of a thread of one fluid inside another fluid, when the appearance of conical interfaces is more pronounced.

3. 3.1.

PHYSICAL EXPERIMENTS SHOWING CONICAL INTERFACES Lava lamps: break-up of a thread of one fluid in another fluid

We see that in the case of a liquid bridge rupturing in air close to the singularity, the slender-body approximation works excellently. We shall now turn our attention to the case when a liquid bridge of one fluid is rupturing while being surrounded by another fluid, which is immiscible with the first. A very beautiful demonstration of this process, familiar to everybody, is the lava lamp. In the experiment (or the lava lamp), a lighter fluid is slowly rising through heavier highly viscous fluid. At some point, the rising bubble of lighter fluid separates so far away from the bulk of it that a neck is formed, which is then ruptured by the surface tension. Watching a lava lamp, we notice that the process of forming a neck is extremely slow, yet the process of rupturing is extremely fast, even when the viscosity of surrounding fluid is high. This makes recording the exact moment of break-up challenging; but these difficulties were successfully resolved in [2]. The photographs taken at the very moment of break-up show the formation of two cones with a common tip and axis of symmetry. Let us suppose that fluid i, i = 1,2 has kinematic viscosity ?].;. The angle of the cones depends on the ratio of viscosities A = v\jvi) but is practically independent of other parameters of the experiment. However, a recent study [10] showed that the break-down of a droplet is not self-similar for certain fluids (water and silicone oil) and remembers the initial as well as boundary conditions. Careful experiments and numerical simulations showed formation of exceptionally thin and slender threads of fluid (with the diameter of the order of several nano-meters with length up to a few millimeters) bridging the gap between the two conical sections drifting apart.

456

Vakhtang Putkaradze

The theory of the phenomenon of two-cone formation was developed by [2] and [11] (see also [12] for the numerical analysis of the process of break-up of a thin liquid thread). The theory and simulations are based on assuming Stokes' flow (low Reynolds number) on both inner and outer fluid. Based on the integral re-formulation of the motion of the interface, the theory provides a dynamic way of predicting the interface position, along with velocity field in space. The formation of two cones at the instant of the drop pinch-off is confirmed by the numerical simulation. The cones' angles also compare favorably with experimental data. In what follows, we shall develop an exact solution of two fluids separated by an interface in the shape of a cone (or two cones) without assuming that Reynolds number is small. We are motivated by the following argument: if the shape of the interface is exactly conical very close to the origin, there is no typical length associated with it, and, as a consequence, no Reynolds number can be defined. We show that it is possible to find exactly self-similar (in space) stationary solutions respecting the conical geometry and taking into account both linear and nonlinear terms. However, in searching for the stationary solutions of this type we sacrifice the dynamic description of the interface afforded by the Stokes flow. Thus, our solutions should be considered as supplementary to those found in [2] and [11]. In our development of exact conical solution, we shall first start with the case when only a single conical interface was present. While we show that mathematically, no exact solution of this problem satisfying the requirements of regularity at the cone's axis and all boundary conditions at the interface can be found, such flows are also of interest as the interface in a shape of a single cone is observed in the experiment of two-fluid selective withdrawal [13, 14]. In the selective withdrawal experiment, a lighter fluid (oil) is withdrawn through a tube which is positioned at a certain elevation from the interface between the lighter fluid and heavier fluid (water). The interface between two fluids is deformed by the flow. For small values of withdrawal rate, only the lighter fluid enters the tube; whereas for high enough values of withdrawal rate, both fluids are withdrawn. At some critical value of the flux, stationary flows are observed for which the interface assumes the shape of an almost perfect cone. The rounding of the cone tip is much smaller than the capillary length, and the appearance of such singular conical shapes makes the phenomenon interesting for our studies. It has also been demonstrated that it is possible to use the thin slender

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

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threads observed close to the bifurcation in two-fluid withdrawal regime to coat microscopical particles used in medical applications [15]. The theory of this phenomenon in the regime of two-fluid withdrawal based on the assumption of slender thread entering the tube is also in preparation [16]. 4. 4.1.

FORMAL DESCRIPTION OF CONICAL FLOWS: SELFSIMILAR SOLUTIONS General considerations

Motivated by appearance of conical interfaces in the two-fluid flow experiments, we shall derive exact solutions of Navier-Stokes equations describing the flow of two immiscible fluids, with the interface separating the fluids being a cone. Solutions of this type have velocities growing without bound at the conical tip as 1/r, where r is the spherical radius from the tip of the cone. Therefore, stress tensor components diverge as 1/r3 at the tip of the cone. Since the curvature of the cone is growing as 1/r at the tip, forces induced by surface tension diverge as 1/r2, i.e., slower than viscous forces. Thus, this solution persists close to the tip of the cone even if the surface tension is introduced. Note that far way from the tip the conical interface will be deformed by surface tension (and gravity in the case of density difference). If r = 0 were the only singularity of the flow, such two-fluid conical solution would be very interesting, not only as a model of what happens in selective withdrawal close to the tip of the cone, but also as an example of a physically realizable solution with finite energy exhibiting singularity in space. Indeed, if the singularity would be present only at r = 0, one could in principle set up an experimental apparatus with prescribed flow at the boundaries, say r = R leading to the singular solution. Unfortunately, as we show below, our exact solution must exhibit infinite velocity not only at the tip of the cone, but also at the all of the cone's axis. We show that the solution without singularities at the axis can only be achieved with unphysical fluids having negative viscosity or density ratio. The singularity at the axis is typical of the three-dimensional solutions of single fluid of this type. A swirling motion can be successfully incorporated into this ansatz as well; however, we shall not use it here, since the formulas become extremely complex and analytical treatment of two-fluid flow is almost impossible. The situation is similar for the case of two-fluid two-cone flows. We

458

Vakhtang Putkaradze

assume that the interface is located at two positions 9 = B\ and 9 = 02, which describes two coaxial conical surfaces. In this case, it is still impossible to achieve a solution with no singularity at the axis 9 = O,vr. Nevertheless, the two-cone flows are still of interest and we shall outline their structure as. The flows described in this section are natural extensions of the singlefluid flows bounded by conical interfaces, which are a class of analytical solutions of Navier-Stokes equations [17] - [21]. One of the important features of the conical flows of this type is that a singularity on the axis of the cone is ubiquitous. Much work has been devoted to the explanation of this singularity. In particular, it was recently proved rigorously [22] that specifying either tangential stress or pressure on the interface makes the regular solution unique. However, it seems to be impossible to satisfy all boundary conditions and still obtain a regular solution. No such result was available for the case of two-fluid flow until now. We will also show that it is possible to derive an exact two-fluid conical solution of Navier-Stokes equation which does not have a singularity on the axis if one of the boundary conditions on the interface is violated. For example, a solution can be achieved if a slip boundary condition on the interface between two fluids is assumed, and we will describe such a solution below for the case of two cones. A solution for a single cone is possible as well, but will not presented here, as it is a simple technical extension of the two-cone case. Instead, we shall choose a more pedagogical approach. First, we discuss the impossibility of a regular solution for the single-cone case satisfying all the regularity requirements. Then, we show that if a slip condition on the interface is assumed, a two-cone solution is possible. We shall just mention that a single-cone regular solution with a slip on the interface is possible (the proof of this fact is relatively easy), but a twofluid conical solution satisfying all the regularity requirements is impossible (which is rather tedious to show) [23]. We shall begin with a derivation of equations, common for both single - and two-cone solutions. 4.2.

Derivation of equations and boundary conditions

We begin by introducing the spherical system of coordinates (r, 9, 0), with the origin being at the cone's tip, and 9 = 0, IT axis aligned with the axis of the cone. Two fluids denoted by a subscript j — 1,2 are separated by

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

459

the interface at 6 — #*. Let us assume that fluid with subscript 1 occupies the area above the interface, i.e., fi < /z*, and fluid 2 is in the area /z > /z*. We assume the velocities of each fluid using the ansatz vr, = VjFjW/r

vej = VjfjW/r

v^ = T^/r,

(18)

where Uj are kinematic viscosities of each fluid and Fj, fj are dimensionless. In this section, we shall put Tj = 0 (no swirling), greatly simplifying the formulas. This ansatz leads to a first order equation of Riccati type, first obtained by Slezkin [24]. We shall repeat the derivation of the Slezkin equation briefly, since the presence of the interface modifies the final result and requires the expressions for stress tensor components in order to match them at the boundary. Incompressibility condition 1 9 , , 1 d . . . x r2Qr

\

>J> rsmQdO

J

leads to a relationship between F.j{9) and fj(6):

which can be simplified introducing the new coordinate /J, = cos 9 and

The incompressibility condition is then simply

If we now define pressure in each fluid as 2

pj(r,n) = ?i£-Pi(li),

(20)

from the Navier-Stokes equations (^-component) we obtain the pressure in the form

Pl

= f-\JLd[i

2 1 - [il

+ Aj,

(21)

where Aj is an integration constant. Substituting (20, 21) in the r-component of Navier-Stokes equations and integrating twice, we obtain the full version of the Slezkin equation for g(fi): (1 - /z 2 )^ + 1

m

+l-g]= -AjU2 + 2BjH + Cr

(22)

460

Vakhtang Putkaradze

Two new integration constants Bj,Cj are introduced in addition to Aj from (21). To solve (22) in each fluid, we have to specify g.j at the interface for each fluid, as well as three unknown constants Aj, Bj, Cj, which gives a total of eight unknowns. In addition, the position of the interface /z* is unknown. In order to complete the system, we have to find the corresponding number of boundary conditions. The kinematic boundary conditions are the continuity of the r and 9 components of the velocity. Moreover, since the flow is stationary, vgj must vanish at the interface 9 — 6>*, or ji — /z*. Thus, the kinematic boundary conditions on the 9 components of velocity are
(23)

The continuity of the r-components of velocity gives Vl(M*) = 02 (/•*»)> which, in view of (22) and (23), can be written as

A(-A lt 4 + 2Blfi* + d) - {-A2/4 + 252/u* + C2).

