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0O X Xa, (pcr XiX0®) Xe Xs ^rit if/ o) coi cos 0) co(r) &>(z) a Q Q Q Q
xxxv
density (kg mr3) number density [N/V) (m~3) space charge density (C mr3) hard sphere radius (m) surface density of brushes (m~2) surface charge density (C m~2) contribution of ionic species i to surface charge (C m~2) surface charge density diffuse layer (C mr2) surface charge density Stern layer (C m"2) standard deviation of x (dim. x ) characteristic time (s) interfacial stress (N m^1) line tension (N) turbidity (nr 1 ) stress tensor (N m~2) interfacial stress tensor (N m"1) flux of x-momentum in y-direction (kg m"1 s~2) = shear stress (N m~2), one of the nine components of the stress tensor rotational correlation time (s) rotational relaxation time (reorientation time) (s) yield stress (N m~2) volume fraction (-) osmotic coefficient (-) p h a s e (-) Flory-Fox constant (-) excess interaction energy p a r a m e t e r (-) critical values of x a n < i
; (rad s^ 1 or s~') solid angle (sr) grand potential (J) from an extensive study on flow in fibrous media \ we find that the Kozeny constant starts to decrease significantly below (j> —0.1. For 0.15, however, C values scatter around 5 ± 1, so indeed [2.2.67] with C = 5 is sufficiently accurate for dense filter cakes of particles of the type shown in fig. 2.1a-c. It is sometimes stated that the KC relation is purely empirical, but this is not quite correct. The pore geometries in a filter cake are clearly too complicated to allow an exact solution of the Stokes equations for viscous flow. The KC relation is an approximation for this solution at the cost of introducing a constant C, which in most cases is indeed empirical. The KC scaling itself, however, is a consequence of Poiseuille-like flow as can be seen as follows41. Consider a packing of particles (as in fig. 2.1) with solid volume fraction xb • ip--(xj for x = 0 ; dp/dx = 0 for x = x m . A 8 ) for clarity.) 0.19, the percolation line (indicated by open squares) is crossed. ' ' M . A . Rutgers, J.H. Dunsmuir, J-Z. Xue, W.B. Russel, and P.M. Chaikin, Phys. Rev. B53 (1996) 5043. 21 H. Verduin, J.K.G. Dhont, Phys. Rev. E52 (1995) 1811. can be substantially higher. The reason is that the smaller particles can plug the gaps between the bigger ones. This has the consequence that, for concentrated systems, the ratio (p/
xxxvi
CP i2° Q(N,V,U)
SYMBOL LIST
interfacial (excess) grand potential (J) interfacial (excess) grand potential per unit area (J m~2) n u m b e r of realizations = microcanonical partition function (-)
1
INTRODUCTION TO COLLOID SCIENCE
Hans 1.1 1.2 1.3 1.4 1.5
Lyklema Becoming acquainted with colloids Some definitions Demarcations and outline of Volumes IV and V Some historical notes General references
1.1 1.7 1.12 1.13 1.15
This Page is Intentionally Left Blank
I INTRODUCTION TO COLLOID SCIENCE HANS LYKLEMA
1.1 Becoming acquainted with colloids Volumes IV and V of FICS will deal with colloids and, by way of introduction, we shall adopt the same procedure as in sec. 1.1 of Volume I, viz., mentioning ten phenomena that all have colloidal roots. Three of these are repetitions from that section. 1. Rivers discharging in seas tend to form shorter deltas than do those flowing out into a lake. 2. In the Royal Institution in London visitors can see a liquid which looks like claret, but which is really a colloidal solution of gold, prepared halfway through the nineteenth century by Michael Faraday. 3. Children and adults are attracted and intrigued by the beautiful changing colour patterns of soap bubbles. 4. Many paints are interesting fluids: if applied by a brush or by spraying, they behave like liquids, but after application they no longer do so (or, at least, they should not!). 5. Very long ago the Egyptians were able to prepare ink of finely divided soot in water, although carbon itself is insoluble in water. 6. Several dyes that are insoluble in water dissolve very well after the addition of surfactants. 7. Under some conditions turbid emulsions, stabilized by a surfactant, become transparent following the addition of a second surfactant. 8. As early as 1200 - 1300 BC people living in what is now Israel and Palestine were able to make glasses with a silvery appearance. 9. Even moderate cooks manage to prepare quasi-solid edible products by adding only a few percent of gelatin to warm mixtures, followed by cooling. 10. Many porous soils have the propensity of salt-sieving, meaning that if seawater is percolated through them, the salt is withheld so that potable water emerges. These ten illustrations may serve as an introduction to the scope and phenomenological wealth of colloid science and its applications. Let us briefly review the respective backgrounds, thereby profiting from the general knowledge already obtained from Volume I and the interfacial science of Volumes II and III. Fundamentals of Interface and Colloid Science, Volume IV J. Lyklema (Editor)
© 2005 Elsevier Ltd. All rights reserved
1.2
INTRODUCTION
1. Besides dissolved matter, rivers also carry along small particles, covering the size range from a few nanometers {nanoparticles) across the coiloidal range (from severalto thousands of nanometers, or microns) to those of macroscopic size. The amount of particulate matter rivers can carry, as well as the size distribution and the nature of the particles, depends on the source, the geology, rainfall, industrial waste, and other factors along their course. Generally, the lower the particle concentration and the smaller the sizes, the more transparent is the river. Most particles have a density exceeding that of water, leading to sedimentation unless the current whirls them up. Bigger particles settle before the smaller ones. Delta formation is an automatic consequence of the reduced flow near the estuary. However, on top of this comes the phenomenon of colloid stability, that is the resilience against aggregation of the particles. Colloid stability can have a variety of origins and in rivers electrostatic and steric repulsions between the particles probably both play a role. It is typical for electrostatically stabilized (so-called electrocratic colloids, a term coined by Freundlich) that they are sensitive to indifferent electrolytes, whose addition leads to coagulation. This is the principle behind observation (1). The Dutch can give a telling illustration. In the Netherlands there is a branch of the River Rhine, called the Ussel. Until the 1930s it debouched into a sea, the Zuiderzee ("Southern Sea") which was salty because it had an open connection to the North Sea. However, the Dutch have the national habit of reclaiming land from the sea, and as part of their activities the connection between the Zuiderzee and the North sea was severed by a long dike. As a result, the salinity dropped and the Ussel delta became more extended. Since then the Zuiderzee has been renamed the "IJsselmeer" {Ussel lake). 2. Colloidal solutions (sols) of gold—and for that matter, all other water-insoluble materials—cannot be made directly simply by mixing large chunks of the solid with water. To obtain particles of the right size, one basically has two options: to apply dispersion (comminution), or condensation. According to the former, large particles are milled to create particles in the colloidal range. The latter approach starts from real solutions, rendering the dissolved matter insoluble by a physical or chemical process, letting the condensate grow until it is in the desired range. The trend is that the former approach is mostly followed in industry, whereas the latter prevails in basic research, because it allows better control. Faraday prepared his sols by a condensation method: he reduced an aqueous solution of gold chloride with phosphorous acid? The technical issues are immediately appreciated: the particle growth has to be stopped once the desired size is attained, and the synthesized particles should be stable against aggregation. Apparently, Faraday was successful in both respects. The ruby, wine-like colour is a consequence of light absorption at a specific wavelength range of the spectrum. Larger particles tend to be more bluish (and less stable, as students of colloid science know by experience). An additional property of sols is their strong lateral scattering of light. This phenomenon is nowadays known as the
INTRODUCTION
1.3
Tyndall effect, (see sec. 1.7.6), although it was discovered earlier by Faraday. When gold sols are electrocratic they can be coagulated by salt addition; when they are sterically stabilized they are (much better) salt-resilient. 3. Colours in soap films result from light diffraction and from the fact that the refractive index depends on the wavelength of the light. Light reflected from one side interferes with that from the other. Changing colours imply changes of thickness. On bubbles one can see that large patches of film of given thickness are moving with respect to those of other thicknesses. From the order of the colours one can even estimate the thickness as a function of time, and in this way follow the thinning or drainage process. By carrying out this analysis one will usually find the thickness to be above the colloidal range. However, for the colloid scientist, things become really interesting for other than aesthetic reasons if the thinning of the film has proceeded until below about 10~6 m. In sec. III.5.3 we have seen that the colloidal interaction forces across it then become relevant; in principle one can measure them in this way, obtaining so-called disjoining pressure isotherms. Thin soap films are therefore excellent model systems. Moreover, the options do not end here, because the way in which the film continues, or ceases, to thin further, determines the stability against rupture. Under the right conditions stable Newton films can be formed, i.e., patches that are so thin that they reflect hardly any light. To the naked eye they behave like black holes and it is historically interesting that, long ago Hooke1' observed them, interpreting them as such. However, real holes in a film are mechanically unstable; they grow very rapidly, leading to rupture. It is concluded that the formation of stable blackfilms is conducive for stability. 4. The paint example takes us to the realm of rheology. Anticipating our systematic treatment, one can say that the desired behaviour requires the paint to be thixotropic (a term briefly defined in sec. III.3b). Thixotropy can be realized if the disjoining pressure isotherm has a shallow minimum: weak attractive bonds keep the particles together, giving rise to a three-dimensional structure at rest, but the application of a shear-force can disrupt these bonds and fluidize the system. After cessation of the applied force the system re-solidifies. 5. As early as 4000-300 B.C. Egyptians were able to write on papyrus with red and black ink, of which the latter appears less perishable over the centuries. Figure 1.1 gives an example. The red pigments were Pb3O4 ('red lead') and HgS ('cinnabar' or 'vermilion'), the black one was soot. None of these is soluble in water. Nevertheless, the Egyptians managed to make ink by stabilizing fine dispersions of them using a vegetable biopolymer from the (gum arabic) bark of the Acacia Senegal or the animal biopolymers casein and albumin as the stabilizer; these also function as adhesives. Some of the stone-age cave paintings (15,000 - 12,000 B.C.) were made without
11
R. Hooke, On Holes in Soap Bubbles, Commun. Roy. Soc. March 28th (1672).
1.4
INTRODUCTION
Figure 1.1. Papyrus from the beginning of our era with Greek handwriting describing a quarrel between Agamemnon and Odysseus (Ilias IV, 340-346). The ink has aged at least as well as the papyrus (Courtesy Allard Pearson museum, Amsterdam, NL.)
binder, but others were stabilized and made to adhere by blood, honey or fats1'. Unwittingly these ancient artists anticipated the modern concepts of steric stabilization by macromolecules. In FICS, polymer adsorption has been discussed at some length in chapter II.5. Basically, the interaction between two polymer-covered particles is repulsive if the polymers are, (1), sufficiently strongly bound to remain on the surface upon interaction and, (2), if the extending tails and loops repel each other across the water into which they are embedded - in thermodynamic language, if water is a good solvent for them. There are various ways in which polymers can affect the stability of colloids. The above-mentioned case of stabilization is called protection. This term is used in particular when sols are made more resilient against coagulation by salts. Another example is that of gold sols, which can be protected by low concentrations of gelatin (below the gelation concentration). In all these cases, the stabilizing polymers are adsorbing hydrophilic colloids, which by their very nature are soluble in the solvent, or have soluble moieties extending into the solution. Gelatin-protected gold sols behave as gelatin sols, though with a heart of gold. However, under other conditions (mostly achieved by adding small concentrations of adsorbing high molecular weight polymers)
11
K. Beneke, Zur Geschichte der Grenzflachenerscheinungen, Gesellschqft, Reinhard Knof (Kiel, Germany) (1994).
Mitteilungen der Kolloid-
1.5
INTRODUCTION
(a)
f i-> i
Figure 1.2. Three modes of polymer-colloid interaction, (a) Two polymer-covered particles. The situation leads to protection if the solvent is good for the polymer; (b) Adsorption flocculation by long polymer molecules, simultaneously adsorbing on more particles; (c) Depletion flocculation caused by the solvent in the gap being sucked out into the solution.
(c)
polymers can also act as Jlocculants, i.e., they lead to aggregation (in this case called Jlocculation. Probably the mechanism is bridging, i.e., the formation of interparticle polymer links. A third mechanism is depletion Jlocculation, which occurs with nonadsorbing polymers. Now the mechanism has an osmotic or entropic origin. As polymers have a certain coil size in dilute solution1', the narrow gaps between approaching particles have to become depleted of polymers; the negatively adsorbed polymer has to find its way in the part of the solution far away from the interacting pair. The ensuing entropically driven tendency of the solution to move from the gap to the bulk drives the particles together. Three possibilities of polymer-colloid interaction are sketched in fig. 1.2. All of this will be discussed in more detail in chapter V. 1. 6. Although hydrophobic dyes are insoluble in water, they do dissolve in the hydrocarbon core of micelles. This process is called solubilization and, as it requires the presence of micelles, it can only be observed above the critical micelle concentration, c.m.c. In fact, solubilization has been invoked as a method for establishing the c.m.c. Micelles belong to the category of association colloids. 7. The disappearance of turbidity suggests that a true solution is formed (or an 11
See the discussion in sec. II.5.2.
1.6
INTRODUCTION
emulsion with droplets having exactly the same refractive index as the continuous medium, a rather hypothetical situation), but further study has indicated that the systems do contain very tiny droplets. Such systems are called micro-emulsions. Between common (or macro-) and micro-emulsions there is a difference of principle. Unlike the former, micro-emulsions are thermodynamically stable. When the ingredients are mixed in the right proportions and at the right temperature, the micro- emulsion forms spontaneously. Another difference is that, unlike macro-emulsions, micro-emulsions tend to be almost homodisperse (all droplets having the same size). Micro-emulsions constitute another category of systems with typical colloidal properties. 8. The silvery appearance of glass beads was achieved under the cross-fertilization of Syrian and Egyptian culture. The Syrians had a strong glass industry whereas the Egyptians were experienced in making faience. The silvery appearance was probably caused by colloidal silver1'. It is not known how our predecessors synthesized such solid-ln-solid colloids, but it is interesting to note that nowadays many ceramic materials are made from (homodisperse) colloids. 9. Solidified gelatin solutions in water are examples of thermo-rerversible gels; at high temperature they are fluid, but upon cooling they solidify and can be cut with a knife. They constitute a system with striking mechanical properties; macroscopically speaking, they behave as a solid, but if the self-diffusion coefficient of the water is measured it Is hardly lower than that in pure water. Apparently the gelatin molecules form cross-links In such a way that a three-dimensional network is formed, into the maze of which the water is phenomenologically immobilized. 10. Salt sieving is a process occurring in dense porous plugs of charged particles. The phenomenon can be explained on the basis of the negative adsorption of electrolytes by charged colloids (the Donnan effect). Theory can be found in sees. II.3.5f and 7e. Briefly, near charged surfaces there is a zone which is depleted of coions, which Is phenomenologically equivalent to a salt-depleted volume. When two such surfaces become so close that the depleted zones overlap, the gap between them becomes depleted as a whole, and therefore electrolytes cannot be transported along these pores. The ten examples given above introduce us not only to the fascinating richness and multifarious appearance of colloids, but also call for systematic treatment. We see that common features recur (for example, the effect of particle interaction in sedimentation, sol stability, and rheology) but sometimes manifest themselves in different ways (disjoining pressures across free- or wetting films are of the same nature as those between particles). To understand this, it helps to recognize common roots between different phenomena. However, one must be continually aware of the complexity of applied colloid science - compare the problems faced by our pre-historic artists and 11
P.S. Zurer, Chem. Eng. News, Feb. 21 (1983) 26.
INTRODUCTION
1.7
producers of ink for modern ink-jet printers. These considerations more or less define the tasks set for Volumes IV and V; to give a systematic treatment of the fundamentals of colloid science, with an eye open for applications. 1.2 Some definitions For various reasons, it is not easy - and perhaps not necessary - to give a simple and comprehensive definition of a colloid. The term itself is a misnomer. It stems from the Greek KoXXa, meaning glue, and was coined by Graham1' because some of his colloidal systems were glue-like. Nowadays we know that most colloids are not glue-ish, but the name has stuck. Over the years the notion has been subject to changes; some investigators give it a wider, others a more restricted meaning. In FICS we shall not adhere strictly to a defined size range but rather consider colloids as a particular state of matter, between true solutions and suspensions. For particulate matter this state is characterized by the relatively large fractions of molecules in the system that are in an interface. One of the fascinating options is that, compared to molecules in true solutions, the interaction between the particles can be tuned, for example by changing the electrolyte concentration. Although definition-wise the size-range of colloids is perhaps not of paramount relevance, in physical processes sizes play important roles, so we have to say something about them. The classic definition is on the basis of size, and the IUPAC definition reads as follows2'. "The term 'colloidal refers to a state of subdivision, implying that the molecules or polymolecular particles, dispersed in a medium, have at least in one direction a dimension roughly between 1 nm and 1 n or that in a system discontinuities are found at distances of that order. It is not necessary for all three dimensions to be in the colloidal range: fibers, in which only two dimensions are in this range, and thin films, in which one dimension is in this range, may also be classified as colloidal." According to this definition, polymer coils in solution are counted as colloids; a claim that most polymer chemists feel as trespassing. We shall not consider polymers as colloids but do treat them in volume V, mainly because of their relevance for steric stabilization. To the lower particle size, in recent years the terms nanoscience and nanotechnology have become fashionable for describing small particles composed of a limited number of atoms. However, nanotechnologists sometimes also claim as nanoparticles particles that are tens-, or even hundreds-, of nm large. We shall not do that, and as a rule only consider particles that are so large that they have their macroscopic bulk properties. Sometimes we have to consider the nanosize range, for
11 21
T. Graham, Phil. Trans. 151 (1861) 183. See the IUPAC manual mentioned in sec. 1.5.
1.8
INTRODUCTION
example in describing the synthesis of colloids (chapter IV.2). To the upper particlesize side one finds the suspensions (for solids), or the sometimes large drops in an emulsion (for liquids). We shall consider these as far as they exhibit colloidal properties. Regarding nomenclature, one should be aware of the fact that, in recent literature, colloidal phenomena are sometimes described under terms such as mesoscopic physics, or mesoparticle science. The origin of these terms stems mostly from the side of physicists who became increasingly conscious of the potentialities of colloids as systems having a scale intermediate between 'molecular' and 'macro'. Recall that in sec. II. 1.6a we have already met the classification of pore sizes in adsorbents as micropores (< 2 nm), mesopores (= 2-50 nm) and macropores (> 50 nm). According to this scheme, mesopores are of the (lower) size range of colloids. So, the prefix 'meso' is basically correct but does not add anything new unless it is further specified. We shall therefore refrain from using such terms. However, the caveat must be made that, because of the differing nomenclature and hence the differences in key words, important papers in the physical literature may escape the attention of colloid scientists and vice versa. The same applies to the fashionable term soft condensed matter.
Figure 1.3. Sols with spherical particles, (a) Homodisperse (very narrow size distribution); (b) Polydisperse (wide size distribution).
INTRODUCTION
1.9
Speaking of sizes implies speaking of size distributions. Sols in which the particles all have exactly the same size and the same shape we shall call homodisperse. Figure 1.3a is an illustration. The term is identical to isodisperse and monodisperse. Linguistically there is no reason to prefer one name over the other since all three have a Greek prefix and a Latin stem. However, we shall avoid the term homodispersed because the perfect participle suggests that the sol was made homodisperse by fractionation. When the particles have different sizes and/or shapes they will be called heterodisperse, or, when we want to emphasize that the size distribution is wide, polydisperse (see fig. 1.3b). In modern times the techniques for preparing well-defined sols have made much progress, but 100% homodispersity is rare. Perhaps biocolloids such as immunoglobulins are the sole illustrations. Very narrow size distributions are observed for hydrophilic colloids such as micro-emulsions, but hydrophobic colloids can nowadays also be made synthetically with such a narrow distribution that upon sedimentation or compression they can crystallize into crystallographically perfectly ordered arraysfsuch as hexagonal packing). See fig. 1.4 for an illustration. Recall that in sec. II. 1.2 we presented some electron micrographs of synthetic model colloids. Figure 1.5 gives an illustration of the other extreme. It is the last-mentioned systems that paint technicians have to deal with, whereas the systems belonging to the category of fig. 1.3a are rather the playground for scientists. We shall have more to say about this in chapter 2 and the following chapters. A colloidal dispersion is a system in which colloidal particles are dispersed in a continuous phase of a different composition, or state. In Faraday's gold sols, and in foams, water is the continuous phase. Sometimes we shall use the term 'colloid' as synonymous with 'colloidal system'. A fluid colloidal system may also be called a sol. Besides Faraday's gold sols, river water with finely dispersed particles in it, surfactant solutions above the am.a, and micro-emulsions are all sols. Emulsions are sols consisting of liquid droplets, dispersed in another liquid with which they do not mix. The latter is the continuous phase. Usually, one of the phases (w), is an aqueous solution, and the other an organic liquid which does not mix with water (o, for 'oil'). So, emulsions can be of the oil-in-water (o/w) or water-in-oil type (w/o). Changing the type of an emulsion (from w/o —> o/w or the other way around) is called inversion. More complicated emulsions also exist, for example o/w/o (oil droplets in water in oil). These are called multiple emulsions. Although emulsion droplets mostly exceed the usual upper size of colloids we shall include them in our discussion (chapter V.8) because of their interesting stability features (in addition to aggregation, droplets can also merge to form a larger one or coalesce), and immense practical interest. Paradoxically, in the jargon of photography, photographic 'emulsions' are not emulsions but particulate sols, whereas such sols are called 'emulsions'. A latex is an emulsion or sol of polymeric particles. The plural is latices (or latexes). Latices are also known as polymeric colloids. As they can be made with narrow size distributions and with specific bulk- and surface properties they are
1.10
INTRODUCTION
Figure 1.4. Electron micrograph of a crystallized homodisperse silica. Marker equals 1 u. (Redrawn from J. Mater. Set. Lett. 8 (1989) 1371; courtesy of A.P. Philipse, Utrecht, The Netherlands.)
Figure 1.5. Electron micrograph of a copper phthalocyanine pigment for automotive paints, made by ball milling. (Courtesy, J. Schroder, BASF, Ludwigshafen, Germany.)
INTRODUCTION
1.11
favoured model systems, both in science and technology. AJoam is a dispersion in which a large fraction of the volume is a gas, dispersed in a liquid, a solid, or a gel. Young foams often consist of spherical bubbles in the continuous phase; bubble Joams or spherical Joams (the latter name is rather unfortunate). Older foams tend to consist of flat lamellae, connected to each other by Plateau borders. They form the familiar polyhedric structures, in which the flat lamellae are of colloidal thickness, whereas the gas compartments are much larger. We shall use the terms foam and froth as equivalent. In some branches of technology one of the terms is more fashionable than the other (e.g., 'froth flotation'). Scientifically speaking, much more has to be said about foams than that they are seen as a set of connected thin films. Therefore, we shall treat thin films and foams in two successive chapters, viz. V.6 and 7, respectively. With respect to their stability in the thermodynamic sense colloids can be subdivided into two groups. (i) lyophobic (hydrophobic if water is the continuous phase) or irreversible colloids, which are thermodynamically unstable. (ii) lyophilic [hydrophilic) or reversible colloids, which are thermodynamically stable. Sols belonging to the former category cannot be made by simply mixing the material to be dispersed and the liquid. A detour is needed and we shall devote an entire chapter to it (chapter IV.2). Such sols are only kinetically stable, in the sense that the particles are kept from contacting each other by special means. Electrical charging is one of the important mechanisms. It leads to double layer formation and two double layers of the same sign repel each other. We have already noted that such electrocratic sols are sensitive to indifferent electrolytes (electrolytes that do not contain chargedetermining ions); when their concentrations exceed a certain critical coagulation concentration (c.c.c.) stability is lost and coagulation ensues. Although there is no sharp demarcation between slow and fast coagulation, c.c.c. values can be established fairly reproducibly. It was already known by the year 1900 that the c.c.c. is drastically lowered (more than proportionally) if the valency of the counterion is increased. This qualitative rule, known as the Schulze-Hardy rulel) was one of the challenges to be met in the development of stability theory (chapters IV.3 and 4). Sols belonging to category (ii) can be made simply by mixing the ingredients. The Gibbs energy of this process is negative. Examples of hydrophobic colloids are:- all inorganic sols, thin films, and (macro-) emulsions; examples of hydrophilic sols include several biocolloids, association colloids, and micro-emulsions. It should be noted that the terms 'hydrophobic' and 'hydrophilic' have meanings 11 After H. Schulze, J. Prakt. Chem. (2) 25 (1882) 431; 27 (1883) 320 and W.B. Hardy, Proc. Roy. Soc. 66 (1900) 110; Z. Physik. Chem. 33 (1900) 3051
1.12
INTRODUCTION
which differ from those used to indicate the relative affinity of water for a surface, as quantified in terms of the water contact angle (sec. III.5.lla) or the heat of immersion (table II. 1.3 in sec. II.1.3f). Even if an ultra-clean gold surface has a zero contact angle for water (table III.A4.1) the Gibbs energy of a collection of gold microcrystals in water is always higher than that of one large gold crystal having the same mass. This list of definitions is not complete. We shall, of course, use all the definitions already given in previous volumes and introduce new ones where needed in specific chapters. 1.3 Demarcations and outline of Volumes IV and V There are several ways of condensing into two Volumes the rich variety that colloid science has to offer; each of these involves choices about what to include and what to exclude. The style of FICS requires a deductive approach, rather than a treatment by kind of system. In a deductive treatment one would choose an initial discussion of principles (such as particle interaction) and typical methods (such as light scattering and rheology), and applying all of this to specific systems in later chapters. On the other hand, a system-oriented approach would call for a discussion of each kind of colloid (hydrophobic sols, association colloids, etc.) and, in passing, considering all applied techniques and their typical properties. Both procedures have their merits and drawbacks. We shall take a kind of hybrid route, in which a number of principles of wide relevance will be treated as such whereas some systems with special characteristics will be allocated to separate chapters. More specifically, in the present Volume we shall emphasize 'particulate1 colloids (lyophobic colloids in which particles can be distinguished), treating relevant properties such as preparation, characterization (size distribution, surface properties), interaction between pairs of particles (electrostatic, and other contributions to the disjoining pressure), with consequences for the kinetics and dynamics (e.g., how fast does a sol coagulate and what are the mechanistic steps in this process?), and concentrated systems (phase separation, fractal aggregates). As the Theological properties of particulate sols primarily depend on the number, properties, and interactions between the particles, a special chapter will be devoted to that. Volume V will contain one methodical chapter (on steric interactions), but otherwise it will deal with 'non-particulate' colloids (polyelectrolytes, biopolymers, association colloids, micro-emulsions, thin films, foams and emulsions), which for easy reference are collectively called 'soft colloids'. In designing these Volumes, a certain restraint must be applied to the degree of complexity to be covered. Among items to be omitted one could think of the fine-tuning of particle interaction models, mixtures of colloids, mixtures of surfactants in association colloids, and polymer-surfactant interactions. These, and other multicomponent interactions lead to a plethora of, sometimes exhilaratingly complex, phenomena, that are abundant in colloid science. We shall also de-emphasize liquid
INTRODUCTION
1.13
crystals and more descriptive features such as ternary phase diagrams, notwithstanding their practical interest. It is simply beyond the scope of FICS to cover all of that, but we shall try and indicate where such extensions and applications come into the picture. We shall mostly consider systems in which a liquid is the continuous phase, meaning that aerosols (colloids in the gas phase) will at most be included cursorily. Overall, these two Volumes will cover the fundamentals of colloid science. 1.4 Some historical notes Colloid science has a relatively long history. Although prehistoric man applied colloidal know-how many thousands of years ago, the cradle of colloid science dates to halfway through the nineteenth century, which is still long before the existence of molecules was irrefutably established. In retrospect, people such as Selmi1' and Graham2' are usually called the founding fathers of colloid science because they were the first to start systematic work on the preparation, characterization and stability of colloids. Selmi had already established the strong detrimental influence that electrolytes have on the stability of sulphur, and other sols. To these workers the name of Faraday 3 , who did his famous experiments with gold sols around 1857, may be added. He also noted the salt effect, and considered light-scattering and the protection by gelatin. He went so far as to realize that changes in the properties of the gold-solution interface play an important role. Had he subjected his sols to an electrical field, he would have discovered that stability and electrophoretic mobility are related. Electrokinetic phenomena have been known since 1809 (see chapter II.4) but the relationship to stability was only established around 1900 by Hardy4'. Perrin, Burton, Kruyt, and many other investigators pursued this line further. One of the main findings was the dramatic difference between (what are now called) hydrophobic and hydrophilic colloids. The latter category is much more resilient towards electrolytes which, in these systems, essentially act as modifiers of the solvent quality (sec. 1.5.4). The discovery of the colloidal state prompted the development of a number of physical techniques. Besides electrokinetics we may mention the discovery of the ultramicroscope by Siedentopf and Zsigmundy5' in 1903, which allowed the direct observation of (particulate) colloids on the basis of their laterally scattered light (the Tyndall effect) which won Zsigmundi a Nobel prize. In this connection, the Nobel prize awarded to Svedberg in 1926 for inventing the ultracentrifuge may also be mentioned.
11
F. Selmi, Nuovi Ann. di Scienze Naturale di Bologna, Ser. II, IV (1845). T. Graham, foe. cit. 31 M. Faraday, Phil. Trans. Roy. Soc. 147 (1857) 145. 41 W.B. Hardy, loc. cit. 5 See R. Zsigmundy, Zur Erkenntniss der Kolloide: fiber Irreversible Hydrosole und Ultramikroskopie. Gustav Fischer (1905); R. Zsigmundi, Colloids and the Ultramicroscope: A Manual of Colloid Chemistry and Ultramicroscopy (transl. by J. Alexander) John Wiley (1909). 21
1.14
INTRODUCTION
Essentially this was an instrument with which hydrophilic colloids, having a small density difference from water, could be fractionated. The quest for a comprehensive theory to account for the stability of hydrophobic colloids dominated the nineteen twenties to -forties and culminated in the development of the now well-known DLVO theory, named after Deryagin and Landau, and Verwey and Overbeek, who developed it independently during the second world war1 2 '. The acronym was coined by Sam Levine. (In passing, and tongue-in-cheek, the American colloid scientists Marjorie and Robert Void referred to it as the Verwey-OverbeekLandau-Deryagin (VOLD) theory.) This theory describes particle-pair interaction insofar as it is determined by electrostatic repulsion and Van der Waals attraction. One of the basic issues that had to be solved was the accounting for the re-distribution of the countercharge upon interaction as a whole, electric double layers are electroneutral, so why would two identical double layers always repel each other at any distance? One of the merits of the DLVO theory was that it could account for the Schulze-Hardy rule. We shall discuss and extend this theory in chapters IV.3 and 4. Of the many other scientists who contributed significantly to this development, we mention Freundlich, who emphasized the relationship between adsorption and stability3'. As we shall show, the desorption of molecules or ions during particle encounter is a key element (although not fully appreciated) in understanding stability phenomena. The other two contributions to pair interaction, steric- and solvent-structure interactions entered quantification in the later half of the twentieth century. Theory for the effect of polymers on colloid stability was initiated by Scheutjens and Fleer4' and de Gennes5', using a lattice theory and a scaling approach, respectively: see chapter V.I. Solvent structure-mediated forces were quantified by several Schools, particularly in the Soviet Union. In FICS they have already been discussed in sec. III.5.3. Modern colloid science has profited greatly from the development of well-defined model colloids and new physical techniques. Special mention must be made of the surface force apparatus, which allows direct measurement of the forces between two surfaces. It was particularly Israelachvili and his co-workers6' who, after much basic work by others, made this apparatus so user-friendly that it can now be found in many colloid chemical laboratories. The modern variant of this is atomic force microscopy or AFM.
11
B.V. Derjaguin (= Deryagin), L.V. Landau, Acta Physicochim. URSS 14 (1941) 633. E. J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier (1948). ^ H. Freundlich, see the reference in sec. 1.5b. 41 G.J. Fleer, M.A. Cohen Stuart, J.M.H.M. Scheutjens, T. Cosgrove and B. Vincent, Polymers at Interfaces, Chapman and Hall (1993). 51 P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press (1974). 6 J.N. Israelachvili, Intermolecular and Surface Forces, 2nd ed. Academic Press (1992). 21
INTRODUCTION
1.15
When one surveys the development of colloid science one may conclude that there has been a continuing cross-fertilization with physical sciences in general, both with respect to the development of concepts and experimental techniques. All of this and much more, will be discussed in the coming Volumes. 1.5 General References 2.5a IUPAC recommendation The general recommendations are: Definitions, Terminology and Symbols in Colloid and Surface Chemistry, prepared for publication by D.H. Everett, Part I, Pure Applied Chem. 31 (1972) 579 and Quantities, Units and Symbols in Physical Chemistry, prepared for publication by I. Mills, T. Cvitas, N. Kallay, K. Homann and K. Kuchitsu, Blackwell (1988). We largely heed their recommendations. 1.5b General texts on colloid science. A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th ed. Wiley (1997). (This well-known textbook also contains some sections on colloids; of necessity more concise than FICS.) A.E. Alexander, P. Johnson, Colloid Science I and II. Cambridge University Press (1949). (Has been, for a longtime, one of the leading books, but is not so quantitative.) A. von Buzagh, Kolloidik; eine Einfiihrung in die Probleme der Modernen Kolloidwissenschqfft. (Of historical interest, by a Hungarian author.) (English transl. Colloid Systems, Technical Press, London (1937).) D.F. Evans, H. Wennerstrom, The Colloidal Domain; where Physics, Chemistry, Biology and Technology Meet, VCH (1994). (About the same level as FICS but more condensed; with an emphasis on association colloids; also contains some interfacial science. Generally well written, but underexposes non-electrostatic contributions to electrical double layers.) D.H. Everett, Basic Principles of Colloid Science, Royal Society of Chemistry (1988). (Concise introduction, also contains some surface science; rather classical.) H. Freundlich, Kapillarchemie, 1st ed. (Leipzig, 1909) and many updates, including the English translation Colloid and Capillary Chemistry, Methuen (1926). (Of more than passing historical interest because it contains a plethora of discriminating experiments and perspective views.) E. Hatschek, The Foundations of Colloid Chemistry. E. Benn (London) (1925). (Contains reprints and English translations of historical papers from the second half of the nineteenth century.)
1.16
INTRODUCTION
P.C. Hlemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, 3 rd ed. Marcel Dekker (1997). (The well-known introductory text by the first author is now revised and expanded. Generally well written and informative. About 400 out of more than 600 pages are devoted to colloid science.) Handbook of Applied Surface and Colloid Chemistry. K. Holmberg, Ed., John Wiley (2001). (Extensive review, grouped into five categories. Surface chemistry in important technologies, surfactants, colloidal systems and layer structures at surfaces. Phenomena in surface chemistry, analysis and characterization in surface chemistry); may be considered the 'applied' counterpart of FICS.) R.J. Hunter, Foundations of Colloid Science. Oxford Science Publ. I (1987); II (1989). (With respect to their level and size, these books may be considered 'primus inter pares' as the comparison of other texts with FICS IV and V are concerned; they are informative; 12 out of the 18 chapters deal with colloids, the others with topics already dealt with in FICS II and III. The level of the various chapters is variable and generally the treatment is less systematic.) Later, Hunter wrote a second edition, published in 2000, in which the material of Volumes I and II was condensed into one volume. Colloid Science, H.R. Kruyt, Ed., Elsevier. Irreversible Systems (1952). Reversible Systems (1949). (Reversible and irreversible = lyophobic and lyophilic, respectively. Very well presented overview; FICS IV and V may be considered their successors.) K.J. Mysels, Introduction to Colloid Chemistry. Interscience (1959). (Introduction for a course; covers most aspects of colloid science. Original, with enlightening explanations and capricious definitions in places.) W.B. Russell, D.A. Saville and W.R. Showalter, Colloidal Dispersions, Cambridge Univ. Press (1989). (Rather advanced, compactly written, requires vector and tensor analysis; emphasis on rheology, diffusion and transport phenomena.) D.J. Shaw, Introduction to Colloid and Surface Chemistry, e.g. 3 rd ed. Butterworth (1980). (Has for a long time been a first introduction to the field.) M.J. Void, R.D. Void, Colloid Chemistry, Addison-Wesley (1983). (Covers our Volumes II - V and is, of necessity, more condensed.)
2
PARTICULATE COLLOIDS: ASPECTS OF PREPARATION AND CHARACTERIZATION
Albert Philipse 2.1
Introduction
2.2
Preparation
2.3
2.2a
Size control
2.6
2.2b
Homogeneous precipitation
2.2c
Precipitation kinetics
2.2d
Particle growth and polydispersity
2.16
2.2e
Particle solubility and Ostwald ripening
2.22
2.3
2.4
2.5
2.1
2.8 2.13
2.2f
Seeded nucleation and growth
2.27
2.2g
Comminution and other preparation methods
2.30
2.2h
Separation and fractionation techniques
2.31
2.2i
Surface modification
2.35
2.2j
Other methods
2.37
Characterization
2.38
2.3a
Visual observations and microscopy
2.39
2.3b
Light scattering
2.43
2.3c
Surface area
2.48
2.3d
Sedimentation
2.50
2.3e
Other methods
2.56
2.3f
Size distributions
2.59
Examples of sol preparations
2.63
2.4a
Silica sols
2.63
2.4b
Sulphur sols
2.65
2.4c
Boehmite and gibbsite sols
2.66
2.4d
Ferrofluids
2.67
General references
2.68
This Page is Intentionally Left Blank
2 PARTICULATE COLLOIDS: ASPECTS OF PREPARATION AND CHARACTERIZATION ALBERT PHILIPSE
2.1 Introduction
Dispersions of inorganic colloids have been prepared and processed since the very beginning of human technology. Already around 7000 BC, about 4000 years before the invention of the wheel, the Near East produced complicated ceramic shapes, which manifested a thorough practical knowledge of concentrated clay dispersions and their processing. Such knowledge is still indispensable in the fabrication of traditional ceramics, such as pottery. The desired outcome of shaping techniques, such as the slip casting of clay dispersions, critically depends on the skilful preparation of colloidal suspensions. Important parameters are the shape and size distribution of particles, their concentration and state of aggregation, which is controlled by ionic strength and polymeric additives. Optimization of these parameters is often a laborious trial and error process and, so, it is not surprising that details of industrial preparation are usually either patented or kept confidential. Another impressive and historical example of dispersion preparation underlies the very pages on which this text is written. Papermaking11 starts with the degradation of wood chips to an aqueous suspension of cellulose fibres with a large percentage of fibres with dimensions in the colloidal size range. Inorganic particles, in the form of silica or bentonite sols, are added to improve the quality and rate of papermaking, a process which comprises the filtering and drying of the mixture of fibres and sol particles on a wire. Dried sheets run out of a papermaking machine at a rate of a few hundred metres per minute, or even faster, and any slight change in the properties and composition of the starting dispersions may have a disastrous effect on this very rapid process. Paper also reminds us of other colloidal fluids, such as paints and ink, with roots nearly as ancient as those of ceramic suspensions. The example of ink preparation by the Egyptians for writing on papyrus is well known21. The Roman author Vitruvius
U
S.G. Mason, Tappi 33 (1950) 440; R.B. McKay (Ed.), Technological Applications of Dispersions, Marcel Dekker (1994). 21
K. Beneke, Zur Geschichte der Grenzflachenerscheinungen, Verlag Reinhard Knof, (1995).
Fundamentals of Interface and Colloid Science, Volume IV J. Lyklema (Editor)
© 2005 Elsevier Ltd. All rights reserved
2.2
PREPARATION AND CHARACTERIZATION
(born around 100 BC) mentions in his De Architectural the deposition of soot on a wall and its manufacture to ink by mixing it with gum (resin). This is an early reference to steric stabilization of inorganic colloids, in this case carbon particles in water. The carbon colloids function as pigments giving the ink its colour. Many other pigments in printing ink, paints and plastics are found in the form of finely ground inorganic oxides or hydroxides. Iron oxides (see also sec. 2.4d) such as red haematite (a-Fe 2 O 3 }, dark brown maghemite (Y-Fe2O3) and black magnetite (Fe 3 O 4 ), were widely applied in ancient painting2' and still belong to the most important pigments31. In paintmaking we probably find the earliest examples of colloid preparation that goes beyond the mere processing of natural materials. The ancient Egyptians, for example, knew how to synthesize the green pigment verdigris, mainly composed of Cu(OH)2, and a silicate pigment known as Egyptian blue with CaCuSi4O10 as the main component41. Also, for these synthetic materials grinding or milling must have been required to obtain the desired pigment dispersion. For direct precipitation of inorganic particles in a liquid phase, or at least reports thereof, we have to make a leap in history. The Flemish chemist van Helmont (1577-1644) fused silica sand with excess alkali to form so-called waterglass and discovered that silica was recovered by treating the waterglass with acid51. Interestingly, waterglass is still a major source for the preparation of silica particles and gels. The method, as will become clear in this chapter, is also a didactic illustration of many aspects of particle formation (see also sec. 2.4a). Another earlier documented example of inorganic colloid synthesis is that of the pigment Prussian Blue (iron (III) hexacyanoferrate (II)). It was discovered in 1710 that when solutions of potassiumferrocyanide and ferric chloride are mixed, deep blue particles precipitate instantaneously61. This beautiful classroom demonstration of colloid formation raises a number of questions for the attentive student, as soon as it is realized that precipitates are actually sub-visible colloidal particles or agglomerates thereof (a by no means trivial insight). What determines the sizes of the colloidal particles and how can they be controlled? What factors determine the growth rate of particles and why is it that nucleation is sometimes extremely fast and sometimes extremely slow? How can we characterize and control the size distribution of particles? What other methods are suitable to monitor properties of the dispersed colloids?
11
Vitruvius, On Architecture, edited and translated by F. Granger, Harvard Univ. Press (193134). Two volumes. 21 W.J. Russell, Ancient Egyptian Pigments, Nature 49 (1894) 374. 31 A. Giltes, Eisenoxid Pigmente in; Pigmente, Ullmann's Enzyklopadie der Technischen Chemie, 3 Aufl., Band 13, Verlag Chemie (1951-70). 41 K. Volke, Kolloidchemie im Altertum, Akademie der Wissenschaften der DDR, Forschungsinstitut fur Auibereitung Freiberg (1989). 51 W.H. Brock, The Fontana History of Chemistry, Fontana Press (1992). 61 Prussian Blue colloids are a true classic: Selmi studied their precipitation as early as 1847. Renewed interest was sparked by their magnetic properties, see S. Choudhury, N. Bagkar, G.K. Dey, H. Subramanian, and J.V. Yakhmi, Langmuir 18 (2002) 7409.
PREPARATION AND CHARACTERIZATION
2.3
Such questions motivate us to study In this chapter several aspects of preparation (sec. 2.2) and characterization (sec. 2.3) of mainly inorganic colloids. The aim is to provide a brief introduction, comprising some basic principles, useful facts and characterization methods, together with references for the reader to pursue a topic in much more depth than is possible or desirable in this chapter. The focus will be on colloids with (approximately) spherical shapes, which simplifies the treatment, and Is also reasonable In view of the widespread study of colloidal spheres. Nevertheless, an occasional reference will be made to anisotropic particles to do some justice to the inorganic colloids in nature and industry with sometimes quite extreme aspect ratios (fig. 2.1), as in the clay dispersions referred to earlier. 2.2 Preparation Insoluble substances, such as metals and their oxides, do not disperse spontaneously in water, so work is needed to bring them into a dispersed colloidal state. One strategy Is prolonged milling and fracturing of minerals In a solution of stabilizing surfactants or polymers until a colloidal system is obtained. A drawback is the broad variety in
Figure 2.1. Examples of random packings of inorganic model colloids with increasing shape anisotropy: (a) silica spheres, (b) haematite spindles, (c) boehmite-silica rods, and (d) imogolite fibres. (Sources, see ref. .) 11 Pictures redrawn from D.M.E. Thies-Weesie, A.P. Philipse, J. Colloid Interface Sci. 162 (1994) 470 (a); D.M.E. Thies-Weesie, A.P. Philipse, and S. Kluijtmans, J. Colloid Interface Sci. 174 (1995) 211 (b); M.P.B. van Bruggen, Langmuir 14 (1998) 2245 (c); and G.H. Koenderink, S. Kluijtmans, and A.P. Philipse, J. Colloid Interface Sci. 216 (1999) 429 (d).
2.4
PREPARATION AND CHARACTERIZATION
Figure 2.2. Some examples of industrial inorganic colloids: (a) cordierite particles prepared by milling used in refractories, (b) kaolinite platelets used in porcelain, (c) magnetite particles in ferrofluids (courtesy of Diona Bica, Timisoara, Romania) and (d) alumina grains in sintered ceramics.
shapes and sizes of the final colloids (fig. 2.2). To achieve better control of the morphology of colloidal particles, a condensation (or precipitation) method is preferred. Here, the colloidal state is approached from a molecular solution in which solute molecules are made to precipitate or polymerize Into large units. The distinction between the two methods, milling and precipitation, can be illustrated by using a glass beaker in the preparation of a silica sol as the starting material. Glass largely consists of amorphous silica and hardly dissolves in water of pH ~ 7. So, to bring the material of the beaker into a colloidal state, we could fracture it and treat the glass pieces in a ball mill in water until a sol is obtained. The condensation alternative would be first to dissolve the glass pieces In a strongly alkaline sodium hydroxide solution to obtain waterglass, which is then diluted to a low weight percent of soluble silica and acidified to neutral pH, during which colloidal silica will precipitate (see also sec. 2.4a). Acidification is needed here to achieve a sufficiently large supersaturation of dissolved silica, exploiting the fact that the solubility of silica strongly decreases below pH —10. The glass milling produces a polydisperse sol, whereas silica polymerization in a waterglass solution can be controlled to yield silica particles with a narrow size distribution in what can be counted as one of the classic sols of inorganic colloid
PREPARATION AND CHARACTERIZATION
2.5
chemistry . In principle, any substance can be brought into colloidal dispersion via precipitation in a supersaturated solution. All that is needed is a method to achieve a sufficiently large supersaturation of the desired material to induce homogeneous nucleation (sec. 2.2b) and prevent or control heterogeneous precipitation (sec. 2.2f). Of course, measures must be taken to ensure colloidal stability of the growing particles, such as increasing the particle surface charge (keeping the pH far away from the isoelectric point) or adding a stabilizing protecting polymer. A high supersaturation can, for instance, be achieved by a chemical reaction which produces a poorly soluble substance. A classic example21 is bubbling hydrogen sulphide through a saturated arsenic trioxide solution to produce an arsenic trisulfide sol As2O3 +3H 2 S-> As 2 S 3 4+3H 2 O
[2.2.1]
Other strategies involve mixing two soluble salts: AgNO3 + KI -» AgU + KNO3
[2.2.2]
the reduction of a metal salt to produce metal colloids H2PtCl6 + BH^ + 3H2O -» Pt i + 2H2 t + 6HC1 + H2BOg
[2.2.3]
Ag2O + H2 -> 2Ag i + H2O
[2.2.4]
and the hydrolysis of metal salts to form oxides or hydroxides 2FeCl3 + 3H2O -> Fe 2 O 3 I +6HC1
[2.2.5]
A1C13 + 2H2O -» A1OOH i +3HC1
[2.2.6]
Precipitation can also be induced by a change in temperature, pH or solvent composition. For example, when water is added to a sulphur solution in ethanol, sulphur particles precipitate because sulphur has a much lower solubility in w a t e r . Metal alkoxides are increasingly used as alternatives for inorganic salts in colloid synthesis. The alkoxides easily hydrolyze to reactive monomers, which polymerize to form discrete particles or gels (networks of particles). The archetypical example is silicium tetraethoxide or tetraethoxysilane (TES), which hydrolyzes as Si(OR)4 + nH2O -» Si(OR)4_n(OH)n + nROH
11
[2.2.7]
R.K. Her, The Chemistry of Silica, John Wiley (1979). For this and other classical examples, see Colloid Science I, Irreversible Systems, H.R. Kruyt, Ed., Elsevier (1952). This is the so-called Von Weimarn sulphur sol.
2.6
PREPARATION AND CHARACTERIZATION
where R is an ethoxy or other alkoxy group. Partially hydrolyzed TES molecules polymerize via condensation reactions such as (OR)3Si-OH+ HO-Si(OR)3 -> (OR)3Si-OSi(OR)3 + H2O [2.2.8] (OR)3Si-OR+ HO-Si(OR)3 -> (OR)3Si-O-Si(OR)3 + ROH Such condensation reactions, depending on the reaction conditions (see e.g. the silica synthesis in sec. 2.4a), may under well-controlled conditions lead to well-defined silica spheres or networks and gels of aggregated small particles. For inorganic colloid syntheses, such control of particle size and structure Is the exception rather than the rule, and is based on in-depth studies as illustrated by Iler's classic study on silica11. 2.2a Size control Dispersed systems, in which all particles have the same or nearly the same size, have always attracted the attention of colloid science. Such monodisperse (also referred to as homodisperse or isodisperse) sols may be of practical Importance; colloidal crystals in photonic materials require uniform particles, and semiconductor colloids in the nanometer size range have specific optical properties, which are very sensitive to particle size2'. The sizes of silver halide colloids for so-called photographic emulsions need to be controlled to less than about 5% to optimize their photographic properties3 ; a demand which implies tight control of particle nucleation and growth. However, for many practical suspensions, such as in paints or ceramic processing, a modest polydispersity is not a serious problem, and is sometimes even beneficial. For example, the random packing density of spheres mixtures is greater than that of monodisperse particles and, consequently41, the viscosity of the mixtures is generally below the viscosity for monodisperse spheres at the same volume fraction. Thus, manipulating the size distribution may be helpful for the processing and densification of sols of ceramic particles. One academic motivation for monodispersity is its requirement of a critical test for theories of colloidal systems or thermal systems in general. Thermodynamically speaking, colloids are nothing but giant molecules but their large sizes allow studying; for example, their (thermo)dynamics via light scattering techniques or microscopy. Preferably there is only one particle size, or a very narrow size distribution, in the sol to keep theory and data interpretation manageable. James Clark Maxwell, unaware of the existence of isotopes, argued51 that the monodispersity of atoms could only be
11
R.K. Her, loc. cit. See for example C.B. Murray, C.R. Kagan, and M.G. Bawendi, Ann. Rev. Maler. Sci. 30 (2000) 545. 31 I.H. Leubner, Current Opinion Colloid & Interface Sci. 5 (2000) 151. 41 For the viscosity as a function of volume fraction, see sees. 6.8 and 6.10. 51 J.C. Maxwell, Nature 8 (1873) 437. 21
PREPARATION AND CHARACTERIZATION
2.7
secured by the Lord himself. The need for divine intervention on the colloidal scale may be disputable, but the preparation of large quantities of nearly identical inorganic colloids is certainly a demanding task, largely due to unavoidable thermal fluctuations in a precipitating solution as will be explained later. The importance of monodispersity was already clearly perceived from the beginnings of colloid science, as witnessed in the work of Jean Perrin11 on the verification of Einstein's theory for Brownian motion and his demonstration of the thermodynamlc equivalence between colloids and molecules referred to above. Einstein derived that the average mean square displacement, (r 2 ) = 6Dt, of a colloidal particle in time t is determined by the translational diffusion coefficient D = kT/J
[2.2.9]
which expresses that Brownian motion, driven by the thermal energy kT, is counteracted in a liquid by the hydrodynamic friction factor / . Einstein's results are valid for particles of arbitrary shape21 but, of course, for an experimental test J must be specified. The obvious choice is the Stokes friction factor, namely / = Qitrp., which is valid for a hard sphere of radius a in a solvent with viscosity rj. Thus, the diffusive displacements of monodisperse spheres with known radii provide a test of the Einstein equations without any adjustable parameter. The well-known outcome of this test by Perrin31 is often considered as the first decisive evidence for the existence of molecules . Perrin realized that this evidence was as strong as his colloids were monodisperse and, so, he and his co-workers undertook a laborious fractional sedimentation procedure to obtain a few hundred milligrams of uniform resin spheres from an initial weight of a kg of gamboge or mastic. This substance was dissolved in methanol and then precipitated by dilution in a large volume of water, resulting in monodisperse fractions of emulsions of spherical particles with a wide variation in size between these fractions. Fractional sedimentation, which in Perrin's case took several months, is not a very practical procedure. An interesting alternative is the addition of non-adsorbing polymers, which cause a depletion attraction (see sees. V. 1.8 an 9) with strength depending on the particle size. The repeated, size-selective, phase separations may produce quite uniform emulsions51. Nevertheless, if possible we would like to avoid fractionation altogether. Realizing that nature provides a very limited source of monodisperse colloids, at least with respect to inorganic particles, we need to understand the essential aspects underlying preparation of uniform particles by precipitation from a solution.
11
J. Perrin, Les Atomes, Alcan (Paris) (1913). A. Einstein, Ann. Phys. 17 (1905) 549. 31 J. Perrin, Ann. Chim. Phys. (8) 18 (1909) 5. 41 M. Kerker, J. Chem. Educ. 51 (1974) 764. 51 J. Bibette, J. Colloid Inter/ace Sci. 147 (1991) 474. 21
PREPARATION AND CHARACTERIZATION
2.8
We note here that the triad in Perrin's approach, namely the preparation of welldefined colloids, the characterization of their size distribution (dispersity), and their eventual application to investigate a physical problem, has served as a model strategy ever since. An example of such an application is the study of concentrated monodisperse sols to be discussed in chapter 5.
Figure 2.3. Schematic phase diagram for a solution, which becomes supersaturated upon cooling; x is the solute mole fraction and T is the temperature.
2.2b Homogeneous precipitation If a substance becomes less soluble by a change of some parameter, such as the temperature decrease in fig. 2.3, the solution may enter a metastable state on crossing the binodal in the phase diagram. In the metastable region, the formation of small precipitates or nuclei initially increases the Gibbs energy; thus, demixing by nucleation is an activated process, occurring at a rate, which is extremely sensitive to the extent of penetration in this metastable region, as will be discussed in 2.2c. In contrast, when we quench the solution into the unstable region on crossing the spinodal (fig. 2.3), there is no activation barrier to form a new phase. This is the so-called spinodal decomposition (briefly alluded to at the end of sec. 1.2.19) in which a spongy phase is formed with a characteristic wavelength1' rather than the collection of particulate colloids formed by nucleation and growth. The morphological contrast is illustrated by fig. 2.4 showing a labyrinth-like silica structure, resulting from spinodal decomposition in a cooling silicate melt21 compared with discrete silica spheres prepared by nucleation and growth in a silica precursor solution (Stober synthesis, see sec. 2.4a). A slow rise in supersaturation by slowly changing temperature or pH in fairly dilute solutions favours the formation of particulate colloids, because we then avoid a deep quench in the phase diagram (unless we are close to the critical point). Well-known examples are the slow precipitation of silica particles in aqueous silicate solutions at near-neutral pH and the nucleation of sulphur colloids upon addition of water to a sulphur solution in ethanol (see also sec. 2.4). 11
J.W. Cahn, Trans. Metall. Soc. qfAlME 242 (1968) 166. H. Xihuai, J. Non-Cryst. Solids 112 (1989) 58; S.G.J.M. Kluijtmans, J.K.G. Dhont, and A.P. Philipse, Langmuir 13 (1997) 4976. 21
PREPARATION AND CHARACTERIZATION
2.9
Figure 2.4. Left: a spongy structure of amorphous silica (so-called porous glasses), prepared by spinodal decomposition of a silica-containing melt. Right: amorphous silica spheres, formed by nucleation and growth in a solution (see sec. 2.4a). The spheres have been imaged in situ by cryogenic electron microscopy (see sec. 2.3 a).
We will briefly recapitulate11 the thermodynamics of homogeneous nucleation, i.e. particle formation in a solution with one solute only, a topic initiated in sec. 1.2.23d. Classical nucleation theory is based on an approximate macroscopic description according to which a precipitating particle (later referred to as a nucleus or cluster) is considered to consist of a bulk phase, containing JVp molecules and a shell with JV? molecules of type i (fig. 2.5). The particle is embedded in a solution containing dissolved molecules i. The volume of this solution is large as compared with that of the particle, so that the former acts as the surroundings of the latter. The Gibbs energy of the particle consists of a bulk part and a surface part GS
_ ^NS
+
yA
[2.2.10].
This follows from [I.A3.8], except that the amounts of substance and the chemical potentials are now written in terms of molecules rather than moles. The surface tension is taken as a constant and, for lack of better insight, equated to its bulk value, which is hardly measurable anyway, see sec. III. 1.13. Implicit is the assumption that the size of the particle is large enough to ignore its influence on y. Unlike the equilibrium state underlying [I.A3.8], characterized by equality of jU{ throughout, we now consider a non-equilibrium situation in which the solution is supersaturated; the activity a( > outsat). As a result, transfer of molecules takes place. We compute the change AGS upon the transport of a small number Ni of molecules from the solution to the particle. Obviously, this consists of two contributions
For an extensive treatment see F.F. Abraham, Homogeneous Nucleation Theory, Academic Press, (1974) and P.G. Debenedetti, Metastable liquids: Concepts and Principles, Princeton University Press, (1996). The last author also discusses spinodal decomposition and the still poorly understood transition from nucleation at very high supersaturation - deep into the metastable region - to spinodal decomposition.
2.10
PREPARATION AND CHARACTERIZATION
Figure 2.5. In classical nucleation theory a nucleus (left) is modeled by a droplet composed of bulk molecules and surface molecules, which have a higher free energy per molecule than the bulk. The nucleus is not necessarily spherical and is modeled here (right) by a spherocylinder.
AGS = AGs(bulk) + AGs(surface)
[2.2.11 ]
Of these, the first is negative (it is the driving force), the second is positive (work has to be carried out against the expansion of the interface). We have, upon withdrawing JV molecules from the solution, transferring them to the bulk of the particle, AGs(bulk) = -N{[jul -/ij-(sat)]
[2.2.12]
where the superindex L refers to the solution. From this AGs(bulk) = -JV 1 kTln[a i /a i (sat)]
[2.2.13]
which can also be written as AGS (bulk) = -NtkT In S
[2.2.13a]
after Introducing the supersaturation ratio S as S^eij/ajtsat)
[2.2.14]
Regarding AGS(surface), we can say that the surface area A is proportional to (JV?)2/3 with a proportionality constant /? depending on the shape of the nucleus. Hence the Gibbs energy increase caused by the transfer is AGS (surface) = y/3N?/3
[2.2.15]
Combination gives AGS =-NkT\nS
+ yfiN2/3
[2.2.16]
where we have omitted the subindex i because there is no confusion. We shall use [2.2.16] as an integrated equation, i.e. with N = IVs, but omit the superscript for typographical reasons. For relatively small clusters the surface area term dominates, whereas AG as a function of JV only starts to decrease due to the bulk term beyond a
PREPARATION AND CHARACTERIZATION
2.11
critical value JV* (see fig. 2.6). This critical cluster size follows from the condition dAG/dJV = O (jV*)1/3= 2 7 j g 1 ' 3/cTlnS which can be used to rewrite the Gibbs energy for formation of a cluster as
[2.2.17]
[2 2 181
^^fsb) j
--
This form is independent of the shape of the cluster and equally holds, for example, for crystalline cubes and amorphous spheres. The maximum in the Gibbs energy is AG* = -A*y;
A* =
fl(N*f/3
[2.2.19]
This maximum is the activation barrier in the formation of colloidal particles by homogeneous nucleation in a supersaturated solution or vapour. Note that the (reversible) work needed to form the surface of the critical cluster equals A * y and that the maximum in AG is only one third of this value because bulk is also formed. This expression for a critical cluster explains why a high supersaturation favours the formation of small colloids; a large S pushes the critical size JV * to smaller values and simultaneously lowers the activation barrier (fig. 2.6). A decrease in the interfacial tension y between colloid and solution, for example by adsorption of surfactants, has, according to [2.2.17] and [2.2.19], a similar effect. This is understandable since a low y cannot compete with the spontaneous bulk formation driving the precipitation, unless the clusters are very small. Colloidal particles, of course, often do not precipitate as well-defined spheres, which is why we left the cluster shape unspecified via the parameter j3 introduced in [2.2.14]. As a specific example of a non-spherical precipitate, we consider a cylinder of length L, capped at both ends by a hemisphere of radius a (fig. 2.5). The number of molecules in the spherocylinder with volume V equals N = VI um , where vm is the molecular volume. The Gibbs energy for the formation of the spherocylinder is
Figure 2.6. Sketch of [2.2.16] for nucleation and growth of a spherical precipitate of radius a in a solution with supersaturation ratio S.
2.12
PREPARATION AND CHARACTERIZATION
AG=4^a2fl + — V-( 4 /3)^a 3 fl + — M M V 2a)
{,
4a)
[2.2.20]
vm
For a sphere AG will always pass through a maximum when the radius is large enough (fig. 2.6), but increasing the length of the spherocylinder does not necessarily produce spontaneous growth at some point. We find ^ dL
= 0
for
a* = - ^ l fcTlnS
[2.2.21]
and that this derivative is positive for a> a* and negative only for a
[2.2.22]
m
with a maximum given by 4 zo AG* = -x(a*) y V 3 ' "
2.V v a* = —ffi-^— kT In S
[2.2.23]
The results in this section for the energetics of nucleation are based on a description, which at first sight leaves much to be desired. Nuclei cannot become arbitrarily small without the macroscopic treatment at some point breaking down2', which is why [2.2.16] contains the inconsistency that AG, the excess Gibbs energy relative to unassociated molecules, does not actually reduce to zero for N = 1. Further, any internal degrees of freedom of clusters, and their translational entropy are not included in [2.2.16]. Finally, the cluster surface is entirely characterized by only one 11 For crystals with faces i, each having an area A{ and surface tension yl, each face contributes Aj/j/3, to the activation Gibbs energy, so the form [2.2.19] remains valid, see R. Defay, I. Prigogine, A. Bellemans, and D.H. Everett, Surface Tension and Adsorption, Wiley (1960). 21 Debenedetti, loc. cit.
PREPARATION A N D CHARACTERIZATION
2.13
surface tension, whereas non-spherical crystalline precipitates may have more than one interfacial tension owing to different crystallographic orientations of the particle surface . (As noted before, shape anisotropy does not change the form of the activation energy [2.2.19]). Granted that only one y suffices to evaluate the activation barrier in [2.2.23], its interpretation is still problematic. Usually y is equated to the surface Gibbs energy of a planar interface at phase co-existence. Thus, / in [2.2.23] is taken to be independent of the activity of molecules in the solution (i.e. the supersaturation ratio S). A numerical evaluation of the activation energy for crystal formation in a hardsphere fluid by computer simulation21 shows that the classical expression [2.2.22] is essentially correct, but that the value of y needs to be adjusted to obtain agreement between [2.2.22] and the numerical results. Extrapolation of the effective y to zero supersaturation yielded the expected surface tension at phase coexistence, but as these surface tensions are experimentally hardly accessible, quantitative predictions from [2.2.22] are in many cases at best conjectural. 2.2c Precipitation kinetics In the precipitation kinetics of colloids in a metastable solution31, we can, in accordance with fig. 2.6, distinguish two regimes. When the colloidal particle is significantly larger than the critical size, it is in the regime of irreversible growth with kinetics to be discussed later. First, we consider the initial regime where small particles struggle with their own solubility to pass the Gibbs energy barrier AG * . This passage is called a nucleation event, which for simplicity we will define as the capture of one molecule by a critical cluster, assuming that after this capture the cluster enters the irreversible growth regime upon which a new colloid is born. This assumption, of course, neglects the finite probability that supercritical clusters may also dissolve. For an estimate of the nucleation rate, however, this simple picture is sufficient. Hence, the number I of colloids which per second come into existence is proportional to c m and c * I = kcmc*
[2.2.24]
where k is a rate constant; c m and c * are the concentrations of single, unassociated molecules and critical clusters, respectively. Note that [2.2.24] predicts second-order reaction kinetics because of our choice to consider only encounters between a critical cluster and one molecule as the rate-determining events. To quantify I, we first evaluate the frequency at which molecules encounter a spherical cluster of radius a by diffusion, following in essence Smoluchowski's diffusion model for coagulation kinetics 11
See for example A.C. Zettlemoyer (Ed.), Nucleation, Marcel Dekker (1969). S. Auer, D. Frenkel, Nature 409 (2001) 1020; Nature 413 (2001) 711. For in-depth studies on various inorganic colloids the work of de Bruyn and co-workers is recommended reading. See for example, J. Dousma, P.L. de Bruyn, J. Colloid Interface Sci. 64 (1978) 154; H.A. van Straten, B. Holtkamp, and P.L. de Bruyn, J. Colloid Interface Sci. 98 (1984) 342; M.J.M. van Kemenade, P.L. de Bruyn, J. Colloid Interface Sci. 118 (1987) 564.
21
2.14
PREPARATION AND CHARACTERIZATION
(see sec. 4.3). The diffusion flux J of molecules through any spherical envelope of radius r is, according to Fick's first law, j
= 47rr2D^[l
[2 .2.25]
dr where D is the molecular diffusion coefficient relative to the sphere positioned at the origin at r = 0 . Each molecule that reaches the sphere surface irreversibly attaches to the insoluble sphere, and we assume that the concentration c m of molecules in the liquid far away from the sphere remains constant c(r = a) = O
c(r^°°)
= cm
[2.2.26]
For these boundary conditions [2.2.25] yields J = 4nDa*cm
[2.2.27]
if it is assumed that J is independent of r, that is, if the diffusion of molecules towards the sphere has reached a stationary state. Such a state is approached by the concentration gradient around a sphere in a time of order a2 / D needed by molecules to diffuse over a sphere diameter. Assuming that sphere growth is a sequence of stationary states, we can identify the nucleation rate / as the flux J multiplied by the concentration c* of spheres with critical radius a* I = 4n;Da*cmc*
[s^rrr3]
[2.2.28]
The concentration c* may be evaluated as follows . Since the reversible work to form a cluster out of JV molecules is the AG from fig. 2.6, the Boltzmann distribution c(JV) = c m exp[-AG/kT]
[2.2.29]
determines the equilibrium concentration of clusters composed of JV molecules. Applying this result to clusters with a critical size, we find on substitution in [2.2.28] for the nucleation rate I = 4KDa*c^lexp[-AG*/kT]
AG* = (4;r/3)(a*)2 y
[2.2.30]
where AG* is the height of the nucleation barrier; the exponent may be identified as the probability (per particle) that a spontaneous fluctuation will produce a critical cluster. The use of an equilibrium Boltzmann distribution in a nucleation flux is perhaps unexpected21, but one can think of a distribution of subcritical clusters from
For an extensive discussion see Debenedetti loc.cit. In the thermodynamics of reversible coagulation an expression can be derived for the distribution of aggregate size which is very similar to [2.2.29]. See D.H. Everett, Basic Principles of Colloid Science, Roy. Soc. Chem. (1994).
PREPARATION AND CHARACTERIZATION
2.15
which critical clusters are removed as soon as they capture additional molecules. Each removal is compensated by the insertion of an equivalent number of single molecules into the metastable bulk solution. In this manner, one can define a steady state nucleation rate for a given supersaturation11. Equation [2.2.30] shows that the nucleation rate is extremely sensitive to the value of a* and, thus, to the supersaturation via [2.2.23]. The maximum nucleation rate at very large supersaturation, the pre-exponentlal kinetic factor in [2.2.30], is of the order 1~
=&
[2-2.31]
as follows from substitution of the Stokes-Einsteln diffusion coefficient D = kTIQnrp.* , where we neglect the size difference between molecules and critical clusters. For an aqueous solution at room temperature with a molar concentration c m = 10~3 M , we find a maximal nucleation rate of order 10 29 m~3sec~1. A decrease in supersaturation to values around S = 5 suffices to reduce this astronomical rate to practically zero. For silica precipitation in dilute, acidified waterglass solutions (see sec. 2.2e), the supersaturation is in order of magnitude close to S = 5 and nucleation may take hours to days. For comparison, the industrial, continuous precipitation of the highly insoluble silver halide colloids21, the basis of classical photographic materials, occurs at a supersaturation, which generally exceeds S ~ 106 . The kinetics of precipitation in a homogeneous solution is notoriously difficult to assess within better than an order of magnitude because of uncertainties in, for example, the interfacial tension that are strongly amplified in [2.2.30]. Nevertheless, the trend predicted from [2.2.30] Is qualitatively correct. Within a narrow range of supersaturation after crossing the binodal in fig. 2.3 the rate of homogeneous precipitation increases from negligibly small to astronomically large. In practice, however, the increase Is limited because experimental nucleation rates often go through a maximum at sufficiently high supersaturation3'. In concentrated solutions, the assumption of freely diffusing molecules underlying the pre-exponential factor in [2.2.30] breaks down, though reduced dlffusivity is unlikely to be the sole cause of any maximum in the precipitation rate. At high solute concentrations, long-time self-diffusion admittedly will vanish but for nucleation only local rearrangements of molecules are required, which may be feasible up to (and possibly even including) close-packing densities. Another factor of importance is that, as already noted in sec. 2.2b, the interfacial Gibbs energy y Is actually not a constant. Simulations of absolute nucleation rates show that,
11
R. Becker, Theorie der Warme, Springer Verlag (1978). 1.H. Leubner, Current Opinion in Colloid & Interface Sd. 5 (2000) 151, reviews nucleation models for silver halides. 31 P. Pusey, in Liquid, Freezing and Glass Transition J.P. Hansen, D. Devesque, and J. ZinnJustin, Eds. 763-931, North Holland (1991). 21
2.16
PREPARATION AND CHARACTERIZATION
in any case for hard-sphere fluids", the maximum in the nucleation rate is indeed primarily due to an increase of y with supersaturation. This increase diminishes the probability that a critical cluster will form on account of [2.2.30]. So, any quantitative prediction for the nucleation rate must at least take this change in y into account. The reader may have noticed that [2.2.30] is very similar to the classical BeckerDdring result21 for homogeneous nucleation in a vapour (see sec. 1.2.23). The difference is the form of the pre-exponentlal kinetic factor, which is obtained here using a diffusion model instead of kinetic gas theory. Consequently, the result [2.2.31] is equivalent to Smoluchowski's expression for the rate of diffusion-controlled coagulation of identical spheres in the initial state of coagulation (see sec. 4.3). In Smoluchowski's treatment, incidentally, there is no activation barrier because of the assumption that colloids irreversibly stick whenever they happen to collide by Brownian motion. However, when attractions are at a level of weakness such that colloidal clusters3' can be disrupted by the thermal energy, the existence of a critical aggregate size can be expected with a rate of formation similar to [2.2.30J. 2.2d Particle growth and polydispersity When no precautions are taken, precipitation from a supersaturated solution inevitably produces polydisperse colloids because nucleation of new particles and further particle growth overlap in time. This overlap is a consequence of the statistical nature of the nucleation process; near the critical size particles may grow as well as dissolve. To narrow down the initial size distribution as much as possible, nucleation should take place in a short time, followed by equal growth of a constant number of particles. La Mer4) pointed out that this can be achieved by rapidly creating the critical supersaturation required to initiate homogeneous nucleation after which particle growth lowers the saturation sufficiently to suppress new nucleation events. It should be noted that La Mer's scheme rests on the extreme sensitivity of homogeneous nucleation rates to supersaturation. An instance of La Mer's scheme is found in the double-jet precipitation of silver halide colloids, in which AgNO3 and NaBr solutions are simultaneously added to an agitated gelatin solution. Here, the number of newly formed crystals quickly reaches a constant value and further addition of reagents causes only further growth of fairly monodisperse cubic crystals51. Another option is to add nuclei (seeds) to a solution with a subcritical supersaturation as when silica particles are added to a saturated aqueous silicate solution (heterogeneous nucleation,
11
S. Aucr, D. Frenkel, Nature 413 (2001) 711. R. Becker, loc.cit. 31 For reversible coagulation sec also J. Groenewold, W.K. Kegel, J. Phys. Chem. B105 (2001) 11702. 41 V.K. La Mer, R.H. Dinegar, J. Am. Chem. Soc. 72 (1950) 4847. 51 J.S. Wey, R.W. Strong, Phologr. Set Eng. 21 (1977) 14; C.R. Berry, Photogr. Set Eng. 18 (1974) 4. 21
PREPARATION AND CHARACTERIZATION
2.17
see sec. 2.2e). The advantage of this seeded growth technique is that the final particle size can be influenced by the concentration of seed particles. A fortunate consequence of particle growth is that in many cases the size distribution is self-sharpening. We will illustrate this effect for colloidal spheres of radius a, which irreversibly grow by the uptake of molecules from a solution according to the rate law ^ p = Jcoan
[2.2.32]
where k0 and n are constants. This growth equation leads either to spreading or sharpening of the relative size distribution, depending on the value of n, as can be demonstrated as follows. Consider at a given time t any pair of spheres with arbitrary size from the population of independently growing particles. Let 1 + £ be their size ratio such that a(l + e) and a are the radius of the larger and smaller sphere, respectively. The former grows according to: — a(l + £) = /coan(l + £)n
[2.2.33]
which can be combined with growth equation [2.2.32] for the smaller sphere to obtain the time evolution of the size ratio: — = kQ a11"1 [(1 + e)n -(1 + £)]
£>0
[2.2.34]
Clearly, the relative size difference e increases with time for n > 1, in which case particle growth broadens the distribution. For n = 1 the size ratio between two spheres remains constant, whereas for n < 1 it monotonically decreases in time. Since this decrease holds for any pair of particles in the growing population, it follows that for n < 1 the relative size distribution is self-sharpening, a conclusion also drawn by other authors21. It should be noted that what applies to the growth kinetics of two spheres also holds for two sufficiently sharp distributions. Thus, [2.2.34] also describes the time evolution of the relative distance of two peaks in a bimodal size distribution. These two peaks are much easier to monitor in time than the width of a single size distribution, which is why growth of a binary sphere mixture is a convenient source of experimental information on kinetic mechanisms, as has been demonstrated for latex3 and silica4' dispersions.
1
' The concentration of molecules is incorporated here in the rate constant kQ and may depend on time because of a generating chemical reaction. Such dependence does not alter the effect of exponent n on the polydispersity because k0 is the same for all particles. 21 J.Th.G. Overbeek, Adv. Colloid Interface Sci. 15 (1982) 251. 31 E.B. Bradford, J.W. vanderHoff, and T. Alfrey Jr., J. Colloid Interface Sci. 11 (1956) 135. 41 A. van Blaadercn, J. van Geest, and A. Vrij, A., J. Colloid Interface Sci. 154 (1992) 481.