(24)

Here, we introduced the viscosity ratio A = V\jv2. Let us now compute the dynamic boundary conditions, which are the continuity of tangential arg and normal ogg components of stress tensors. Using the velocity ansatz (18), expression for the pressure (21), equation for g (22) and vanishing of g on the interface (23) after some algebra leads to the following answer: _

P]v)

/ - 2 % i . + 2£?A

a

«* - ~ ^


\—I-M2

2 P

-^{-g'^)

) ' + A3).

(

}

(26)

If we introduce 5 — {viIv%){p\/Pi), the dynamic boundary conditions at the interface become 8(2Axn* - 2Bi) = 2A2li, - 2B2

(27)

for the rB component and

KJM

~ M) = 92(»*) ~ M

for the 99 component. Again, the 90 stress equation can be simplified using (22), becoming 5(Ai - 2J3i^ - Cx) = A2-

2B2[i* - C2.

(28)

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

4.3.

461

Single cone: regularity requirements and singular solutions

Equations (22) together with the boundary conditions (23, 24, 27, 28) specify five equations for nine unknowns. Additional equations can be obtained if certain conditions on regularity of solution are imposed at /j, = ± 1 , the axis of symmetry of the cone. It is clear that vg must be zero on the axis [i = ± 1 . In addition, we require that vr is bounded when \i —> ± 1 . Thus, we infer regularity conditions on g.y. gi{n) -» 0

g[(fi) bounded

/j, ->• - 1

(29)

g2(/i) -> 0

g'2(n) bounded

/i-»l.

(30)

We shall now show that enforcement of these regularity conditions leads to the unphysical value for the interface position //», and so regular solution to this problem cannot exist. From Slezkin's equation, regularity conditions on g\ (29) is achieved only if the right-hand side of (22) is proportional to (1 + //) 2 , i.e., Bx = -A,

d = -A,.

(31)

Similarly, regularity conditions on g2 require that B2 = A2

C2 = -A2.

(32)

Then boundary conditions (24) and (28) become \Ax{\ + ii\) = A2{\-[il)

(33)

<JA1(l + //,) = A 2 ( ^ - l ) ,

(34)

and boundary condition (27) becomes identical with (28). Dividing (34) by (33), we find the following expression for the position of the interface

/x* =

T

r.

(35)

0 — A

Clearly, for A > 0 and 5 > 0 we have |/i*| > 1. But since /i* = cos#*, I/i* I < 1 must be satisfied. This shows that we cannot find a solution to a single-cone problem satisfying regularity requirements at /x = ± 1 . On the other hand, if we impose weaker regularity conditions, removing the requirement on the boundedness of g'j(fi) at ji = ± 1 , we can now find

462

Vakhtang Putkaradze

Figure 1: An example of single-cone solution with g'(fi) having logarithmic singularity on the axis /j, = ±1.

a solution. Instead of four conditions imposed by (31) and (32) we have just two conditions: at \i = 1. -Ax

+ 2Si + Cx = 0.

and at /i = — 1. - 4 2 - 2 5 2 + C2 - 0. Since we have dropped the number of constraints by two, we can choose the value of the interface angle /z* arbitrarily. We see that a solution is possible for any /i*, and is parameterized by two additional parameters Ai and ^42. Both vr and v$ components diverge as 1/r as r —> 0, and vr diverges when //—>±1, or#—>-0,7r. The total energy is finite in any ball of finite radius. Thus, we see that it is impossible to have a stationary solution with exact conical singularity at the origin, while remaining regular everywhere else. However, we can still interpret the solution described in this section as the outer solution of the problem, which requires regularization close to the origin, presumably at length scale less than lv. The regularization can also smoothen out the apparent singularity at the symmetry axis \x = ± 1 . Let us note that there is no condition on the critical angle /i*. This, perhaps, is physical as the inner solution at the origin determines this angle. An example of solution g{\i) with the singularity ± 1 is shown on Fig. 1. Another way to obtain a meaningful solution is to drop one of the boundary conditions at the interface. We shall derive equivalent solutions for the

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

463

two-cone case under the assumption that the continuity of tangential velocity is violated. The single-cone case can be solved in a similar fashion. In this case, the regularization in the form of a thin shear layer at the interface is required.

4.4.

Two cones: Regularity requirements and singular solutions

The theory of self-similar conical solution can also be applied to the case when the interface consists of two cones having a common axis. In this case, the interface is positioned at \i = ji\ and [i = ^2, so we must consider three domains. Fluid 1 is at —1 < // < n\ (domain 1) and \ii < \i < 1 (domain 3), whereas fluid 2 is ii\ < {i < ^ (domain 2). In each domain, equation (22) holds. As we mentioned before, a regular solution satisfying all the boundary conditions at the interface is impossible. In our further study of twocone flows, we shall assume that the fluids can slip at the interface. The continuity of velocity does not have to be enforced then. Being forced to drop one of the boundary condition, we have decided to admit the discontinuity of tangential velocity at the interface as it is perhaps the least dangerous sacrifice, at least from the physical point of view. First, one can assume that the addition of certain surfactants can allow the fluids to slip. Second, even if the true boundary condition is no-slip, we can still imagine a thin boundary layer between the fluids which accommodates the difference of tangential velocities. In this boundary layer, the flow is not self-similar, but outside this thin boundary layer the flow is well described by our self-similar ansatz. The problem of the flow of a plane jet consisting of two immiscible fluids was considered recently by Herczynski et al [25], and an extension of this work to conical geometry is possible. On the other hand, if we drop continuity in tangential or normal stress at the interface, we see no physical way to explain that a finite force is acting on an infinitely thin (or even finite, but very thin) layer of fluid adjacent to the interface. Thus, we shall assume that the continuity of tangential velocity is violated and proceed with the calculation. Notice that fluid 2 can now have singularities at the axis \i = ±1, as it does not extend there. Thus, for fluid 2 there is no condition on A^^B^ and Ci- Let us first seek solutions which are regular at /i = ± 1 . In domains 1 and 3, having regularity condition at the axis requires that Bx = -Ax

Ci = - A i

(36)

464

Vakhtang Putkaradze

at fx = — 1 (domain 1) and B3 = A3

C3 = -As

(37)

at fj, = 1 (domain 3). Similar to the one-cone case, on each interface H = nu i = 1, 2, we enforce the continuity of normal and tangential stress as follows. For the r6 component we have now 5(2Alfjll - 2Bi) = 2Am - 2B2 [M>

5{2A3ii2 - 2B3) = 2Am - 2B2. The continuity of 69 component requires

5{Al - 2Bm - d) = A2- 2B2JM - C2 [ 5(A3 - 2B3fi2 - C3) =A2- 2B2[i2 - C2. ' Equations (37,36,38,39) specify eight equations for eleven parameters A t , Bi and C% (i = 1,2,3) and the angles fj,\^2. Additional constraint comes from the fact that there is a solution of (22) in domain 2, which we call g2(/j,), must solve boundary-value problem 92^ = Mi) = 0

92^ = to) = 0.

(40)

Thus, we expect the final solution to depend on two parameters, let us say Ai and n\. We have also assumed that the fluids are given, which fixes the value of 8. Analytical progress can be achieved under the assumption that in domain 2 (i.e., in ambient fluid), the flow is sufficiently slow, so nonlinear terms g2/2 in (22) are small. Then, general analytical solution of this linearized version of (22) is given as

g2(ri = C(l-ii2)

+ A15F(ri,

(41)

where C is arbitrary for now and F

^)

=

I ^U(^ TX ( ^ - J ) - ^ v (Mi + 1)0^1 -to)

+ ^ + 2^2M+

(l + Mi)(M2-l)(l-/i 2 )log[^]).