2.18
PREPARATION AND CHARACTERIZATION
The requirement n < 1, for self-sharpening, is in practice a realistic one. For example, when the growth rate is completely determined by a slow reaction of molecules at the sphere surface, we have ^ p = koa2
[2.2.35]
implying that da/dt is a constant, so n = 0. The opposite limiting case is growth governed by the rate at which molecules reach a colloid by diffusion. The diffusion flux for molecules with a diffusion coefficient D, relative to a sphere centred at the origin at r = 0 , is given by [2.2.25]. We assume that the saturation concentration is maintained at the particle surface, neglecting the influence of particle size on c(sat) (the Kelvin effect, see sec. 2.2e), and keeping the bulk concentration of molecules constant11 c(r = a) = c(sat)
c(r -> °°) = C(«)
[2.2.36]
For these boundary conditions, the stationary (i.e. r-independent) flux towards the sphere equals (see [2.2.27]): J = 4«-Da[cH-c(sat)]
[2.2.37]
showing that the rate at which the colloid intercepts diffusing molecules is proportional to its radius and not to its surface area. Suppose every molecule contributes a volume vm to the growing colloid, then for a homogeneous sphere the volume increases at a rate ~*a3
= Jvm
[2.2.38]
which on substitution of [2.2.37] leads to — = Du m [cH-c(sat)]a" 1
[2.2.39]
with the typical scaling a2 ~ t as expected for a diffusion-controlled process. Thus, the exponent in [2.2.32] for diffusion-controlled growth is n = - l , and consequently the relative width of the size distribution decreases in time. This conclusion is based on a diffusion flux, which assumes a steady-state diffusion of molecules towards colloids, which grow independently from each other. Reiss21, however, has shown that also when these assumptions are invalid, diffusional growth still sharpens the size distribution. Diffusion-controlled growth of a homogeneous sphere was first studied by Langmuir3', who introduced a formula very similar to [2.2.39], albeit for the evaporation of 11
A decrease in c M due to exhaustion of a finite bulk is treated in A. Philipse, Colloid Polym. Scl. 266(1988) 1174. 21 H. Reiss, J. Chem. Phys. 19 (1951) 482. 31 1. Langmuir, Physical Rev. 12 (1918) 368.
PREPARATION AND CHARACTERIZATION
2.19
a sphere for which the derivative in [2.2.39] is negative. Langmuir used a diffusion model to explain the evaporation rate of millimetre-sized iodine spheres in quiet air. He found that the rate of weight loss of the spheres confirmed diffusion control, and obtained from the rate a reasonable value for the diffusion coefficient of iodine molecules in air. Equation [2.2.39] is also useful to estimate colloidal growth rates. Molecular diffusion coefficients in water at 25°C are of the order D~ 10" 5 cm 2 s" 1 and taking a typical volume fraction t>m[c(°°)-c(sat)] = 0.01 of reactive molecules we find from [2.2.39] that for diffusion-controlled growth the surface area increases in time as da 2 1 At ~ 20(//m}2 s~ ! . This implies a nearly instantaneous growth of submicron colloids, which indeed is observed in, for example, the precipitation of magnetite (see section 2.4d). Whenever particle growth is much slower, the kinetics may be determined by a slow reaction step at the surface of the colloid, or by the slow production of precipitating molecules via a chemical reaction as in the case of sulphur sols (see sec. 2.4b). This is not the place for in depth refinement1 of diffusion-controlled kinetics beyond a flux of the form [2.2.37], but we cannot totally ignore the involvement of charged species in the precipitation of inorganic colloids. Hence, an electrostatic interaction may be present between the growing colloids and the molecules they consume, which will either enhance or retard the growth, depending on whether colloids and monomers attract or repel each other. From the classic studies of Kramers21 and Debye31 on diffusion in a force field, we can infer that the diffusion coefficient D of the monomers in the diffusion flux J has to be replaced by an effective coefficient of the form D
eff= —
[2.2A0]
a J e -u(r)/fcT r -2 dr
a
where u(r) is the interaction energy between molecule and colloid. The same type of integral, incidentally, appears in the theory of slow coagulation in sec. 4.3b. Suppose the molecules are ions with charge ze and that the colloidal sphere has a surface potential y/° . To obtain an upper estimate of the effect of the ion-colloid interaction on the growth kinetics, we consider the low salt limit where the interaction is unscreened. Then u(r) is obtained from Coulomb's law as u r
<)
a
ze
W°
n
,o o ,.1 ,
= "„— u =—— = zy° 2.2.41 y kT °r ° kT where uQ is the colloid-ion contact interaction energy and y° = ey/° /kT , as before.
11
D.F. Calef, J.M. Deutch, Ann. Rev. Phys. Chem. 34 (1983) 493. H.A. Kramers, Physica 7 (1940) 284. 31 P. Dcbye, Trans. Electrochem. Soc. 82 (1942) 265. 21
2.20
PREPARATION AND CHARACTERIZATION
Thus, this Coulomblc interaction [2.2.40] yields Deff=D-^-
[2.2.42]
So, for colloids that have to grow by a diffusion flux of like-charged ions, the growth kinetics is slowed down exponentially by the Coulombic repulsion; when y/0 = 75 mV the effective diffusion coefficient for divalent ions is about 0.01 D. Added salt screens the colloid-ion interaction and, therefore, moderates the influence of yo on the growth kinetics. The interaction between monomers and the growing colloid, within the approximations underlying [2.2.42], does not change the growth equation [2.2.39] and, hence, does not affect the conclusion that diffusional growth sharpens the size distribution. We will investigate whether this conclusion still holds when we drop the assumption that the growing sphere is a homogeneous object of constant mass density. It is well known that diffusional growth may produce heterogeneous structures with an internal density profile. A familiar example is the precipitation of silica at low pH, where ramified clusters are formed rather than the fully condensed SiO2 particles at alkaline pH. The difference is due to the low reactivity of silanol groups towards condensation at acid pH, which obstructs the densification of a cluster. Suppose a monomer volume fraction profile ip(x) is present in the growing colloid, where x is the distance to its centre. Then, the rate of growth is, instead of [2.2.39], given by 0(a)^- = Du m [c(~)-c(sat}]a- 1
[2.2.43]
because each monomer contributes a volume vm / 0(a) to the growing colloid upon arrival at its surface at x = a . When this volume contribution increases with the colloid radius, i.e. when the average mass density of the colloid decreases, the large particles in the size distribution have a gain in growth rate. This scenario will occur for the fractal clusters produced by diffusion-limited aggregation23' (DLA). Precipitation by DLA forms an interesting, purely kinetic contrast to classical nucleation and growth, where the excess surface Gibbs energy provides the nucleation barrier, as well as the driving force for further growth by ripening (see sec. 2.2e). The kinetics of fractal growth will be treated in sec. 4.5c; here, a further completion of [2.2.43] will suffice. Consider monomers with volume p 3 which diffuse towards a single spherical cluster with total radius a. The number of monomers, IV , in the cluster scales as41
11
R.K. Her loc. cit. T.A. Witten Jr., L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. 31 P. Meakin, Faraday Discuss. Chem. Soc. 83 (1987) 1. 41 L.G.B. Bremer, Fractal Aggregation in Relation to Formation and Properties of Gels, Ph.D. thesis, Wageningcn Agricultural University, The Netherlands (1992). 21
PREPARATION AND CHARACTERIZATION
(
2.21
\ c 'f
—
[2.2.44]
PJ where df is the fractal dimensionality. The average monomer volume fraction in the cluster is accordingly
a3
{p)
assuming, as usual in DLA models, that the monomers are spheres. However, the cluster may also be composed of randomly oriented fibers or platelets. They significantly reduce the average density of a cluster, but do not necessarily change its fractal dimension11. The local volume fraction at a distance x from the cluster centre is df(x)dl~3 «Xx)~-M3 KPJ
[2.2.46]
as can be checked by its substitution into the definition of the average density a
((/>) = —^- \(j)(x)A7tx2dx 4;ra J •>
[2.2.47]
Thus, the volume fraction at the edge of the cluster is
[2.2.48]
which on substitution into [2.2.43] leads to the following scaling of the growth rate of the outer radius of the cluster ^~a2"df dt
a~tl/d<-1
[2.2.49]
For df = 3 , we recover the familiar square-root time dependence of diffusional growth of a homogeneous sphere. A fractal dimensionality, df < 3 , enhances the growth rate, but as long as df > 1 , the form of da/dt
is such that self-sharpening will occur.
Clusters with a fractal dimensionality df = 1 are rather unlikely for colloidal growth as they should consist of straight spikes of length a growing from a common source into a radial direction, such that the mass increases linearly with the radius [N ~ a in [2.2.44]). Since for three-dimensional DLA, the fractal dimensionality certainly exceeds unity, a value as high as df = 2.49 has been reported , it follows that self-sharpening will occur for diffusional growth of both homogeneous and heterogeneous clusters. Two comments should be made here. First, we have disregarded the 'fingered' surface structure of a fractal cluster by assuming that a monomer will stick whenever it arrives 1
' Heterogeneous structures and gels of inorganic fibres arc reviewed in A. Philipse and A. Wicrcnga, Langmuir 14 (1998) 49. 21 M. Fleischmann, D.J. Tildesley, and R.C. Ball, Fractals in the Natural Sciences, Princeton University Press (1990).
2.22
PREPARATION AND CHARACTERIZATION
at a distance a from the cluster centre, I.e. the cluster surface is uniformly sticky. Taking structural details into account may change the scaling relations
[2.2.49].
Second, consistent with the approach discussed elsewhere in this chapter we look at clusters growing independently in a bulk. In a later stage of growth, it is clear that aggregation of fractal clusters themselves becomes the kinetically dominating event in the formation of large aggregates and space-filling gels, a topic extensively treated elsewhere. The size distribution resulting from the precipitation and growth of inorganic particles is often (and also here) regarded as a purely kinetic phenomenon. The colloids simply stop growing when supersaturation has dropped sufficiently. The size distribution may further change in time due to cluster-cluster aggregation and Ostwald ripening (see sec. 2.3e) and, of course, by coagulation due to Van der Waals forces, but there is no evolution towards a thermodynamically stable size distribution. That, at least, is the classical view for inorganic colloidal dispersions. It is well known, however, that other dispersed systems exist, such as microemulsions, in which the dispersed phase is actually in a state of thermodynamic equilibrium. This is a consequence of very low interfacial tensions owing to the adsorption of surfactants. There is no a priori reason why inorganic colloids could not be thermodynamically stable due to adsorbed layers, which puts a Gibbs energy penalty on further decrease of the surface area. Various authors have alluded in the past to this possibility11, and recent experiments on the preparation of silver halide sols21 provide clear examples of inorganic colloids with thermodynamic control of the size distribution. Understanding this control and its occurrence is undoubtedly important for further improvement and extension of preparation methods for well-defined colloids. 2.2e Particle solubility and Ostwald ripening For colloidal spheres of given radius a and surface tension y, there is one solute concentration c{a) at which the colloids have a critical size and reside at the Gibbs energy maximum in fig. 2.6. This concentration, c{a), also called the equilibrium solubility of the colloids, follows from [2.2.23], which in the context of solubility usually is written as ln[c(a)fc(sat)] = 2/vm/a*kT
[2.2.50]
which is a result known as the Gibbs-Kelvin equation; c(sat) is the equilibrium solubility of a flat surface as we already encountered before, see [1.2.23.24]. Note that we have replaced the activities in the supersaturation ratio S by concentrations, so [2.2.50] is valid only for dilute solutions. The increase in solubility for small spheres
11
J.Th.G. Overbeek, Faraday Discuss. Chem. Soc. 65 (1978) 7. I.L. Mladenovic, W.K. Kegel, P. Bomans, and P.M. Frcderik, J. Phys. Chem. B107 (2003) 5717. 21
PREPARATION AND CHARACTERIZATION
2.23
Figure 2.7. Equilibrium solubility of a sphere with radius a according to the Gibbs-Kelvin equation [2.2.50]. For silica particles with a radius around a = 1 nm, the solubility in neutral water is about 300 ppm (see text for discussion).
(or in the vapour pressure of small droplets) Is equivalent to an enhanced, excess Laplace pressure Ap = 2y/a, which can be recognized in [2.2.50]. An important feature of the Gibbs-Kelvin equation is its generality; its form does not depend on the assumption of amorphous spheres characterized by only one surface tension. For the equilibrium solubility of a crystal, the equation applies to each of the faces of the crystal11
rL=r2=jL
= J^_]n_^L
[225l]
Here, yi is the surface tension of face i, which is at a distance r( from the centre of the crystal; only faces of the equilibrium polyhedron are taken into account and corners and edges are ignored. In these Wulff relations, the ratio yi I r; for a crystal face plays the same role as y/a for a spherical droplet. Alternatively, one can also interpret [2.2.51] as describing polydisperse spheres with different surface tensions such that they are all in equilibrium with the same solution with concentration c{a). The Wulff relations fix the equilibrium shape of the crystal since in the equilibrium form there exists one centre such that these relations are satisfied. The increased solubility according to [2.2.50], also referred to as the Gibbs-Kelvin effect, is not easy to quantify for sols of inorganic particles as their surface tension is difficult to determine experimentally (see sec. III. 1.13) and has to be indirectly obtained from adsorption data. For an order of magnitude estimate of the Gibbs-Kelvin effect, we consider the case of amorphous silica in water (fig. 2.7). Her21 reports the range y = 0.05-0.1 N/m, corresponding to an equilibrium radius in the range a*ln[c(a)/c(sat)] = l-2nm
[2.2.52]
See for a derivation and further discussion: R. Defay, I. Prigogine, A. Bellemans, and D.H. Everett, Surface Tension and Adsorption, Wiley (1966). 21
R.K. Her, loc. cit.
2.24
PREPARATION AND CHARACTERIZATION
at T = 298 K, for a molar silica volume of vm = 27.2 cm 3 /mol. At neutral pH the bulk solubility c(sat) of silica Is about 100 ppm, which implies that in an aqueous waterglass solution containing c(a) = 300 ppm of soluble silica the equilibrium radius is in the range a* = 0.9-1.8 nm. At alkaline pH the bulk solubility of silica rapidly increases to a value of c(sat) = 300 at about pH = 10. Thus, in the 300 ppm waterglass solution at pH = 10 the critical particle size of [2.2.23] tends to infinity, so precipitation of silica particles in this case is extremely unlikely. The expected significant solubility of silica particles in the nanometer size range is indeed observed in practice, as is the drastic effect of pH. It is not a coincidence, but a consequence of the Gibbs-Kelvin effect, that in aqueous silica sols particle radii are usually at least a few nm and that the pH of the sols is usually set below 10. The interfacial tension in the Gibbs-Kelvln equation [2.2.50] is, of course, not a fixed parameter and may be intentionally decreased to enhance particle solubility. A classical example is Agl, which is hardly soluble in water. However, when excess silver or (in particular) iodide ions are added the solubility increases dramatically as manifested by the formation of complexes or small silver iodide clusters, which apparently have a reduced interfacial tension. It has been shown that these small particles (up to ~1 nm in size) form spontaneously1'. Clearly, such reversible dissolution of Inorganic salts is a possibility whenever ions or other species strongly adsorb on the material In question. Modification of a particle surface, of course, may also decrease the solubility or rather reduce the rate of particle dissolution due to a protective layer of insoluble material. Even much less than monolayer coverage may be sufficient for this purpose as in the case of the solubility of silica, which is drastically reduced by adsorption of aluminate species in minute quantities21. Returning to the Gibbs energy maximum in fig. 2.6, we note that it presents an unstable equilibrium, which can be maintained only for critical particles of exactly the same size. For polydisperse particles (with the same surface tension), there is no single, common equilibrium solubility; particles either grow or dissolve. Clearly, the largest particles have the strongest tendency to grow owing to their low solubility. This coarsening of colloids, i.e. the decrease of specific surface area in time, is also known as Ostwald ripening and it is an important ageing effect, which may occur in any polydisperse system of sufficiently small particles. It is observed in emulsions and aqueous sols, as well as colloidal metal catalysts in a high temperature gas (decrease of catalytic activity in time). An obvious consequence of Ostwald ripening is loss of specific surface area, which may proceed quite rapidly for small, highly soluble particles. Illustrative examples are aqueous sols of nanometer-sized silica particles (fig. 2.8), which immediately after preparation undergo a rapid decrease in surface area on a time scale of hours to days, followed by a much slower decay, which may continue to
11 21
I.L. Mladenovic et al., toe. cit. R.K. Her, loc. cit.
PREPARATION AND CHARACTERIZATION
2.25
Figure 2.8. Aqueous sols of very small silica particles, freshly prepared by acidifying a waterglass solution, exhibit a rapid, initial decrease in specific surface area (measured following the Sears method) due to Ostwald ripening. On a time scale of months the surface area saturates at a value typically in the range of 700-800 m 2 /g . (Courtesy of Kenneth Larsson and Bo Larsson, EKA Chemicals, Sweden). stabilize at 700-800 m 2 g" 1 after several months. The area is still high and on TEM pictures very small clusters can be seen. Ageing is accompanied by a slight increase of pH. The behaviour of small, highly soluble particles, as in fig 2.8, illustrates that in the condensation method there is actually no sharp distinction in time between 'sol preparation' and 'ageing.' In the initial precursor solution, the specific surface area decreases as soon as precipitates start to grow. The surface area, of course, is a macroscopic quantity, which loses its meaning for particles with radii down to molecular size. Note, however, that we introduce a precipitate surface area A via [2.2.11] and, therefore de facto,
also a specific surface Ag which continuously
decreases upon the growth of the particles of fig. 2.6. We will now briefly outline the kinetics of dissolution and ripening. In a polydisperse sol, the bulk concentration c(~) is not constant, but slowly decaying in time due to the gradual disappearance of small, soluble particles. At any moment in time there is one sphere radius a0 , which is in metastable equilibrium with the bulk concentration c(oo) = c (sat)exp
2yvm
[2.2.53]
where c(sat) is again the equilibrium solubility of a flat surface. If the local solute concentration near a sphere with radius ai is also fixed by the Gibbs-Kelvin equation, the steady state diffusion flux for sphere I is
2.26
PREPARATION AND CHARACTERIZATION
J, =4,Da i c,sat)[exp[^]-exp[^]j
[2.2.54]
It is clear that spheres with radii a{ < aQ dissolve because J < 0 , whereas for a{ > a0 , the particles grow. The average particle radius and, of course, the critical radius a 0 increase in time, so that the exponents in the diffusion flux can be linearized at a later stage of the ripening process. In that case, we can write for the growth (or for the dissolution rate) of sphere I the approximate result 2r —af = 6Da iC (sat)^S-
dt
l
l
kT[aQ
n [2.2.55]
at\
One limiting case of Ostwald ripening allows for a simple analytical solution, namely monodisperse spheres with radius a, from which dissolved matter is deposited on very large particles or a flat substrate. If this substrate controls the bulk concentration, a 0 is infinitely large and consequently =-eDclsat)^- 23 -
[2.2.56]
Thus, for this case the particle volume decreases at a constant rate. To go beyond such a bidisperse model and evaluate the time evolution of a continuous size distribution of spheres, growing and dissolving according to [2.2.55], is a demanding task, dealt with in the classical studies of Lifshitz and Slezov11 and Wagner21 (LSW theory). We quote the essential results, referring to reviews for more discussion on the principles31 and applications41 of the LSW theory. The assumptions in this theory are the same as those underlying [2.2.55]: there is only transport due to diffusion, the sphere solubility is so low that the Gibbs-Kelvin equation can be linearized and there is no interaction between the spheres other than that their growth rates are coupled by the average bulk concentration. The LSW theory predicts for large times the asymptotic result ^~« dt
D c ( s a t )
9
2< kT
[2 . 2 .57]
i.e. in a late stage of the ripening process, the average particle size increases as t 1 / 3 . Further, the supersaturation correspondingly falls as t~ 1/3 and the number of spheres as r"1 . A remarkable finding of the LSW theory is that due to Ostwald ripening the size distribution approaches a certain universal, time-independent shape, irrespective of the initial distribution. The LSW theory appears to work well for emulsions '; for
11 I.M. Lifshits, V.V. Slezov, Zhur. Eksp. Teor. Fiz. 35 (1958) 479. The names are also transcribed as Lifshitz and Slyezov or Slyozov, for instance in J. Phys. Chem. Solids 19 (1961) 35. 21 C. Wagner, Z. Elektrochem. 65 (1961) 581. 31 W. Dunning, Particle Growth in Suspensions, A.L. Smith, Ed., Academic Press (1973). 41 P. Taylor, Adv. Colloid Interface Set 75 (1998) 107.
PREPARATION AND CHARACTERIZATION
2.27
inorganic particles, a comparison with experimental data is less straightforward1'. It is, in any case not correct to use the t~ 1/3 scaling as the general hallmark for Ostwald ripening in view of the restrictive validity of the LSW theory, also see sec. V.8.3b. For example, close to the nucleation stage when many highly soluble particles are present, linearization of the Gibbs-Kelvin equation and the assumption of non-interacting particles will be invalid. Another important factor is the topology, which for grain growth in a polycrystalline material (fig. 2.2d) or bubble growth in a foam21, is obviously very different from the isolated spheres in the LSW theory. 2.2/ Seeded nucleation and growth So far, we have assumed that particles nucleate and grow in a solution of only one solute. In practice, strict homogeneous precipitation is difficult to realize because of the omnipresence of contaminants, dust, motes and irregularities on the vessel wall, which may act as nucleation sites for the new phase. This so-called heterogeneous nucleation may have a dramatic effect on the kinetics as can be observed after opening a bottle of beer or champagne when carbon dioxide bubbles rapidly nucleate on the glass surface. See also sec. V.7.2a. The reader may wish to verify the effect of adding extra nucleation sites in the form of sugar or sand grains. Heterogeneous nucleation, however, is not necessarily a nuisance. Actually, it is an important strategy to decrease size polydispersity. This was first exploited by Zsigmondy31 who used the extremely fine Faraday gold sol4) as a seed solution for the preparation of quite monodisperse gold colloids. The seed can also differ chemically from the precipitating material, leading to the formation of core-shell colloids. Of the many examples, we mention the growth of silica on gold cores51, and other inorganic particles61 and the preparation of core-shell semiconductor particles71. Such well-defined composite colloids are increasingly important in materials science, in addition to their use in fundamental studies. The efficiency of seeds or a container wall to catalyze nucleation is due to the reduction of the interfacial Gibbs energy of a precipitating particle. As a simple but illustrative example81 we consider a phase a, which nucleates as a spherical cap of radius a on a flat seed substrate /? immersed in a liquid L. The cap wets the substrate with a contact angle 8 as shown in fig. 2.9. As in the case of homogeneous nucleation 1
' W. Dunning, loc. cit. N. Rivier in: D. Bideau and A. Hansen, Eds., Disorder and Granular Media, North Holland (1993). 31 R. Zsigmondy, P.A Thiessen, Das Kolloide Gold, Akad. Verlag. Leipzig (1925). 41 M. Faraday, Phil Trans. Royal Soc, 147 (1857) 145. Faraday prepared gold particles with a diameter around 3 nm by reduction of a gold salt with phosporus in ether. 51 L.M. Liz-Marzan, M. Giersig, and P. Mulvaney, Langmuir 12 (1996) 4329. 61 F. Caruso, Adv. Mater. 13 (2001) 11. 71 H. Weller, Quantized semiconductor particles, Adv. Mater. 5 (1993) 88. For a detailed treatment of heterogeneous nucleation, including other nucleus shapes, see the chapters by R.A. Sigsbee and A.G. Walton in Nucleation, A.C. Zettlemoyer (Ed.) Marcel Dekker (1969). 21
2.28
PREPARATION AND CHARACTERIZATION
Figure 2.9. Heterogeneous nucleus of substance a in the form of a spherical cap on a planar seed P immersed in a liquid L. The precipitating phase a partially wets the seed with a contact angle 6 . in 2.2.b, it is assumed here that a nucleus is a macroscopic piece of structureless 1) bulk matter to which equilibrium thermodynamics can be applied. The Gibbs energy change due to the formation of the cap in fig. 2.9 then equals AG h e t = 2na2(\-cos0)yah+7ra2sin2
e[ya^ - 7 p L ) - — x a 3
J{6) Vm
[2.2.58]
f{0) = - (1 - cos 0)2 (2 + cos 9] 4 This expression comprises the surface area with interfacial tension y aL between a and the liquid, the interfacial area between a and the substrate times the difference yap - 7pL , and the decrease in Gibbs energy due to the volume of the cap. The geometrical factor J(0) is the ratio of the volume of the spherical cap in fig. 2.9 to that of a sphere with the same radius. The interfacial tensions in [2.2.58] are related by Young's equation, which is valid when 6 is the contact angle for equilibrium with respect to the horizontal force components r?L = r^ + raLcos0
12-2.59]
Substituting this result into [2.2.58], we find from the condition d(AG het )/da = 0 that the critical radius equals a'
het
^VZaL. kT lnS
[2.2.60]
which is the same as the critical radius in [2.2.23] for homogeneous precipitation in the absence of a substrate. The energy barrier can be written in terms of the homogeneous energy barrier AGhom in [2.2.22] AG
het = AGhom/(0>
[2.2.61]
showing that J(0) quantifies the catalytic effect of the substrate. The presence of this substrate does not change the critical radius of the sphere, but only reduces the Gibbs energy maximum due to complete or partial wetting by the newly precipitated phase.
11 We disregard here any effect of crystal structures and their (miss)match in epitaxial growth, see Zettlemoyer loc. clt.
PREPARATION AND CHARACTERIZATION
2.29
Note that for complete dewetting, when 0= 180° and J(G) = 1, the Gibbs energy maximum equals AGhom ; then, precipitation proceeds as if no seeds or substrates are present. For any contact angle in the range 0 < 9 < 180°, the substrate lowers the activation energy for nucleation because 0 < J{9) < 1 . If the contact angle is nearly zero, it will be impossible to maintain supersaturation in the presence of the substrate. In view of the strong dependence of the nucleation rate on the activation energy in [2.2.30], it is clear that seeds may speed up precipitation kinetics considerably. The heterogeneous nucleation rate r het will have a form similar to that of the homogeneous rate J hom in [2.2.30], and the pre-exponential factor [2.2.30] will remain the same in order of magnitude. Thus 'het ~I h o m exp[AG^ m (l-/(e))/fcT]
[2.2.62]
For contact angles 8 < 30°, J[6) is practically zero and the nucleation rate is enhanced by a factor of exp(AG^om /kT). The independence of the critical size of the presence of a substrate in [2.2.60] is perhaps unexpected and, in any case, unlikely to be a general feature of heterogeneous nucleation. Here, this independence is a consequence of geometry assumed in fig. 2.9; the radius of curvature of the cap on the smooth, flat substrate is the same as for the spherical nucleus in a homogeneous solution. Therefore, neither the solubility nor the critical radius changes because these are fixed by the curvature. When the cap is deformed, or when the substrate is curved or structured, the situation is evidently more complicated. Steps and kinks on the substrate may act as active sites because they enable more of the surface of the nucleus to be in contact with the substrate, which lowers its surface excess Gibbs energy. One extreme case is a cavity in the substrate, which allows maximum contact for a (non-cylindrical) nucleus as discussed elsewhere1'. See also fig. V.7.5. Such a cavity is actually a simple example of a template, which may direct the morphology of a growing cluster. For homogeneous nucleation in a solution, the possibilities for controlling particle shape are very limited. There is, admittedly, an impressive variety of methods for synthesizing anisotropic colloids in a bulk solution21, but in most cases the outcome of a method can rarely be anticipated. Particle shape is sensitive to various parameters, such as pH, temperature, reactant concentrations, nature of anions and organic additives that may block certain faces. Thermodynamics, of course, only provides us with an equilibrium shape for crystals according to the Wulff relations [2.2.51]. However, growth of the equilibrium form is rarely encountered31 and the effect of the experimental parameters mentioned earlier on the growth kinetics of
D.R. Uhlmann, B. Chalmers in Nucleation Phenomena, D.E. Gushee, Ed., Am. Chem. Soc. Publ. (1966). 21 E. Matijevic, Chem. Mater. 5 (1993) 412. 31 W. Dunning, loc. cit.
2.30
PREPARATION AND CHARACTERIZATION
the faces of a polyhedral particle is hard to predict. So, In this respect heterogeneous nucleation and growth on (or In) a host Is an attractive alternative to achieve shape control. Potential hosts, such as micelles and other self-assembled structures, are reviewed elsewhere1'21. 2.2g Comminution and other preparation methods In condensation methods, colloidal particles are prepared from molecular species, whereas in dispersion methods the colloidal size range is reached by breaking down a macroscopic phase into progressively smaller parts. A well-known example of the latter is emulsification, which comprises the dispersion of one liquid in the presence of another (see sec. V.8.2). Sometimes shaking or stirring suffices to obtain an emulsion, and in other instances strong mechanical forces from 'colloid mills' are needed. Only when the interfacial tension between the two liquids is very low may the thermal motion of the molecules provide the energy required for emulsification, a well-known practical example being the spontaneous emulsification of agricultural chemicals in water. Mostly, this situation leads to microemulsions. Emulsification is also an ingredient of increasing importance for the preparation of inorganic colloids; a liquid reactant is emulsified and then polymerized to form a solid particle31. This strategy is being used, among others, for the synthesis of very small, monodisperse silica particles41. Instead of an emulsion, one can also start from an aerosol; here, the dispersion step is the formation of airborne liquid droplets, which contain some inorganic precursor. In spray drying (an important method in the ceramic industry), such droplets are dried in a flow of hot air to produce inorganic powders. In other aerosol methods water is not removed, but deliberately added to obtain inorganic particles, as in the case of titanium (IV) ethoxide aerosols that react with water vapour to yield spherical, amorphous titania colloids. One can say that from the viewpoint of inorganic colloid synthesis, emulsification and aerosol methods are nothing but condensation methods, except that the 'reactor' has been formed by a dispersion technique. The dispersion of inorganic material itself, also called comminution51, is the process of mechanical fracture in a ball mill. Such a mill is a rotating cylindrical vessel, containing inorganic materials (crystals, aggregates of particles) and tungsten carbide or alumina balls. The ease with which a mineral or clay can be ground depends on the surface tension y and the mechanical strength of 11
M. Pileni, Nature Materials 2 (2003) 145. J.H. Adair, E. Suvaci, Curr. Opin. Colloid Interface Sc(. 5 (2000) 160. 31 The procedure reminds of monomer droplets which polymerize to latex colloids by the addition of a monomer-soluble initiator. See e.g. R. Buscall, T. Corner, and J.F. Stageman, Polymer Colloids, Elsevier (1985). 41 K. Osseo-Asare, reviews microemulsion-mediated synthesis of inorganic colloids in the nanometer range in Handbook of Microemulsion Science and Technology, P. Kumar and K. Mittal, Eds., Marcel Dekker (1999) 549. 51 See the review by De Castro and Mitchell, mentioned in sec. 2.5a. 21
PREPARATION AND CHARACTERIZATION
2.31
the solid. The minimum work needed to break a column of material of cross-sectional area A equals the surface excess Glbbs energy AG = 1yA . If the newly formed surfaces are immersed in a solution, any adsorption will lower the surface tension according to the Gibbs adsorption equation [1.2.13.8] dy = -RTTd\na
[2.2.63]
where F is the surface concentration, a the activity of the adsorptive. Thus, in view of the Glbbs Isotherm, we can understand why comminution is carried out on solids submerged in a solution rather than in air. In aqueous solutions, the species adsorbed on inorganic surfaces Include ions which form an electrical double layer. Alternatively, often surfactants or polyelectrolytes are added to further reduce the tendency of particles to adhere to one another after comminution. However, most of the energy input is not invested in increasing the interfacial area, but dissipated as heat in the agitation process which must achieve a very high energy density close to the particles to break them down. The applied stresses must overcome the mechanical strength of the particles. There are several ways to achieve that. Agglomerates can be dispersed by impact on a surface (of the vessel or of balls that are added), or the particles are forced to undergo pressurization and decompression in rapid cycles. Brittle particles are better dispersed by the impact mechanism, elastic particles rather by shear. A variety of mills are commercially available, both for dry and for wet milling. It may be added that the Van der Waals attraction is also reduced by the solution, in comparison to dry powders in air or another gaseous atmosphere. The notorious 'caking' of dry powders, Incidentally, is partly caused by capillary forces when the powders are actually not dry enough. A similar attraction is observed for colloids in a liquid mixture, with one component preferentially wetting the colloids. The comminution method produces particles with a broad distribution in shape and size and, in general, the (relative) distribution becomes wider for a longer milling time. The lower end of the size distribution Is a particle size of O(/an) for a typical milling time of hours to days. A much higher degree of dispersion is difficult to achieve because very high shear forces are required to fracture solids in the sub-micron range. Milling, nevertheless, Is frequently applied in industrial practice for the preparation of dispersions, not only to break up particle aggregates but also to simultaneously and intensively mix particles and polymeric additives prior to processing the dispersions. 2.2h Separation andjractionation techniques Preparation of a sol is usually followed by a separation procedure, which may serve various purposes. Distillation is the obvious method to remove volatile components, such as ammonia from a silica alcosol (see sec. 2.4a). It may also be used to concentrate sols (at reduced pressure) or to transfer particles to other solvents. Dialysis and electrodialysis are employed to remove low-molecular compounds and contaminating electrolytes, which migrate out of the sol across a semipermeable membrane into a
2.32
PREPARATION AND CHARACTERIZATION
liquid reservoir. A simple but effective dialysis setup is a flexible cellulose bag, filled with sol and suspended in a flow of demineralized water. (Such membranes, incidentally, are not chemically inert because cellulose fibres may hydrolyze.) Often the sol level in the cellulose tube rises due to the increasing osmotic pressure. The loss of electrolyte also manifests itself in a viscosity increase due to the electroviscous effects, see sec. 6.9b. Solute diffusing across the cellulose membrane may also stem from dissolving colloids because of the Gibbs-Kelvin effect. For example, prolonged dialysis of silica sols produces a notable weight loss due to the continuous removal of soluble silica, which promotes dissolution of small particles. Dialysis against a salt solution is equivalent to ion exchange. Ion exchange resins are available for simultaneously exchanging cations for H+ and anions for OH". These 'mixed-bed' resins are used for preparing nearly salt-free sols with a large Debye length. To separate the colloids themselves from the liquid phase, either filtration or sedimentation is required. These two methods will be discussed in some more detail; for filtration below, for sedimentation in 2.3d. In a filtration process, the colloids are separated from the suspension by their accumulation on a filter or membrane, which is permeable for solvent and lowmolecular species11. Liquid flow is driven by a pressure difference, which for vertical filtration is due to the weight of the liquid itself, plus a piston or external gas pressure. The liquid transport through the membrane and the growing packing of colloids, with typical microstructures as in fig. 2.1, is an example of flow in a porous medium. Thus, important trends, such as why small particles are difficult to separate, can be explained in terms of d'Arcy's law [1.6.4.36] for viscous flow of an incompressible fluid through a porous medium, which we write here as u = -—Vp
[2.2.64]
n Here u is the average flow velocity of the liquid with viscosity r] in a medium with permeability B, driven by an average hydrostatic pressure gradient Vp. The porous medium in filtration is the layer of deposited colloids, which grows in time. Thus, the draining liquid experiences an increasing drag, which retards the filtration rate, as is observed in processes such as slip casting of inorganic sols and water purification. To quantify this retardation, consider a sol with solid volume fraction <j>, which forms a filter cake with volume fraction
[2.2.65]
Using d'Arcy's law [2.2.64], we can eliminate u as -(B/ T))[Ap/L{t)), which can be 11 For an Illustrative example on poly(styrene) colloids, see K. Bridger, M. Tadros, W. Leu, and F. Tiller, Sep. Sci. and Techn. 18 (1983) 1417.
2.33
PREPARATION AND CHARACTERIZATION
substituted in [2.2.65] and integrated to obtain the filtration law, which quantifies the rate at which colloids are separated L2(t) =
——Apt
[2.2.66]
c ~
Here, C is the Kozeny constant11, which for a random sphere packing is about C ~5. The KC relation is very useful because a value of C ~ 5 also holds for dense packings
&
a
~± d
~ A d
~ 2L d
Figure 2.10. Specific surface area Ag related to a or d for a variety of particle shapes. For high aspect ratios, A« only depends on the smallest dimension d. 11
J. Kozeny, Sitzber. Akad. Wissensch., (Wien) {Ila) 136 (1927) 271.
2.34
PREPARATION AND CHARACTERIZATION
of non-spherical particles (including fibers11), as well as mixed particle sizes2'31. The specific surface area (fig. 2.10), therefore, provides a reasonable estimate of B and, consequently, of the filtration rate of a particular dispersion of particles. We note here that C may vary with the concentration
P
A
t><,
A
g
where A is the specific surface area of the particles, here51 defined as the surface area per particle volume. The liquid permeability has the dimensions of a length squared (for example, a tube radius squared in the case of the Hagen-Poiseuille law, see sec. I.6d sub (2)). Hence, the permeability of the pore space scales as Bpore-^) Af
[2.2.69]
In a filtration experiment, however, we do not measure fluid flow in the pores, but the flow rate averaged over the whole filter cake, including the solid phase. Since inside a solid the flow rate is zero, the overall permeability that determines the filtration rate is BJlZ^tA-2
[2.2.70]
which is the KC scaling in [2.2.67]. It is clear that B and, hence, the filtration rate, is determined by the smallest dimension of a particle, e.g. the thickness of a platelet, 11
E.J. Wiggins, W.B. Campbell, and O. Maass. Can. J. Res. B17 (1939) 318. P.C. Carman, Trans. Inst Chem. Eng. 15 (1937) 150; J. Soc. Chem. Ind. 57 (1938) 225. 31 G.W. Jackson, D.F. James, Can. J. Chem. Eng. 64 (1986) 364. 41 D.M.E. Thies-Weesie, A.P. Philipse, J. Colloid Interface Sci. 162 (1994) 470. 51 in [2.3.16] and elsewhere Ag is in m 2 per gram. 21
PREPARATION AND CHARACTERIZATION
2.35
which determines the specific surface area (fig. 2.10). Particles with all dimensions in the micron range are easy to separate by filtration, whereas any dimension in the nanometer range necessitates very high pressures. We refer here to the filtration of stable sols; aggregation of small particles to large clusters will enhance the filtration rate11. Sedimentation is not a suitable alternative to separate stable particles, which are too small for filtration because settling and filtration rate have the same particle size dependence, as will become clear in section 2.3 d on sedimentation. An interesting aspect, lastly, of the filtration of colloidal suspensions is that the density of the particle deposit strongly depends on particle shape. For example, for rigid fibers or rods21 with high aspect ratio L / d » 1, the packing density of randomly oriented particles asymptotes towards zero as
11
The filtration rate can be used to monitor the effectiveness of polymeric flocculants. An entrance to relevant literature is J. Gregory, A.E.I, de Moor, ACS Symp. Ser. 240 (1984) 445. 21 S.R. Williams, A.P. Philipse, Phys.Rev. E 67 (2003) 051301. 31 E.P. Plueddeman, Silane Coupling Agents (Plenum) (1991); D.E Leyden, Ed., Silanes, Surfaces and Interfaces, Gordon and Breach (1986). Also see sec. 2.4. 41 A. Philipse, A. Vrij, J. Colloid Interface Sci. 128 (1989) 121.