(42)

Then, the condition (40) yields a homogeneous linear system for C and Ai, which is solvable if and only if the determinant A of this system is zero, i.e., A( M l , fi2) = 6 [(1 - f4)F(to)

- (1 - vl)F(vi)}

=0

(43)

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

465

Figure 2: fi2 versus n\ for linearized flow in ambient fluid (domain 2).

with F(fi) given by (42). It is important to keep in mind that if second fluid is absent, S = 0 so (43) does not pose any condition connecting fi\ and /i2-

Notice that the solution of (43), defining an implicit relationship between (i\ and [i2 is no longer dependent on the parameter 5 for 5 ^ 0 (and, naturally, on the amplitude A\), making it very convenient for analysis. Equation (43) has a trivial root ^2 = A*i, and another non-trivial root which is of interest to us. We present the dependence H^i^i) of Fig- 2. As is clear from this figure, (43) can only be solved for a limited range of — 1 < Hi < 0.30. Note that a solution described here was obtained for linearized flow only in ambient fluid (domain 2). In domain 1 and 3, the flow remains fully nonlinear. Let us finish our analysis by showing that a fully nonlinear solution can be obtained as well. If we do not assume linearity of (22) in domain 2, we cannot connect boundary conditions at fi = HI and /i = [ii easily and a fully nonlinear numerical solution must be sought. (We shall mention here that a solution of (22) can be sometimes expressed in terms of hypergeometric functions, but such analysis is more complex than the one presented here and does not add in either clarity or simplicity). Given 5 and for any A\. Hi and /J,2, we can find the rest of the coefficients A t , B{ and Q and proceed by shooting from ji = Hi to /i = ^i- Then we numerically find Hi such that 52(^2) = 0. Notice that we have to repeat this process iteratively, as

466

Vakhtang Putkaradze

Figure 3: Nonlinear solution g{ji) satisfying regularity conditions at fi = ±1.

the coefficients A,, B{ and d depend on fi2. It is interesting that while the solution of the iterative process does depend on A\, /ii and 5, the main part of the solution is already correctly predicted by the linearized result in Fig. 2. The difference between the linearized and fully nonlinear solution for (i2 never exceeds 1 %. The limit of 5 —> 0 is of interest as well, as it describes the conical flow with free surface when the second fluid is inviscid and not moving, or simply absent. This limit also give us a physical insight into the region of validity of the linearized solution. In Fig. 3 we present the result of fully nonlinear solution for A\ = 1, 5 = 0.2,/xi = —0.3. The oil-water mixture frequently used in the experiment has 5 ~ 10~2, however we have used a somewhat larger value of 8 in Fig. 3 for demonstration purposes. As S —> 0, the solution in domain 2 converges to 0 uniformly. If we assume that A\ remains of order one, it can be seen from the boundary conditions that A2, Bi and C2 are of order 5, Then, necessarily, g2 = 5G2(fx) + O{52) where G2 is of order one and satisfies the linearized equation (22). We then proceed analogously to our discussion of the linearized solution in the domain 2: an exact solution of this linearized Slezkin equation can be found, yielding the same connection between \x\ and \i2 as shown on Fig. 2. The behavior of solutions for large values of Ai, A\ ~ 1/5, remains unclear but will be addressed in future studies. Presumably, this question is of mostly academic interest, as large values of A\ will presumably correspond to unstable flows. However, since solutions described here are exact solutions of Navier-Stokes equations, a detailed information about the behavior in all regimes may be of importance for checking advanced numerical codes for simulation of fluid flows with an interface. Having completed the study of conical flows, let us now turn our at-

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

467

tention to a similar problem of radial two-fluid flow in two dimensions [26]. Also in this case, a complete analytical solution of the problem can be found. Such a solution can be realized when a point where two cones merge is artificially stretched out in space (say, by the effect of external forcing) so the interface between two fluids will now consist of two sheets joining along a line. 5. 5.1.

TWO-DIMENSIONAL TWO-FLUID FLOWS General Considerations

The physical picture considered here is the following: one fluid is being injected through a line source into another fluid. Such flow may be realized when a cracked oil pipe leaks oil when being submerged in water. We shall consider a slightly more general case: two viscous, incompressible fluids being ejected from the same point source in a plane. Two fluids are assumed immiscible and are separated by an interface, which is denoted by the two straight dashed lines originating at the source. We assume that the fluid j , j = 1, 2 is flowing with the flow rate Qj and has density pj and kinematic viscosity Vj. The effect of gravity is neglected. The only case relevant for practical applications is Q\ ^ 0, Q2 = 0. It appears, however, that arbitrary Q2 ^ 0 can be considered without extra effort, and it introduces valuable insight into the structure of the solutions. Thus, in the rest of this section we shall consider arbitrary values of Q\ and Q2The flows developed here are based on the classical Jeffery-Hamel solutions, which describe the outflow and inflow of viscous incompressible fluid in a linearly expanding channel (wedge) with a given angle between solid walls, as formulated by [27] and [28]. One dimensionless parameter of the problem is R = Q/v, Q being total flux and v being kinematic viscosity, and another dimensionless parameter is the angle of the aperture, a. Behavior of the solutions is very interesting. It was shown in [29] that there is an infinity of solutions and each solution undergoes a bifurcation at some R = Rc, where the critical Reynolds number Rc depends on the number of oscillations in the velocity profile of the solution and the angle of the wedge. In spite of a long history, many important results concerning the structure of Jeffery-Hamel solutions and their stability are quite recent, see [30]-[39].

468

Vakhtang Putkaradze

Several generalizations of Jeffery-Hamel flows have been made. An extension describing the flow of fluid in free space without boundaries was first considered in [40] but the analysis was not complete. Goldshtik and Shtern [32] were the first to point out the exact bifurcation values: it was shown there that there is an infinity of solutions with fc-fold symmetry, k = 1,2,... and each such solution exists if R < vr(fc2 — 4). This type of flows was subsequently analyzed in [33]. Unaware of that earlier work, Putkaradze and Dimon [41] re-derived the same results using analytic structure of solutions (Weierstrass P-functions). The analytical structure of solutions described in [41] will be crucial for exhaustive description of two-fluid case, as we demonstrate below. Another generalization of JefferyHamel solutions which includes a vortex and a sink at the origin was first done by Goldshtik, Hussain and Shtern [33]. Voropayev and co-authors extended Jeffery-Hamel flows to include a quadrupole at the origin, see [42] [45]. Bourgin and Tahiri [46] studied Jeffery-Hamel type flows for the case when velocities on the interface are prescribed. Moffatt [31] posed different boundary conditions at the surface of the wedge and solved Jeffery-Hamel flow in the limit of small Reynolds numbers. Jeffery-Hamel flows were also used as a basis for the study of fluid flows in channels with curved walls in [47] and [48].

5.2.

Equations and Boundary Conditions

The derivation of governing equations in this section is similar to that for Jeffery-Hamel flows. We shall therefore only go through the derivation briefly. For details, see [24]. We assume that two immiscible, incompressible fluids are ejected from a point source in two dimensions. The fluids have kinematic viscosities v\ and V2 and densities p\ and pi. If we assume that the fluids are flowing purely radially, i.e., ty, = 0 in each fluid, the incompressibility condition dr(rvr) = 0 gives vrJ = u ^ .

(44)

Here and below, the subscript j = 1,2 indexes the fluid. The prefactor v:l in (44) makes the velocity function Uj((f>) dimensionless. For convenience, substitute

uj(<£) = - 6 / j ( 4 0 - 2 ,

(45)

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

469

then each of the functions fj() satisfies a single ordinary differential equation:

ff = Aft-arfi-bj,

(46)

where the constants a,j and bj, j — 1,2 are to be determined to satisfy boundary conditions at the interfaces. The prime denotes the derivative with respect to 0. We assume that the interface is located at (p = —a/2 and (p = a/2. If we denote A = vi/1/2 the viscosity ratio and 5 — A2pi/p2, the boundary conditions at the interface become: A(6/i(-a/2) + 2) - 6/ 2 (-a/2) + 2 A(6/i(a/2) + 2) = 6/ 2 (a/2) + 2 <J/{(-a/2) = / 2 ( - a / 2 ) 5f[(a/2) = /2(a/2) <J(3ai - 4) = 3a 2 - 4

(47)

/_°52(-6/i( = —«o/2 and 4> = OLQ/2. We have discovered, however, that the answer strongly depends on the values of a 0 and chosen initial profile, showing that several solutions of this problem are possible. Thus, even though the continuation method is a possible way to obtain a two-fluid Jeffery-Hamel solution, it does not necessarily converge to the desired solution. In the remainder of this section, we present an alternative method of studying the solutions, based on the Weierstrass Elliptic

470

Vakhtang Putkaradze

Functions. The method allows the complete classification of the hierarchy of solutions. In addition, our method yields a type of solution which is impossible to obtain by Newton's method with continuation (SE solutions, see below). The idea of our method consists of finding the solutions satisfying the boundary conditions at (j) — Oi/2 automatically provided they satisfy the boundary conditions at
5.3.

Analytic Properties of the Solutions

Equation (46) can be interpreted as a description of motion of a particle in a cubic potential U(f) = — 4 / 3 + a,jf + bj, which shows that the solutions of (46) are periodic with some period, let us say pj, with pj being real. Upon the change of variables 4> —> i<j), equation (46) describes motion of a particle in a cubic potential —[/(/), which shows that the solution fj( + 2ZTJ) = fj((/>)• This is a consequence of a more general fact that solutions of (46) - Weierstrass Elliptic Functions - are doubly periodic functions in the complex plane [49]. Periods are connected with the invariants through the formulas a

j = 6 0 E' n , m ( n Pj + m 2iT3y4 bj = i4OTltn,m(npj+m2iTj)-6,

(AS) K }

where the prime denotes that the summation is taken over n, m such that n2 + m2 > 0. In general, the periods may be two arbitrary linearly independent complex numbers. The solution of (46) can be represented either in terms of invariants, i.e., a.j and bj, or in terms of periods, i.e., pj, 2iTj. These representations are completely equivalent from the mathematical point of view. Dependent on the case, however, it is advantageous to use either the invariants or the periods representation. To distinguish between these representations, we shall denote f(z) = V{z; a,j, bj) if f(z) is represented in terms of invariants a,j and bj, and use curly brackets after the semicolon f(z) = V(z; {pj, 2ITJ}) if f(z) is represented in terms of periods pj, 2%rr Also, we always explicitly mention which representation is used in each particular case to avoid confusion.