2.36
PREPARATION AND CHARACTERIZATION
the large variety of functional moieties, which can be attached to the particle surface, and even buried inside particles when the processes of silica synthesis and modification are mixed11. SCAs hydrolyze to reactive silanols, which graft themselves onto silica via formation of siloxane linkages. As an illustration, immerse a hydrophilic glass slide for about 30 minutes (or even less) in a solution of typically one percent of SCA2) in ethanol, with some acid or base added to facilitate hydrolysis of the SCA. Next, rinse the slide with pure ethanol to remove free SCAs and dry it in hot air. The slide is now poorly wetted by water as a manifestation of its surface modification. In a similar manner, one obtains hydrophobic sand or the functionalized silica grains used in affinitychromatography. For the particles in a silica sol, the surface modification chemistry is basically the same as for macroscopic silica, but an important additional challenge arises and that is to avoid coagulation during the modification procedure. Once reactive oligomers or polymers attach to a colloidal core, the core-shell particle behaves as one kinetic unit with an average kinetic energy of (3/2)fcT. This energy has to be weighed against the replacement of a large number of solvent molecules by the adsorbed species. Even a very small Gibbs energy penalty per replacement may suffice to produce aggregates that do not break apart by thermal motion. Such aggregation can also be induced by minute changes in the nature or composition of the solvent, a subtle effect that is often difficult to predict or to explain afterwards. The fact is that any small change in composition involves a large number of low-molecular species, with a net enthalpy change that easily compensates the entropy loss due to aggregation of large colloids. Thus, the image of colloids coated by reactive molecules while diffusing around in an inert, neutral solvent background is clearly inappropriate. All molecular interactions must, in principle, be accounted for, a challenge which we cannot meet yet. One obvious counterexample to this neutral background is any solvent adsorption on (modified or unmodified) colloids. Water adsorption on silica is well known, see sec. 3.13b, but polar organic solvents such as dimethylformamide or triethylphosphate also adsorb in significant amounts on bare silica particles, often sufficient to prevent their coagulation. One could make a case that accurate characterization of a colloidal dispersion includes measurement of the immersion enthalpy of particles in their solvent, see sec. II.2.3d. Surface modifiers have to compete with solvent adsorption, which will lower the grafting density31. We have indicated several reasons why transfer of charged colloids in polar liquids to modified particles in stable organic sols is a tricky process, which often has to be
11
A. van Blaaderen, A. Vrij, Langmuir 8 (1992) 2921. Take a coupling agent with a hydrophobic group, such as the TPM in sec. 2.4a. 31 A.M. Nechifor, A.P. Philipse, F. de Jong, J.P.M. van Duynhoven, R.J.M. Egberink, and D.N. Reinhoudt, Langmuir 12 (1996) 3844. 21
PREPARATION AND CHARACTERIZATION
2.37
optimized by trial and error11. Small particles, it should be noted, also have a kinetic disadvantage, because the number densities of nanometer-sized particles are high. Therefore, any coagulation will occur rapidly, since the coagulation rate is proportional to the square of the number density. For modified, stable colloids, of course, the small particle size becomes a benefit in view of the many functional groups per gram. Lastly, one attractive option, which should be mentioned, is the simultaneous synthesis and modification of inorganic colloids by their nucleation and growth in the presence of the modifying agent, which also influences (and perhaps even controls) the particle size. Examples are the formation of small magnetic particles21 by thermolysis of metalcarbonyl-precursors in surfactant solutions, and the synthesis of extremely small gold colloids by reduction of gold salts in the presence of silane coupling agents31. 2.2/ Other methods Preparation methods mentioned so far certainly do not exhaust the routes for obtaining colloidal sols. Metal colloids, for example, can also be formed by electrical disintegration methods. Here, an arc is passing between electrodes under water, vaporizing electrode material to a metal gas that subsequently condenses into particles of colloidal dimensions. The formation of aqueous metal sols by electrical dispersion techniques was pioneered by Bredig and others 4 ' but has been replaced by more convenient alternatives such as the reduction of metal salts or the thermal decomposition of metal-carbonyl compounds and metal ions complexed by chelating agents (for example, triethanolamine)51. Another preparation strategy already explored in the early days of colloids science, however, has been more lasting, and even evolved into a separate branch of materials research, often referred to as sol-gel processing6'. Thomas Graham7' not only coined the term 'colloids' but also the terms 'sol' and 'gel' to denote, respectively, the initial and final state in the coagulation of a liquid dispersion to a space-filling solid-like material. Graham studied this transition for silica, alumina and other inorganic substances in water as well as ethanol. He found, for example, that water-glass in alcohol may change from an 'alcosol' to an 'alcogel' with nearly the same volume at already very low silica concentrations (see also section 2.4a). Sol-gel transitions and other coagulation phenomena became a classical topic of colloid science; the field of sol-gel processing is broader and aims to cover the whole route from a liquid sol via gelation, drying and sintering to the final solid state, usually 11
C. Pathmamanoharan, PhD. thesis, Utrecht, The Netherlands (1998). T.W. Smith, D. Wychick, J. Phys. Chem. 84 (1980) 1621. 31 P.A. Buining, B.M. Humbel, A.P. Philipse, and A.J. Verkleij, Langmuir 13 (1997) 3921. 41 G. Bredig, Z. Angew. Chem. 11 (1898) 951; T. Svedberg, Die Methoden zur Herstellung Kolloider Losungen Anorganischer Stqffe, Verlag von Theodor Steinkopff (1909). 51 For an update on electrical methods see Delplancke's review in sec. 2.5. 61 C.J. Brinker, G.W. Scherer, Sol-Gel Science, Academic Press (1990). 71 T. Graham, Phil. Roy. Soc. London 151 (1861) 183; J. Chem. Soc. August (1864) 618. 21
2.38
PREPARATION AND CHARACTERIZATION
a ceramic material11. The process may start with a precursor solution of aqueous salts or metal-organic compounds; in particular metal alkoxides are widely used in sol-gel research (sec. 2.4 provides examples of both types of precursors). In addition to molecular precursors, sol preparation may also employ dry powders synthesized by vapour-phase methods at the high temperatures produced by a furnace, flame, plasma or laser. Advantages are the high purity of the powders, and the possibility of atomicscale mixing in the vapour phase. Metal alkoxide precursors are convenient because they readily react with water to form metal (hydr)oxides at room temperature. In a limited number of cases (silica, titania and zirconia) monodisperse spheres are formed21. These amorphous spheres have a significantly lower mass density than the corresponding bulk oxide and contain solvent as well as residual organic groups. The specific surface area is generally much greater than expected from particle dimensions measured with TEM, and at least for silica it is well known that the spheres noticeably shrink when exposed to the vacuum in an electron microscope. These features clearly show that hydrolysis of metal alkoxides does not produce massive spheres but rather spherical sponges having sufficient internal cross-linking to maintain their shape, and sufficient porosity to allow permeation of solvent and small molecules. The variety of preparation schemes In sol-gel processing involving either precursors or powders, is enormous, see the extensive literature survey in ref. *'. Sol-gel literature is, consequently, often a useful information source on inorganic colloid synthesis, and the chemistry of hydrolysis and condensation of metal ions in solution31 or reactions involving metal alkoxide precursors1'. However, what has been said in section 2.2f on the limited predictability and control of particle size and morphology remains true. We may rightfully look with some envy to the controlled colloid formation In biomineralization4'. Examples are the single crystals of magnetite (Fe3O4) and other minerals made by bacteria, the monodisperse ferrihydrate (5Fe2O3.9H2O) colloids in the ironstorage protein ferritine, and the beautiful silica structures sculped by diatoms51. Further study of the still poorly understood in vivo preparation methods used by organisms may provide new ideas for man-made colloids. 2.3 Characterization After synthesizing a colloidal dispersion and performing the required purification or separation techniques, as described in the previous section, we wish to characterize 11 U. Schubert, N. Hiising, Synthesis of Inorganic Materials, Wiley-VCH (2000); C.J. Brinker, G.W. Scherer, loc. cit. 21 C.J. Brinker, G.W. Scherer, loc. cit. 3) For this intricate chemistry see J.P. Jolivet, Metal Oxide Chemistry and Synthesis; from Solution to Solid State, Wiley (2000). 41 S. Mann, J. Webb, and R. Williams, Biomineralization, VCH (1989). 51 L. Addadi, S. Weiner, Angew. Chem. Int. Ed. Engl. 31 (1992) 153.
PREPARATION AND CHARACTERIZATION
2.39
the colloidal particles. Nowadays, sophisticated techniques are available to investigate colloids in nanometric detail, in real as well as reciprocal space [vide infra). A discussion of all measurable parameters, with techniques to match, requires an encyclopaedia, so we only present a selection, referring for a more extensive coverage to the provided literature entries. 2.3a Visual observations and microscopy A great deal of information can already be obtained from visual inspection of a sol, aided by a torch or small laser. Some trends, to which no doubt exceptions may be found, are listed below, starting with optical properties. (i) Coloursv For colloids, which do not absorb light at visible wavelengths, the turbidity is only due to light scattering. A bluish appearance in this case is due to Rayleigh scattering of particles with a typical diameter on the order of 100 nm or smaller. This bluish Tyndall effect can be clearly observed for dilute dispersions of latex particles and several metal (hydr)oxide colloids, such as boehmlte and silica. A milky white appearance may be due to anything that shortens the mean free path of photons in the dispersion: large particle size, high refractive index and high colloid concentration. Multiple scattering is easy to demonstrate as it spreads an incoming narrow beam of laser light. A white appearance sometimes manifests aggregation; the bluish Tyndall effect for small aluminum hydroxide or silica colloids changes to white turbidity when the particles coagulate. Inspection of a (either stirred or shaken) sol with a light beam between crossed polarizers reveals optical birefringence when the dispersed particles have an anisotropic (plate or rod-like) shape. This birefringence, a mosaic texture of dark and light patches of sol regions with different optical axes, is quite spectacular for concentrated sols of tungsten oxide platelets, and can also be observed for vanadium oxide or boehmite fibres (see sec. 2.4c). Optical birefringence is caused by particles, which align in a shear flow, and when stirring is stopped the mosaic pattern usually rapidly decays by rotational Brownian motion. For very concentrated, strongly interacting particles a permanent birefringence may result from the inability of the plates or fibres to reorient. Colour effects due to absorption are too numerous to discuss here. Identification of particle composition on the basis of absorption is not always straightforward; witness, for example, the variety in yellow, brown and red colours of the iron (hydr)oxide colloids2'. Another important issue is the particle size dependence of absorption spectra
11
How informative colours and turbidity can be in assessing particle sizes is illustrated well for the case of polymer latex dispersions in E.I. Franses, L.E. Scriven, W.G. Miller and H.T. Davis, J. Am. Oil. Chem. Soc. 60 (1983) 1029. For particle sizes In surfactant systems the authors even present a diagnostic guide based only on perceptions of transparency and colour. 21 U. Schwertmann, R.M. Cornell, Iron Oxides in the Laboratory, VCH (1991)
2.40
PREPARATION AND CHARACTERIZATION
and the quantum size effect extensively discussed elsewhere11. A well-known observation here is the blue shift caused by coagulation of an initially red sol of stable gold particles. (ii) Settling When particles settle significantly within a few days it is worthwhile to estimate the effective Stokes radius, which would produce the order of magnitude of the observed settling rate. If this radius is much larger than the expected colloid size, either this expectation is wrong or the colloids are aggregating (or both). A sharp interface between sedimenting suspension and supernatant does not necessarily imply monodispersity (see further 2.3d). The sediment on the bottom should also be observed when the vessel is tilted; stable colloids tend to flow like a liquid (be it very slowly when the particles are densely packed), whereas aggregated particles form sediments or gels with a yield stress. Stable colloids produce diffraction colours when they form ordered sediments with spacing on the order of optical wavelengths. Well-known examples are the colloidal crystals observed in sediments of repulsive spheres, but rod-like particles may also produce Bragg reflections, such as in the so-called Schiller layers of /9-FeOOH rods, which may settle into a smectic structure21. In the latter case the visual appearance is a dark brown sediment, which exhibits specular reflection with iridescent colours depending on the angle of reflection. With a ruler one can already easily estimate an informative number, namely the particle volume fraction cH/h in sediment with height h formed in a suspension with height H and initial volume fraction
H. Weller. loc. clt. Y. Maeda, S. Hachisu, Colloids Surf. 6 (1983) 1. 31 H. Sonntag, K. Strenge, Coagulation Kinetics and Structure Formation, Plenum Press (1987). 41 R. Buscall, Colloids Surf. 5 (1982) 269. 51 A. Philipse, A. Wierenga, loc. cit. 21
PREPARATION AND CHARACTERIZATION
2.41
dering structures on the glass surface characteristic of buttermilk. It should be noted that stirring or shaking might considerably enhance the rate of coagulation due to an autocatalytic effect. The largest particle clusters are the most efficient in capturing particles in a shear flow and, therefore, grow fastest. This is orthokinetic coagulation, to which we return in sec. 4.5b. That is why stirring a sol is not without risk; it may coagulate a sol, which has marginal stability. Exposing a sol sample to high shear rates in a rheometer is a severe test for stability because any small floe, which is 'harmless' in a quiescent disperion, will grow rapidly in the shear field in an autocatalytic fashion. For the influence of particle interaction on sol rheology, see sec. 6.13. The onset of coagulation or phase separation sometimes announces itself clearly by the so-called critical opalescence, i.e. a strong increase in the light scattering on approach of a critical point due to the occurrence of large fluctuations in density, and, hence, in refractive index. Whenever such fluctuations can be observed in a gently shaken sol (their texture is reminiscent of the flow-induced birefringence mentioned earlier), it is pretty sure that the sol will gel or phase-separate soon thereafter. Observing what happens when an acid or base is added to a charge-stabilized dispersion is always informative. The pH at which a sol coagulates will in general be at its isoelectric point (i.e.p.), though there is the notorious counter example of silica, which is often quite stable at its i.e.p. of about 2 but rather coagulates near pH ~ 8, see sec. 3.13b. The sediment volume of settled floes is expected to reach its maximum at the pH where particles most strongly attract each other. The charge sign of colloids at various pH values can be checked simply by inserting the poles (or Pt-wires connected to them) of a battery and observing at which pole deposition takes place. (iv) Rheology The viscous and elastic properties of suspensions will be dealt with in chapter 6. Here we only mention some easy visual checks. Very concentrated stable dispersions, as in sediments of filter cakes, display shear thickening, which makes them hard to process. Squeezing such a filter cake, we notice that it falls dry due to dilatancy. When we observe shear thinning, i.e. lowering in effective viscosity when stirring or shaking a dispersion, the colloids are attractive, a hypothesis that can be checked by observing the increasing viscosity when leaving the dispersion quiescent for a while. The origin is the breakdown and reestablishment of coagulate networks. The latter process may take some time. Air bubbles are convenient markers for viscoelasticity. When quickly rotating a vessel, the bubbles are slightly out of phase with the oscillations of the fluid, an effect that can be clearly observed with a bottle of salad dressing. A gradual trapping of air bubbles accompanies the growth in yield stress in a gelling dispersion. When a concentrated dispersion gradually turns into a stiff gel with a high yield stress, as in the case of commercial silica sols on a time scale of months, a low-frequency response is heard upon gently tapping the vessel containing a gel, known as a ringing gel.
2.42
PREPARATION AND CHARACTERIZATION
(v) Microscopy Optical microscopy is, of course, a valuable extension of visual inspection. One can, for example, observe gel structures, growing clusters in a phase separation or using polarized light, liquid crystals or tactoid formation of anisotropic colloids. To observe colloids in bulk, confocal microscopy11 is a versatile method, which in the footsteps of Perrin (sec. 2.2a), is used to study Brownian motion in concentrated dispersions21. This method, however, is not (yet) a routinely available characterization technique, but rather belongs to the category of research tools, which falls beyond the scope of this text. This category also comprises the rapidly expanding field of scanning probe microscopy of colloids in the nanometer size range31. The microscopy, without which no characterization of colloids is complete, is, of course, electron microscopy41. Transmission electron microscopy (TEM) is employed to determine sizes, size distributions and particle shapes. The number-average particle size and the spread around this average can be used to predict averages found from other techniques, as explained in appendix 1. Particle sizes can be made absolute, in principle, by adding calibrated latex spheres to the dispersion. The average colloid-latex number ratio on the TEM grid provides a rough estimate of the initial colloid number density. TEM has the disadvantage that the samples have to be dried (which may produce aggregation) and subsequent exposure to a high vacuum may distort or shrink the particles. Therefore, one should be very careful about drawing too many conclusions about the colloid structure in the wet state from TEM images. Particle topography can be imaged with scanning electron microscopy (SEM). The additional advantage of SEM is elemental analysis by energy dispersive X-ray analysis41. Also here, sample preparation and high vacuum exposure may give rise to artefacts. Two techniques that are presently in a state of development should be mentioned that circumvent this problem. Environmental scanning electron microscopy (ESEM) images colloids, which are kept in an environmental chamber in a water vapour atmosphere with adjustable pressure and temperature. There is no need to coat the particles with conducting film as In conventional SEM. In this way, hydrated colloids can be characterized in their native state, as has been shown for latex spheres as well as inorganic particles . Cryogenic TEM images a vitrified film prepared by a fast temperature quench (usually in liquid ethane) of a liquid dispersion film. In principle, vitrification preserves
11
T. Wilson, Confocal Microscopy, Academic Press (1990). W.K. Kegel, A. van Bladeren, Science 287 (2000) 290. 31 See e.g. A. ten Wolde, Ed., Nanotechnology, The Netherlands Study Centre for Technology Trends (1998), and B. Bhushan, Ed., Springer Handbook of Nanotechnology, Springer (2004). For a useful literature entrance, also for the various types of optical and scanning probe microscopies and their applications to colloidal dispersions see: E. Kissa, Dispersions; Characterization, Testing and Measurement, Surfactant Series 84, Marcel Dekker (1999). 51 R.H. Ottewill, A.R. Rennie, Eds., Modern Aspects of Colloidal Dispersions, Kluwer (1998). 21
PREPARATION AND CHARACTERIZATION
2.43
the particle distribution and morphology of the structures in the liquid film1'21. Inorganic, iron colloids with radii as small as 2 nra have been imaged in this way31. Cryo-TEM characterization is without doubt an important complement to scattering techniques. The latter have the advantage of probing very large numbers of particles, in a 3-dimensional bulk, on a variety of length scales. Cryo-TEM studies fewer particles in a quasi 2-dimensional film, but directly visualizes any structure formation, shape and size details, which are usually difficult to obtain unambiguously from scattering data in reciprocal space. 2.3b Light scattering To characterize colloids with scattering techniques, visible wavelengths as well as neutrons and X-rays are employed. The choice of the wavelength is determined by the length of scales to be probed, but also by the (complex) refractive index of the colloids. For example, concentrated colloidal dispersions are usually too opaque for classical light scattering methods to apply . Light scattering, nevertheless, is a versatile characterization method for many colloid and polymer solutions. Restriction of the treatment to light is further justified by the fact that scattering by other radiation is not fundamentally different51, so that the general form of the equations to be derived remain applicable, mutatis mutandis. Light scattering is extensively treated in several excellent reviews56'71. The aim here is to give a brief description restricted to the characterization of homogeneous, non absorbing spherical particles in a dilute suspension. For a discussion on light- and other types of scattering in concentrated systems, see chapter 5. Elsewhere81, the principles outlined here are generalized to spheres of variable composition, rods, polymers etc. The topic was introduced in chapter 1.7. (i) Static light scattering (SLS) When the refractive index rij of a colloid differs from the index n 2 of the solvent, the electric field of an incident light beam induces an oscillating dipole in the colloid, which causes scattering of light in all directions. We assume that the electric vector of the incident light with wavelength A is polarized perpendicular to the scattering plane, and we detect the scattered photons with the same polarization at an angle 0. A sphere
11
Y. Talmon, Ber. Bunsenges. Phys. Chem. 100 (1996) 364. P.M. Frcderik, W.M. Busing, J. Microscopy 144 (1986) 215. 31 K. Butter, P. Bomans, P. Frcderik, G. Vroegc, and A. Philipsc, Nature Materials 2 (2003) 88. 41 Multiple scattering, however, is exploited in diffusive wave scattering methods, sec E. Pike, J. Abbiss, Light Scattering and Photon Correlation Spectroscopy, Kluwcr (1997). 51 M. Kerker, The Scattering of Light and other Electromagnetic Radiation, Academic Press (1969). 21
61
71
C. Tanford, Physical Chemistry
of Macromolecules,
Wiley (1961).
B.J. Berne, R. Peeora, Dynamic Light Scattering, Wiley (1976). K.S. Schmitz, An Introduction to Dynamic Light Scattering by Macromolecules, Academic Press (1992).
2.44
PREPARATION AND CHARACTERIZATION
of radius a behaves as a Rayleigh (point) scatterer when a IX « 1 , producing a scattered intensity 7 at a distance r from the sample (see also [1.7.7.8]).
Hi 4
[nf+n^j
This is Rayleigh's famous equation. I is independent of the scattering angle, and increases with a 6 , because the scattered field amplitude (the square root of the intensity I) is proportional to the polarizability and, hence, to the volume of the sphere. Note also the well-known Rayleigh law I ~ A~4, which accounts for the blue sky and also for the bluish appearance of sols of small particles. For a sphere with radius a comparable with X, the electric fields scattered from different regions of the sphere have different phases. The resulting interference decreases the intensity measured by the detector relative to the intensity I, according to [2.3.1 ]. This reduction can be accounted for by a factor P(q), I
RGD = / x P ( £ / )
[2.3.2]
Here, 'RGD' refers to the Rayleigh-Gans-Debye limit (sec. I.7.8d) (nl-n2)a/X«l
[2.3.3]
where the incident light is hardly distorted by the sphere. Further, P(q) is the form factor, which for homogeneous spheres is given by p ( q ) = r3[sin(qa)-qacos(qq)]f
L
[ 23 4 ]
M
Note that the form factor is normalized such that P(0) = 1, whereas P{q) < 1 for finite values of the scattering vector q = (4;z7/t)sin(6V2)
[2.3.5]
The zero values of the form factor occur when tan(qa) = qa with roots qa = 4.493, 7.725, etc. So, the location of minima in the angular scattering profile directly provides the sphere radius. This determination is only accurate in the RGD limit for sufficiently monodisperse spheres because polydispersity washes out the details of the form factor. Conversely, one can conclude from sharp intensity minima that the spheres under study must be quite monodisperse. It should be noted that even for monodisperse spheres, the form factor minima may fade due to multiple scattering if the sol is not sufficiently dilute. A well-known method to determine a sphere radius at small scattering angles is to employ the so-called Guinier approximation1^
11
A. Guinier, G. Fournet. Small-angle Scattering o/X-Rays, Academic Press (1955).
PREPARATION AND CHARACTERIZATION P(q) = exp[-qa 2 /5]
2.45 [2.3.6]
Equation [2.3.4] reduces to [2.3.6] for sufficiently small values of qa . Note that this method only requires relative intensities because the particle radius is obtained from the initial slope of a Guinier plot of In l(q) against q2 . The Guinier radius, obtained from [2.3.6], is quite sensitive to deviations from monodispersity because of the strong a-dependence in [2.3.1 and 6]. The average scattered intensity for polydisperse noninteracting spheres in the Guinier region is proportional to 'RGD~(a6exP[-92a2/5])
[2' 3 - 7 l
where the angular brackets denote a number average as defined in 2.3f. We can rewrite [2.3.7] to 7
RGD ~exp[-q 2 a2 /5]
a 2 = (a 8 )/(a 6 )
[2.3.8]
where the occurrence of the so-called Guinier radius ac shows that large spheres contribute heavily to the averaged SLS particle size. The apparent radius can be converted to the number-averaged radius (a) a2. s ( l + 13s2)(a>2,
for s 2 « 1
[2.3.9]
Here, sa is the relative polydispersity defined in app. 1, which is assumed to be small. This assumption also requires the absence of particle aggregates and contaminants, such as dust and air bubbles, which strongly contribute to the scattered intensity at small q. If these requirements are met, Guinier plots may be extrapolated to q = 0 to obtain the molar mass M of the colloids. For JV identical scatterers in a total scattering volume V, we can rewrite [2.3.1] to »« Vc \ =M
n/ M;
-noz M
r 2 2 4 N A y P 2 |_n 1 2 +2R 2 2 J
[2.3.10]
where c is the weight concentration of the particles with mass density p and JVAv is Avogadro's number. This form shows that the forward scattering intensity indeed provides a molecular mass since the other parameters can, in principle, be measured. This method, of course, necessitates absolute scattering intensities; Huglin explains calibration procedures11. The effect of polydispersity on the absolute SLS intensity at q = 0 can be found from a generalization of [2.3.10]. The molecular mass is a weight average and the corresponding apparent radius equals a 3 = (a 6 )/(a 3 ), which can be simplified further using the moment expansion from appendix 1. Inspection of [2.3.10] makes clear that the scattering intensity can be reduced by
M.B. Huglin, Light Scattering from Polymer Solutions, Academic Press (1972).
2.46
PREPARATION AND CHARACTERIZATION
lowering the optical contrast, i.e. the difference between the refractive indices n{ of the colloids and n 2 of the solvent. For perfectly homogeneous particles the scattered intensity vanishes at zero contrast, whereas inhomogeneous colloids have a residual scattering intensity even when the solvent matches their average refractive index. Measuring the intensity as a function of the solvent refractive index, also referred to as contrast
variationl],
is useful to characterize the internal structure of colloidal
particles. Contrast variation is, in particular, sensitive to refractive index changes at the surface of the colloids due to, for example, grafted or adsorbed polymers, as discussed in detail elsewhere21. (ii) Dynamic Light Scattering (DLS) Particle characterization by SLS relies on time-averaged scattering, so it is immaterial whether the suspended particles are stationary or not. To characterize the Brownian motion of the colloids by a determination of the diffusion coefficient in [2.2.10], one can employ the fluctuations of scattered light in time using dynamic light scattering (DLS). The essence of a DLS experiment has been explained in sees. I.7.6c,d, 7 and 8, see also fig. 1.7.10. Here we briefly summarize the method. The coherent light of a laser, illuminating a periodic grid, produces a static diffraction pattern on a screen, but when the grid is replaced by a colloidal suspension the pattern changes continually. We observe flickering bright spots due to constructive interference of light scattered by individual colloids and dark patches manifesting destructive interference. The time-dependent intensity fluctuations in this speckle pattern are, of course, caused by perpetual Brownian motion, and it is clear that somehow the dynamics of these fluctuations contain information on the translational
diffusion
coefficient D. One way to harvest this information is to determine the time correlation function
C{[t) of the scattered light field31. This function comprises a characteristic
time t
needed for a significant change in the speckle pattern for a given value of the
wave vector q. For times t « t whereas, for t » t
, the intensity pattern has not significantly decayed,
the speckles are uncorrelated. The characteristic time is defined as
tq=\/Dq2
[2.3.11]
which can be interpreted as the typical fluctuation time of the speckle pattern at the detector, or roughly the time taken by a particle to freely diffuse a distance q"1 , in accordance with Einstein's law for quadratic displacement by Brownian motion. For the simplest case of a sol of identical, non-interacting spheres a DLS experiment
Contrast variation is also a versatile method for X-rays and neutrons, see e.g. R. Ottcwill. in
Colloidal Dispersions, J.W. Goodwin, Ed., Roy. Soc. Chem. (1982). 21 A. van Helden, A. Vrij. J. Colloid Interface Set 76 (1980) 4)8. Time correlation functions were introduced in Lapp. 11.
PREPARATION AND CHARACTERIZATION
2.47
yields 11
Cf(£) = exp[-t/t q ]
[2.3.12]
i.e. a single exponential function of the correlation time t, which provides t and ultimately the hydrodynamic sphere radius ah via the Stokes-Einstein equation, D = kT I Qjirja-^. This radius is usually larger than the actual particle radius due to factors that slow down diffusion, such as the presence of an electrical double layer or adsorbed (solvent, surfactant or polymer) molecules. The friction factor J determined via DLS measurements can also be used, in combination with ultracentrifugation, to determine the molar mass of the colloids. This classical procedure, further explained in sec. 2.3d, is quite general since the particle shape need not be specified. For polydisperse, non-interacting spheres, [2.3.12] is generalized to21 (lexpl-t/t Cf[t) = ±
])
q /
[2.3.13]
where the brackets denote an average over the distribution of particle radii. It is seen that C{{t) is now a sum of exponentials, each weighted by the intensity scattered by the pertinent species; for RGD spheres I is given by [2.3.2]. When the distribution is so narrow that the delay times t
in [2.3.11] are close, one can expand the exponential
about a mean value to find lnCf(t) = -D a p p q 2 t + o(q 4 t 2 )+...
[2.3.14]
with an apparent diffusion coefficient If X
Dapp=—
a h =(aB ) / { a 5 )
[2.3.15,
So, from a fit of the logarithm of the measured correlation time to [2.3.14], we obtain at small t a hydrodynamic radius a h , which can be converted to a number average using the moment expansion, discussed in appendix 1. Our rudimentary sketch of DLS on dilute sphere suspensions neglects many complicated but important issues, such as particle interaction at finite concentration through long-range electric and hydrodynamic forces and scattering by non-spherical colloids or flexible polymers, which have extra terms in Ct(t) due to rotational and internal motions. When it is known a priori (from electron microscopy) that the colloids under study are spherical and that they are non-interacting RGD scatterers, one obtains a hydrodynamic radius with an accuracy of a few percent, if all works well. A significant wave vector-dependence of the apparent diffusion coefficient obtained
K.S. Schmitz, loc. cit.: P. Pusey, R. Tough, in Dynamic Light Scattering and Velocimetry, R. Pccora, Ed., Plenum (1982). 21 P.G. Cummins, E.J. Staples, Langmuir 3 (1987) 1 109.
2.48
PREPARATION AND CHARACTERIZATION
from [2.3.14] may harbour a variety of aggregation or interaction effects, which are studied in various monographs11. 2.3c Surface area The specific surface area Ag of colloidal particles is an important characteristic for many applications, for instance in catalysis and adsorption. It also determines the rate at which particles can be removed afterwards by filtration using [2.2.67]. For a particle with volume V and mass density p , the specific area is defined as: Ag = A/pV
[2.3.16]
For a given amount of mass the sphere has the minimum surface to volume ratio; any shape deformation at constant volume increases A . For anisometric colloids, such as clay platelets or vanadiumpentoxide fibres, A is largely determined by the particle thickness (see fig. 1 .d). For sufficiently thin platelets or long fibres, the length (distribution) and detailed shape is irrelevant. Only when the relevant particle dimension in fig. 2.10 is inhomogeneous, will dispersity affect the value of As . We generalize [2.3.16] to in
5>i A i
PLNK i=l
where JV; is the number of particles with area A{; the brackets denote a number average. For polydisperse spheres, it follows that the specific surface area is given by Ag = 3/pas as={a3)/{a2) [2.3.18] The apparent sphere radius as can be estimated from the sphere dispersity as will be explained in app. 1. Liquid permeability measurements have been widely applied to determine A for spheres21 and non-spherical particles up to the extreme aspect ratios encountered in paper and fibrous media . Carman introduced using [2.2.67] to obtain the surface area of powders, and found that the method is not affected in accuracy if the powder contains mixed sizes of particles and particles of irregular shape. A liquid permeability (or filtration rate) measurement, however, becomes impractical for colloids in the nanometer size range because this would require very high hydrostatic pressures, as discussed in 2.2h. One option here is to dry the sol and determine the surface area by the well-known BET method introduced in sec. II.1.5f. For very small particles, it should be noted that techniques of gelling and drying a sol generally
K.S. Schmitz, loc. cit.: R. Pccora, Dynamic Light Scattering: Application of Photon Correlation Spectroscopy, Plenum (1983). D. Thics-Wccsie ct al., loc. cit. 31 G. Jackson, D. James, Can. J. Chem. Eng. 64 (1986) 364.
PREPARATION AND CHARACTERIZATION
2.49
produce some area loss by coalescence at particle-to-particle contacts and/or sintering. For aqueous silica sols the specific surface area can be determined, following Sears11, by measuring the amount of alkali adsorbed from solution as the pH is raised from 4 to 9. To increase the amount of adsorbed base and to eliminate any effect of unintentional small amounts of electrolytes in samples, the titration is carried out in a saturated sodium chloride solution (about 200 g NaCl/litre). Sears standardized this method using a number of silica powders with a specific surface area known from BET ( N 2 ) adsorption, which allows a direct conversion of titer volume to specific surface area. Of course, for this conversion any other base-consuming species must be removed or corrected for. Silica is remarkably stable at such high ionic strength, as long as the pH is low and only when sufficient 0.1 N sodium hydroxide is added, such that pH ~ 8 sols start to slowly coagulate and become turbid (see sec. 3.13a). However, hydroxyl groups from the NaOH are still able to reach all surface silanol groups in the fresh particle aggregates, so coagulation does not affect the outcome of the Sears titration. The method, also used in fig. 2.8 to monitor the decrease of surface area in time due to Ostwald ripening, is routinely applied in industry as a rapid area check for freshly prepared sols21. It would be interesting to know whether sols of other inorganic colloids can also be rapidly characterized with a standardized acid-base titration. The Sears method is just one member of a family of adsorption techniques to determine surface areas, employing adsorption of ions, nitrogen, water vapour and organic dyes, such as methylene blue. Some illustrations can also be found in sec. 11.2.7c. Information on this family is easy to locate in the literature . Mercury porosimetry for porous surfaces has been introduced in sec. II. 1.6b; the method can also be used for particle size analysis41. Lastly, one often-overlooked aspect should be mentioned here and that is the effect of surface roughness, which is disregarded in the scaling A ~ a™1. For compact spheres with a surface fractal dimensionality df , the specific surface area scales as51 Ag~A2-dfadi-3=-(-)f g a\A)
df>2
[2.3.19]
1
where X is the diameter of the probing molecule, which is used to measure the surface area. The particle size dependence of the specific surface area is only reciprocal for a smooth object (d f = 2), whereas the dependence is weaker for fractal surfaces and even disappears for df = 3. Soil particles form a well-documented example of the 11
G.W. Sears, Anal. Chem. 28 (1956) 12, 1981. This is one of the oldest illustrations of a colloid titration. 21 K. Andersson, B. Larsson, and E. Lindgren, Silica Sols and Use of the Sols, US Patent 5, 603, 805 (1997). 31 S.J. Gregg, K.S.W.Sing, Adsorption, Surface area and Porosity, Academic Press (1982). 41 D.M. Smith, D.L. Sternmer, Powder Tech. 53 (1987) 23. 51 M. Borkovec, Q. Wu. G. Degovics. P. Laggner. and H. Sticher, Colloids Surfaces A73 (1993) 65.