471

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

Figure 4: Velocity function u(
5.4.

Satisfying boundary conditions at 0 = —a/2 and (j) = a/2

We are now ready to describe different types of solutions obeying boundary conditions (47). As was mentioned earlier, we seek solutions satisfying boundary conditions at (p — a / 2 provided that boundary conditions at 0 = —a/2 are satisfied. The first possibility is to seek solutions f\{4>) which are periodic with the real period p\ = a/k\ and f2{4>) with the real period p2 — (2TT — a)/k2 for some integer numbers k\ and k2. Using our mechanical analogy, solutions of this type represent a particle oscillating in a cubic potential [/(/) = - 4 / 3 + a,/ + bj. The velocity function u() = - 6 / ( 0 ) - 2 for such solution is illustrated on Fig. 5.4. We distinguish different periodic solutions by their periodicity. We denote solutions with fi(4>) being k\ periodic and f2(4>) being A;2 periodic as P(ki)P(k2) (periodic-periodic). In the analysis of P{k{)P{ki) solutions we will use the period representation. The P{k\)P{k2) solutions are given by fcW)

= V{<\> + ir, + 9h {PJ,2IT3}).

(49)

The unknowns are 6j (phase-shifts) and Tj (imaginary periods). Thus, it is advantageous to use the period representation for P(k\)P{k,2) solution. The second possibility to satisfy boundary conditions at 0 = ± a / 2 is to require the flow to be symmetric with respect to the reflection around t h e line (f> = 0, more precisely, f\(-) = f\{4>), h(ir

- 4>) = h{^

+ 4>)-

Since there is no condition on the periods, the representation of solutions in terms of invariants a,j,bj is advantageous. Solutions of this kind are given by: fx{4>) ="P(0 + ZTi(ai,6i);ai,&i) / 2 (0) = p( 7 r + 0 + ? r 2 (a 2 ,6 2 );a 2 ,6 2 ).

l

'

472

Vakhtang Putkaradze

The unknowns are the invariants a-j and bj. The imaginary half-periods Tj(a,j,bj) can be expressed in terms of the invariants (a,j,bj) by inverting (48). Even though no requirements on real periods are made, we shall also enumerate these solutions in terms of two integer numbers, k\ and &;2. This implies that the real periods pj satisfy fci -

^! —

fci+1

/t;i\

We shall denote such solutions E(k\)E{k2) ("even-even"). In the study of E(h)E(k2) solutions, it is best to use invariants as unknowns. The third possibility is to have one of the solutions fj{4>) be singular. From the properties of equation (46) we conclude that singularities of /,() are periodically spaced second-order poles: f3 ~ (z — npj)~2, where n is any integer number. These solutions are physical, if the singularity occurs outside the domain of the solution, i.e., in another fluid. For example, if f\{4>) is such singular solution, we need the singularities of f\(4>) to lie outside the region —a/2 < ) cannot be periodic, since that would necessarily mean having singularities inside the domain —a/2 < < a/2. Thus, f\{4>) has to be a symmetric function of ) — /i(0)- A solution of this type is illustrated on Fig. 5.4. Then, f2(4>) has to be a symmetric function of (p also. From the boundary conditions one can deduce that f2() has to be a non-singular symmetric function, i.e., of type E{k2)- Therefore, the third type of solution is given by putting together a singular symmetric solution S and non-singular symmetric solution E. This can be done in two possible ways: SEfa) (fi() being singular) and E(ki)S (f2(4>) being singular). We shall only consider the SE{k2) solutions - the E(k])S type is completely analogous. For SE(k2) solutions, we use representation in terms of invariants: / i ( 0 ) = W ; 02,62) / 2 (0) = P(7r + 0 + zr2(a2)62);a2,62).

, , l

;

Again, the unknowns are the invariants dj, bj and a. There are two more possibilities to satisfy the boundary conditions automatically, which correspond to degenerate types of solutions (in the sense that they are realized for the set of parameters (R\, i?2) of measure zero). The fourth possibility is to have one of the functions fj constant, and another function oscillatory. One can see that if, for example, / 2 = const,

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

473

Figure 5: Velocity function u(4>) in the singular domain of SE(k2) solution. The poles of velocity are occuring outside the physical domain

/i(0) must be periodic with the period a/k\. We shall call these solutions P{k\)C solutions if f2 = const and CP{k2) if A = const. Finally, the most trivial solution possible is the one with f\ = const and f2 = const. We shall refer to these solutions as CC solutions. As was demonstrated in [26], three cases P{k\)P{k2) , E{k\)E(k2) , SE(k2) are realized for the set of parameters (i?i,i?2) of positive measure. Other types of solutions, involving at least one C in their notation, can only be realized for sets of codimension at least one in the parameter space (i?i,jR2)- Therefore, the analysis of P(ki)P(k2) , E{ki)E{k2) and SE(k2) type of solutions is physically most relevant. However, we shall mention P(k\)C and CP{k2) solutions later as they play an important role in bifurcations. To specify a two-fluid Jeffery-Hamel solution, we must first determine the type of solution {P{k\)P{k2) , E{k\)E{k2) or SE(k2) ) as well as periodicity of the velocity profiles k\ and k2. Once these are chosen, we can analyze the behavior of the system on the parameter plane (Ri, R2). The most interesting bifurcation diagram is exhibited by the P{ki)P{k2) solutions. The introduction of an extra Reynolds number as a free parameter yields a highly complex bifurcation picture, typical of bifurcations of degeneracy two [50]. In particular, if we take a line through the parameter space (Pti, R2) (denoted by Reff) and compute the value of u'(a/2) (proportional to the shear stress on the interface), we obtain a boomerang-shape diagram typical for the umbilical catastrophies [50], as shown on Fig. 6. Such a bifurcation diagram is only understandable if two parameters Ri and R2 are considered. The boomerang structure persists even if we take the line R2 = 0 in the parameter plane, corresponding to no flux in the second fluid (which is the physical regime). Such 'ghost entrance'

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Vakhtang Putkaradze

Figure 6: The value of non-dimensional shear stress on the interface u'(aj2) taken along a line on the parameter plane Ri,R2.

of the second parameter R2 into this problem is highly non-trivial and unexpected, and shows that the free surface affects the flow beyond the apparent complexity of the formulas. As we have seen, we must continue the study into the physically unrealizable regime R2 ^ 0 in order to fully understand both the physics and mathematics behind of the flow. The bifurcations of E{k\)E(k,2) and SE(k2) solutions, while not showing the same umbilic character of the bifurcation as the P{k\)P{k2) , are still sufficiently complicated. It was shown in [26] that E(k\)E{k,2) solutions, as well as P{k\)P{k2) solutions, exist on large areas of the parameter plane (Ri,R2), whereas SE(k2) type can only be found in relatively small (in measure) areas. The boundaries of the existence domains are formed by the P{k\)C (or CP{k2)) solutions, or, alternatively, by merging of a solution with its mirror image. Finally, one can demonstrate that the SE(k2) solutions cannot be obtained by the method of continuation in principle, as the singular part exhibits a pole-type singularity on real line. While this singularity is hidden by the fact that it appears in the domain of the second fluid, it must appear on real line if the starting point of the continuation is the two fluid being identical (i.e., no free surface). It is essential therefore to employ the method of P-functions described above to achieve a complete classification of solutions. We shall also note that this method includes, as a particular case, the solutions when second fluid is inviscid and not moving or simply absent. In

The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads

475

this case, only solutions of the periodic or even types are possible (the singular solution must necessarily have poles in the physical domain, yielding infinite flux, and must therefore be ruled out). Such solutions (fan jets) may play important role in some applications. The question of the selection mechanism for the two-fluid solutions is still unresolved. It may be argued that the E(1)E(1) solution looks like two parabolic profiles joined together and therefore is the most probable one to be chosen by the flow. It is also conceivable to conjecture that the profiles with high number of oscillations and alternating regions of strong inflow and outflow are unstable due to Kelvin-Helmholtz-type instability. While these arguments are plausible, a convincing argument for selecting one of the solution is still missing. Moreover, it has been demonstrated [51] that the stability of radial flows strongly depends on the upstream and downstream boundary conditions on velocity perturbations. Thus, perhaps it is possible that several solutions can be realized by altering the upstream and downstream boundary conditions in numerical simulations or experiments. 6.

CONCLUSIONS

In this chapter, we have outlined the role conical flows may play in the process of droplet break-up in two and three dimensions. While most of the previous work in droplet fission is based on the Stokes approximation, which allows treatment of moving interface, our approach utilizes the Landau-Squire or Jeffery-Hamel exact stationary solutions of Navier^-Stokes equations. Thus, our approach cannot treat moving interface and should be seen as complimentary to the previous studies using Stokes methods. Nevertheless, some of the interesting effects caused by the combination of inertia and free surface (multiplicity of solutions, coexistence of inflows and outflows, intriguing bifurcation diagrams, etc) should persist when time dependence is taken into account. The exact introduction of time dependence is impossible in the purely analytical framework outlined here. Thus, numerical simulations of the full Navier-Stokes equations and detailed measurements of velocity profiles in experiments are of paramount importance for future progress.

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7. ACKNOWLEDGEMENTS The author acknowledges fruitful and inspiring discussions with W. Zhang and J. Eggers. The idea of using two-fluid radial flow (originally in Stokes approximation) was suggested to the author in a discussion by H. Stone and M. Brenner. This work was partially supported by Petroleum Research Grant (Type AC). The hospitality of the Department of Mathematics, Ibaraki University (Mito, Japan), where this work was completed, is greatly acknowledged, as well as financial support of the Japanese Society for Promotion of Science (JSPS). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

J. Eggers, Rev. Mod. Phys., 69 (3) (1997) 865. I. Cohen, M. Brenner, J. Eggers, and S. Nagel, Phys. Rev. Lett., 83 (6) (1999) 1147. X. Shi, M. Brenner and S. Nagel, Science, 265, (1994) 219. J. Eggers and T. Dupont, J. Fluid Mech., 262 (1994) 205. F. J. Garcia and A. Castellanos, Phys. Fluids, 6 (1994) 2676. J. Hoppe Lectures on integrable systems Springer, Berlin 1992. J. Keller and M. Miksis, J. Fluid Mech, 232, (1991) 191. M. Brenner, J. Lister and H. Stone, Phys. Fluids, 8(11) (1996) 2827. A. U. Chen, P. K. Notz and O. A. Basaran, Phys. Fluids, 8(11) (2002) 2827. P. Doshi, I. Cohen, W. Zhang, M. Seigel, P. Howell, O. Basaran and S. Nagel, Science, 302 (2003) 1185. W. Zhang and J. Lister, Phys. Rev. Lett., 83 (6) (1999) 1151. J. Lister and H. Stone, Phys. Fluids, 10, (1998) 2759. I. Cohen and S. Nagel, Phys. Fluids, 13 (12) (2001) 3533. I. Cohen and S. Nagel, Phys. Rev. Lett., 88(7) (2002) 074501. I. Cohen, H. Li, J. Hougland, M. Mrksich and S. Nagel, Science, 292 (2001) 265. W. Zhang, In preparation. B. Squire, Phil. Mag., 43 (1952) 942. A. F. Pillow and R. Paull, J. Fluid Mech., 155(1985) 327. M. A. Goldshtik, M and V. N. Shtern, J. Fluid Mech., 218 (1990) 483. V. N. Shtern and F. Hussain, J. Fluid Mech., 309 (1996) 1. V. N. Shtern and F. Hussain, Ann. Rev. Fluid Mech., 31 (1999)537. C. F. Stein, J. Fluid Mech., 438 (2001) 159. V. Putkaradze, in preparation. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1991. A. Herczynski, P. D. Weidman and G. I. Burde, Phys. Fluids, 16 (4) (2004) 137. V. Putkaradze, J. Fluid Mech., 483 (2003) 1. G. B. Jeffery, Phil. Mag 29, (1915) 455. G. Hamel, Jahresbericht Deutsch. Math. Verein, 25 (1916) 34. L. Rosenhead, Proc. Roy. Soc. A, 175 (1940) 436. W. H. H. Banks,P. G. Drazin and M. A. Zaturska, J. Fluid Mech., 186, (1988) 559. H. K. Moffatt and B. R. Duffy, J. Fluid Mech., 96 (2), (1980) 299. M. A. Goldshtik and V. N. Shtern, Fluid Dynamics, 24 (1989) 191. M. A. Goldshtik, F. Hussain and V. N. Shtern, J. Fluid Mech., 232 (1991) 521.

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[34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

477

L. E. Fraenkel, Proc. Royal Soc. Lond. A, 267 (1962) 119. G. A. Georgiou and P. M. Eagles, J. Fluid Mech., 14 (1988) 259. P. M. Eagles and G. A. Georgiou, Phys. Fluids, 30 (12) (1987) 3838. P.M. Eagles, J. Fluid Mech., 24(1) (1966) 191. S. C. R. Dennis, W. H. H. Banks, P. G. Drazin and M. B. Zaturska, J. Fluid Mech., 336 (1997) 183. F. J. Uribe, E. Diaz-Herrera, A. Bravo and R. Peralta-Fabi, Phys. Fluids, 9 (9) (1997) 2798. R. Berker, Handbuch Der Physik, VI1I/2, (1963) 20. V. Putkaradze and P. Dimon, Phys. Fluids, 12 (1), (2002) 66. S. I. Voropayev and Ya. D. Afanasyev, J. Fluid Mech., 236 (1992) 665. S. I. Voropayev and Ya. D. Afanasyev, Phys. Fluids, 6 (1992) 2032. S. I. Voropayev, H. J. S. Fernando and P. C. Wu, Phys. Fluids, 8 (2) (1996) 384. S. I. Voropayev and H. J. S. Fernando, Phys. Fluids, 8 (9) (1996) 2435. P. Bourgin and N. Tahiri, in P. H. Gaskell, M. D. Savage and J. L. Summers (eds.), Proc. of the First European Coating Symposium, University of Leeds, (1995) 23. L. E. Fraenkel, Proc. Royal Soc. Lond. A, 272 (1963) 406. I. J. Sobey and P. G. Drazin, J. Fluid Mech., 171, (1986) 263. K. Chandrasecharan, Elliptic Functions, Springer-Verlag, 1984. R. Thorn, Structural Stability and Morphogenesis, Benjamin Inc, 1975. V. Putkaradze and L. Romero, in preparation.

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Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.

Chapter 12

Statistical mechanics of dense emulsions D. N. Petsev Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131 1. INTRODUCTION Emulsions are a specific type of colloidal systems where both the disperse media and disperse phase are liquids. Typical examples are dispersions of oil in water or water in oil. In most cases the dispersed phase (oil or water) is in the form of spherical droplets of different sizes. The shape (usually spherical) is enforced by the interfacial tension while the size distribution is usually dependent on the emulsion preparation protocol and, as time progresses, on the dynamics of flocculation and coalescence. The latter, on the other hand, is a function of the droplet direct and hydrodynamic interactions. The focus of this review will be mostly on ensembles of submicrometer sized monodisperse emulsion droplets often referred to as miniemulsions. An important property of miniemulsions is that they exhibit Brownian motion, which has a strong impact on their dynamic and thermodynamic properties. Therefore, such droplet ensemble is amenable to statistical mechanical analysis and this is the purpose of this article. Emulsions are usually polydisperse but taking into account this fact will complicate the analysis too much without contributing to the main results and conclusions. However, a procedure for preparation of monodisperse emulsions is available [1], which means that the theoretical treatment outlined here could be always related to a particular experimental system. When comparing emulsions to solid liquid dispersions (e.g. hard spheres, charged hard spheres, etc.) it becomes evident that there are some important differences following from the fact that the droplet interfaces are fluid and flexible. Often the interfacial fluidity is suppressed by surfactant adsorption and the interfacial flexibility might be unimportant because the interfacial tension renders the droplets effectively rigid. However, there are many other cases where one or both of these conditions do not apply. In general, for droplets with size of the order of a few hundreds of nanometers and interfacial

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D.N. Petsev

tension of about a few dynes/cm (usually below 10), one may expect that droplet deformability will impact both the direct (e.g., electrostatic, van der Waals) and hydrodynamic interactions. This impact is due to the fact that all these interactions are sensitive to the specific shape of the interacting surface. Besides, interfacial deformation itself contributes to the interaction energy since work is done against the forces governing the droplet shape (interfacial tension, bending energy). As a result two droplets will deform upon approach and acquire a shape that could be reasonably approximated as truncated spheres, see Fig. 1. For large macroemulsion drops (between a few micrometers to a few millimeters) the effect of the deformation becomes so great that one may consider only the thermodynamic and hydrodynamic interactions within the plane parallel film that forms between the surfaces and neglect the contribution of the surrounding regions. Particular cases of oil-water-surfactant mixtures are microemulsions, which are formed when the interfacial tension is low [2-4]. Contrary to ordinary emulsions, microemulsions are thermodynamically stable. The interfacial deformability of the microemulsion droplets is governed primarily by the bending rigidity rather than by the interfacial tension. Microemuslions can be sorted in three groups based on the phase state [2,3]. When the surfactant in the solution has high hydrophilic-lipophilic balance the microemulsion exists in the form of small oil droplets (normally with diameter below a 100 nm) in a continuous aqueous phase. If excess oil is present in the

Fig. 1. Sketch of two deformed droplets (a) in absence of long ranged repulsive forces and (b) in the presence of long range repulsion (e.g., electrostatic). If not long range forces are present the droplets deform upon contact of their surfaces. Otherwise a liquid film with thickness h is formed.