2.50
PREPARATION AND CHARACTERIZATION
effect of surface roughness. Their surface dimension is close to df = 2.4, as follows from several independent surface area studies1 . Clearly, for an accurate characterization of surface area and data interpretation, information on the surface structure is needed. Here it is convenient to employ SAXS, where at high magnitudes of the scattering vector q the surface dimension follows from a log-log plot of scattering intensity versus q (Porod's law2)). 2.3d Sedimentation The settling of colloids under gravity or in a centrifuge is a rich (but surprisingly little consulted) source of information on their size, shape, and interactions. The equipment varies from an analytical ultracentrifuge, which records a sedimentationdiffusion equilibrium profile with a high resolution to a vessel for studying settling under gravity. In the latter case, the descent of the boundary between supernatant liquid and settling sol is measured. The observation of an initially sharp boundary, which gradually spreads in time, may manifest polydispersity, back-diffusion of the particles, or both. A boundary, which stays sharp, is consistent with the settling of monodisperse particles (with negligible diffusion), but certainly not proof of it. Even a polydisperse system may produce a sharp boundary due to a strong decrease of the settling rate with increasing concentration. Particles at the low concentration side of a boundary then catch up with the slower moving colloids in the high concentration region. The possibility of such a self-sharpening boundary necessitates additional tests before it can be concluded that a sample is monodisperse. It is often thought that the presence of several sedimenting boundaries ('layered sedimentation') manifests a mixture of particles, which is fractionated during the sedimentation process. Layered sedimentation, however, may occur in any system due to small temperature gradients that induce convective rolls31. Convection is suppressed by letting the settling proceed in narrow tubes or capillaries, but in larger vessels convection must be expected to occur unless strict temperature control is applied. We also note that layered sedimentation may manifest a thermodynamic demixing (see chapter 5), in which gravity pulls different phases apart. If the descending boundary provides the sedimentation velocity v{t) of non-interacting colloids, we can obtain their mass m on the basis of Newton's second law d / x m— v{t) = (m-mo)g + Jv(t)
[2.3.20]
Here, mQ is the mass of displaced solution or sol, g is the acceleration of gravity a n d / is the hydrodynamic friction factor. The effective colloid mass can also be written as
M. Borkovcc et al., loc. cit. For a review of small-angle scattering by fractal systems see P.W. Schmidt, J. Appl. Crystallogr. 24 (1991) 414. 31 D. Mucth. J. Crocker, E. Esipov, and D. Grier, Phys. Rev. Lett. 77 (1996) 578. 21
PREPARATION AND CHARACTERIZATION
2.51
m - m 0 = m(l - PQV ) for colloids with partial specific volume V
in a solution with
mass density pQ . For rigid (inorganic) colloids the inverse mass density usually is a good measure of the specific volume, but this is not so for drainable, porous particles or polymers. For a particle, which is initially at rest relative to the solvent, the solution of [2.3.20] is u(c) = u[l-exp(-t/r)]
T=m/J
[2.3.21]
where v=
- ^ L
=
-{i-pQVp)g
[2.3.22]
is the stationary sedimentation velocity reached when the particle weight and the frictional force Jv(t) exactly balance. To find the effective colloid mass m-m0
we
need the friction factor, which can be obtained from the diffusion coefficient D = kT/ J measured in a separate, dynamic, light-scattering experiment. Then, the colloid mass follows from the Svedberg v s =- = g
{m-mr.)D -— kT
equation: [s]
in which we have also introduced the sedimentation
[2.3.23] coefficient
s , a mobility defined
as the sedimentation velocity per unit of the applied acceleration, either from gravity or a centrifuge. The quantity s is actually the viscous relaxation time of a particle with mass m - mQ, i.e. the time taken by the particle to dissipate its kinetic energy when the acceleration is switched off. This, of course, is very similar to the time r in [2.3.21] needed to reach a stationary state. A typical value is r ~ 5 x 10~ 9 s for a silica particle with radius a = 100 nm sedimenting in water. Hence, there is clearly no need to worry about inertia in a sedimentation experiment. The Svedberg equation [2.3.23] is valid for particles of arbitrary shape. Instead of a measured diffusion coefficient, one can also insert a theoretical friction factor in [2.3.22] when the shape of the colloids is known. Results are available for oblates 11 , prolates, rods and a variety of other nonspherical particles. We only quote here the well-known outcome for spheres, also known as the Stokes value of the sedimentation coefficient s=2£-Poa2
[2
9 n Here, r\ is the solvent viscosity and p- p0 the mass density difference between particle and solvent. When sedimentation coefficient and molecular mass are known, one directly obtains the friction coefficient of the colloidal particles. This provides only limited information about their shape. From the specific colloid volume V
we
calculate a particle volume mV , and if we assume that the colloid is a sphere with
J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics. Prentice-Hall (1965).
2.52
PREPARATION AND CHARACTERIZATION
radius a , Stokes law f = 6nrja predicts the minimum value of the friction coefficient of the colloid in question. A larger, experimental, friction factor may be due to a hydration layer or a deviation from the spherical shape. The effect of the shape is modest for nearly spherical colloids; when a sphere is deformed at constant volume to become oblate or prolate, an aspect ratio of nearly 10 is needed11 to increase the hydrodynamic friction by 50%. Since this increase is nearly the same for both shapes", it is clear that additional information is needed to extract a particle dimension or shape from the hydrodynamic friction factor. (i) Sedimentation-diffusion
equilibrium
Colloidal particles settle under the influence of gravity until a sedimentationdiffusion equilibrium is established. This equilibrium is the balance between the downward particle flux due to gravity and a back flux due to diffusion, which opposes the concentration gradient created by gravity. The equilibrium concentration profile c{x) may also be seen as the isothermal balance between a gradient in osmotic pressure 77 and the particle weight per volume of sol + c(x)[m-mQ)g
=0
[2.3.25]
CuC
Here, x is the distance to the bottom of the vessel at x = 0 . For ideal particles, for which van 't Hoff s law 17 = ckT applies, we find the exponential (or barometric) height distribution
( -x/U
\ 8
'
L= 8
ZcT
[2.3.26]
(m-mo)g
where cQ is the particle concentration at the bottom and I is the so-called gravitational length, which is a measure of the thickness of the profile ( I is actually the average height of the colloids relative to x = 0 ). The equilibrium profile, in principle, provides the effective mass of the colloids. However, an accurate determination of the concentrations decay c[x) is far from straightforward. Vessels should be rigorously thermostatted because the concentration profile is very susceptible to liquid convection. Convective rolls may induce layering or completely homogenize the sol, even for minute temperature gradients. Nearly inevitably, concentration effects also come into play because approaching the bottom of the vessel the concentration rises and at some point van 't Hoffs law may have to be replaced by a virial series, such as in [1.7.8.10], At sufficiently high altitude, of course, the concentration profile approaches the exponential in [2.3.26]. For particles of known mass, it is possible to quantitatively investigate the concentration effects just mentioned. If we succeed to determine c{x), the equation of state follows from [2.3.25] by the integration
11
Sec K.E. van Holde, loc. cit. p. 81
PREPARATION AND CHARACTERIZATION
2.53
h
/7 p =/7 h +(m-m o )gjc(x)dx
[2.3.27]
P
Here, h is an altitude that is sufficiently high for the pressure to obey van 't Hoff s law. By changing the integral's lower boundary p, the pressure 77 as a function of colloidconcentration c
is recovered. The main experimental challenge is to find a way to
determine colloid number densities. For example, monitoring the optical turbidity as a function of height has the disadvantage that for higher concentrations the signal is nonlinear in the colloid concentration. Piazza, and co-workers11 employed spheres with a crystalline anisotropy, of which the number can be counted by depolarized light scattering, allowing them to retrieve the equation of state over a wide concentration range. Such a quantitative characterization of particle interactions is unfortunately unfeasible in many practical cases, and even though the sedimentation-diffusion profile is equivalent to the osmotic pressure, it often only provides qualitative information. For example, when for monodisperse spheres the (visually observed) profile is much more extended than the gravitational length L, we can at least conclude that osmotic pressures are much larger than expected from van 't Hoff s law. This may be attributed to a charge on the colloids (see below) or to significant repulsive interactions. Attractions between the particles should shrink the equilibrium profile, though attraction between the colloids may also lead to voluminous non-equilibrium gels (sec. 6.14). The sedimentation-diffusion equilibrium is quite sensitive to the dispersity, primarily because the particle mass enters into the Boltzmann exponent in [2.3.26]; small particles are pushed to high altitudes, whereas very large particles remain in the vicinity of x =0. For spheres of species i the ideal equilibrium distribution is Ci(x)
= c O i exp(-x/i g i )
c 0 i =JV t o t i /A; g i
[2.3.28]
We employed here the normalization that the total number of particles i in the height distribution in a vessel with cross-sectional surface area A is NtoU = AJCi[x)dx o
[2.3.29]
It is obvious from [2.3.28] that heavy particles (small I ) contribute mostly to the concentration at the bottom, whereas the lighter ones (large I ) dominate at high altitude. For non-interacting particles, the total number density decays exponentially, with a gravitational length that provides the number-averaged colloid mass ; g =fcT/(l-p o y p )(m)g
11
R. Piazza, T. Bellini, and V. Degiorgio, Phys. Rev. Lett. 71 (1993) 4267.
[2.3.30]
2.54
PREPARATION AND CHARACTERIZATION
a result that follows from summing the forces [2.3.25] for all species i. A profile of the total weight concentration yields the corresponding weight averaged mass m w = (m2)/{m), whereas application of the Schlieren optics11 produces the z-average, mz = (m 3 )/(m 2 ). As always, the type of average depends on the experimental method used to investigate a sample. For polydisperse colloids, the various averages may differ considerably (see also appendix 1), whereas their identity is a clear proof of monodispersity. (ii) Sedimentation of charged particles In comparison to uncharged colloids, fairly little is known about the SD-equilibrium of charged particles, although it is clear that charge effects may already be substantial for ideal colloids . A striking example is shown by charged colloids at low external salt concentration. The Donnan osmotic pressure for non-interacting colloids in this case has the limiting form n = {z + l)ckT
[2.3.31]
where z is the number of free counterions produced by each colloid . It is assumed here that the counterions dominate the external salt; when sufficient salt is added, the pressure gradually decreases to 77 = ckT. On substitution of [2.3.31] in the force balance [2.3.25], we find
c(x)=coexp[-(x/ig(l + z))]
[2.3.32]
showing a gravitational length which, compared with the uncharged state, has been increased by a factor (1 + z), which is quite substantial since z may be of order 1000. The physical meaning of the (1 + z) term is that the practically weightless counterions tend to form a homogeneous distribution for entropic reasons, whereas the colloids are pulled down by gravity. Electroneutrality, however, couples colloids and counterions and the net result is an increase in the colloidal gravitational length. The 'entropic lift' due to counterions is actually equivalent to a homogeneous electric field, which is inevitably present in an equilibrium density profile of charged particles, and reduces the effective colloid mass, as discussed elsewhere in detail2'41. Thus, to determine the mass (i.e. the gravitational length) of charged colloids, sufficient salt should be added such that the Boltzmann profile reduces to [2.3.26].
11
K.E. van Holde, R.L. Baldwin, J. Phys. Chem. 62 (1958) 734. R. van Roij, J. Phys. Condensed Matter 15 (2003) S3569; A.P. Philipse, J. Phys. Condensed Matter 16 (2004) S4051. Extensive discussions on the fraction of counterions that is free will follow in chapter III and V chapter 2. 41 M. Rasa, A. Philipse. Nature 4 2 9 (2004) 860. See also R. van Roij, loc. cit. 21
PREPARATION AND CHARACTERIZATION
2.55
Figure 2.11. Schematic of an ultracentrifugation experiment (not to scale). The colloids move radially to the bottom of the sector-shaped cell with an apparent weight (m — m.Q)apT at a distance r from the axis, which rotates at an angular velocity 0) . (Hi) Analytical
ultracentrifugation
Characterization of colloids via settling or sedimentation-diffusion
equilibrium
under gravity is only possible for a restricted class of particles, which have a suitable value of gravitational length in the range of mm to cm. Also, in view of the mentioned convection and detection problems, an analytical ultracentrifuge is an important, if not indispensable, characterization tool. There is an extensive, mainly biomolecular literature, on centrifugal analysis 11 . We will briefly discuss the methods to determine a colloid mass. A spinning rotor exerts a centripetal force on the sedimentation cell, which is directed towards the rotation axis. The corresponding centripetal acceleration of the cell at a distance r from this axis is a = co2r , where a> is the angular rotor velocity in radians per second. The colloids move towards the bottom of the cell (fig. 2.11), experiencing an effective weight increase, which is completely equivalent to an enhancement of the gravitational acceleration from g to co2r ; the colloids at some position r cannot judge whether their weight is due to a centrifugal field or to gravitational pull. The Svedberg equation [2.3.23] remains, therefore, exactly the same, with the sedimentation coefficient s = vI co2r . The determination of s is as follows. Suppose the boundary between sol and the supernatant moves at a rate u = d r b / d £ . Integration of fi)2rs = d r b / d t yields ln-^—= or2 s(t-tQ)
[2.3.33]
Analytical Ultracentrifugation in Biochemistry and Polymer Science. S. Harding, A. Rowc. and J. Horton. Eds.. Roy. Soc. Chem. (1992).
2.56
PREPARATION AND CHARACTERIZATION
where rb{t) is the position of the boundary at time t. The sedimentation coefficient, therefore, follows from a graph of the logarithmic term in [2.3.33] versus ( t - t 0 ) . The boundary, of course, does not remain infinitely sharp as it traverses the cell because of diffusional spreading. Then, the question is 11 which point should be used as rb in [2.3.33]. This point turns out to be the second moment of the curve for the concentration gradient
T-2 = Jr2(ac/3r)dr/J(3c/ar)dr
[2.3.34]
where both integrations include the boundary, i.e. from a position in the homogeneous solvent to a position in the plateau region in the homogeneous sol. To determine the colloid mass from a sedimentation equilibrium profile, one uses a rotor speed, which is smaller than that used for a velocity experiment; packing of all colloids near the bottom of the cell has to be avoided. Instead, it is desirable to achieve a profile, which is sufficiently extended for data fitting, in particular of the dilute tail of the profile where colloidal interactions are insignificant. The ideal profile follows from the centrifugal force F = {m- mo)a)2r, corresponding to the potential energy of a colloid at position r r
-J Fdr = (m-mQ)co2-(a2-r2),
[2.3.35]
a relative to the meniscus at a. The Boltzmann distribution for ideal particles is, therefore,
[
r2
_n2 1 5Z -
2X
J
A2
=7
frf
\~T
(m-m o j<» z
[2.3.36]
Note the analogy with the barometric height distribution [2.3.26]. The thickness of the profile, set by the length X , can be adjusted by changing the rotor speed co. A graph of In c(r) versus r 2 will yield the length X and, therefore, the effective mass of the colloids. This mass determination, which is in principle quite accurate, has been fruitfully (and frequently) checked for monodisperse biomolecules (proteins, viruses, DNA fragments); molecular masses generally match the values known from elemental compositions very well. The existence of extensive literature on data analysis and instrumental issues21 shows that, nevertheless, for most investigators the analytical ultracentrifuge is anything but a simple black box, just as the engine of a car is for most drivers. 2.3e Other methods For practical dispersions such as paints or ceramic suspensions, their application will largely determine the choice of characterization techniques, in addition to those
21
Sec for a pointed discussion: K.E. van Holdc, Physical Biochemistry, Prentice Hall, (1971). S.E. Harding, A.J. Rowc, and J.C. Horton, loc. cit.
PREPARATION AND CHARACTERIZATION
2.57
mentioned in previous sections. For inorganic colloids processed to an eventually dried compact, as in ceramic shaping techniques, one can largely appeal to the usual methods of powder technology. These methods include X-ray analysis to identify crystalline components, mercury-intrusion to measure porosities of 'green' or sintered bodies, and thermal analysis to investigate temperature-dependent properties. The last mentioned analysis comprises, among other things, differential thermal analysis (DTA) and thermal gravimetry (TG). DTA exposes a material to a controlled temperature increase as a function of time and records release or uptake of heat due to phase transitions (including melting points or melting trajectories), chemical reactions, and any other endothermic or exothermic process. TG monitors the weight of the sample in the course of the temperature-time scan and detects, for example, the loss of water which was adsorbed on particles or generated by condensation of hydroxyl groups, as are often found on oxidic materials. A combination of DTA and TG is certainly also useful for inorganic model colloids, for example to determine the weight fraction of organic material due to a leftover of a surface modification (see sec. 2.i). The latter will produce an endothermic peak and simultaneous weight loss roughly in the range 400-600°C, the temperature range over which organic molecules are burnt off. In addition, physically adsorbed water will be detected as an endothermic loss already below 100°C, and release of water or other low molecular solvents at higher temperatures is indicative of porous colloids with internal silanol or alkoxy groups, as occur in the silica spheres prepared by the Stober process. Exothermic peaks at temperatures around 1000°C or higher may manifest any of the many phase transitions found in alumina and silica containing (clay) materials1 . Thus, DTA-TG, in combination with chemical analysis results for elemental percentages, contributes to a fairly complete material picture of colloidal particles. Spectroscopy (NMR, Infrared, etc.), of course, provides even more chemical detail on colloids and their surface coverage. One important 'application' of model colloids is their use in critical test of theories. Then the primary concern is not so much knowledge of the chemical composition of colloids (useful as it may be), but rather the surface parameters which appear explicitly in the theory under study. For charged colloids these are in any case the double layer parameters. Two of such parameters offer themselves, the surface charge density <7° and the electrokinetic potential C, . The former follows from colloid titrations when the charge-determining mechanism is known, the latter from electrokinetics. Principles and elaborations can be found in various places of FICS, especially sec. I.5e (titration), chapter II.3 (composition of double layers), and chapter II.4 (measurement and interpretation of electrokinetic potentials). It is good to keep in mind that by titration and electrokinetics very different double layer parameters are measured. In fact, for a full characterization of the double layer composition both techniques should be simul" F.H. Norton, Fine Ceramics, Kricgcr (1987).
2.58
PREPARATION AND CHARACTERIZATION
taneously applied to the same system. Only in this way can the composition of the inner part of the double layer be established. We note that the difference between the point of zero charge (p.z.c.) and the isoelectric point (i.e.p.) is a measure of specific adsorption (sec. II.3.8). In many cases only ^-potentials are available. Experience has shown that for situations of not too strong double layer overlap these potentials are satisfactory characteristics to be substituted in equations for the Gibbs energy of interaction (chapter 3). For sterically stabilized particles, information about the amount and distribution of attached polymers is needed, see chapter V.I. Often one is interested in measuring a concentration dependence, which brings on a characterization problem that is often swept under the carpet, namely the issue of the specific particle volume, which here deserves some more discussion. A theoretical concentration dependence is usually expressed in terms of particle volume fractions, whereas one measures, say, a diffusion coefficient or low-shear viscosity, as a function of colloid weight concentrations. How should they be converted to volume fractions? Clearly, a measurement of the mass density or specific volume of the colloids is needed. For rigid hard spheres, one option is to measure the intrinsic viscosity and to find the specific volume that produces agreement with Einstein's value of 2.5 for the coefficient of the volume fraction. (Viscosity is a better option here than sedimentation or diffusion, because the volume fraction enters on the level of single, non-interacting particles). Factors such as porosity, softness and surface charge of particles and deviations from the spherical shape, can be couched into an effective specific volume which matches the Einstein result. For further information on the viscosity of particulate matter, see sees. 6.9, 10 and 13. However, this procedure yields an effective hydrodynamic volume fraction which may be inappropriate for equilibrium measurements such as the osmotic pressure or the static structure factors from light scattering. To find thermodynamic volume fractions one can also choose the specific volume such that the colloidal hard spheres start to freeze at the theoretically expected volume fraction, a procedure which, of course, is only feasible for the limited class of colloids which form colloidal crystals. We note here that such crystals in principle produce the particle mass from the location in reciprocal space of Bragg peaks, analogous to the counting of atoms in a unit cell in X-ray diffraction. Direct measurement of particle mass densities in solution by weighing dispersions as a function of concentration requires more material than is often available in the case of model colloids. Commercial equipment is available for this weighing on a small scale by measuring the resonance frequency of capillaries filled with dispersions. The latter method is very accurate for pure liquids, but for dispersions prone to uncertainties due to, among other things, the sensitivity to details of cleaning procedures of measuring cells. An alternative is using quartz crystal microbalances (QCM's). Sedimentation profiles from ultracentrifugation (see sec. 2.3d) provide the buoyant mass and therefore still require a separate specific volume measurement. For biomolecules centrifuga-
PREPARATION AND CHARACTERIZATION
2.59
tion in a salt gradient is employed: the molecules stay suspended at a height at which their buoyant density is exactly matched by the salt solution. This method, which provides an accurate and well-defined thermodynamic specific volume, is suitable for mass densities below about 1.8 g/cm3 , the maximum density of the salt solution (usually CsCl2 ). This mass-density range includes polymer colloids, but excludes many inorganic particles. The latter could be handled by measuring sedimentation velocities in solvent mixtures and extrapolation to zero velocity, a method which apparently has not been exploited yet. To conclude, the conversion of weight concentration to volume fraction (or particle number density) is usually not straightforward and needs to be made explicit in the characterization of colloids under study. 2.3/ Size distributions Characterization of colloidal particles is incomplete without specification of their size distribution. For this, various options are available, including the ultracentrifuge, a method discussed by Harding et al.11. Advantages of light scattering methods include measurement speed and the very large number of particles that are sampled. The procedure, however, is far from simple. The main problem is the inversion of the measured field autocorrelation function [2.3.13] to obtain the intensity-weighted contribution of each particle species. This inversion has no unique solution when the measurements are contaminated by noise21. In addition, many subtleties in sample preparation and data analysis need to be addressed, as discussed extensively by Provder3'. The direct determination of a large number of diameters by electron microscopy is accurate and simple, in particular for inorganic colloids, which usually maintain their integrity during drying on a grid and exposure to vacuum. For easily deformable latices or emulsion droplets, other techniques such as confocal microscopy may be used (see chapter V.8). Another useful (but yet little employed) option is cryogenic electron microscopy, a technique discussed briefly in 2.3a. When colloids are sufficiently small, say with radii below 100 nm, quite a large number of them can be simultaneously imaged in the glassy cryo-TEM film. When the colloids are repulsive due to surface charge or a polymer coating, which has a low contrast for TEM, one may observe clearly separated particle cores4 (c.f. fig. 2.4), which form a convenient input for image analysis software; the S-distribution of fig. 2.11 has been obtained in this manner. Though extensive single-particle imaging is the best option to obtain a reliable size distribution without a priori assumptions about the colloids, it is not always possible in practice or convenient for routine analysis. Often one relies on fractionation methods
" s . E . Harding, A.J. Rowe, and J.C. Horton, loc. cit. P.G. Cummins, E.J. Staples, Langmulr 3 (1987) 1109. T. Provder, Ed., Particle Size Distribution: Assessment and Characterization, ACS Symposium Series 332 (1987); Particle Size Distribution II, ACS Symposium Series 472 (1991). 41 A.P. Philipsc, G.H. Kocnderink, Adv. Colloid Interface Set 100-102 (2003) 613. 21
2.60
PREPARATION AND CHARACTERIZATION
in which the distribution is broken up into classes making use of some particle property. Sieving of powders is the classical method for separation based directly on particle size; most other methods rely on the response of particles to external fields or a change in particle interactions. An example of the latter is the fractionation of iron oxide particles by repeated phase separation induced by the addition of salt, which preferentially removes the larger particles11. This fractional distillation is expected to work for any interparticle attraction, which is size-dependent. The procedure reminds one of fractionating a polymer solution by slow addition of a poor solvent upon which molecules with high molecular weights precipitate first. We will now briefly explain some fractionation methods, which employ external fields. Magnetic particles can, in principle, be fractionated by an external, inhomogeneous magnetic field B . The magnetic force on the particles is 2) F = (m-V)B
[2.3.37]
where m is the magnetic moment of the particle, which is proportional to the particle volume. To separate small, paramagnetic colloids, large gradients are needed. They can be produced by magnetizing a steel wool matrix; near curves and edges of the filter large gradients exist, which capture particles from the dispersion '. By increasing the magnetization of the matrix, fractions with increasingly smaller particles can be captured. This high-gradient magnetic separation has important applications in the removal of iron oxides from clay dispersions and wastewater. However, its potential for quantitative fractionation is much less developed than for techniques based on sedimentation. The disc centrifuge photosedimentometer
(DCP) separates spheres, which sedi-
ment radially outward past a detector with a velocity determined by Stokes' law. The technique appears to be robust and sufficiently accurate, for example, to resolve the various components in mixtures of standard polystyrene spheres31. For non-spherical colloids, the analysis (as always) is less straightforward than for spheres. We note here that for particles with high aspect ratios, the sedimentation rate is determined mainly by the smaller dimension (c.f. the surface areas in fig. 2.10). For example, for thin rods with diameter d and length L, the (orientationally averaged) friction factor is41 fo=3miL/ln[2L/d)+a
[2.3.38]
where or is a number of order unity. Consequently, the sedimentation coefficient of the thin rods is
" V. Cabuil, R. Massart, J. Bacri, R. Perzynski, and D. Salin, J. Chem. Res. (S) (1987) 130. J. Svoboda, Magnetic Methods for the Treatment of Minerals, Elsevier (1987). 31 T. Provdcr, 1987, loc. cit. 41 S. Broersma, J. Chem. Phys. 32 (1960) 1632. 21
PREPARATION AND CHARACTERIZATION
2.61
s=
7^-pv^^{p-poH^)
[2 3 391
--
So here only the distribution in diameters is of importance, which could simplify the DCP analysis. Equation [2.3.391, incidentally, warns us that fractional sedimentation is not useful to decrease the polydispersity in length of rods or width of platelets. Sedimentation Jield-Jlow Jractionation (sedimentation FFF) fractionates particles in a flow channel with a field acting perpendicular to the stream direction11. The (centrifugal) field forces particles to accumulate at one wall of the channel where the viscous drag is large so that downstream displacement of particles is retarded. The distance to the wall depends on the particle size, which leads to size fractionation in the flow direction. The method is quite sensitive and mixtures of well-defined spheres can be analyzed with good resolution. For information on still another fractionation method, hydrodynamic chromatography, we also refer to Provder21, whose analyses include a comparison of the various particle characterization methods applied to one and the same series of monodisperse PMMA latices. Once a sufficiently large number of particles have been sampled in each fraction, it may be useful to compare the result with one of the standard mathematical distribution functions, some of which are given below. For a continuous distribution the n moment is defined as (a n }= fa n P(a)da o
;
|p(a)da = l o
[2.3.40]
Here, P[a) is the normalized probability distribution for the radius a and P(a)da is the probability for a radius to be in the interval a, a + da . Note that P(a) has the dimensions of reciprocal distance, which is why it is also called the probability density. The normal (or Gauss) probability density has the familiar, bell-shaped function and obeys P(a) =
i=exPr-(a-
[2.3.41]
in which (a) is the number-averaged radius and aa is the {absolute) standard deviation defined by cj2=((a-
[2.3.42] s a defined through
s 2 = cr2 /(a) 2 in appendix 1. Fairly narrow distributions, as for silica and latex spheres, are often fitted reasonably well with a Gauss model as illustrated for silica in
11 21
J.C. Giddings, F.J.F. Yang, and M.N. Myers, Anal. Chem. 46 (1974) 1917. T. Provder 1987 toe. cit.
2.62
PREPARATION AND CHARACTERIZATION
fig. Al.l. For many other colloidal systems, however, the size distribution is asymmetric. This may be due to various factors, such as a milling process, secondary particle nucleation, the growth mechanism or the tendency of the larger particles to aggregate. For the y-Fe2O3 colloids in a ferrofluid (fig. Al.l), the asymmetric distribution often fits a log-normal probability distribution reasonably well
P{a) = — . l exp - ^ — — aV2;rlnz 21nz
'—
[2.3.43]
where a1 z = l + — ° - = i +s2 (a) 2
[2.3.44]
in which, as before, s 2 is the relative polydispersity. The normalized radius moments of the log-normal distribution are given by
l
a]
2
a
for s 2 « 1 . So, from a measured or estimated polydispersity, one can compute the higher radius moments and predict the apparent radius obtained by a particular characterization method. As an example of a discrete probability distribution we mention the Poisson distribution P(a) = ^ - e x p [ - ( a > ]
a = 0,1,...
[2.3.46]
This probability function is skewed, but rapidly becomes more symmetrical upon increasing the average (a) . The Poisson distribution is especially useful when the number of events (here, particle radii) is small. For a large number of random variables, the Poisson distribution is fairly well approximated by normal distribution21. In many cases, a fit of experimental counts to a theoretical probability distribution will be poorer than in fig. Al.l. The data may simply disobey the chosen distribution, for instance, because the distribution is bimodal or the number of counts may be insufficient to draw a clear conclusion anyhow. Luckily it is possible to approximate the moments in [2.3.40] only on the basis of a measurement (or choice) of the relative polydispersity without presupposing any particular size distribution. The approximation is actually the truncated expansion [2.3.45], which is valid for any not too broad size distribution as shown in appendix 1.
" P.N. Pusey, H.M. Fijnaut, and A. Vrij, J. Chem. Phys. 77 (1982) 4270. A. Papoulis, loc. cit.
21
PREPARATION AND CHARACTERIZATION
2.63
2.4 Examples of sol preparation This section provides a few commented preparation methods for various inorganic sols. There are several reasons for such a presentation. Firstly, there is no substitute for learning about practical colloid chemistry than going into the lab to make your own colloids. Secondly, the examples illustrate that quite well-defined colloidal dispersions, used in state-of-the-art colloid research, can be obtained with simple methods. No chemical equipment is needed beyond what is used in a freshman chemistry course possibly with the exception of the autoclave for the boehmite synthesis. Thirdly, the examples also illustrate that this simplicity in method may be misleading. The outcome of a colloid synthesis is often difficult to explain or to adjust in a predictable manner. Elements of art and surprise remain. The selection below is biased by the author's hands-on experience; many more synthesis examples useful for teaching or research can be found in the general references of sec. 2.5. 2.4a Silica sols Silica sols are usually prepared in aqueous solutions from waterglass or in ethanol from the precursor tetraethoxysilane (TES). The formation of colloidal silica by acidification of waterglass is extensively documented elsewhere1'21. Here we only describe an instructive experiment in which silica supersaturation is generated by a change of solvent, instead of a change of pH3). A stock sodium-silica solution (Na2O . SiO2 , 27 wt% SiO2 ) is diluted with double distilled water to 0.22 wt% SiO2 . Under vigorous stirring, 0.2 ml of this dilute waterglass solution is rapidly pipetted into 10 ml absolute ethanol. A sudden turbidity increase manifests the formation of small, smooth silica spheres with a diameter around 30 nm and a typical dispersity of 20-30%. The solubility of silica in ethanol is much lower than in water, and it is estimated that in this experiment the supersaturation ratio due to the alcohol addition is on the order of S = O(10), which, in view of section 2.2b, should indeed produce very rapid homogeneous nucleation. The preparation of so-called Stober silica spheres from the precursor TES in an ethanol-ammonia mixture is well documented41. To obtain spheres with a radius of about 60 nm and a typical dispersity of a ~ 10-15%, the procedure is as follows. TES (60 ml, freshly distilled to remove any polymeric species) is injected under the liquid level of a thoroughly stirred mixture of 200 ml ammonia (25%) and 3 litres (preferably distilled) absolute ethanol in a vessel, previously thoroughly cleaned by multiple rinsing with, subsequently, distilled water and absolute ethanol. The TES solution is
11
R.K. Her loc.cit. K. Andersson, B. Larsson, and E. Lindgren, Silica Sols and the Use of Sols. US patent 5603805 (1997). 31 P.A. Burning, L.M. Liz-Marzan, and A.P. Philipsc, J. Colloid Interface Sci. 179 (1996) 318. 41 W. Stobcr, A. Fink, and E. Bohn, J. Colloid Interface Sci. 26 (1968) 62.
2.64
PREPARATION AND CHARACTERIZATION
gently stirred in the closed vessel; after about 15-30 min., an increase in turbidity manifests the formation of silica spheres in the alcosol, which grow to their final size over a time scale of hours. Silica growth can be continued by adding small portions of TES to control the final radius . This seeded growth method has the risk of introducing secondary silica nucleation, so samples should be checked with TEM. Secondary particles are usually small enough to be separated from primary spheres by repeated sedimentation. They generally do not disappear by Ostwald ripening because of the very low silica solubility in ethanol. Stober silica spheres can be easily silanized by surface modification with 3-methacryloxypropyl-trimethoxysilane (TPM) as follows. TPM (about 1-3 ml per gram of silica) is added to the alcosol, after which the solvent is distilled to reduce the alcosol volume by about 30%. Unreacted TPM is removed afterwards by repeated sedimentation-redispersion cycles. The non-desorbing TPM layer, with a hydrodynamic thickness of a few nm, improves the stability in various organic solvents. Silanes are also very useful to modify silica with fluorescent or phosphorescent dyes, as discussed in refs.2'31. With respect to storage of silica sols, the following points should be noted. Aqueous silica sols generally show aging effects; the specific surface area decreases (fig. 2.8) and the pH tends to increase, probably due to sodium hydroxide leaching. Commercial silica sols are usually quite stable, as manifested by a constant (Newton) viscosity. However, over longer periods of time (say one year) the viscosity gradually increases and space-filling gels are often formed. Stober alcosols may aggregate in the course of time, especially for larger particles with a relatively high ammonia concentration. Removal of ammonia by bubbling nitrogen through the alcosol is one remedy. Another option is distillation together with silanization of the particle surface (see above), leading to TPM-coated silica spheres, which in absolute ethanol have practically unlimited stability. Hydrophobic silica spheres have been used extensively for the study of hard-sphere colloids in apolar solvents . A suitable surface modification in this respect is the esterification of surface silanol groups of silica spheres under vacuum distillation in a pure octadecyl alcohol melt at about 180°C. A more convenient procedure is to add a Stober alcosol directly to an excess solution of octadecyl alcohol in triethyl phosphate51. After distilling all ethanol and ammonia, the solution is stirred for several days at 140-160°C under a flow of dry nitrogen. The resulting octadecyl-coated
11
A. Philipse. A. Vrij, J. Chem. Phys. 87 (1987) 5634. A. van Blaadercn, A. Vrij, loc. cit. 31 M.P. Lettinga, M. van Zandvoort, CM. van Kats, and A.P. Philipse, Langmuir 16 (2000) 6156. Some illustrations follow in chapter 5. 51 A.M. Nechifor, A.P. Philipse, F. dc Jong, J.P.M. van Duynhoven, R.J.M. Egberink, and D.N. Reinhoudt, Langmuir 12 (1996) 3844.
PREPARATION AND CHARACTERIZATION
2.65
particles can be transferred to a stable dispersion in cyclohexane using sedimentationredispersion cycles". 2.4b Sulphur sols An often quoted example of the formation of a monodisperse sol is La Mer's method2'3 for preparing sulphur colloids, in which S is gradually formed by the reaction of thiosulfate with acid S2O§~ + 2 H + ^ S O 2 + H 2 O+S
[2.4.1]
This is a slow reaction, such that growth of sulphur particles occurs on a time scale of hours. La Mer's method is as follows. One (1.00) ml of 1.50 N H2SO4 is added to 995 ml of double distilled water in a one-liter volumetric flask that is thermostatted at 25°C. One (1.00) ml of 1.5 N Na 2 S 2 O 3 is added rapidly, after which the flask is quickly made up to 1 liter, mixed thoroughly and returned to the thermostat. Within 12 h a weak scattering can be observed from a hand laser (a Tyndall beam), manifesting growing sulphur particles. They continue to grow over a period of about 24 hours after which they settle, presumably because the sulfur colloids become quite large though colloidal instability may also play a role (the Van der Waals attractions must be substantial in this case). Sulphur growth can be stopped by titrating unreacted thiosulfate with an iodine solution in potassium iodide, according to I3 + 2S2O2- -> 31" + S 4 O|"
[2.4.2]
Because iodine solutions have an intense yellow to brown colour, even at high dilution, iodine can serve as its own end point indicator. Titration is continued until a barely perceptibly pale yellow sol remains. By applying iodometry after various time intervals on a number of acidified thiosulfate solutions, sols with various particle sizes are obtained. The sulphur sols are very suitable for a demonstration of the angular dependence of light scattering. When a beam of plane-polarized white light is viewed with the eye in a plane perpendicular to the polarization, spectral colours may be observed at angles, which depend on particle size51. This is a clear indication for a narrow size distribution of the sulphur colloids or, to be more precise, of the colloids, which dominate the lightscattering intensity; the presence of small, weakly scattering sub-particles cannot be excluded. Electron microscopy or atomic force microscopy of dried samples of sulphur
11
A.M. Nechifor ct al., toe. cit. V.K. La Mer, M.D. Barnes, J. Colloid Scl 1 (1946) 71. 31 A.B. Levit, R.L. Rowcll, J. Colloid Interface Set 5 0 (1975); for a preparation based on H 2 S. sec: G. Chiu, E.J. Meehan, J. Colloid Interface Set 6 2 (1977) 1. 21
41
LA. Vogel, Textbook of Quantitative Chemical Analysis, Longman (1989).
51
V.K. La Mer, M.D. Barnes, loc. cit.
2.66
PREPARATION AND CHARACTERIZATION
sols does not yield images of well-defined spheres (the attempt failed in the author's laboratory), but a variety of morphologies (indeed, containing elemental sulphur), including raspberry-like submicron particles composed of much smaller units. The imaging is hampered by the fact that sulphur colloids easily melt or deform; moreover, crystallizing salts in the drying TEM specimen complicate the picture. So far in the literature, making direct images of them has not supported the presumption that La Mer's method produces monodisperse spheres. 2.4c Boehmite and gibbsite sols One method to synthesize rod-like colloids employs aqueous aluminium alkoxide solutions to form elongated, crystalline A1OOH (boehmite) particles. The alkoxide is first hydrolyzed1' at room temperature in an aqueous HCl-solution, followed by a hydrothermal treatment at about 150°C in an autoclave. By varying the pH and the type and concentration of alkoxide, the length of the boehmite needles can be adjusted in the range of 100-400 nm; the needle thickness is 10-20 nm. The starting aluminiumalkoxides are Al(OBus)3 (aluminium tri-sec-butoxide, ASB), a volatile, colourless liquid, which hydrolyzes easily due to air moisture and Al(OPrM3 (aluminium triisopropoxide, AIP), a white powder, which is less reactive towards moisture. An aqueous HCl solution is made by pouring a concentrated HCl stock into water (never the other way around). The HCl solution must be titrated if its molarity is not precisely known. To a stirred mixture of 2900 ml of double distilled water and 22 ml of HCl (37%), 59.8 ml ASB is added after which a white precipitate, presumably aluminum hydroxide, is formed. (If the stock ASB is not clear but yellowish, it should be purified by distillation from hydrolysis products). Next, 46.0 g AIP is added, which dissolves within a few hours. The solution is gently stirred at room temperature in a closed vessel for a week. Then the now clear solution is autoclaved for 22 h at 150°C. One option is to heat the solution in partly filled metal pressure vessels with a Teflon inner core, which are slowly rotating inside an oven, as described by Buining et al.11. Commercial equipment for hydrothermal treatment is available. After this treatment, the vessels are allowed to cool to room temperature. Note that hydrolysis of the alkoxides produces alcohols, which increases the pressure in the autoclaved vessels. The Teflon inner cores, see ref.11, should not be removed unless they are completely cooled to room temperature, otherwise they will no longer fit in the pressure vessels. The somewhat turbid, easily flowing boehmite dispersion is dialyzed in cellophane tubes for 1-2 weeks against demineralized water to remove alcohols and salts. The dialyzed dispersion is highly viscous due to the strong, double-layer repulsion in the now nearly salt-free dispersion. Inspection of the dispersion between crossed polarizers reveals permanent birefringence. After dilution, the dispersion exhibits streaming
11
P.A. Buining, C. Pathmamanoharan, J.B.H. Janscn, and H.N.W. Lekkerkerkcr, J. Am. Ceram Soc. 74 (1991) 1303.