Statistical Mechanics of Dense Emulsions

481

system, it resolves as a separated macroscopic phase. This type of microemulsion is often referred to Winsor I type. For surfactants with low hydrophilic-lipophilic balance, the opposite situation could be observed when water droplets form in the oil phase and the excess water (if any) exists in separate macroscopic phase. This is Winsor II type of microemulsion. By varying some experimental parameters (temperature for nonionic and electrolyte concentration for ionic surfactants) one may switch from Winsor I to Winsor II and vice versa. By doing that, the microemulsion goes through a region of extremely low interfacial tension [4]. Based on the composition oil/water/surfactant a third phase may resolve between the oil and water where both the oil and water are interconnected forming a bicontinuous network [2-4]. This type of microemulsion is called Winsor III. A thermodynamically stable microemulsion could be mechanically stirred to form a macroemulsion which may exhibit very interesting properties in terms of stability due to the coexistence of small microemulsion droplets (below a 100 nm) and large macroemulsion drops, which could about micrometer size [5]. In the case of sub-micrometer miniemulsion droplets both the plane parallel film region and the surrounding curved regions contribute to the droplet energy of interaction and it is often incorrect to ignore one in comparison to the other. Hence, for such emulsions one has to be able to account properly for the energy of interaction of deformable droplets in order to be able to model and analyze their macroscopic properties and stability. The same argument is valid for the interactions between microemulsion droplets. The main difference is that the contribution of the interfacial tension is often negligible while one should properly take into account the bending rigidity. Recently, extensive studies have been undertaken to derive expressions for the pair energy, [6-12] see also Chapter 8. This chapter is organized as follows: the next Section presents a brief overview of the statistical mechanical treatment of dense fluid systems based on the application of the integral equation theory. Section 3 elucidates the implications in the approach due to the fluidity and deformability of the droplets, Section 4 presents some computation results for the macroscopic structure and properties of emulsions. The particular case of extremely dilute emulsions is very important since it provides a reference state for normalizing the radial distribution function. That is why it is discussed in Section 5. Section 6 summarizes the concluding remarks. 2. STATISTICAL MECHANICS OF COLLOIDAL DISPERSIONS A BRIEF OVERVIEW Statistical mechanics is a valuable tool, which relates microscopic parameters to the macroscopic properties of a system that consists of a large number of

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D.N. Petsev

molecules or colloidal particles. It is important and useful to be able to deduce the structure and the thermodynamic parameters of such a system from the interaction energy between the molecules or particles that can be often determined or calculated from elsewhere. A convenient approach to achieve this goal is using integral equation theories [13-17]. Integral equation theories in general aim to relate the pair interaction energy between two molecules (also colloidal particles or Brownian emulsion droplets) to the macroscopic thermodynamic properties of the whole system as osmotic pressure or compressibility. There are a number of integral equation based approaches that employ different approximations and are therefore applicable to different particular cases. Central concepts in all versions of the integral equation theories are the pair correlation and/or radial distribution functions, which are discussed briefly in this section. 2.1. Correlation functions, structure and thermodynamics Our consideration of the topic of correlation functions, structure and their relation to thermodynamics will be restricted to briefly recalling some of the basic ideas and definitions since there are many excellent texts that deal with this matter in detail [13-17], see also Chapter 1. When considering colloidal systems and emulsions in particular it is important to emphasize the fact that there are two levels at which the distribution and correlation function formalism could be applied. The first level is molecular - the pertinent statistical ensemble consists of the molecules of the droplets, the solvent surrounding them and all species that might be present (electrolyte, surfactants, etc.). This level provides valuable information about the structure of the droplets, their interfaces and the solvent in close vicinity to the interfaces at a molecular scale and is discussed in greater detail in Chapter 1 of this book. The second level is particle where the ensemble consists of the dispersed colloids, or droplets in the case of emulsions. The molecular structure of the drops and the disperse medium that surrounds them are not taken into account explicitly but indirectly by means of macroscopic parameters that modify the particle interactions like dielectric permittivity, ionic strength, Hamaker constants, etc. Although less detailed than the molecular level, this approach allows successfully treating complex colloidal systems and offers some useful insights about their structure at the particle length scale, as well as their overall macroscopic thermodynamic properties. Let us consider a fluid of N particles that occupies a volume V and has temperature T. The probability of having any two of these particles to be in volumes dv\ around rj and dr2 around r2 while the remaining N-2 particles are allowed to be anywhere in the rest of the volume Fis

Statistical Mechanics of Dense Emulsions

g

483

( r " r ») = V(jV_2)i;Jo -Jo exp[-#/,(rp...,rN)]*,...*N J^

poo

poo

~ — J o ...Jo exp[-/3t/w(r1,...,rN)jJr3...JrN where o

...J o exp[-/?t//v(r1,...,rN)]
(2)

is the configuration integral, [/^(r,,...,!^) being the TV-particle potential energy. The multiplier (3 = \lkBT is the inverse thermal energy, where kB is Boltzmann constant and T is temperature. The transition between the first and second line in Eq. (1) is justified for large number of particles, N. The quantity g (r,,r 2 ) is the two particle distribution function and in case of an isotropic system (e.g., fluids) depends only on the distance between the particles r = r, — r2 and g{2)(rt,r2) = g(r)

(3)

where g[r) is the radial distribution function. Its meaning is probability to have one particle at the origin and another at a distance r from it. As it could be seen from Eq. (1), the two particle distribution function and hence the radial distribution function depends on the //-particle energy UN(r{,...,rN). This quantity is very difficult to obtain but for particles, interacting via central forces, and assuming pairwise additivity the TV-particle energy becomes

l/ff(r,,...,rN) = i $ > f o )

(4)

where M ( ^ ) is the pair interaction energy between particles i andy, separated by a distance rtj. Knowing the radial distribution function, one may calculate the macroscopic properties of the system of dissolved molecules or suspended colloidal particles (including droplets) as the osmotic pressure, Pos, or the isothermal osmotic compressibility, Xr respectively

484

D.N. Petsev

^ . = 1-4/30 P ^ r ) — f d ? , J

p

(3

(3 dpos

o

r = r/2a

(5)

^

^ >

df

^ oy

T

>\

where a is the particle radius, p is the particle number density and <> / = 4iraip/3 is the volume fraction. The latter quantity is very convenient when dealing with colloidal dispersions, includingly emulsions. The radial distribution function, g(r), is directly related to the structure factor of the system, S(q) by a Fourier transform (q is the wave vector and q is its magnitude)

S(q) = l + 6^ f"g(f)exp(-iq-r)df,

r = rl2a,

q = 2qo

(7)

7j- JO

or for isotropic systems

S{q) = \-mr\\-g{r)p^r2df, Jo

l

J

q = 2qa.

(8)

qr

The structure factor can be determined experimentally by light or neutron scattering experiments. Knowing the radial distribution function allows for calculation of all thermodynamic quantities of interest (for details one may refer to Refs.) [1317]. Therefore this approach allows for theoretical analysis of phase eqilibria. The conditions for "gas-liquid" phase equilibrium, when the system separates into two regions with low and high density, are given by 7} =7},, P,=Pn,

n,=nu

(9)

where T, P and u, are the temperatures, pressures and chemical potentials in the low density (subscript L) and high density (subscript H) phases respectively. Recently it was shown that the radial distribution function could also be used for computation of the freezing boundary in the phase diagram [18-20], see Ref. [16]. These authors have shown that the freezing point can be determined by the following equation se-s2

=0

(10)

Statistical Mechanics of Dense Emulsions

485

where sex is the excess entropy multiparticle correlation expansion of the entropy of liquids [21]

«=2

sn is the partial entropy contribution of n particles and sex has the meaning of excess entropy per particle due to multiparticle interactions and correlations. The pair term is given by

s2=-12 J o ~[g(r)lng(r)- g(r) + \]r2dr .

(12)

This simple freezing criterion has been tested against molecular dynamics simulations for a variety of potentials and proved to be remarkably accurate [16,18-20,22-25]. 2.2. Ornstein-Zernike integral equation for the radial distribution function Obtaining the radial distribution function for a multiparticle system (like colloidal dispersion) is not simple. The reason is that the interaction between each two particles is affected by the presence of all the others in the ensemble. That is why all equations for determining the radial distribution function introduce assumptions and approximations. There are a number of excellent reviews where the equations for the radial distribution function are discussed and compared in detail [13-17]. Below we present a very brief description of the Ornstein and Zernike [26] approach and some of the pertinent approximations that are often applied. This equation has to be solved numerically and the best current algorithm was suggested by Labik et al. [27]. The Ornstein-Zernike equation reads [26], see also Refs. [13,14,16,17]

h{r) = c(F) + ^JQnJc(\f-r'\)h(f')dr'

(13)

where /z(r) = g(r) —1 is the total correlation function and c(r) is the direct correlation function. Eq. (13) that means the total correlation between two particles could be separated into a direct, c(f), and an indirect term given by the integral. The indirect term on the other hand is also expressed by direct correlations, averaged over the positions of the remaining particles. Hence, the Ornstein-Zernike equation has a very simple and clear physical meaning. Still, it is not sufficient because in order to obtain the radial distribution function

486

g(r)

D.N. Petsev

[or the total correlation function h(f), which is equivalent] one must

know the direct correlation function c(r). Unfortunately it is not known and one has to make an assumption about its form. There are different models for the direct correlation function, also called closure approximations. A very good discussion of the different closures is given in Ref. [16]. Below we will consider only a few that are most relevant to colloidal dispersions and emulsion systems. 2.2.7. Percus-Yevick approximation The Percus-Yevick approximation for the direct correlation function reads [13,14,16,17,28]

c(r) = {l- exp[-/3«(r)]}g(r).