PREPARATION AND CHARACTERIZATION
2.67
birefringence, which confirms the presence of non-aggregated needles oriented by a flow field. The birefringence is destroyed by addition of some ammonia, which coagulates the boehmite particles. The boehmite dispersion, stored in a plastic bottle, may be stable over a time scale of months to even several years. Storage in glass vessels is not recommended because of possible deposition of soluble silica on the positively charged boehmite. TEM micrographs reveal somewhat irregular rods with an average length probably around 180 nm and a width close to 10 nm; the relative size dispersity is typically 30-40%. Shorter rods with a length of about 100 nm can be prepared by starting with 2850 ml water, 9.7 ml HC1 (37%) and 156 ml of ASB. The reproducibility of the dimensions of the boehmite crystals is modest and details of, for example, the hydrothermal treatment may significantly affect the sizes and shapes of the final particles. The temperature of the hydrothermal treatment greatly influences the particle morphology. At lower temperatures (T~135°C), mainly hexagonal gibbsite platelets are observed after 22 h, which apparently recrystallize to boehmite needles at higher temperatures or longer times. These platelets also form when the alkoxide solution (see above), instead of being autoclaved, is stored for several months at room temperature. The fairly monodisperse gibbsite hexagons (typical diameter 150 nm, thickness 13-15 nm) are useful model colloids, in particular, because they can be grafted with polymers to produce organosols of uncharged platelets1'21. Boehmite rods have also been coated by silica3', see also fig. 2.1c. 2.4d Ferrqfluids Ferrofluids are stable colloidal dispersions of single-domain magnetic particles41, which behave as liquid ferromagnets; the fluid moves towards a magnet and may adopt exotic equilibrium shapes . Most ferrofluids are based on magnetite (Fe3O4) particles, which oxidize to maghemite (^-Fe2O3), without losing their magnetic properties. The colloids are usually sterically stabilized by a grafted layer of oleic acid and dispersed in non-polar solvents, such as cyclohexane; aqueous sols of magnetite particles are more prone to aggregation when stored over longer periods of time. The traditional method for synthesizing non-aqueous ferrofluids consists of extensive milling of magnetite minerals in an organic solvent in the presence of adsorbing surfactants. Instead of this comminution technique, which may take weeks, a fast condensation route may be used on a laboratory scale. Here, magnetite particles precipitate upon alkalization of a FeCl 2 /FeCl 3 solution in what must be an instance of rapid, homogenous nucleation. Particle formation already starts before pH gradients have 11
A.M. Wierenga, T.A.J. Lenstra, and A. Philipsc, Colloids Surf. A134 (1998) 359. F. van dcr Kooij, E. Kassapidou, and H. Lckkcrkerker, Nature 406 (2000) 868. 31 A.P. Philipse, A.M. Nechifor, and C. Pathmamanoharan, Langmuir 10 (1994) 4451. Magnetic pair particle interactions will be discussed in sec. 3.10c. R. Rosensweig, Ferrohydrodynamics, Cambridge University Press (1985). 21
2.68
PREPARATION AND CHARACTERIZATION
disappeared by stirring; there is considerable overlap of nucleation and growth, which partly explains the poor control of the particle size distribution in this otherwise convenient synthesis. The following procedure is based on1'21. In 380 ml demineralized water, FeCl2 4H2O(3.29 g, 16.5 mmol) and FeCl3 -6H2O (8.68 g, 32.1 mmol) are simultaneously dissolved. (The hygroscopic properties of anhydrous salts make it more difficult to achieve the correct Fe 2 + /Fe 3 + ratio). Under vigorous stirring at room temperature, 25 ml ammonia (25%) is added; a dark precipitate immediately forms. This magnetic precipitate is collected with a permanent magnet and, after decantation of the supernatant, is mixed with 40 ml 2M HNO3 , which brings the pH below the isoelectric point of iron oxide, and repeptizes the precipitate. After 5 min. of stirring, the oxidation to maghemite is completed by adding 60 ml of an aqueous 0.35 M Fe(NO3)3 solution and subsequent refluxing of the stirred solution at its boiling point for 1 h. On a permanent magnet, the maghemite settles as a reddish sediment. After decanting the supernatant and washing the precipitate twice with 100 ml 2 M HNO3 (decant the acid as much as possible), the precipitate is redispersed in 50 ml demineralized water to a stable, black maghemite sol with a typical solid weight concentration of 5-6 g/1. The maghemite particles can now be grafted with oleic acid on a small scale at room temperature. To that end, 2 ml of the aqueous sol is diluted with 50 ml demineralized water, coagulated by adding a few drops of ammonia (25%) and sedimented on a magnet. After decanting the supernatant and washing with 50 ml water, 100 ml water is added to the gently stirred precipitate, followed by the addition of 6-8 ml oleic acid. Within a few minutes, all maghemite colloids migrate into the oil phase where, after separation from the colourless aqueous phase, they are washed three times with 10 ml ethanol to remove water and any excess surfactant. After drying in a nitrogen flow, the oleic acid-coated maghemite particles are easily redispersed in a few milliliters of cyclohexane to form a stable dispersion, which can be manipulated quite effectively with a magnet. At this point, a liquid ferromagnet has been obtained. TEM micrographs show somewhat irregular maghemite crystallites with an average diameter of typically 10 nm and a relative dispersity around s ~ 30% (see also figs. 2.2 and Al.l). Thermogravimetry and infrared measurements ' indicate the presence of covalently bound oleic acid molecules occupying an average surface area of 0.28 nm 2 . 2.5 General references 2.5a Preparation Polymer colloids have not been addressed in this chapter. For a suitable entry to the literature on their preparation and characterization, see A. Elaissari, Colloidal Poly11 21
A. Bee, R. Massart, J. Magn. Magn. Mater. 149 (1995) 6. G.A. van Ewijk, G.J. Vrocgc, and A.P. Philipse. J. Magn. Magn. Mater. 201 (1999) 31.
PREPARATION AND CHARACTERIZATION
2.69
mers, Synthesis and Characterization, Marcel Dekker (2003). The literature on inorganic colloid synthesis dates back to the beginnings of colloid science. A very useful entrance, in particular, to the older literature is H.R. Kruyt (Ed.), Colloid Science I: Irreversible Systems, Elsevier, (1952). Also see: J.Th.G. Overbeek, Monodisperse Colloidal Systems, Fascinating and Useful, Adv. Colloid Interface Sci. 15 (1982) 251-277. (A pointed review on monodisperse colloids and growth mechanisms.) Useful texts on nucleation and growth: F.F. Abraham, Homogeneous Nucleation Theory, Academic Press (1974). M. Baraton, Synthesis, Functionalization and Surface Treatment of Nanoparticles, Americal Scientific Publishers (2003). (Deals with many aspects of particle functionalization and its applications.) The Colloid Chemistry of Silica, H.E. Bergna, Ed., American Chemical Society (1994). (Reviews developments in the study of silica sols and gels since the appearance of Iler's book in 1979.) M.A. Brook, Silicon in Organic, Organometallic, and Polymer Chemistry, Wiley (2000). (Overview of silicon chemistry, including many aspects of silica and functional silanes.) R.M. Cornell, U. Schwertmann, Iron Oxides in the Laboratory, VCH (1991); The Iron Oxides, VCH (1996). (The essential text on the topic.) C.L. De Castro, B.S. Mitchell, in Synthesis, Functionalization and Surface Treatment of Nanoparticles, M.I. Baraton, Ed., American Science Publishers (2003). P.G. Debenedetti, Metastable liquids: Concepts and Principles, Princeton University Press (1996). (A good review of nucleation theory.) J. Delplancke, in Synthesis, Functionalization and Surface Treatment of Nanoparticles, M.I. Baraton, Ed., American Science Publishers (2003). Colloid Gold: Principles, Methods and Applications (three volumes), M. Hayat, Ed., Academic Press (1989). J.P. Jolivet, Metaloxide Chemistry and Synthesis, Wiley (2000). (Discusses condensation mechanisms for aqueous cations, and surface chemistry of colloidal oxides.) J. Livage, M. Henry, and C. Sanchez, Sol-Gel Chemistry of Transition Metal Oxides in Progr. Solid State Chem. 18 (1988) 259. (Reviews many molecular precursors and their polymerization to inorganic oxides.)
2.70
PREPARATION AND CHARACTERIZATION
E. Matijevic, Preparation and Properties of Uniform Size Colloids, in Chem. Mater. 5 (1993) 412. (Reviews monodisperse colloids with a broad spectrum of morphology and composition.) Technological Applications of Dispersions, R.B. McKay, Ed., Marcel Dekker (1994). (Reviews, preparation and properties of colloids used in a variety of applications including paints, paper, ceramics, and plastics.) A.E. Nielsen, Kinetics of Precipitation, Pergamon Press (1964). S. Oden, Der Kolloide Schwefel, Thesis Upsala University (1913). (This still seems to be the latest monograph on sulphur colloids; see also S. Oden, Kolloid-Z. 8 (1911) 186.) M. Ozak, Preparation and Properties of Well-defined Magnetic Particles, MRS Bulletin (December 1989) 35. Fine Particles Science and Technology, E. Pelizzetti, Ed., Kluwer (1996). (Provides numerous references to many aspects of colloid synthesis.) Particle Growth in Suspensions,
A.L. Smith, Ed., Academic Press (1978).
(Proceedings of a Symposium.) Fine Particles; Synthesis, Characterization and Mechnlsms of Growth, T. Sugimoto, Ed., Marcel Dekker (2000). (A comprehensive text on the formation routes and mechanisms of inorganic as well as polymeric colloids.) T. Svedberg, Die Methoden zur Herstellung Kolloider Losungen Anorganische Stoffe, Theodor Steinkopff Verlag (1909). (Probably the first monograph on inorganic colloid synthesis.) A.G. Walton, The Formation and Properties of Precipitates, Interscience Publishers (1967). H. Weiser, Inorganic Colloid Chemistry (two volumes), Wiley (1933). (A rich and still relevant source of detailed preparation methods.) Nucleation, A.C. Zettlemoyer, Ed., Marcel Dekker (1969).
Separation techniques Standard texts on flow in porous media are: R.E. Collins, Flow of Fluids through Porous Materials, Reinhold (1961). For a treatment of colloidal filtration see also W.B. Russel, The Dynamics of Colloidal Systems, University of Wisconsin (1987).
PREPARATION AND CHARACTERIZATION
2.71
A.E. Scheidegger, The Physics of Flow through Porous Media, University of Toronto Press (1974). 2.5b Characterization An instructive overview of various averages is: J.T. Bailey, W.H. Beattie, and C. Booth, Average Quantities in Colloid Science, J. Chem. Educ. 39 (1962) 196-202. S.E. Harding, A.J. Rowe, and J.C. Horton, Analytical Ultracentrifugation in Biochemistry and Polymer Science, Roy. Soc. of Chem. (1992). (Comprehensive coverage of analytical ultracentrifugation of (bio)polymers; no reference is made to inorganic particles.) M.B. Huglin, Light Scattering from Polymer Solutions, Academic Press (1972). (Includes many practical aspects.) For a recent overview of characterization methods see: E. Kissa, Dispersions; Characterization, Testing and Measurement, Surfactant Series Vol. 84, Marcel Dekker (1999). R. Pecora, Dynamic Light Scattering; Applications of Photon Correlation Spectroscopy, Plenum (1983). (Still a basic text.) A.P. Philipse, Colloidal Sedimentation and Filtration, Current Opinion Colloid Interface Sci. 2 (1997) 200. (Literature entrance for colloidal spheres and nonspheres.) E.R. Pike, J.B. Abbiss, Light Scattering and Photon Correlation Spectroscopy, Kluwer (1997). (Overview of both experimental and theoretical developments. For more reviews of light scattering see footnotes of sec. 2.3b.) For extensive information on characterization and fractionation of polydisperse colloids see: Particle Size Distributions; Assessment and Characterization, T. Provder, Ed., Am. Chem. Soc. Symp. Ser. (1987); Particle Size Distributions II. T. Provder, Ed., Am. Chem. Soc. Symp. Ser. (1991). For further types of characterization (electrokinetics, surface charge, steric stabilization, etc.), see the relevant chapters in FICS.
This Page is Intentionally Left Blank
3
PAIR INTERACTIONS
Hans Lyklema 3.1
Colloid stability: definition of terms and issues
3.1
3.2
Electric interaction. Basic principles; homo-interaction
3.6
3.3
Interaction between identical parallel flat diffuse double layers
3.4
at constant potential
3.10
3.3a
Charge and potential distribution
3.11
3.3b
Gibbs energy
3.16
3.3c
Disjoining pressure
3.21
3.3d
Interaction in the Debye-Hiickel approximation
3.24
Interaction between identical parallel flat diffuse double layers at constant charge
3.5
3.6
3.7
3.8
3.9
3.28
3.4a
Basic mechanism
3.28
3.4b
Elaboration
3.30
Interaction between flat identical Gouy-Stern double layers. Regulation
3.34
3.5a
The issue
3.34
3.5b
Chemical and electrical regulation
3.36
3.5c
Introduction of surface charge regulation models
3.43
3.5d
Charge regulation in the literature
3.46
3.5e
Intermezzo: status quo
3.49
Hetero-interaction
3.50
3.6a
Basic phenomena
3.50
3.6b
Theory. Diffuse layers only
3.53
3.6c
Elaboration on the Gouy-Stern level
3.58
Interactions for on-planar geometry
3.65
3.7a
Interaction between spheres. Basic issues and definitions
3.65
3.7b
Methods and approximations for isolated particles
3.67
3.7c
The Deryagin approximation
3.70
3.7d
Numerical results
3.72
3.7e
Analytical expressions
3.76
3.7f
Intermezzo. Comparison of decay functions
3.78
3.7g
Miscellaneous
3.79
Other contributions to the pair interaction
3.84
3.8a
Van der Waals (dispersions) interactions
3.84
3.8b
Hamaker constants. Update and extension
3.90
3.8c
Solvent structure-mediated interactions
3.94
Extended DLVO theory: DLVOE
3.95
3.9a
Updating the DLVO model
3.95
3.9b
General features
3.96
3.9c
Influence of the Hamaker constant and retardation
3.9d
Influence of the Stern layer thickness (d)
3.100
3.9e
Influence of the particle radius (a)
3.101
3.99
The Schulze-Hardy rule
3.10
Interaction at constant crd
3.105
3.9h
Interaction between regulated surfaces
3.106
3.9i
Lyotropic (Hofmeister) series
3.108
3.9j
Overcharging; charge reversal
3.112
3.9k
Coagulation by electrolyte mixtures
3.115
3.91
DLVOE theory. Conclusion. Alternatives?
3.117
Forced pair interaction
3.119
3.10a
3.119
Gravity
3.10b Electrical and mechanical forces
3.11
3.12
3.13
3.102
3.9g
3.121
3.10c
Magnetic forces
3.123
3.10d
Optical forces
3.130
Pair interactions in non-aqueous media
3.130
3.11a
3.131
Classification
3.11b Apolar media (group I)
3.132
3.1 lc
Low polar media (group II)
3.137
3. l i d
Polar liquids
3.138
Measuring pair interactions
3.139
3.12a
3.140
Obtaining pair interactions in multiparticle assemblies
3.12b Interaction between macrobodies, SFA
3.145
3.12c
Individual particles near surfaces, SFM
3.152
3.12d
Springless measurements
3.155
3.12e
Measuring techniques and conclusion
Case studies: Oxides and latices
3.159 3.159
3.13a Titanium oxides
3.160
3.13b Silica
3.169
3.13c
3.176
Latices
3.14
Applications
3.183
3.15
General references
3.186
3 PAIR INTERACTIONS HANS LYKLEMA
The stability of colloids has been a central issue in colloid science and will remain so for a long time. We only have to ponder questions such as: (i) Why can colloids be stabilized electrostatically, although double layers as a whole are electroneutral? (ii) Why do sedimenting colloids sometimes form open structures, but in other cases, dense sediments? (iii) Why is it possible that electrically repulsive colloids can, under certain conditions, phase-separate? These questions help us to realize how many-faceted and challenging the issues are, and imply a wide range of practical applications. The consideration of stability phenomena will pervade Volumes IV and V. With the present chapter the systematic treatment starts. 3.1 Colloid stability; definition of terms and issues Stability is an 'umbrella term', encompassing a variety of phenomena, including that of instability. Because of this wide scope of manifestations, different investigators will use this term with different meanings. The problem becomes better defined when we ask: 'stability against what?' We can, for instance, consider stability against agglomeration of particles, against spontaneous growth by Ostwald ripening, against phase separation, against sedimentation, against shear and so on. One can imagine systems that are stable against aggregation, but not against Ostwald ripening or against sedimentation. Emulsions may be unstable against aggregation, but stable against coalescence. A painter will call his paint stable when the pigment does not settle, irrespective of the question whether or not the particles aggregate. Obviously, we have to define the nature of the stability that we are going to discuss, prior to embarking on systematic theories. Of this rich collection of stability phenomena, particle-particle interaction is the heart of the matter and is the theme of the present chapter. Pair interaction means: interaction between two particles, embedded in an infinitely large amount of electrolyte solution acting as the environment. Basically, we consider the components of the Gibbs or Helmholtz energy, or the grand potential of interaction and the Fundamentals of Interface and Colloid Science, Volume IV J. Lyklema (Editor)
© 2005 Elsevier Ltd. All rights reserved
3.2
PAIR INTERACTION
disjoining pressure, respectively, quantifying them as far as possible for a range of geometries (e.g., plates, spheres, ...) and conditions (such as long and/or short distances, and similar or dissimilar particles). We might recall that the notion of disjoining pressure was introduced in sec. 1.4.2. For two parallel plates, at a distance h apart, Fl(h) is the amount by which the normal component of the pressure tensor exceeds the outer pressure. Thermodynamically, depending on the process conditions (p, V or ju), Ga(h), Fa{h) or &a(h) in J m~2 is the isothermal reversible work of bringing these two surfaces from an infinite distance to distance h apart. Recall that the subscript 'a' means 'per unit area'. From that we find, for parallel flat plates mh) = -\—^-!-\
I
dh
[3.1.1a]
)P,T
or (dFAh)} I7[h) = - —S— V dh )v,T
[3.1.1b]
or
™ =- ( ^ l V
dh
13.1.1c] )M,T
For isolated pair interactions in incompressible systems, these three functions are identical. In the present chapter, we shall generally consider the quantity Ga because of the process conditions chosen. On the other hand, if we want to consider pair interactions in confined geometries, or the interaction between a pair selected from a large collection of particles, Fa is the appropriate choice. Then, Ga and F a may differ significantly. For multiple interactions, as in concentrated systems, I2a appears more appropriate (see chapter 5). For two infinitely large parallel plates, Jl[h) is the force between unit area of the one plate and the other, infinitely large plate. For finite systems, such as two spheres, it is more convenient to consider the total Gibbs or Helmholtz energy of interaction {G{h) or F{h), in J), and instead of I7{h), the total force F{h) or F(h) (in N), depending on our wish to emphasize its vectorial character. The conversion of Ga for plates into G{h) for spheres etc., requires mathematical procedures. Well known is the Deryagin approximation, already used for Van der Waals interactions, (fig. [1.4.13]); see sec. 3.5c. In chapter 4 the dynamics and kinetics of coagulation will be discussed. What happens when particle pairs are sufficiently destabilized so as to coagulate at a measurable rate? Also, what types of coagulates are formed; compact aggregates or more open structures? We shall use the term 'dynamics' for the time dependence of pair interactions, and 'kinetics' more specifically for the mechanism and rate of growth of the ensuing aggregates. In a sense, chapter 4 bridges chapter 3 (pairs of particles) to chapter 5 (on concentrated colloids). In chapter 5 multiparticle inter-
PAIR INTERACTION
3.3
actions and the ensuing macroscopic phase behaviour will be discussed. Not surprisingly, chapter 6, on rheology, also deals extensively with concentrated systems. Unless stated otherwise, we assume in this chapter that the interacting particles are continually at equilibrium. We now ask what forces may operate between colloidal particles dispersed in a liquid? In the previous volumes of FICS, some types of forces have already been discussed in some depth. We shall now briefly review this matter, repeating some of the earlier material where appropriate, to achieve some comprehensiveness in this chapter. 1) London-Van der Waals, or dispersion interaction. We have treated these in detail in chapter 1.4 and shall summarize and extend the results in sec. 3.8. These forces are ubiquitous; they depend on the nature of the particles and the medium, and on the geometry of the particles. As a first approximation, the Van der Waals contribution to the Gibbs energy of interaction, with Ga v d w = Fa v d w = Ua v d w between two particles, a (shortest) distance h apart, can be written as Ga,vdw = -A 12(3) /(geometry, h)
[3.1.2]
where A12(3) is the Hamaker constant for the interaction between particles of nature 1 and nature 2, respectively, across the medium 3. In chapter 1.4 we have given examples of the function J. Hamaker constants were collected in table I.A.9 and an update is given in app. 3. For homo-interaction (material 1 identical to material 2), with Hamaker constants of the type Aj^gj, G a V d w < 0 (attractive). For hetero-interaction the Hamaker constant can, in a few situations, be negative (table I.A.9). In practice, such situations occur most often when one of the components is a vapour; see sec. III.5.3, on wetting films. 2) Electrostatic interactions, mentioned before, but not yet treated. The origin is double layer overlap. The thermodynamics and structure of isolated double layers have been discussed extensively in chapter II.3. One of the most striking features of double layers is the very strong influence that indifferent electrolytes exert; they reduce y/^ , the potential of the outer Helmholtz plane, oHp, (i.e., the potential of the diffuse part of the double layer), and compress that layer (i.e., the Debye length K~1 is reduced). As electrostatic interaction is mainly determined by the diffuse parts of the double layers, this synergistic electrolyte effect makes itself strongly felt in the stability of hydrophobic colloids. This is the origin of the Schulze-Hardy rule (sec. 4.9e). The trend is that two isolated particles with the same charge sign repel each other. An exception to this rule occurs when one particle is highly charged, but the other only slightly. In this case, upon approach, even when both charges have the same sign, the higher charged one may induce a reverse charge onto the other, followed by attrac-
3.4
PAIR INTERACTION
tion. For a systematic analysis, see sec. 3.6 on hetero-interaction. Unlike Van der Waals attraction, at a given charge or potential, Ga el is independent of the nature of the particles; on the other hand, Ga v d w is virtually insensitive to electrolytes and, for that matter, insensitive to the presence of a double layer. Equations for Ga el vary widely, depending on the geometry of the system, strong or weak overlap, high or low electrolyte concentration, etc., but for weak overlap and low potentials many of them have this shape: G
a,el = /(DL)( ! / 1 ) 2 e-' rh
[3.1.3]11
where /(DL) contains properties of the two double layers, solution- and geometrical quantities (such as the dielectric permittivity and particle size). Here, i//d is the potential of the diffuse part of the double layer. Equation [3.3.18] is an illustration of such an equation (for low potentials, hyperbolic tangents may be replaced by their arguments). Obviously, the exponential factor stems from the exponential potential decay of the isolated diffuse double layer. The distance h is that between the two oHp's; this distance is shorter than h in [3.1.2] by an amount of twice the Stern layer thickness. From G ae |, 77el can be obtained by differentiation with respect to h, but there are also ways to compute /7 el directly (sec. 3.3c). Note that equations such as [3.1.3] often contain iffA rather than the surface potential y/°; this is so, because it is the overlap of the diffuse parts which is most important. This has a historical background. In the original theory, as developed independently by Deryagin and Landau, and by Verwey and Overbeek, the conscious assumption was made that, upon interaction, the surface potentials on the particles would remain constant and equal to their values at infinite separation of the particles. As these authors ignored Stern layers, their surface potential \fp is often replaced by our yfd, which explains the appearance of this potential in [3.1.3]. At the same time, the distances h in [3.1.2] and [3.1.3] were set equal. In reality, the process is much more complicated; upon interaction, the charge- and potential distribution over the Stern- and diffuse parts will change. This process is called regulation. We shall return in detail to this matter in sec. 3.5, and to the dynamics in chapter 4. The term hetero-interaction is used for the interaction between particles of different nature if G a V d w is addressed, whereas in G ael the term hetero-interaction refers to different values of the potential and/or charge, irrespective of the nature of the particles and solvent. As this term can also be used for particle pairs of different shapes, we shall have to be specific. It is historically interesting that, long ago, von Smoluchowski2', in his famous theory for the kinetics of coagulation, considered
11
Note that in some theories the distance-dependence is written exp(-2ich), where h is the half distance. We shall not follow this habit because later we shall discuss hetero-interaction, for which the half distance loses its meaning. 21 M. von Smoluchowski, Z. Physik. Chem. 92 (1917) 129.
PAIR INTERACTION
3.5
interactions between pairs of different sizes, but not of different double layer properties. For two particles of different, but not too high, diffuse double layer potential, at large Kh G a e l =/(DL)i^yd e -*-h
[3.1.4]
In practice, diffuse double layer potentials are not directly measurable. However, experience has shown that the replacement of i//d by the electrokinetic potential ( is often warranted, where the potentials (^d and f are those for isolated particles. Generally speaking, electrostatic interaction is an important feature, and we shall pay much attention to it, starting with the next section. Colloid stability, determined only by Van der Waals attraction and electric repulsion, will henceforth be referred to as the DLVO regime. We introduced this name in sec. 1.4. Although the elaborations in sees. 3.3 through 3.7 often surpass those by DLVO's own analyses, we continue to use the term DLVO regime, although the term DLVO theory in the stricter sense will be restricted to the original elaboration. Sometimes, interactions not belonging to this category are called non-DLVO interactions, and the term extended DLVO theory (DLVOE) will be used for DLVO theory to which specific other interaction contributions are added (see sec. 3.9). 3) Steric interactions are interactions caused by macromolecules and can be repulsive or attractive. Three basic modes of action are illustrated in fig. 1.2. The full treatment follows in chapter V. 1, but we anticipate that, by making two remarks: (i) Quantitatively, G a s t e r can be very high, tending to outweigh electrostatic repulsion, depending on its range of action. Particularly in systems with weak double layers (as for dispersions in non-aqueous media of low polarity) steric stabilization is often the sole mechanism that keeps particles apart. On the other hand, depletion flocculation is relatively weak. (ii) Steric, electric, and dispersion forces are not additive. Polymer trains modify the composition of the Stern layer, and hence iff6-. For random (homo)polymer adsorption the volume fraction in loops and tails is usually low enough for us to ignore its influence on the diffuse part of the double layer. Further, enrichment of polymer on surfaces modifies the Hamaker constant and the effective h because a third phase is introduced. 4) Magnetic interactions represent a special case, but when such forces are operative they often outweigh other interactions. It is very difficult to stabilize colloids against magnetic attraction. We shall discuss magnetic colloids in sec. 3.10. 5) Solvent structure-mediated interactions. We have used this, admittedly somewhat clumsy, term to cover all interaction phenomena caused by the structure of the intervening liquid, insofar as it is modified by the presence of a surface. (See, for example, sees. 1.5.4 and III.5.3). Structural modification near a hard wall includes the
3.6
PAIR INTERACTION
familiar density oscillations (sec. II.2.2), reorganization caused by hydrogen bonding to the solid, or by hydrophobic dehydration. In the literature these phenomena come under a hotchpotch of names, reflecting the specific interpretation the various authors have in mind, such as 'water structure forces', 'structural forces', 'hydration forces', or even 'acid-base interactions'. Sometimes these names reflect the inability to interpret certain observed phenomena quantitatively in terms of well-understood interactions. Solvent structure-mediated interactions are current subjects of study. Some aspects are reasonably well understood (for example, the density oscillations have been reproduced in the surface force apparatus, see fig. II.2.2), others have alternative interpretations (attraction between hydrophobic surfaces at ranges far exceeding that of molecular interactions is not caused by hydrophobic bonding). In the present chapter we shall have to discuss this matter more than once. For the quantitative formulation, we recall [III.5.3.13] on p. HI.5.35 /7
solv.str(h» = - K e " h M
I 3 ' 1 -51
This is an empirical equation, in which X is of the range of molecular interactions in the solvent. We shall return to this matter in sec. 3.8c. Are these forces additive? The answer is not unequivocal, because it depends on the kind(s) of forces involved and on the dynamics of interaction. For practical purposes, electrostatic and dispersion forces are additive. By the term, 'practical purposes,' we mean that, in practice, interaction forces can rarely be measured with an absolute accuracy of better than 5%, and that we therefore do not have to worry about non-additivities smaller than that. (Note also that such an accuracy requires extremely well-defined, uncontaminated surfaces, that are molecularly flat or homogeneously curved.) The most obvious deviation from additivity arises when there is steric interaction in combination with double layer overlap, because the ionic charge distribution, and its dynamics will be affected by the adsorbed train and loop segments. We shall return to this topic in ch. 5.1. Finally, we introduce the term forced interaction to account for those cases where an interaction is influenced by an external force, including gravity, an applied electric field, shear, or acoustic waves. Experiments with the surface force apparatus and AFM also belong to this category. Recall that we have already introduced the term, 'forced wetting' (fig. III.5.5). In the present chapter we shall begin with spontaneous interactions; external forces will be considered in sec. 3.10. 3.2 Electric interaction. Basic principles; homo-interaction We start by considering the simplest situation, of two identical particles, each carrying identical electric double layers, embedded in a solution of fixed concentrations (i.e.,
PAIR INTERACTION
3.7
having fixed chemical potentials) of an electrolyte, containing charge-determining ions and an indifferent electrolyte. The particles are assumed not to settle, but to move randomly by Brownian motion. When they meet upon a chance encounter, repulsion is felt. We may ask, 'Why'? The answer is not as obvious as may appear at first sight. The most direct, but oversimplified reply, 'because they are charged, and equal charges repel each other', is immediately parried by the equally oversimplified counter-statement that the double layers do not interact at all electrostatically because, as a whole, they are uncharged. In the nineteen thirties this issue occupied the minds of several colloid scientists; there are even papers concluding that the electric interaction between identical particles is repulsive at certain distances but attractive at others. Had the diffuse double layers been spatially fixed, then one could imagine a repulsion at long distance (because of overlap of the extreme parts of these layers, carrying charges of the same sign) and attraction at shorter distance (because the surface charge of the one particle starts to attract the countercharge of the other). However, diffuse double layers are not static. They can, and will, regulate their structures upon overlap. Such a structural change involves a change in entropy which also contributes to the Gibbs energy of interaction. Deryagin and Landau, and Verwey and Overbeek resolved this issue by computing the Gibbs energy G of the system at any h by an isothermal reversible charging process. For one flat isolated diffuse double layer we have already implemented this scheme, leading to [II.3.5.20]; a
a
r [
2
J
For a Gouy-Stern double layer the result is [II.3.6.65] AG* =
^ =^ - ^ - ^ L s h f ^ l - l l
,3.2.2,
In these equations, the A refers to the difference with respect to the reference state, i.e., the uncharged surface, and the subscript 'a' means that G is counted per unit area. The quantities <7° and a^ are the surface and diffuse charge densities, respectively [a° + od+(j1=0 because of electroneutrality, where o1 is the specifically adsorbed charge). C} and C\ are the inner and outer Helmholtz layer differential capacitances, respectively, y is the dimensionless potential (y d = Fy/A IRT, y° = Fy/° IRT , etc.) and z is the valency of the (symmetrical) electrolyte. Therefore, if one is able to compute the Gibbs energy for two interacting double layers, subtraction of twice [3.2.1] or twice [3.2.2] yields the Gibbs energy of electric repulsion per unit area. Upon interaction, variations of these AG^'s as a function of h have to be considered. In order to avoid undue additional A 's, we shall denote the result simply
3.8
PAIR INTERACTION
as Ga e l , realizing that for the two interacting double layers the reference state is identical. Recall that AG? and AGJ are negative, as they should be, because the double layers formed spontaneously. The Gibbs energy of two double layers in interaction is also negative, but less so than for the two isolated double layers. Consequently, Ga el > 0 . Deryagin and Landau only considered diffuse double layers but Verwey and Overbeek devoted a section to Stern corrections without deriving our [3.2.2]. As we are now interested in the basic principles, let us for the moment assume that both double layers are entirely diffuse, i.e., y° = y d and a° = -cfi . One of the questions that had to be addressed was: 'what happens to the values of y° and a° upon interaction?' Deryagin and Landau took y° to be constant. Verwey and Overbeek did the same but gave their choice some justification. They had in mind the (at that time already well studied) silver iodide system, for which y° was assumed to be fixed, because the concentration of charge-determining ions remains constant, so that Nernst equilibrium would be retained (see sec. 1.5.5c). The argument also applies to oxides and some other types of relaxed double layers. When y° is fixed (at its value for h —> oo), o~° should decrease upon overlap; in V-O language, by desorption of charge-determining ions. The reason for this decrease is that the proximity of the second surface with the same charge makes it unattractive for such ions to be on the surface. Eventually, in the limit h —» 0 , a° —> 0 . With this in mind, it becomes evident that at least part of /7el is of a chemical nature. Double layers in isolation form spontaneously by adsorption (and/or desorption) of charge-determining ions. Hence, when owing to the approach of a second particle, the adsorption of such ions is inhibited, work has to be done against their chemical affinity. Stated otherwise, the particles repel each other. Usually, this mechanism is called, interaction at constant potential. It is realized that such a type of interaction requires adjustment of the surface charge. We shall call this process surface charge regulation. Anticipating sec. 3.5 we note that the less specific term 'charge regulation' occurs in the literature with a somewhat less general meaning. The alternative, interaction at constant charge, applies to systems with fixed surface charges, such as poly(styrene sulfate) latices or the plates of clay minerals. In this case, upon overlap y° shoots up and the corresponding contribution to Ga el is of a purely electrical nature. In fig. 3.1, a sketch is given of the changes in potential and charge for these two types of interaction. Interaction at constant surface charge requires surface potential regulation. As far as the author is aware, this term does not yet occur in the literature. Not only because of the neglect of the Stern layer, but also on dynamic grounds, something can be said against these mechanisms. For many systems with initially relaxed double layers, surface ions simply do not have the time to desorb during a Brownian encounter. Then there are two options: (i), the system behaves as a system of constant charge or, (ii), the surface charge proper will not decrease, but it is made ineffective by adsorption of counterions. The latter mechanism requires a Stern layer
PAIR INTERACTION
3.9
Figure 3.1. Difference in charge-potential behaviour between interaction at constant charge (A) and constant potential (B). In case A, \j/° rises upon interaction, whereas in case B, a° - » 0 . The surface charge density is indicated. The dashed lines show (d(i//dx) x _ 0 which, by virtue of Gauss' law [II.3.6.23], is proportional
to a°.
over which the countercharge is regulated; this will be discussed in sec. 3.5. Intermediate cases can also be imagined, depending on the nature and magnitudes of the ion fluxes and their yields on the time scale of a Brownian collision. In the following chapter these dynamics will be discussed extensively. It is enlightning to look at this issue from a thermodynamic (phenomenological) viewpoint. Consider two relaxed flat parallel double layers, originally far apart (h-»<*>)• embedded in a solution of constant composition, i.e., with p,T and all chemical potentials fixed. Adsorption equilibrium is assumed to prevail. We want to know the change in Gibbs energy per unit area when the distance between the plates is reduced from °° to h. The only changes that occur are those in the two surface excess Gibbs energies, G° , for each plate. Hence, G3iel =2[AG£(h)-AG£(~)] = 2[G£(h)-G£(°°)]
[3.2.3]
The A 's may be dropped because both terms refer to the same reference state of no adsorption. For G° we derived the following; see [I.A5.4] or [III.A2.4] G
a=-2>iri
[3 241
-
For the sake of rigour we note that a convention underlies this equation (see sees. 1.2.6 and 1.2.10). Without digressing into this matter we state that the following argument also holds for the alternative convention. We have added the minus sign to indicate that the adsorbate forms spontaneously. Let us be explicit and take by way of example an oxide dispersed in an aqueous solution containing HNO3 , KNO3 and an
3.10
PAIR INTERACTION
adsorbable organic substance A. Then, for two identical double layers, G
a = -2(/'HNO3rHNO3 +/'KNO3 rKNO3 + ^ ^ 1
I3-25'
Basically, this means that the reference state is a pristine interface in a solution having the given ,«HNO , /^KNO a n d MA • T n e s ° l u t i o n i s dilute; all surface concentrations are referred to water, and rw/iw is not in the equation. For the arguments on this, see sees. 1.5.6b or II.3.4. Now consider the variation as a function of distance;
All differentiations are taken with p, T, and all /u's constant. We use the abbreviation, s, for KNO3 (salt). We now identify the three terms. r H N O equals a° /F (we neglect 7"KOH). Upon reduction of h, /"HNO g o e s down, so the first term is repulsive (it makes AG° less negative). On the other hand, Ts < 0 ; after all, this is the Donnan expulsion term (negative adsorption). Because this negative term becomes less negative upon overlap, dFs I dh > 0. In words, overlap in combination with depletion (or negative adsorption) leads to attraction. This is the basis for the phenomenon of depletion coagulation, or depletion jlocculation. Finally, the last term is also repulsive if overlap of adsorbates leads to desorption. This phenomenological analysis demonstrates that there is no difference in principle between the first and the third term: in both cases work is done against the (chemical) binding Gibbs energy of HNO3 and A, respectively. However, with charged interfaces the first term does not come on its own; it is accompanied by a second, attractive term, which is of an electrical nature. Models are needed to quantify these, but for pair interactions the repulsive term always exceeds the attractive one. For diffuse double layers this is well known: only part of the surface charge is compensated by negative adsorption of co-ions. It remains a matter for further discussion whether this also applies to concentrated sols. Treatments similar to our above approach have been elaborated by Hall11, Ash et al.2), Ruckenstein31 and Pethica41. 3.3 Interaction between identical parallel flat diffuse double layers at constant potential This type of interaction rarely occurs in practice because there is no physical reason for y d to remain fixed upon interaction. Only in the absence of a Stern layer may y d
11
D.G. Hall, J. Chem. Soc. Faraday Trans. (II) 68 (1972) 2169. S.G. Ash, D.H. Everett, and C. Radke, J. Chem. Soc. Faraday Trans. (II) 69 (1973) 1256. 31 E. Ruckenstein, J. Colloid Interface Set 82 (1981) 490. 41 B.A. Pethica, Colloids Surf. 20 (1986) 156. 21
PAIR INTERACTION
3.11
be replaced by y° which may remain constant as far as it is determined by Nernst's law. However, as we have extensively discussed in chapter II.3, double layers that are completely diffuse exist only at very low surface potentials and low electrolyte concentration. It is not appropriate to limit the discussion to this idealized case. In sec. 3.5 we shall show that in the more realistic situation of overlap between two Gouy-Stern double layers, regulation across the Stern layer never leads to constancy of y d . Nevertheless, we shall start the elaboration for the simple case of fixed y d because it contains a number of relevant principles and steps that recur in later derivations. Physically speaking it means that for the moment we ignore Stern layers and dynamic issues. Essentially we follow Verwey and Overbeek1' with a number of modifications. We consider only one symmetrical (z-z ) electrolyte at fixed p and T.
Figure 3.2. Schematic representation of the (dimensionless) potential between two identical parallel flat diffuse double layers. The dashed curves are the profiles for the two double layers far apart. The spatial variable x is counted from the left hand side plate; xm = h / 2 is the midpoint value.
3.3a Charge and potential distribution A sketch of a possible potential-distance curve is given in fig. 3.2. Because of the overlap y(x) between the two surfaces (drawn curves) is increased above the value it would have had for one single double layer (dashed). As the potentials at the surface (y d = U° in this case) are assumed to remain fixed, the slopes (dy/dx) near the two surfaces decrease. Because of Gauss' law, [1.5.1.22b] or [II.3.6.23] we have for the left double layer
m
=-*?-
[3.3.1,
quantifying the reduction of the surface charge in terms of the slope, i.e., in terms of
11
E.J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophoblc Colloids. The Interaction of Sol Particles having an Electric Double Layer, Elsevier (1948), chapter V. Also available as a Dover reprint (2000) which lacks the subtitle. Henceforth, this book will be abbreviated as V-O.
3.12
PAIR INTERACTION
the electric field adjacent to the surface and W
{dx)x_h
=- ^ -
[3.3.1a]
RTeoe
for the r.h.s. double layer. For homo-interaction the minimum potential y m = Fy/m/RT is halfway between the two plates. (For hetero-interaction this symmetry is lost; then the minimum is displaced towards the surface with the lower potential or there is no minimum at all.) At the minimum the field strength is zero, meaning that the total charges, including those on the surfaces, between x = 0 and x = xm , and between x = xm and x = h are zero. However, the potential at the minimum is not zero. An outer force is needed to maintain it at the increased value. In principle, measurement of this force is one of the ways of measuring disjoining pressures. When double layer overlap is not strong, y m is sometimes assumed to be equal to the sum of the two individual potentials. This is the linear superposition approximation (LSA)11. In fig. 3.2, this is assumed to be the case. Of course, linear superposition only applies for the range around xm . To find the distribution we must integrate the Poisson-Boltzmann equation for the range between x = 0 and x = h . This equation reads [II.3.5.9]2' ^ ^ dx2
= K2 sinh(zy)
[3.3.2]
The equation can also be written as d2(zy)l d(Kx)2 , rendering it dimensionless. In sec. II.3.5a we demonstrated how it can be integrated. The result was (also see [II.A2.20]) -M.\ =-—\cosh(zy) +const] dx) zz The integration constant can be found from the boundary condition
[3.3.3]
f—1 = 0 for y = y m \dx) We find const. = -cosh(zy m ), so that
[3.3.4]
— = + - J2 Tcosh(zy) - cosh(zym)] [3.3.5] J dx zv L For 0 < x < xm , we need the minus sign because y is a decreasing function of x . For the right half, xm < x < h , the plus sign is needed. To find y m a second integration is needed, either between the limits x = 0 and x = xm , or between y = yd ( = y° ) and y = ym . The integral is 1
' This approximation goes back to S. Levine, J. Chem. Phys. 7 (1939) 831. ' Diffuse double layer theory is replete with hyperbolic functions. Definitions, properties and important relationships are collected in II. app. 2. Often we shall refer to equations from this appendix.
2
PAIR INTERACTION *=xm
3.13 y=ym
,
- J 6iKX) = -f= $ U ^ d 2 m =o 4 V [cosh(zy)-cosh(zy )J x y
13.3.6,
We only Integrate over one half of the x -range; the other half gives the same result, the required factor 2 being already accounted for in [3.2.3]. Elaboration leads to an elliptic integral of the first kind, for which tables are available. So, an exact numerical solution is available. However, let us first consider the limiting case that the potentials are low enough to replace the hyperbolic cosine by the first two terms of its series expansion. Essentially, this is the Debye-Hiickel approximation (coshx = 1 + x 2 /2! + x4 /4! + ..., see [II.A2.8]). This approximation is acceptable when x 4 /4! « x2 /2! or zyd « Vl2 = 3.46 corresponding with i//1 < 85 mV for z = 1. This condition is fairly often satisfied. In that case,
_K_H= r
zd,
i£ + X £ ) -1
[3.3.7,
This equation can be simplified by taking the cosh of both sides, using definition [II.A2.8] and realizing that elny = y . It leads to
«-(?)•£ or d
ym =
2 = ydsech(x-h/2) [3.3.8a] W coshkh/2) W which is an analytical expression for y m as a function of the separation, h . Had we not made the low potential approximation, the result would have been X-h
= 2e-y m / 2 [Ffe-y m / 2 ,|)j-[FJe-y m / 2 ,arcsine-(y d -y m '/ 2 )]
[3.3.9]
in which
F
(M=f-i===5= J
z
o
I3-3-10!
z
jl-k sin x
is an elliptic integral for which tabulations are available. Figure 3.3 gives results. For the conversion of the abscissa axis into real distance see table 1.5.2. For (1-1) electrolytes and c = 10~5,10~3 and I C H M , X-*1 =96.1, 9.61 and 0.96 nm respectively; for (2-2) electrolytes the corresponding values are half as much. The limiting law overestimates y m at given h , but remains a good approximation if y d is not too high and if overlap is not too strong. To assess what this means in practice, realize first that every unit of y corresponds with 25.6 mV of potential at room temperature. Diffuse double layer potentials are rarely above 150 mV, and under conditions close to coagulation they are much lower. In addition, at high salt
3.14
PAIR INTERACTION
Figure 3.3. Midway potential as a function of separation for two interacting flat double layers according to the limiting law [3.3.8 or -8a] and to [3.3.9]. The latter results are sketched from Verwey and Overbeek's table IX .
concentrations K increases but y d decreases. With respect to the region of low kh (strong overlap), where [3.3.8] becomes increasingly defective, it may be added that in this range also other problems emerge, challenging the very applicability of pure diffuse double layer theory such as specific adsorption, ion size effects and dominance of other interaction forces. So, it makes little sense to consider diffuse double layer theory extensions for this range. The practical route we are taking is that, as far as electrostatics are concerned, we subsume (most of) these corrections in Stern layers (sec. 3.5). The advantages are that the conditions for the remaining diffuse part of the double layer are such that Debye-Huckel approximations such as [3.3.8] are acceptable. Let us at this stage introduce another analytical expression for y m , valid for arbitrary yA but for such a weak overlap that y m is determined by linear superposition of the two constituting potentials. In this LSA approximation, deformation of double
11
V-O p. 69.
PAIR INTERACTION
3.15
layers upon overlap is ignored. Hence, it applies only to very weak overlap, Kii/2 » 1 . When this approximation holds, there is no difference between the electric interaction at constant potential (this section) and at constant charge (sec. 3.4). It depends on h and, hence, on the type of measurement whether the LSA is satisfactory. Recall from [II.3.5.22] that for a single double layer the potential decay is given by tanh(zy(x)/4) = {tanh(zyd/4)}e"'fx
[3.3.11]
If applied to the half-way situation y(x) —> y m , KX —> Khli and as a large Kh corresponds with a low y m the hyperbolic tangent on the left-hand side may be replaced by the first term of its series expansion, viz. zy m /4 , see [II.A2.9]. Hence, in the LSA zy m =8tanh(zy d /4)e-^ m = 8tanh(zyd /4)e~Kh/2
[3.3.12]
We shall use this equation in some of our coming analyses. One of the features is that it shows y d to increase linearly with y d if the latter potential is low, but to become independent of y d when y d is very high (the hyperbolic tangent strives for unity at high value of the argument, see fig. [II.A2.1 ]). The reduction of the surface charge (c° = -o^ in the present approximation) follows from [3.3.1] and [3.3.5], applied at x -» 0 . We find ad(h) =
£°£(M]
= (-sign y) ^ ^ h [cosh(zyd) - cosh(zym)]
[3.3.13]
where we have used K1 = 2F2cz2 / eoeRT (see [II.3.5.8]). For y m = 0 this equation reduces to the surface charge density for two isolated double layers, 2a^ = 2crd(h = °o), with , (zuA\ 2eneicRT (zud) (jd = -^8eo£cRT sinh =2— = ° sinh\^—\ ri
\ 2 )
zF
[3.3.14]
v ^ J
which, for aqueous solutions at 25°C, equals a6- = -11.73>/c sinh(-0.0195 zf/1)
(//Ccm"2)
[3.3.14a]
with c in M, and iffi in mV. See [II.3.5.13 and 13a]. The conversion of the hyperbolic functions requires [II.A2.40]. In [3.3.13] o^fh) depends on h because the difference between y d and y m is distance dependent. As this relationship is known (fig. 3.3), CTd(h) is also accessible. In fig. 3.4 some typical results are given. It is seen that the decrease of a^ sets in only when r h / 2 < 2 , to become substantial only below ich/2 ~ 1. All three parameters y d , c and z influence the decrease. The top curve is representative for a stable colloid, the other two apply to unstable systems. For higher valencies, substantial reduction of
3.16
PAIR INTERACTION
Figure 3.4. Reduction of the diffuse surface charge as a result of decreasing plate distance. The dashed levels indicate I(3d according to [3.3.14],
3.3b Gibbs energy The electrical contribution to the Gibbs energy of interaction Gael is obtainable as the difference 2[G£el(h)-G£el(°°)L see [3.2.3]. Of these, G£e,(°°) is available, see [3.2.1]. Since the two Gibbs energies are both referred to the same reference state (isolated uncharged surfaces) we again drop the A sign. Gaey{h) can now be found by a charging process along the lines used for Gael(°°), see the derivation of [II.3.5.20]. The only difference is that the integration does not take place from 0 (zero) to y d , but from y m to y d . As y m is known (sec. 3.3a), the integral can be solved. However, this requires a laborious mathematical procedure for which we refer to V-O1'. The result is Gae,(h) = -2cRT J (coshy-l)dx-£°£{R^] o
J (^-) dx
[3.3.15]
o
which can be converted into an integration over the potential Gaei(h) =
_^Z:|^L(3eym_2
+
e-ym) + 2 N /2(c O shy d -c O shy m ) + y
r
+
11
V-O, p. 77-81.
J ym
e-y-eym —=dy ^ 2 (cosh y - c o s h y m )
[3.3.16]
PAIR INTERACTION
3.17
Figure 3.5. Electric contribution to the interaction Gibbs energy for two flat parallel diffuse double layers for different values of (constant) y d . Monovalent electrolyte. Scaling: for K = 1, 10" 1 , 10~2, ... nm" 1 , G ael expressed in 1, 10" 1 , 10~2, ... mJm~ 2 , respectively. Region of strong interaction.
Figure 3.6. As fig. 3.5; region of weak interaction.
3.18
PAIR INTERACTION
As with [3.3.6], this expression can be converted to elliptic functions, for which tables are available. For h —> °° it can be shown that the expression reduces to twice [3.2.1]. We shall present the results in graphical form. These results are derived from a table1' for the function ./(y m >y ) =
Z2
^ a el
[3.3.17]
In figs. 3.5 and 3.6 G a e l is given as a function of distance for the regions of strong and weak overlap, respectively. It is seen that this Gibbs energy decreases rapidly with distance. The top range in fig. 3.5 is of academic value only because upon strong overlap the double layers do not remain diffuse: counterions are then transferred to the Stern layer, and high values of y d are therefore automatically obviated. Also considering the range of K -values, encountered under the usual experimental conditions, Ga el is rarely higher than 1 m j m~ 2 , i.e. it is only a small fraction of the interfacial (Helmholtz) energy, or grand potential, as judged by the interfacial tension.
Figure 3.7. As fig. 3.5, but now semi-logarithmically. Drawn curves, exact results, dashed: according to approximation [3.3.18].
11
V-O, p. 82.
PAIR INTERACTION
3.19
Given the large range of Kh -values covered, It also is revealing to give the same plots semi-logarithmically. See fig. 3.7. Linearity in this plot, observed for Kh/2 > 1 indicates an exponential decay. In this range the curves are more or less parallel; moreover, they become less dependent on y d when y d becomes larger. The reason for this has already been explained in connection with [3.3.12]. The deviations from linearity are negative for low y d and positive for large y d . We repeat that in practice the former deviations will prevail. In fig. 3.7, the dashed curves refer to the following approximate analytical expression. G
a.el = - ^ ^ [ t a n h ( z y d / 4 ) ] 2 e - ^
[3.3.18]
This equation has a fair validity range, as can be judged from figs 3.7 and 3.8. The hyperbolic tangent enters through [3.3.12]; it quantifies the independence of y d at large y d and the proportionality with (yd)2 if y d is low, already anticipated in [3.1.3]. Moreover, it predicts an overall linearity of lnG ae j with Kh . For low Kh [3.3.18] overestimates Ga ej at low y d , but underestimates it at high y d . As the derivation of [3.3.18] is easier via the disjoining pressure than via the Gibbs energy, we postpone this until the following subsection.
Figure 3.8. Gibbs interaction energy as a function of distance for three values of v; y d = 6 , (1-1) electrolyte. The dashed curves refer to [3.3.18].
3.20
PAIR INTERACTION
In practice, for instance in the surface force apparatus, one usually measures interaction Gibbs energies or forces not as a function of tch but as a function of h for different, but fixed, values of the electrolyte concentration, i.e. at fixed K . Three such curves are shown in fig. 3.8. They demonstrate the dual effect of the electrolyte concentration. For h -> 0 , G ae] increases proportionally with Vc , i.e., proportionally with K . On the other hand, higher K implies stronger screening, i.e. a steeper descent of the curves. For that reason the curves for different K cross each other. Approximation [3.3.18] works better if K and h are higher. For low electrolyte concentration the approximation is relatively poor but then the absolute value of Ga el is negligibly small. Therefore, all told, [3.3.18] is a reasonable equation for practical purposes. The influence of the valency is complicated: increase of z increases Ga el at very short distance, but the decay is steeper, just as it is for K in fig. 3.8. However, the valency also occurs explicitly in [3.3.17], beyond the effect it exerts upon K . Perhaps the dominant effect is that at given surface potential y° multivalent counterions tend to lower iyd more than do monovalent ions. This is a feature that acts beyond the diffuse layer, which is now under discussion. To illustrate the effect of z for a purely diffuse layer we present in fig. 3.9 a comparison between a (1-1) and a (3-3)
Figure 3.9. Gibbs energy of interaction between flat diffuse double layers at constant y° Comparison between a (1-1) and a (3-3) electrolyte. Drawn curves, z = 3 ; dashed curves, z = 1 The potential y" is given.
PAIR INTERACTION
3.21
electrolyte. For a realistic comparison one should compare situations of large yd and z = 1 with those for low y d and z = 3 . The combined effect of lowering y d and compression of the double layer upon increase of valency is at the root of the extremely strong influence of z on the stability of hydrophobic colloids, as already expressed in the Schulze-Hardy rule.
3.3c Disjoining pressure There are two ways to obtain /7 e l . The first is by differentiation of the Gibbs energy with respect to the distance,1' as in [3.1.1a]. In this way, [3.3.16 or 18] can be differentiated, to obtain the exact solution or a good approximation, respectively. The second approach is by identifying the force required to keep the plates at a certain distance. Prior to the development of DLVO theory, this latter approach was taken by a number of scientists, including Langmuir and Deryagin. The basic idea is that overlap leads to an increase of the counterion concentration between the two plates, and hence to an osmotic pressure. This pressure depends on h , and at given h it also depends on the position x between the plates. In addition to this osmotic pressure there is also a pressure resulting from the electrostatic field known as the Maxwell stress, which also depends on x at given h . The sum of these two pressures must be independent of x, otherwise the system would be mechanically unstable, i.e., this sum is only a function of h . We call it p(h) and the difference between p(h) and p(~) = p(bulk) is the disjoining pressure we are looking for. Let us now elaborate this thermodynamically. Consider again the two identical flat plates, a distance h apart. What is behind the plates does not matter. The intervening liquid must be in contact with an infinitely large reservoir of fixed composition, which can absorb any electrolyte which is expelled from between the plates if h is reduced, and which also acts as a buffer for expelled electrolyte (i.e. which keeps the chemical potentials constant). In this scheme no volume work and no work to change the //'s is involved. In that case, Gibbs and Helmholtz energies are identical. Anticipating chapter 5, we note that for concentrated sols these conditions have to be reconsidered. Let us call the (hydrostatic) pressure in the reservoir p(°°), the °° sign indicating 'infinitely far from the plates'. This pressure is identical to the outer pressure p . Now the plates are brought from infinite distance to a state where the double layers overlap. To keep the plates at distance h in mechanical equilibrium upon an infinitesimal displacement the change in the pressure dp plus the change in the electrical pressure, which can be written as pdif/, must be zero. In formula, dp + pdy/=O
[3.3.19]
Integration to obtain /7(h) has to take place between the middle of the plates (where
11
V-O, p . 9 1 .
3.22
PAIR INTERACTION
the electric field is zero, so that only the osmotic contribution remains) and a position in the surroundings; for the potential this means integration over y/ from zero to y/m . Hence, h vm /7el(h) = p ( h ) - p M = J d p = - J pdy/ [N m~2] [3.3.20] o zFc Asp{x) = zF[c+(x)-c_[x)] = Integration yields
sinh[zy(x)], see sec. 11.3.5a, this can be substituted.
77el(h) = 2cRT(cosh zym -1) = 4cRTsinh2 [zym 12)
[3.3.21 ]
For the last transition we have used [II.A2.2.38]. Equations [3.3.21] are deceptively simple, but solution requires a model to find y/m, just as in the derivation for Gej(h). The h-dependence of /7el stems from the h-dependence of y/m . Before continuing, let us make a few remarks. First, about the equality [3.3.19]. Substituting Poisson's law for p,
f!P_£ A.^ = dP_£o£Aff i Zf , dx
° (k 2 dx dt
p - - 2 - 1 — I = const. 2 Vdx)
0
[3.3.22]
2 dx I dx J
(independent of position)
[3.3.23]
This is another way of formulating the mechanical equilibrium. The second term on the l.h.s. is the Maxwell stress, mentioned before in this subsection, and already generally formulated as [1.4.5.25 and 26] without mentioning the name. Halfway the plates, where d(e7dx = 0, p becomes equal to the osmotic pressure p(h) for x = h 12 . Subtraction of the external pressure p(°°) = p gives /7 e l . In the second place, let us come back to the discussion following [3.2.6], where it was stated that there are two competing contributions to /7 el : a repulsive one, caused by the excess of counterions and an attractive one, caused by the depletion of co-ions. The former always exceeds the latter. We can now make this quantitative. Following an argument similar to that used in sec. II.3.5b, but now referring to the midway situation, we can say that according to Boltzmann the excess concentration of z-valent counterions there is c(ezyln -1), and that of co-ions c(e"zyln -1) (assuming y m > 0), so that the total midway excess concentration is c\(ezym -l)+(e- z y m -l)l = 2c(coshy z m -l)
[3.3.24]
Multiplication with RT gives the ensuing osmotic pressure, which is identical to
PAIR INTERACTION
3.23
[3.3.21]. This is essentially along the lines of Langmuir's approach1'. Thirdly, [3.3.21] can also be derived from the argument that adjacent to the surfaces the osmotic part of /7el(h) vanishes, so that only the Maxwell stress part remains. Hence, /7ej(h) is simply the difference between the Maxwell stress at (either) surface at distance h and at distance °° . This was essentially Deryagin's approach2'3'. In formula, 2 2 naW=?£\(*E) MJ*Z) J
[3.3.25]
The difference between the slopes (d(//dx)0(°°) and {di/// dx)Q{h) corresponds to the difference between the surface charges, according to [3.3.1]. So, [3.3.25] is related to the change of the square of the surface charge upon interaction. For d ^ / d x we have [3.3.5]; squares have to be taken for ym{h) and y m = 0, the difference yields immediately [3.3.21]. In passing, it should be noted that the Maxwell stress also occurs explicitly in the second term of the r.h.s. of [3.3.15]. The fourth remark is that [3.3.21] has a simple form and that it is general in the sense that it also applies for interaction at constant charge. Between these two boundary conditions the difference is in the way in which the potential distribution, and hence the way in which the force, varies with h. In sec. 3.4, this will be elaborated. Let us now consider the case of low midway potentials. In that case, we can replace the hyperbolic cosine by the first two terms of its series expansion, leading to coshzy m = 1 + (zy m ) 2 / 2 : /7el(h) = cRT(zym)2
[3.3.26]
in which [3.3.12] can be substituted for zy m . The result is /7el(h) = 64cRT[tanh(zyd /4)] 2 e ^
[3.3.27]
Upon integration with respect to h this leads to [3.3.18] for the Gibbs energy of interaction. So, at least for this approximated case is the equivalence proved between the two methods for describing the repulsion. For the validity of [3.3.27], similar things can be said, as given below [3.3.18]4'. The above discussion summarizes the essentials of working with disjoining pressures. We shall now conclude this section by deriving some approximate analytical
11
I. Langmuir, J. Chem. Phys. 6 (1938) 893. B.V. Deryagin, Izv. Akad. Nauk. SSSR, Ser. Khim. (1938), No. 5, 1153. 31 B.V. Derjaguin (= Deryagin), Trans. Faraday Soc. 36 (1940) 203. Note that in all equations where the distance dependence is given by exp(—rh) simply 21
G
a , e l = Klle\ •
3.24
PAIR INTERACTION
equations that are based on this analysis and that may appear useful for certain purposes. First, instead of using [3.3.12] for y m in [3.3.26] to find [3.3.27] we can substitute [3.3.8], which is valid for not too high potentials, see above [3.3.7], The result is
nel=
C K 2
y
2
[3.3.27a]
cosh 0th/2) and the corresponding Gibbs energy becomes h/2
G
,=-2cRT(yd)2 ael
f =± = -2cRT(yd)2 2 J cosh (/rh 72)
. iSSL^f [, - ,anh(f)] . «SL W ^
h/2
[ sech 2 (jtfi 7 2)dh' J
aJLmu!]
Equation [3.3.27c] exhibits the usual exponential decay at large Kh , but shows that for low Kh the decay is less steep. Later we shall compare some decays graphically (fig. 3.31). Replacing in [3.3.27a, b or c] (yd)2 by (16/z 2 )tanh 2 (zy d /4) extends the potential range without affecting the trends in the decay. The results are 16cKTtanh 2 ( Z y d /4) z2cosh(x-h/2)
G ael =^^ I tanh 2 f^l^V= a.el
K
=
[3.3.28b,
^ 4 J 1 + e~Kh
^fLtanh2 ^ j ^ t a n h ^ j
13.3.28c]
This equation is generally valid provided the interaction is not very strong. Regarding the orders of magnitude, Gibbs energies have the same dimensions as interfacial tensions but are mostly much smaller. Illustrations of 77(h) curves will be given later for more elaborate types of interaction, see for instance figs. 3.15 and 3.18. Their order of magnitude is O (N cm" 2 ). 3.3d Interaction in the Debye-Hiickel approximation In subsecs. 3.3a-3c, general solutions have been given, but some approximate analytical equations valid for low potentials and/or weak overlap were also included. Cases of low y everwhere, belong to the domain of the Debye-Hiickel (DH) theory. We shall now discuss derivations in which the DH premises are accepted from the very beginning. The mathematical advantage is that it is often easier to arrive at analytical expressions. Although physically the results are generally inferior to those in the previous subsections the DH approximation may be helpful to describe (i) the diffuse part of Gouy-Stern layers, where the high-potential part is
PAIR INTERACTION
3.25
sequestered; (ii) cases where other problems prevail (surface roughness); (iii) situations of weak overlap; (iv) convex spherical double layers. Recall fig. II.3.7, indicating the quality of the DH approximation in y(x] curves. The starting equation is obtained by replacing sinh(zy) in the Poisson-Boltzmann (PB) equation [3.3.2] by the first term of its series expansion % = K2y 6.x2-
[3.3.29]
We have used the equivalent in spherical geometry in the DH theory for strong electrolytes, see [1.5.2.9]. Now y = y{x). The general solution of [3.3.29] is y = Acosh(Kx) +Bsinh{Kx)
[3.3.30]
which can be verified by substitution in [3.3.29]. Alternatively, [3.3.29] can also be solved by the 'multiplication by 2(dy/dx)- trick' used to solve the PB equation, see after [II.3.5.9]. In passing, for a single double layer the general solution of [3.3.29] is y = const. exp(-K-x), with const. = y d . In this case, there are no exp(+x-x) terms in the hyperbolic functions. In fig. II.3.7 this DH approximation was compared with the full PB expression. The constants in [3.3.30] can be found from the boundary conditions, see fig. 3.2. For x = 0 , y = yA , cosh(/rx) = 1, sinh(x-x) = 0, hence A = yd . For x = h, y= y d cosh(K"fi)+ B sinh(rh), from which d [l-cosh(vh)l
,3.3.31]
sinh(x-h) Hence, -4- = cosh(x-x) + 1 ~ c o s h ( r h ) sinh(yx) a
y
[3.3.32]
sinh(x-h)
For the midway potential y m (ich\ l-cosh(x-h) ,(ich\] Hf y m = y d cosh — + —^sinh — y
y
I
\ 2 )
sinh(K-h)
[3.3.33]
V 2 JJ
which can be reworked by using the relationships between hyperbolic functions (II.app.2)
(coshx = l + 2sinh 2 (x/2);
sinhx = 2sinh(x/2)cosh(x/2);
cosh 2 (x/2)-
sinh 2 (x/2) = 1) to give ud m y (h) = 2 [3.3.34] cosh(/fh/2) So, we see that in this approximation the same result is obtained as before, see [3.3.8a], and the dashed curves in fig. 3.3, from which it can be inferred how satisfactory [3.3.34] is.
3.26
PAIR INTERACTION
For the diffuse charge ad{h) , from [3.3.1] ^{h)
= -^£J^)
[3.3.35]
The slope (dy/dx) is obtained directly from [3.3.29], Multiplication of the two sides by 2 d y / d x leads to f^-|
= K2y2 + const.
[3.3.36]
where the constant can be found from (dy/dx) = 0 at y = y m . Hence, ^ = + /fVy 2 -(y m ) 2 dx
[3.3.37]
the minus sign applies to the left of the minimum, the plus sign to the right. This is the DH equivalent of [3.3.13]. From [3.3.35 and 37], od ( h ) = T f o £ ^ I ^ ( y d ) 2 _ ( y m ) 2
= T
^ £ ^ ( y d , 2 _(ym)2
[3
3
38]
which is the DH approximation of [3.3.13]. In these equations y d and y m are generally functions of h. Substitution of [3.3.34] gives ad{h) = + —
— sinh(*rh/2)
[3.3.39]
For the disjoining pressure, we have in this approximation [3.3.26]. Using either [3.3.34] or [3.3.39] we can express 77el(h) in terms of y d or &1 . The results are ei
cosh 2 (Khl2)
and nel(h) =
—^ 2£o£sinh2(x-h/2)
[3.3.41]
respectively. If we want to work with a dimensionless charge we can, as before, see [II.3.5.14] introduce the abbreviation p = (8eoe cRT)"1/2
[m2 CT1]
[3.3.42]
yielding el
4<*T
Equations [3.3.41 and 43] are equivalent. Both give the force per unit area at distance h for a given state of diffuse charge, either dictated by y d or a^ . At this stage, there
PAIR INTERACTION
3.27
can be no difference between interaction at constant potential (the case considered so far) and at constant charge (sec. 3.4)), which only tells us by what mechanism the final situation was reached. However, if we now integrate [3.3.40] and [3.3.41 or 43] to obtain the Gibbs energy of interaction this difference comes into the play. The former yields G^ el , the latter G ^ where the superscripts indicate the mode of interaction. The results are Gg, = 2cz2RT^2
[i-tanhOtfi/2)]
[3.3.44]
AT
and G ^ = 4cKT(pcrd)2[l-tanh(K-h/2)]
[3.3.45]
respectively. In fig. 3.31, these and other decay functions will be compared graphically. It is noted that there are a posteriori attempts to improve the validity range of the equations containing the potential by substituting (16/z2)tanh2(zyd/4) for (yd)2 . The DH approximation can also be used to obtain simple equations for the Gibbs energy directly. At very low potentials the diffuse double layer capacitance of a single double layer Cd = EOEK is independent of the potential, see [II.3.5.18], and diffuse charge and potential are then proportional to each other: da^ /dy/^ =<7d/y/1 = C d , od = —2
yd
[3.3.46]
F so that, for a single double layer,
o
which can also be obtained from [3.2.1] by replacing the hyperbolic cosine by the first two terms of its series expansion. Equation [3.3.47] can also be written as
AG =
° " I F adyd
[3 3 481
--
The symmetry with respect to charge and potential is sometimes handy if interaction at constant y d Is compared with that at constant cfi (with the caveat that the required low potentials are rare). For [3.2.3) we can write G
a,eI=4 G i° )(h) - G i° )M ] = - i r j
with, as sub-cases,
d
^r~dh
I3 3 491
--
3.28
PAIR INTERACTION
and
for interaction at constant potential (sec. 3.3) and at constant charge (sec. 3.4), respectively. When a^ and y d both vary, we come in the regulation regime (sec. 3.5). Finally, as the DH approximation is relatively better for spherical double layers, we shall return to this in the pertaining sec. 3.7. The collection of analytical equations for the interaction Gibbs energies and disjoining pressures in appendix 2 also contains results in the DH approximation. However, in practice there are almost always Stern layers, and then the constancy of y d and a^ depends on the way in which charge is regulated over the diffuse part, the Stern layer and the surface. We shall consider that in sec. 3.5 and now only consider the limiting case of fixed a^ and a purely diffuse double layers. This also is a restrictive constraint. 3.4 Interaction between identical parallel flat diffuse double layers at constant charge We now consider the case that upon interaction cfi remains fixed, as a result of which iffi has to increase. As little as in the previous section, we do not yet worry about the mechanisms by which charge constancy can be achieved in a purely diffuse double layer. Physically, 'constant charge' situations will be met for parallel clay particles and, for spherical symmetry, for latices with covalently bound sulphate groups provided
PAIR INTERACTION
3.29 [3.4.1] 11
AG°(~) = AG°(chem) + AGf (el) in which AG^(chem) = -(Td((/d
[3.4.2]
a*
AGf(el)= f y^'dcr*
[3.4.3]
o
Carrying out the required partial integration, this led to our AG°(~) = - J (jd'dy/*'
[3.4.4]
o
in [II.3.2.3]. The dashes in [3.4.3 and 4] refer to values assumed during the charging process. Of this, [3.2.1] is the elaboration for a single diffuse double layer and [3.2.2] for a Gouy-Stern double layer. Hall21 elaborated the thermodynamics in some detail, without invoking Stern layers. The term 'interaction at constant potential' is shorthand for 'interaction maintaining adsorption/desorption equilibrium at the surface'. This condition implies adjustment of the surface charge upon approach of the particles (figs. 3.1b and 3.4). Equation [3.2.6] is typical in describing the thermodynamics. For interaction of diffuse double layers at constant charge, no adsorption or desorption takes place upon overlap, although the adsorption contribution [3.4.2] to [3.4.4] is of course retained. So, upon overlap we only need [3.4.3] and the corresponding integral at distance h . Introducing, as before, dimensionless potentials and restricting ourselves again to purely diffuse double layers, we arrive at AG
i°ei=^r- J yt'OiMo*'- J y^Mdo* o
[3.4.5]
o
where (3d is the charge, as it varies during the charging process from 0 to o^ (in both integrals) and y d is the increase of the diffuse double layer potential if the double layer is charged and the two surfaces are kept a distance h apart. The superscript (a) to G indicates 'at fixed charge'. The equation tells us that we must charge the (diffuse) double layer twice, first when the two surfaces are a distance h apart, then when they are isolated ( h —> °°). In both cases, the charging takes place from 0 to a^ , i.e. to the (diffuse) charge for isolated particles. The two integrals differ because the relation between y/d and (3d is different for different overlap distances. In [3.3.49b], Recall that the superscript a refers to 'surface excess for an isolated double layer', (a) for 'interaction at fixed charge' and, where needed (v|/) for 'interaction at constant (chemical) potential'. All equations in sec. 3.3 are derived for constant potential even though there are no superindices. 21 D.G. Hall, J. Chem. Soc. Faraday Trans. (II) 71 (1975) 937.
3.30
PAIR INTERACTION
we already had this equation in the Debye-Hiickel approximation (after changing the integration variable). 3.4b Elaboration
There is no need to pass through all the steps discussed in the constant potential case because by a relatively straightforward transformation it is possible to reduce the present problem to the previous one. To that end, all integrals in [3.4.5] are partially integrated to give yd(h)
[ AG
ali=^y-
d
d
d
^lh)dyA'w
o {a (h)-y w}- J o
yd(h)
ydM
+ J od'Mdyd'M + J t^Vldyd'Coo)
[3.4.6]
yd(h)
o
where we have split the integral for h = °° into two parts11. The sum of the first and second integral in [3.4.6] is nothing else than the Gibbs energy for the constant potential case, so
G& = G ^ + ^ O d { y d ( h ) - y V ) } + ^
!
' j ^Mdy^M
[3.4.7]
yd(h)
where G ^ , is identical to Gael in sec. 3.3b. In this equation, we may use [3.3.14] for a^ and, realizing that the integral equals twice [3.4.4], for which we already have |3.2.1], the equation becomes
+L s h [
Z (h)
^
]-Cosh[^]||
[3.4.8,
Regarding the signs, recall that [3.3.14] has a minus sign because a positive value of y d gives rise to a negative cfi . In the literature this sign is sometimes omitted, either because of sloppiness or because a° is meant. The product y d sinh(zyd/ 2) is always positive. As yd(h) > yd(°°) (at constant charge the potentials rise due to overlap) the first term in square brackets is positive. So is the second. Hence, G^\ l s always larger than GjQ < meaning that interaction at constant charge is more repulsive than that at constant potential. The difference vanishes at such weak overlap that yA(h) -> y d (~), i.e. when the LSA applies. Transformations like this have been proposed and elaborated by G. Frens, (Ph.D. thesis, Utrecht, 1968) and by E.P. Honig, P.M. Mul, J. Colloid Interface Sci. 3 6 (1971) 258.
PAIR INTERACTION
3.31
Equation [3.4.7 or 8] can be evaluated if yd{h) is known. This function is different for different CT*1's. According to sec. 3.3b exact numerical solutions are available for that. See fig. 3.3, which has to be renormalized to various values of a^ rather than for y d . As a result, G^ is obtainable numerically.
Figure 3.10. Electric contribution to the interaction Gibbs energy at constant o^ between two identical flat double layers. Scaling: for r = 1. 10"1, 10~2 nm~ *, for which G^ ej corresponds to 1, 10~', 10~ 2 , ... mJra"^ , respectively. The curves are drawn for certain values of y°(oo), which can be converted into values of o^foo), using [3.3.14].
Figure 3.10 gives an Illustration taken from Frens, loc. cit. Tables can be found In Honig and Mul's paper. The curves are drawn for given values of y d (h = °°), rather than for a^ [h = °°) because in practice mostly the zeta potential is measured, which can be converted into a^ (h = °°), using [3.3.14]. The two axis scales and the conversion factor depend on K . The curves can be compared with those for interaction at constant potential, figs. 3.5 and 6. It is readily verified that interaction at constant charge substantially exceeds that at constant potential, particularly at strong overlap. The physical reason is that in the former situation the charge refuses to seep away when the force, exerted by the second double layer, is imposed. In this way, the Gibbs energy per unit area is established. We recall from sec. 3.3c that [3.3.21 ] for the disjoining pressure remains valid in the constant charge case.