(14)

This approximation allows for obtaining an analytical solution of the OrnsteinZernike equation for hard sphere fluid where u(r) = oo for r<\,

u(r) = 0 for r>\.

(15)

In some cases the hard sphere model represents very well emulsion systems. For example, oil in water emulsion with high interfacial tension. In general, the Percus-Yevick approximation works well when the pair interaction energy is short ranged although it tends to overestimate the first peak in the radial distribution function. Besides the hard sphere model it gives an analytical solution for the sticky hard sphere model where the attraction is infinitely short ranged [29]. However, the Percus-Yevick approximation suffers from thermodynamic inconsistency [13,16,17]. The hard sphere equation of state obtained from the pressure virial expression (5) is different from the one derived using (6). Similar problems occur for the sticky sphere model [29-32]. In this chapter, however, we will concentrate on emulsions where the interactions are more complex than in the simple hard or sticky sphere cases. 2.2.2. Hypernetted chain approximation The hypernetted chain closure approximation [13,14,16,17,33-37] for the direct correlation function is given by c(r) = h{r)-f3u(r)-\ng{r).

(16)

The hypernetted chain approximation is known to work better for long ranged interactions. Contrary to the Percus-Yevick model, the hypernetted chain

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487

approximation underestimates the height of the first peak in the radial distribution function. There are a number of modifications of the hypernetted chain approximation [38-41], see also Ref. [16]. 2.2.3. Rogers-Young approximation Rogers and Young [42] suggested a closure approximation that imposes thermodynamic consistency. To achieve this, they interpolated between the Percus-Yevick and the hypernetted chain approximation using

where f(r) = \-exp(-ar)

(18)

with a being an adjustable parameter. The Rogers-Young approximation is currently the best closure for systems with repulsive interactions. Its application, however, for attractive particles is questionable [43]. Still, our analysis of moderately attractive emulsions [11,44] showed that this closure gives reasonable results. 2.2.4. Other closure approximations There are other closure approximations that have been suggested in the literature. For more details the reader may refer to the excellent reviews of Caccamo [16] and Nagele [53]. As discussed above the choice of closure approximations should be based on the specific system under consideration and taking into account the specific assumptions involved in each of them. A good practice is to test the solution against Monte Carlo or Brownian dynamics simulations. 3. DISTRIBUTION AND CORRELATION FUNCTIONS IN ENSEMBLES OF EMULSION DROPLETS: FEATURES DUE TO THE INTERFACIAL DEFORMABILITY This Section presents analysis of the possible impact that the interfacial flexibility and deformability may have on some of the basic statistical mechanical quantities. The effect on the pair interaction energy is also briefly discussed (see Chapter 8).

488

D.N. Petsev

3.1. Configurational intergral, pair correlation and radial distribution functions Similarly to hard particle dispersions, emulsions could be represented by a system of N oil droplets suspended in water. The droplets may be stabilized by surfactants (ionic and/or nonionic) while the continuous phase may contain some background electrolyte and is considered as structureless. Below we present a brief outline of the impact the droplets deformability has on the statistical mechanical analysis. This subject has been presented in more details in Refs. [44,45] and [11]. The main difference between solid dispersions and emulsions (with low interfacial tension) is in the fact that droplets may deform upon approaching each other [6-11,46]. The deformation is due to the pair interaction energy between the droplets but at the same time the pair energy itself depends on the deformation, see Chapter 8. As a consequence, the positions of the mass centers of the droplets are not the only integration variables in the configuration integral (2). For each given distance between two droplets there could be a number of different configurations corresponding to different droplet deformations, (or formation of films with different radii - see Fig. 2).

Fig. 2. A pair of interacting droplets may exhibit different deformations at the same mass center to center distance. Therefore for a given configurational variable r there are multiple states corresponding to different film radii.

Statistical Mechanics of Dense Emulsions

489

Therefore, the film radii, R formed between the droplets are additional degrees of freedom that have to be accounted for when defining the configuration integral. It becomes [44]

o -Jo Jo "Jo ex P[-^(^'-^M;^-^)]^,-^^...Jr N This is the configuration integral for a system of N droplets that may form M films. If two droplets are far and do not form a plane parallel film, its radius is essentially zero and the configuration is still properly counted. The pair correlation function for such a system is g{2)(Ri;rl,r2)=constx—-

V2 Z

»

.../ Q

%J 0

/ ... *J 0

*J 0

. (20) exp[-(3U{Rl,...,RM;rl,...rN)]dR2...dRMdrr..drN

As mentioned before, the energy of droplet interaction,L^(/?,,...,i?M;rl,...rjV), depends on the deformation. The "const" multiplier has to be determined by the requirement that the correlation function becomes equal to one for infinite separation of the droplets, |r, — r2| —> oo. As we will discuss later, it depends on the configurations generated by droplet deformations at infinite distances, i.e. without interactions. The number of films (including those with zero radius) is difficult to determine. However, this number is not important [44]. In fact, one can integrate (20) over the film radius R\ and derive an analog of Eq. (1)

g

(2)

V2 (r,,r 2 )=co^x — Z

. (21)

N

.../

I ...I

exV[-pU(Rv...,RM;rl,...rN)]dRr..dRMdrr..drN

There is an important difference between Eq. (1) and Eq. (21) and there are additional configurations and variables, which contribute to the averaging. This is actually an application of the potential distribution theorem [47]. The constant multipliers in Eqs. (20) and (21) are not the same. Similarly to the case of solid particle dispersion, we can simplify the problem further by assuming a pairwise additivity of the interactions, including the effects of the droplet deformation

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D.N. Petsev

U{Rl,...,Ru;r[,...,rN) = ±-J2u(R
(22)

where Ry is the film radius, formed between droplets / and/, separated by a distance ry. Expressions for uyR^r^ are available in the literature [6-11], see also Chapter 8. Using approximation (22) is most helpful for obtaining the averaged pair energy at infinite dilution as well as the normalization constant in Eq. (21). Thus for vanishing volume fraction <j> —> 0 we define g°(R;F)~exp[-0u(R;r)},

R = R/2a

(23)

and g°(r) = constx f """g°(R;r}dR^ constx Pexp\-/3u[R;r)\dR.

(24)

Eq. (24) gives the radial distribution function for an infinitely diluted system of droplets, averaged over all possible deformations. Note that in the second integral the integration limit Rmax is replaced by infinity. This is justified since the interaction energy ui,R;r\ strongly increases with the film radius R and hence the integrand in (24) rapidly decreases and tends to zero. Therefore, large values of R have negligible contributions to the configurations of the droplet pair and to the radial distribution function. Eq. (24) allows defining a pair potential of mean force (averaged over all possible film radii), viz.

/?w(r) = ln[g°(r)].

(25)

This pair energy, w(r), is the one to be used in the closure approximations (14,16,17) instead of simply u(r) when deformable liquid droplets are considered and not solid dispersions of spheres. The normalization constant in Eq. (24) can be determined by the condition that the radial distribution function becomes equal to one at infinite separation of the droplets,

const = { r~exp[-/?«(£;/:r-»oo)ldKl . [Jo

L

^

More details are given in Section 5.

J

(26)

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491

3.2. Pair interaction energy between droplets Emulsion droplets with high interfacial tension behave very much like a solid colloidal dispersion. The interaction energy and stability of solid dispersions has been studied for a long time and there are excellent sources on the subject, for example see Refs. [48-53], which by no means represent a full list. In this review, however, we are interested mostly in emulsion systems with low interfacial tension (usually below 5 mN/m) where the approach and interaction of two droplets may lead to deformation and film formation due to both thermodynamic and hydrodynamic forces. 3.2.1. Interfacial extension and bending contributions Deformation modifies the colloidal interactions like electrostatic, van der Waals, steric, depletion, hydration, etc. It also gives rise to additional contribution due to the deformation of the interface itself, which is not present in solid dispersions. This contribution is due to the extension of the droplet surface and/or change of the local curvature upon deformation [6-11,44-46,5456], see also Chapter 8. For small deformations the surface extension energy is given by «J(/?) = 47ra 2 (2 7 ( ) ^ 4 +£ G ^ 8 ),

R = R/2a

(27)

where 70 is the interfacial tension of a spherical, undeformed droplet and EG=(dj/d\nS)s=s

=(d"ild\nR2}

_ is the Gibbs elasticity of the surfactant

monolayer stabilizing the droplet. Often the second term in Eq. (27) could be ignored when compared to the first one, and this is the case for most emulsion systems. For systems with extremely low interfacial tension, however, the Gibbs elasticity term becomes dominant. Such systems often tend to form microemulsions [4,5,57] in which the bending rigidity of the interface also becomes important u, (R) = -8ira2B0H v

;

1

— R2 = 32irkc —f 1 - ^ U 2H0 ao{ 2a)

2

(28)

where Bo is the bending moment and kc is the bending elasticity constant [58], H = — 1 / a and H0 = — \/a0 are the local interfacial and spontaneous curvatures respectively while a and a0 are the droplet and the spontaneous curvature radii. The energy contributions given by Eqs. (27) and (28) do not depend on the distance between the droplets explicitly. The deformation (given by the radius R) is determined by the colloidal interactions that are present and they are distance dependent (see the discussion below, Chapter 8 and Refs. [6-11,44-

492

D.N. Petsev

46,54,55]). However, all colloidal type of interactions like van der Waals, electrostatic, depletion, steric, salvation, etc., decay with the droplet separation and become zero for infinite distance. The interfacial deformation related contributions do not necessarily disappear at infinite droplet separations and therefore may still generate statistical configurations that need to be properly taken into account. That is accomplished by the condition (26), see also the discussion in Section 4. 3.2.2. Colloidal interactions and droplet deformation The typical colloidal forces such as electrostatic, van der Waals and others induce the droplet deformation and are modified by it but at the same time. That is why the shape of a droplet doublet is determined by minimizing the free energy of interaction while taking into account all possible contributions [6,8-11]. Since detailed analysis of many particular cases has been presented elsewhere [6-11,44-46,54-56], in this review we will restrict ourselves to a few examples based on the Derjaguin-Landau-Vervey-Overbeek (DLVO) theory [48,50,51], which includes electrostatic repulsion and van der Waals attraction. The latter is accounted for in the framework of the model of Hamaker [59]. The electrostatic repulsion between two deformed droplets is given by the following expression u(h,R) v

'

= ^^tanh>[P^}eW(-2Kah)LRi K (3K

{

4

)

'\

+±-\

naj

h = h/2a

(29)

where h and R are the film thickness and radius respectively (see Fig. 1), Cel is the number electrolyte concentration, e is the elementary charge, \t 0 is the droplet surface potential and K is the Debye screening parameter defined by J =2 ^

C

"

.

(30)

z is the background electrolyte charge number and eoe is the dielectric permittivity of the solvent. In absence of any deformation, the film radius is zero and Eq. (29) reduces to the well-known nonlinear superposition approximation for the electrostatic repulsion between two colloidal solid spheres [48,50,51,53]. The van der Waals attraction between two deformed spheres with shapes approximated by those of truncated spheres could be determined by the following expression

Statistical Mechanics of Dense Emulsions

/ - ~\ A u (h,R) = -^L

493

1 1 hil + h) —+„, l ,,+21n-^ / L

J

(31)

2

4R

h2(l + h)\2 + h)2 where, again for no deformation (R = 0), the expression for the van der Waals energy reduces to that for two undeformed solid spheres [59], see also [48,50,51,53]. The coefficient AH is known as the Hamaker constant although it is not truly a constant and may depend on the separation distance between the droplet surface h due to the electromagnetic retardation [60-63]. If retardation is unimportant (like for small particles below one micrometer [53])

4=A^'A'"''l. 413 £ , - £ ,

(32,

16^2 ( „ » + „ > ) "

ex and e2 are the dielectric permittivities, while n\ and n2 are the refractive indices for the drops and the surrounding medium respectively. hp is Planck's constant, co is the fluctuation frequency of the interacting molecular dipoles and c is the speed of light. Hence, in absence of retardation, the Hamaker coefficient is a true constant. For larger particles however electromagnetic retardation becomes substantial and the Hamaker coefficient becomes

^=^-^^ 2 + ^4^4 2 i. 4/3 ex — e2

4im2

(33)

n, + n2 h

Eqs. (32) and (33) are approximations and there is a more complete theory [63]. However, it is too complicated to be introduced into a statistical mechanical model. A relatively simple interpolation formula between the non-retarded and fully retarded cases has also been suggested (see for example [53], and Chapter 8 of this book). The retarded Hamaker coefficient decays with the distance between the interacting surface and therefore the van der Waals energy decreases even faster with the separation. In addition to the retardation, the Hamaker coefficient could decrease even further if there is a presence of extremely high (~1M) electrolyte concentrations [53,62]. For lower

494

D.N. Petsev

concentrations, the electrolyte effect on the Hamaker constant could be safely ignored. The typical numerical values of Hamaker constants for the cases of oil/water/oil or water/oil/water systems (like most emulsions) are between 3 and 4xlO"21 J[52]. 3.2.3. Total energy of pair interaction We assume that the total energy of interaction between two droplets is a sum of all the contributions described above u(h,R) = umv(h,R) + uel(h,R) + us(R) + ub{R).

(34)

This is the energy to be introduced in Eq. (24) to obtain the pair energy averaged over all possible deformations. It is convenient, therefore, to rearrange the separation variable from film thickness, h (distance between the drop walls, see Fig. 1), to center to center distance, r using the simple geometrical relationship

I H* r = h + 2a.\\ V

r,

or

- /

r = h + yll-4R2

a

.... •

(•)•>)

r = r/2a, h = h/2a, R = R/2a Using (35) we can transform u\h,R\ into

u(f,R\.

4. EXAMPLES In this section we present some computation results for the radial distribution function, the structure and macroscopic properties of emulsion and microemulsions. The model system that is considered is a monodisperse ensemble of Brownian droplets suspended in quiescent fluid. It is important to remind at this point that monodisperse emulsion systems can be obtained [1] and have been for extensive experimental studies [54,64-72]. 4.1. Radial distribution function, structure factor and macroscopic properties of monodisperse droplets with low interfacial tension In this section we present some results for radial distribution functions and structure factors of emulsions and microemulsions. The computations were performed using the Rogers-Young closure approximation [42] as implemented in the numerical algorithm developed in Ref. [27]. Although Rogers-Young closure works best for repulsive potentials, our comparison to Brownian simulations shows that it is also applicable to systems where there might be an

Statistical Mechanics of Dense Emulsions

495

attractive component in the interaction (e.g., van der Waals attraction). An obvious effect of the droplet deformation is that the radial distribution function will exhibit non-zero values for interdroplet distances less than their diameter. This is shown in Fig. 3a where the radial distribution function for emulsion droplets (full line) is plotted together with that for hard spheres with the same droplet diameter (dotted line). The interaction between the droplets includes only the surface extension energy governed by the interfacial tension and infinitely short length surface repulsion that prevents the droplets from coalescing. This type of interaction implies that there is only one value of the film radius (if film is formed) for each center to center separation, see Fig. 1 a. Using Eq. (35) we obtain [11,45] us(F) = 27ra2lo(l-F)2

for

r <\

= 0 for r > 1 This form of the interaction potential provides that g(r —»oo) = 1 without the explicit application of the normalization procedure described above, see Eqs, (23-26). The volume fraction used in this calculation is 0 = 0.42 and ira2j/kT = 760. Besides the effect of non-zero values at separation less than the droplet diameter, it can be seen that the peaks are lower and less developed than the referent hard sphere case. This means that the local ordering decays more rapidly with the distance between the droplets compared to hard sphere suspension with the same volume fraction. This is also evident from the plot of the structure factor of the same systems, see Fig. 3b. Hence, the fluidity of the droplet interface decreases the propagation of the structure induced by the excluded volume interactions. It is instructive to compare the radial distribution function for emulsions (where the surface deformation is controlled by the interfacial tension) and microemulsions in which the interfacial properties are governed by the bending elasticity. Such and example is shown in Fig. 4. The energy of interaction is [11,45] u* r) = — ^ 1 - ^ 1-r, aQ 2j = 0,

f>\

r<\,

ao=a0/a .

(i/)

496

D.N. Petsev

Fig. 3. Radial distribution functions (a) and structure factors (b) for emulsion droplets (solid curves) and hard spheres (dotted curves). The parameters for the calculation are Pna2^ — 760 and (/> = 0.42. Reproduced by persmission of the American Institute of Physics, D.N. Petsev and P. Linse, Phys. Rev. E 55 (1997) 586.

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497

Fig. 4. Radial distribution functions (a) and structure factors (b) for microemulsion droplets (solid curves) and hard spheres (dotted curves). The parameters for the calculation are kc=\l(3, a0 = 1 and ^ = 0.42. Reproduced by permission of the American Institute of Physics, D.N. Petsev and P. Linse, Phys. Rev. E 55 (1997) 586.

498

D.N. Petsev

Eq. (37) was derived using similar geometric arguments as for (36). Again the normalization at infinite separation is fulfilled. The parameters for this calculation are = 0.42, the bending constant is kc =kT and the radius of spontaneous curvature a0 = a . The fluidity of the droplets, although governed by different parameters, has the same effect on the structure of the single drop phase microemulsion. This is evident by comparing the radial distribution function, Fig. 4a, and the structure factor, Fig. 4b, to the respective plots for hard sphere suspension. The effect of the droplet deformability for these particular examples (emulsion and microemulsion) is strong enough to be detected by experiments like measurements of the structure factor using light scattering [43].

Fig. 5. Radial distribution functions for uncharged (solid curve) and charged (dotted curve) emulsion droplets. The volume fraction in both cases is