3.32
PAIR INTERACTION
Obviously, in deriving expressions for G^, and 77el one can also start from the Poisson-Boltzmann equation, carrying out the integration at constant (3d . When we want information on the potential distribution upon interaction at constant charge (i.e. the equivalent of fig. 3.3), we can proceed as follows. Equation [3.3.5] is also valid for the present case. If we apply it to x —> 0, we find for the left-hand side double layer =+- k[coshfzyd)-cosh(zym)1
(Mj
[3.4.9]
where we need the minus sign if y < 0 and the plus sign for y > 0 . Because of [3.3.1 ] the l.h.s. equals F(J° I RTeo£ = -F&11 RTeo£ so that od = ±^^^2cosh(zy d )-cosh(zy m )
[3.4.10]
in which y d and y m both depend on h . For h —> °° this expression reduces to that for two isolated double layers. For any cfi , yd can be expressed in y m . This equation is formally equivalent to [3.3.13] but the difference is that there &1 adjusts itself whereas now it is fixed. For not too high potentials, as quantified above [3.3.7], [3.4.10] reduces to od =
2Fcz^^(yd)2_(ym)2
[3.4.11]
as we had before (see [3.3.38]). Hence, ym=yj(yd)2
-(cr d ) 2 /2e o eRTcz 2
[3.4.11a]
Regarding approximate analytical expressions, recall that for the disjoining pressure in the DH approximation we already have [3.3.40, 41 and 43]. All 77's are insensitive to the mode of charge regulation; they just represent pressures at a given distance and state of the surface. For the Gibbs energy at constant charge we derived [3.3.45]. Another approach is choosing (dy/dx) as the variable, rather than y. Given [3.4.9] and Gauss' relation between ad(h) and {dy/dx)x=0 there is some logic in this. The resulting differential equation can be obtained by eliminating y d between [3.4.9] and [3.3.14]. This is easily achieved by making explicit cosh(zyd) in [3.4.9], converting it into a hyperbolic sine, using cosh2 x = 1 + sinh2 x . The result is
[^ W +2cosh(zym) f =4 [ 1 + 4 i L ^ l 2 _| K \dxj
J
[
Kzdx\dxJ\
[34 l2]
As in the constant potential case, integration leads to a rigorous solution in terms of elliptic integrals. When the potentials are low enough to use the same approximation as in [3.4.11] and [3.3.26], the integration can be carried out over dy/dx, running
PAIR INTERACTION
3.33
from 0 at x = x m to dy/dx given by [3.3.1 ]. According to Ohshima11 the results are I2
nil/2
[ 1 + f^f)
-!5££I L-**
I, zFcf1 ) j
(3.4.13,
zFo 4
i7el(h)=jrG^1(h) [3.4.14] for the Gibbs energy and the disjoining pressure, respectively. Equation [3.4.13] may be considered the constant charge equivalent of [3.3.18]. Equation [3.4.13] can be condensed somewhat by realizing that 2eoeicRTIzF = p~l, see [3.3.42] G
L%™ =i^^{(pcr d ) 2 - ^ - ( p ^ ) 2 ) }
[3.4.15]
Ohshima did not compare his results with the exact values. Another approach for finding analytical expressions is to exploit the fact that conservation of the (diffuse) surface charge implies conservation of the countercharge which, for each infinitesimal layer dx is related to the second derivative of the potential via Poisson's equation £n£RT (d2u) p(x)dx = —2—— f dx
[3.4.16]
Gregory21 has used this route for relatively low initial potentials. He finds
[
/ hM 1/2
r
\l + (yd)2 sech2 ^ - H
I and
GW
(h) = ^L K
v^)\
] -1
[3.4.17]
J
L d l n f B + ^ C O t h f h / 2 ) ] - ln{(y*f + coahWi)+ BBtohWijU J y
1 +y
J
[
J J [3.4.18]
with B = [l + (y d ) 2 cosec 2 (x7i/2)] 1/2
[3.4.19]
Gregory shows that this set of equations differs insignificantly from the exact numerical result (our fig. 3.10) f o r y d < 2 , except at very strong overlap, where [3.4.18] underestimates the Gibbs energy. For instance, at y d = 2 and Kh = 0.02, [3.4.18] is lower than the exact results by 9%. As at such short distances other features become operative as well (Van der Waals forces, solvent structure-mediated forces, consequences of surface rugosity, etc.), it is questionable whether this very low Kh range in the equation is practically relevant. The conclusion is that [3.4.17 and 18] 11 21
H. Ohshima, Colloid Polym. Set 252 (1974) 158. J. Gregory, J. Chem. Soc. Faraday Trans. (II) 69 (1973) 1723.
3.34
PAIR INTERACTION
are useful for practical purposes. Regarding the application of the equations for Ga el to real systems, the following can be said about the substitution of experimental characteristics. (i) For Gg°^[, if written in terms of cfi , this value is equal to a^(h = °°) and may be converted to the (measurable) C, -potential using [II.3.5.14]: —^ = y d =-ln["-p
[3.4.20]
(ii) For Gg gj, if written in terms of y , f can immediately be substituted. If wanted, o^ (h = <») is obtainable from [3.3.14], (iii) /7el(h) is, at a given y d , o^, independent of the mode of regulation. However, conversion of these yd(h) and cfiih) values to those at h = ~ do require the choice of such a mode. See further appendix 2. 3.5 Interaction between flat identical Gouy-Stern double layers. Regulation 3.5a The issue Not much methodical thinking is needed to realize that the two models of interaction, described in sees. 3.3 and 4, are idealizations, rarely met in practice. Let us first consider the, already rare, situation of such low surface potential and low electrolyte concentration that the double layer is purely diffuse. By what mechanisms can y d (= y° in this case) or a^ = -o° be kept constant? The four godfathers of DLVO theory reasoned that for systems like colloidal silver iodide (at that time a model colloid) y/° would be related to pAg through Nernst's law, see [II.3.7.3a]. As upon overlap pAg remains constant, so should y/°. We have seen, however, that upon particle approach at fixed y/°, a° should decrease. So, charge-determining ions (say I~-ions) must desorb. The issue assumes a dynamic nature: a collision between two particles is a fast process, and the question is whether the desorption of I~ -ions from the surface is fast enough to follow suit. We shall discuss dynamics extensively in chapter 4, but in anticipation it is realized that adsorption of counterions to the adsorbed I^-ions is an efficient alternative for reducing a^ . Often it appears simpler to move counterions normal towards the surface than to move surface ions out of the narrow gap between the two particles by desorption and lateral transport. Obviously this process requires a Stern layer; without a Stern layer, it is impossible to reduce &1 if the surface charge has no time to relax. Another way of arriving at the same conclusion is: assume that y/° and \jA are both constant and let the surface and the oHp be separated by a charge-free gap of thickness d. Then, reduction of a^ implies that the capacitance C of the Stern layer must decrease. As for a charge-free Stern layer C1=eoe1/d this is virtually impossible, because f' and d are determined by the liquid structure. As far as the
PAIR INTERACTION
3.35
electrostatics are concerned, the Stern layer cannot be charge-free, although one can still discuss where this charge banks up. Interaction at constant surface charge is the limiting situation of this latter case. A typical illustration of a constant (surface) charge is that of the plates of clay mineral particles. As explained in sec. II.3.10d, this charge is caused by isomorphic substitution of metal ions inside the solid matrix by ions of a lower valency. The resulting net negative charge does not relax, and certainly not during brief particle encounters. A second example is that of a poly(styrenesulphate) latex in which the strong sulphate groups on the surface are fully dissociated at ambient pH and covalently bound to the polymer, constituting the particle. It is obvious that in neither of these systems surface charges can relax upon double layer overlap. The only way in which a^ can be reduced is by transfer of cations from the diffuse to the Stern layer. So, whatever the reasoning, we arrive at the task of formulating an interaction model involving diffuse and non-diffuse double layers, in which charge redistribution over these two double layer parts, i.e. regulation, is accounted for. There are two ways of approach. The first, to be considered now, is that of reversible overlap, i.e., assuming the double layers to be continually at equilibrium, that is, continually relaxed. Interaction of relaxed double layers is sometimes referred to as interaction at constant chemical potential. The expression is a bit sloppy because it does not mean that the chemical potentials in the environment are fixed (they may even change!) but that all ^ 's in the double layer are always identical to those in the environment. Alternatively, the issue can be considered from the dynamics side, taking into account the various ionic fluxes taking place in the double layers. Such dynamics will be discussed in chapter 4. Before embarking on such charge regulation models we recall that there are a variety of advanced theories in which the inner double layer part is treated statistically, considering ions as discrete charges, taking ionic correlations and image charges into account (sec. II.3.6b) . Such integral theories do not need models with smearedout charges and, when all goes well, they automatically account for the role of the solvent, although specific adsorption remains enigmatic. Intellectually speaking, these approaches are appealing, but they are computationally demanding and the plethora of equations tends to obscure the physical and chemical phenomena. However, in sees. II.3.6c and following, we have shown that under the ambient measuring conditions the Stern layer model, either or not in conjunction with a site binding model, can adequately account for double layer properties. Therefore, we shall now also present interaction at this level, first considering two identical flat double layers, postponing hetero-interaction to sec. 3.6.
11
For updates see P. Attard, Curr. Opinion Colloid Interface Sci. 6 (2001) 366, and M. Quesada-Perez, E. Gonzales-Tovar, A. Martin-Molina, M. Lozada-Cassou, and R. HidalgoAlvarez, Chem. Phys. Chew. 2003 234.
3.36
PAIR INTERACTION
3.5b Chemical and electrical regulation Just as it is mandatory to distinguish chemical and electrical contributions to the Gibbs energy of double layer formation, one must make this distinction in describing changes of Gibbs energies of ad/desorption of charge-determining ions on/from the surface and of (small or big) specifically adsorbed ions at/from the iHp. In addition, we can also distinguish spatial and surface regulation. The former implies ion fluxes from one part of the double layer to the other, the latter only considers charge adjustment in one layer (the surface and/or the iHp). The latter cannot proceed without the former. Let us now consider the following realistic situation: there is a double layer of which the surface charge o° is fixed during particle encounter, either because it is strongly bound, or because it cannot desorb and escape to the bulk during the collision. Double layer overlap leads to a reduction of a^ , which can be achieved by transporting counterions from the diffuse to the Stern layer. Although a° remains fixed, a° + ai , acting as an 'effective' surface charge, is reduced. The extent to which this reduction takes place depends on the capacitances of the Stern and the diffuse layer. In this connection, the term 'capacitance' should be taken in its general meaning as indicating its purely electrical capacitance, determined by dielectric permittivities and thicknesses, and its chemical capacity, determined by ion uptake by specific binding. A Stern layer, together with the layer containing the surface charge, has a high regulation capacity when it can absorb much charge without greatly affecting the potentials in it. The converse is true for a low regulating capacity. The ion uptake capacity is therefore determined by inner and outer Helmholtz layer capacitances, C| and C\ respectively, and an equation for the inner layer charge density cr1. The chemical part of the regulation capacity is determined not only by the specific adsorption Gibbs energy of the ions, but also by the degree of occupancy 0. Stern layers for which a1 = a1 (sat) have a lower capacity than those for which the inner layer is almost empty. The higher the Stern layer regulation capacity is, the better the 'constant y d ' limit is attained. Anticipating discussions of experiments, it is interesting to recall that the phenomenon of repeptization is often invoked as an indication of 'constant charge' interaction. Repeptization is the redispersion of hydrophobic colloids that are coagulated by the addition of electrolyte, upon washing out the salt. Several systems, including silver iodide,, the classical prototype of a 'constant potential' system, exhibit some degree of repeptization. The argument is that, had interaction taken place at constant potential, no charge on the surface would have remained, and hence there would be no driving force for redispersion. However, as argued before, in many situations it is more likely that during a collision the charge on the surface remains essentially intact, the diffuse charge moving to the Stern layer. Hence, peptization is perfectly in line with a reduction of a° + a{, and it is therefore an extra reason for analyzing interaction of Gouy-Stern layers, rather than just surfaces with a diffuse layer.
PAIR INTERACTION
3.37
What happens with such layers is that upon overlap charge and potential are regulated, i.e. they both adjust as a function of h . We note that such regulation also includes adjustment of the co-ion distribution, i.e. regulation of the negative adsorption. An embryonic analysis along these lines was already given by Verwey and Overbeek11, who assumed y/° fixed and a simple Stern layer. A later analysis by Melville and Smith21 is more in line with our approach. In passing we note that in the literature the term 'charge regulation' is often used in a more restricted sense, i.e. a combination of a site binding models on the surface, connected to a diffuse layer. In many of these elaborations, the Stern layer model is rudimentary, or absent. See sec. 3.5d. Let us review the equations that we have at our disposal. They have to be solved simultaneously. It is logical to start with the disjoining pressure as in [3.3.21 ] 77el(h) = 2cKT[coshy m (h)-l]
[3.5.1]
because this expression is generally valid. We have made the h -dependence explicit, because that is the issue at hand. Once I7(h) is known, we can integrate to find the Gibbs energy h
G ael (h) = -J/7 e l (h')dh'
[3.5.2]
where h' is the integration variable. We used this equation before, see [III.5.3.5]. The relation ym[h) is rigorously available in terms of elliptical integrals, see [3.3.9] and fig. 3.3. However, as we want to derive an analytical expression we shall use [3.3.8a] y m =
^ cosh(x-h/2)
[3.5.3]
which is fairly generally valid, see above [3.3.7]. An alternative is [3.3.12]
zy m =8tanh(zy d /4)e- r f l / 2
[3.5.4]
which is more restricted, because it requires the linear superposition approximation. In either way, we relate ym{h) to yd{h). Next, we use [3.3.13] c7d(h) = [sign(-y d )]^^-J2{|cosh[zy d (h)]-cosh[zy m (h)]|}
[3.5.5]
where the variation of the diffuse charge with distance is now constrained by the charge balance cr° + cri(h) + od(h) = 0
11 21
V-O, sec. VII.5. J.B. Melville, A.L. Smith, J. Chem. Soc. Faraday Trans. (I) 70 (1974) 1550.
[3.5.6]
3.38
PAIR INTERACTION
Figure 3.11. Identification of the various planes and potentials for interaction between GouyStern layers at constant surface charge. h = oo ; h finite. When the equations derived for two flat diffuse double layers are used h applies to the distance between the two oHp's. Constancy of a° implies that the slope dy/dx in the inner Helmholtz layer is fixed. where er° does not depend on h by choice of conditions. We also chose cMh) to be located at the inner Helmholtz plane. For yl(h) we have (see [II.3.6.27a and b]) Fa°
y'(h) = y ° ( h ) - — —
[3.5.7]
Upon interaction the absolute value of the surface potential must increase (y° depends on h ) because a° does not depend on h ; however, the increase is less than in the case of constant charge and purely diffuse double layers. Cj and CJ, are the capacitances of the inner and outer Helmholtz layer, respectively. They are equal to £oe\/j3 and e^^ly, respectively, where e\ and e^ are the relative dielectric permittivities of the inner and outer Helmholtz layers. In fig. 3.11, the various parameters are illustrated. We assume e', £?, • P a n d 7 to be independent of h. This is very realistic except at extremely low h. As the two Helmholtz layers are charge-free, the integral and the corresponding differential capacitances, C1 and CJ,, respectively, are constant and identical. The overall capacitance of the Stern layer is not constant,
PAIR INTERACTION
3.39
though, because it changes with a1, see [H.3.6.31]. For more background information see sec. II.3.6c and d. Finally, we have the adsorption isotherm for ions at the iHp, for which we derived [H.3.6.36]
oi =
«.^.VW-y>
l + XjKjexpf-Zji/1)
13.5.9,
where Ns is the number of sites per unit area in the iHp into which ions i can adsorb, Xj is the mol fraction of i and Kj = exp(-A ads G mi /RT)
[3.5.10a]
pK; = A a d s G m i /2.303 RT
[3.5.10b]
or
account for specific adsorption, if any. In the absence of specific adsorption K{ = 1 . The number JVS may, but does not need to, be identical to the number of sites on the surface. With minor changes, all of this can be modified to attain the case of constant surface potential y° . In that case a° becomes a°[h) in [3.5.6 and 8] whereas in [3.5.7] y°(h) becomes the constant y° . So, we now have a set of equations from which we can derive the potential and charge distribution, Gael and /7 e l , all as a function of h . Figure 3.12 gives a first pair of results. Panel a illustrates the reduction of the diffuse part of the countercharge with decreasing distance for different values of er°
Figure 3.12. Spatial charge and potential regulation in a Gouy-Stern layer. Constant surface charge. Parameters: C\ = 120 fiF cm" 2 , C{2 = 20 //Fern" 2 , Ns = 5 x 1014sites cm" 2 , Kt (= K a n i o n ) = 2 , electrolyte cone. 10"' M, monovalent symmetrical electrolyte (ic~l = 0.96 nm ). Panel a, distance dependency of the diffuse charge for various a° , panel b, the same for the four potentials at a° = 10 fjC cm" 2 .
3.40
PAIR INTERACTION
(indicated). The fraction that is not compensated by cfi comes on the account of a1. For large distances the fraction Ic^/cr0] decreases with increasing a°. Upon overlap the diffuse charge reduces, and even strives toward zero, implying that then the surface charge is completely compensated by
(a)
(b)
Figure 3.13. Gibbs energies of interaction, corresponding to fig. 3.12. Panel (a), influence of the surface charge; panel (b), influence of CJ, at fixed a° = lOfiC cm" 2 .
Corresponding Gibbs interacting energies are shown in fig. 3.13. Panel (a) illustrates the (expected) influence of the surface charge, panel (b) that of C^. The increase of Ga el with Cg results from the increase of the charge in the diffuse part in this direction. Under the present parameter choice, condition C\ is invariant because (dy IAx)h=0 is fixed at fixed a° , through Gauss' law [3.3.1 ]. Figure 3.14 shows the influence of specific adsorption at the iHp on the interaction Gibbs energy. The novel feature is that, as a function of Ks, Ga ej passes through a minimum. At low K{ the decrease of Ga el at any h results from the increasing specific (anion) adsorption, which leads to a lower fraction of a° that is compensated by a^ (not shown). When K{ continues to grow, specific adsorption can become superequivalent, o^ changes sign and when a1 has become very strong the limiting case of interaction at constant (a° + c^) is approached. For the parameter set leading to fig. 3.14 the minimum of G ael is situated at about KL = 100 , corresponding to a
PAIR INTERACTION
3.41
Figure 3.14. Spatial charge and potential regulation in a Gouy-Stern layer. Given is the interaction Gibbs energy as a function of the specific adsorption Gibbs energy a° = 10,uC cm" 2 , Kj variable (indicated). Other parameters as in fig. 3.12. The dotted curves refer to K; values above those to produce the minimum of Ga.el •
Figure 3.15. Spatial charge and potential regulation; disjoining pressures. Parameters as in fig. 3.12a, K, = K3nioa = 2 . Interaction at constant surface charge a° (indicated).
specific adsorption (Gibbs) energy A ads G m i of about -4.6 RT. This is a very realistic value; such chemical adsorption energies can compete with an electric repulsion felt by a monovalent ion at a potential of about 114 mV. In this connection, a link can be made to a phenomenon of colloid stability, known as irregular series, i.e. sol stability which, with increasing electrolyte concentration, first decreases until an instability minimum, beyond which it rises again. The phenomenon is observed for specifically adsorbing counterions, including ionic surfactants. Basically, the interpretation is the same as in fig. 3.14, viz. superequivalent counterion adsorption occurs. Direct proof is the sign reversal of the £"-potential. The difference with the theoretical picture of the figure is that in the latter case superequivaleney is achieved by increase of K{ at given c whereas in the irregular series it is the other way around. Disjoining pressures are represented in fig. 3.15. As compared to the Gibbs energy of interaction (fig. 3.13a) there are no qualitatively new features.
3.42
PAIR INTERACTION
Figure 3.16. Spatial charge and potential regulation in a Gouy-Stern double layer at fixed surface potential. Panel a, lj/° = +120 mV ; panel b, y/° = +60 mV . Other parameters are as in fig. 3.12. Figure 3.16 gives charge (panel a) and potential regulation (panel b) for interaction at fixed surface potential ( y/° = +60 mV). Upon interaction, the diffuse countercharge approaches zero, some of this charge moves to the iHp (a1 becomes slightly more negative), the other part helps to reduce <J° . However, the latter charge does not attain the zero value (except for h —> 0, not drawn) because the specifically bound charges remain present and fully compensate CT° at very short h. In figs. 3.17 and 18 the corresponding Gibbs energies and disjoining pressures are given. The trends are according to expectation. The new element is the influence of the inner layer capacitance in fig. 3.17b. (In the case of constant surface charge and fixed ei and d, CJ is invariant.) Gibbs energies increase with Cj because high values of this capacitance keep y' high.
Figure 3.17. Gibbs energy of interaction at fixed surface potential for Gouy-Stern double layers. Panel a, influence of the surface potential at variable a and all other parameters fixed as in fig. 3.12. Panel b, influence of Cj at fixed \f = +80 mV .
PAIR INTERACTION
3.43
Figure 3.18. Disjoining pressure for the conditions of fig. 3.17a.
Let us finally consider orders of magnitude and briefly anticipate experimental verification. The curves for G ael and 77el are results that could in practice be measured, considering that the purely diffuse version of DLVO theory has only very limited applicability. It is a pity that no experiments exist to check figures such as 3.12 and 16. For instance, there is no unambiguous way of measuring y d = £ as a function of h. So, in practice the quality of a certain model can only be judged by how well the Gatl(h) and/or /7el(h) curves can be described over the entire h-range by one of the basic assumptions [o° or y° fixed) and the optimum set of parameters. Given the available leeway in these parameters and some experimental uncertainty, the result is often not unique. As a trend, interaction at fixed o° remains stronger than that at fixed y°, depending on parameters. Regarding the orders of magnitude, Ga el values can amount to a few m j m" 2 , but under practical conditions tend to remain much lower than that, particularly at Kh'k.i and moderate values for o° and y°. Then they become comparable with the values of interfacial tension (see the tables in III.App.l). For large h, Gaej « y; upon decreasing h, Ga el and y become comparable. However, at very short distances structural changes beyond those discussed so far also enter the game, so that such comparisons lose their sense. 3.5c Introduction of surface charge regulation models Charge regulation upon overlap of double layers is the rule. The exception is that all charges remain in position with respect to the surface to which they belong. Most interaction models therefore involve tacitly or explicitly some displacement of charge, i.e. of charge regulation. Even interaction at constant surface charge a° involves spatial ionic transports, notably from the diffuse to the Stern layer as illustrated in the
3.44
PAIR INTERACTION
previous subsection. However, it may be noted that in the literature the term 'charge regulation' is mostly used in a much more restricted sense, viz. in that a° becomes a°(h) because the charge forming equilibria shift upon interaction. Often for this a site binding mechanism (see sec. II.3.6e), ignoring Stern layers, is assumed. In order to retain contact with the current literature we shall now add an elaboration in which we assume that the regulation of the surface charge a° is governed by a site binding mechanism. So, we now relax the condition that a° or y° is fixed, but let these two parameters adjust themselves as a function of h, in addition to the transport across the Gouy, Stern and surface layer. Obviously this is a multiparameter issue: to the parameters in sec. 3.5b at least twopK 's have to be added, to account for the charge-forming equilibria. For amphoteric surfaces with only one dissociable group, two pK 's are needed, one for the acid function and the other for the base function. The planar surface site binding makes the regulation capacity sensitive to the pH (for oxides) or to the pAg (for silver halides). In the literature several conventions can be found to define the two charge forming reactions and, hence, the two equilibrium constants. There also is a 'one K' model, involving half charges on the various surface groups. By colloid titration only it is impossible to discriminate between these mechanisms, so it is essentially a matter of taste or habit, which one to choose. In passing, it is noted that 'two K' models do not have a parameter more than the 'one K' model because only combinations of the two K's are accessible. Of course, for surfaces containing more types of groups (like proteins) the number of pK 's increases in keeping. It is beyond the size of this section to treat all of this exhaustively. Rather we shall elaborate a relatively simple situation to indicate relevant trends and discuss some alternatives in sec. 3.5d. We consider a surface containing hydroxyl groups (ROH) that can either become positively charged by adsorption of a proton (to form ROHJ ) or negative by losing one (leading to RO~ groups). Acid-base colloid titration of oxides is a means of determining the difference between the total positive and negative charge through a° = F ( [ R O H £ ] - [ R C T ] )
[3.5.11]
but we cannot establish [ROHJ] and [RO~] individually (see I.5.6e). Square brackets indicate surface concentrations in mole m~2 . Positive and negative charges are assumed to reside in the same plane, identified as the surface. Specific adsorption is ignored, because it enters our analysis via [3.5.9] and hence automatically participates in the spatial part of the regulation. For the moment we do not worry about the finiteness of numbers of sites (for most oxides the experimentally accessible range covers only a fraction of the maximum). We briefly repeat and extend the site binding theory for this case from sec. II.3.6e. The acid and base dissociation constants follow from
PAIR INTERACTION
3.45
R O H ^ ± R O " + H+
K =
ROH + H 9 O?=>ROH++ OH-
Kh =
2
a
2
b
[R
° ' xs , [ROH] H+ [ROHi] X O H - [ROHi] K 2__OTL = 2 w_
[3.5.12a]
[3.5.12b]
[ROH] x w [ROH] x^+ Here, xf is the mole fraction of charge-determining ion i in the surface; Ka and Kb are dlmensionless. (As stated, in the literature other defining equations can be found, and even pK 's in which the K is not dimensionless.) When the (dimensionless) potential in the surface is y° , x s + can be related to its bulk value via Boltzmann's law, *H+=Ve~y°
[3 5 131
- -
which, in turn, can be related to the pH. x H+ = V m e- 2 - 303 P H
[3.5.14]
Here the molar volume Vm of the water enters to correct for the dimensional problem incurred by the definition of pH as - 1 0 logc + , i.e. as the logarithm of a dimensionhaving quantity. The surface charge follows from [3.5.11-13] as \Khx e-y°
K
e y°l
3
}
[3.5.15]
We had this equation before, see [II.3.6.43]. The various K 's are related to each other and to the point of zero charge: ( o° = 0 , x , = x° and y° = 0 (reference point for ti
H
the surface potential as it occurs in [3.5.13]) (K
V /2
K
l3 5 161
**=[-IC)
--
It is convenient to introduce the (dimensionless) Nernst potential yN =-2.303 (pH-pH°) via * H + = * °n + e
-2.303 lP ( p H - pP H ° )
,,N
' = * °n + e y
[3.5.17] [3.5.18]
Substituting this in [ 3.5.15 ], using [ 3.5.16 ], leads to g° = F[ROH]
(K a Kb V / 2
I
K
2sinh(y N -y°)
[3.5.19]
w J
It is noted that (KgKb)1/2 acts as a capacity factor: the larger this product, the higher the surface charge is at given potential, or, the less sensitive the potentials are to changes in the charge. On the other hand, it follows from [3.5.14] applied at the p.z.c.
3.46
PAIR INTERACTION
and [3.5.16] that the point of zero charge pH° is related to the difference ApK = pKa -pK b or to the quotient (Ka/Kh)1/2 pH° = logVm + iApK + ipK w
[3.5.201
This is as expected: the larger ApK , the lower the proton concentration must be in the solution to render the surface electroneutral. For ApK = 0 pH° = 7 . Had we not allowed the number of sites to be infinite, see above [3.5.12], equation [3.5.19] would have become more complicated1':
N (Ka a Kb h /Kww ) 1 / 2 -2sinh(y -y°) y .,_1 / 2 ., N l + (K a K b /K w ) -2cosh(y -y°)
[3.5.21]
We do not give the derivation. It is seen that o° depends on h through y° and on pH via jyN and [3.5.18], Figure 3.19 gives an elaboration. The parameter set is similar to that leading to figs. 13a and 17a, except that er° or y° are not fixed. The new feature is that G ael now is pH-dependent. At the p.z.c. it is zero, because there is no charge on the surface; the more distant pH is from pH° , the larger G a e l . This picture can be more or less modified at will by adjusting, Cj, and JVS , or extended by allowing for specific adsorption at the iHp. Figure 3.19. Spatial charge and potential regulation combined with planar surface regulation. Chargefree Stern layer. Parameter values: q =120 nF cm" 2 , JVS = [ROH] = 5 x l 0 1 4 cm" 2 , Ka = 1 0 - 1 , K b = i o - 3 , (pH° = 4.26), 0 = 10"' M. The pH is indicated. Curves with pH > pH° are dashed.
3.5d Charge regulation in the literature The treatment of sec. 3.5c captures all essential features of charge regulation. It can be elaborated, for instance, by considering more types of surface groups. Anticipating sec. 3.6 on hetero-interaction, we shall offer in this section a brief extract of the rather lf
D. Chan, T.W. Healy, and L.R. White, J. Chem. Soc. Faraday Trans. (I) 72 (1976) 2844.
PAIR INTERACTION
3.47
extensive literature, to give some feeling for what is available and for pointing to alternative models and elaborations. The oldest example is, to the author's knowledge, Verwey and Overbeek's treatment of interaction between Gouy-Stern layers11. This treatment refers to spatial regulation, as does the work by Melville and Smith, mentioned in sec. 3.5b. One of the oldest illustrations of surface charge regulation only goes back to Bierman2'. Basically, this paper handles interactions between non-identical surfaces. Stern layers are ignored. However, the assumption is made that y° becomes y°{h) because the occupancy of the surface by charge-determining ions changes upon overlap according to a Langmuir-type equation. Otherwise stated, a° is regulated to become
[3.5.22]
which resembles our [3.5.21]. Apart from the replacement of F[ROH] by eNs [Ns is the total number of dissociable surface groups, which may, but does not need to be identical to JV in [3.5.9]) their 'capacity coefficient' 8 is written as
"v-O, sec. VII.5. A. Bierman, J. Colloid Interface Sci. 10 (1955) 231. 31 B.W. Ninham, V.A. Parsegian, J. Theor. Biol. 31 (1971) 405. 4 Recall that a functional is a function of a function. We met functionals in sec. III.2.6, and III.app.3 introduces the variational calculus, i.e. the method for finding the function which minimizes the (Helmholtz) energy. 51 D. Chan, J.W. Perram, L.R. White, and T.W. Healy, J. Chem. Soc. Faraday Trans. (I) 71 (1975) 1046. Also see T.W. Healy, Pure Appl. Chem. 52 (1980) 1207 and I.M. Metcalfe, T.W. Healy, Faraday Discuss. Chem. Soc. 90 (1990) 335. 21
61
D. Chan, T.W. Healy, and L.R. White, J. Chem. Soc. Faraday Trans. (I) 72 (1976) 2844.
3.48
PAIR INTERACTION
S=2jK_/K+
[3.5.23]
which differs from our 2jKaKil / Kw
because they define the charge determining
reactions differently, viz., through [ROH]cs+ K. = — + [ROH+]
[RO"]cs + K = — [ROH]
[3.5.24]
So, apart from the dimensions (surface concentrations c s instead of mole fractions) their K+ resembles K^1 whereas K_ = Ka . In this convention the capacity factor is related to ApK whereas the p.z.c. equals ^ ( p K + + p K J . The caveat is that in comparing results of models the basic assumptions must also be verified. In the further elaboration, this model was applied to the interaction between dissimilar surfaces. A variety of situations can be accounted for, even though Stern layers are ignored. On the basis of this model, Carnie and Chan11 formulated a generalized theory that could also be applied to other particle geometries but with the restriction of linearization, a restriction that was partly relaxed by Ettelaie and Buscall21, although Stern layers were still assumed absent. Behrens and Borkovec31, using the lpK model, did include an embryonic Stern layer of zero thickness. In fact, accounting for Stern layers of finite thickness is a recurrent issue in site binding models. Sometimes the Nernst potential yN and the actual potential y° are thought to have a different position, (the 'Nernst plane' as opposed to the 'surface plane'). Relatively best is the triple layer model but even by this it remains difficult to account for the value of y d for isolated particles41, let alone to describe yd{h] accurately. Healy and White51 gave a brief description of a 2pK site binding model with specific adsorption of counterions and a zeroth order Stern layer (as in fig. 3.11 but with y=0). Several models, plus their own elaboration in terms of Gibbs energy functions, have been reviewed by Reiner and Radke61. Stern layers were accounted for explicitly in an appendix to that paper. Usui71 published an approach similar to ours, in that Stern layers were explicitly accounted for. His model was simpler than ours in that he did not discriminate between the inner and outer Helmholtz potential, but more advanced in that he considered two double layers of unequal potential or charge. Various applications can also be found in the literature, for instance with pen11
S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci. 161 (1993) 260. R. Ettelaie, R. Buscall, Adv. Colloid Interface Set 61 (1995) 131. 31 S.H. Behrens, M. Borkovec, J. Phys. Chem. B103 (1999) 2918. 41 See, for instance, J.A. Davis, R.O. James, and J.O. Leckie, J. Colloid Interface Sci. 63 (1978) 480, from which fig. II.3.63 is taken. 51 T.W. Healy, L.R. White, Adv. Colloid Interface Sci. 9 (1978) 303. 61 E.S. Reiner, C.J. Radke, Adv. Colloid Interface Set 47 (1993) 59. 71 S. Usui, J. Colloid Interface Sci. 97 (1984) 247. In a more recent paper the iHp and oHp were distinguished (J. Colloid Interface Sci. 280 (2004) 113). 21
PAIR INTERACTION
3.49
etrable surfaces, as may occur in biological systems, or with surfaces along which charges can move laterally. We mention a paper by Hsu and Kuo,l) in which the rates of charge-regulating processes are made explicit, thus anticipating the dynamics of chapter 4. 3.5e Intermezzo: status quo The previous pages reviewed the fundamentals of electrostatic double layer interaction. We discussed and elaborated it for three different modes of interaction (constant surface charge, constant surface potential and regulation). Generally stated, the issue is to identify the charging mechanisms, the charge and potential distribution, and the way in which these features change upon interaction. So far, the treatment was restricted to two identical semi-infinite plates, a system that is not the most common in practice. Rather, colloidal particles are spherical, if not irregularly shaped. In addition, interactions do not necessarily occur between identical double layers. For example, adhesion phenomena (sphere-plate interaction) and AFM measurements almost always involve different double layers. This is also the case for wetting layers (sec. III.5.3) where the GL and the SL interfaces mostly carry double layers with different regulation capacities. Obviously, generalizations of the previous sections to hetero-interaction and/or surfaces of different geometries are demanded, but equally obviously the number of situations to be described grows beyond bounds. Even if we would confine ourselves to sphere-sphere and sphere-plate interaction, each surface can have three interaction modes, independent of the other. This leads to at least six cases, and a multiple of this if we are going to derive equations for special situations with respect to tea, Kh , y° and y d . It is easy to get lost in a forest of equations, so for reasons of economy we must restrict ourselves to the fundamentals. In elaborating theory, we also must remain realistic in the sense that rarely enough experimental evidence is available for profound testing. Mostly, only f-potentials of isolated particles are at our disposal, i.e. yd(h = ~) and crd(h = «>). With this information, we cannot do more than apply equations for Gael(h) and /7el(h) at fixed y d or fixed cfi and find out which one describes the results best over the entire range of distances and salt concentrations. Such results are not necessarily unique. For instance, this procedure tells us nothing about regulation, let alone about the dynamics. One is better off if, in addition to f, the surface charge o° is also known, both as a function of pH and the electrolyte concentration c. In that case one could model the Stern layer and estimate the parameters Ka , Kb , K{, C\ and C\ , which are required to quantify regulation capacities. As paper is patient, there is a great temptation to invent a variety of new phenomena on the basis of a limited number of interaction studies, with
11
J.P. Hsu, Y.C. Kuo, J. Chem. Soc. Faraday Trans. 91 (1995) 4093.
3.50
PAIR INTERACTION
unexpected results. Returning to practice, we shall limit our systematic discussion to - interactions between dissimilar double layers, including regulation, but only for flat surfaces, in sec. 3.6. We shall call these hetero-interactions, see above [3.1.4] - interaction between spheres and a few other geometries, in sec. 3.7. Special features, like the effects of ion valency, will be interspersed where convenient. To compensate for this space-imposed limitation we shall collect a number of relevant equations in appendix 2. 3.6 Hetero-interaction 3.6a Basic phenomena Consider two interacting charged parallel plates. The two electric double layers are different. We are interested in the changes in the structures of these double layers, the Gibbs energy of interaction and the disjoining pressure, all as a function of the distance h between the two outer Helmholtz planes (oHp's), i.e. the planes beyond which the countercharge is diffusely distributed. By 'different' we mean that the (dimensionless) potentials y° , y' and y d = F£7 RT and the surface charge densities o°, (J1, a*1 ~ o^k may be different with respect to signs and/or magnitudes. Qualitatively new features may occur. When two surfaces with different potentials, but of equal sign approach each other, the long-distance interaction is always repulsive, but at short distance it may become attractive because of induction: the surface with the higher potential may impose a potential with opposite sign on the other. With interactions at fixed charge, this cannot happen. Whether or not in practice such a reversal takes place depends of course on the regulation capacities of the two Stern layers, a phenomenon already recognized by Prieve and Ruckenstein1'. Hall gave it a thermodynamic footing21. When two unequal double layers overlap a variety of phenomena can occur, determined by the two surface potentials yl and y2 • and/or surface charges
D.C. Prieve, E. Ruckenstein, J. Colloid Interface Sci. 63 (1976) 317; 76 (1980) 539. D.G. Hall, J. Chem. Soc. Faraday Trans (II) 73 (1977) 101.
PAIR INTERACTION
3.51
inner layer regulation capacity
Figure 3.20. Hetero-interaction between flat plates. Schematic picture of potential- and charge distribution in the overlapping diffuse double layers, (a) Infinite distance; (b) overlap at high inner layer regulation capacity; (c) ibid, at low capacity. Discussion in the text.
As compared with homo-interaction the other new element is the asymmetry of the y[h) profile between the two surfaces. The minimum potential y m is no longer located at the half-distance ( x m =h/2, see fig. 3.2} but shifted towards the surface with the lower y d . Figure 3.20 gives a sketch of what may happen with the diffuse double layer parts. In this scheme, the inner layers of thickness d are indicated, but we have refrained from drawing y(x) lines in these parts because there is a plethora of options for those. Rather, the overall capacity of these layers, including the two solid surfaces, to absorb charges is assumed high in case b, but low in case c. In case b the
3.52
PAIR INTERACTION
Figure 3.21. Hetcro-interaction leading to attraction. The potential reverses sign at a specific value of h.
minimum is more to the right than in case c. Situations can also be imagined in which the minimum disappears completely, as illustrated in fig. 3.21. This phenomenon is a typical illustration of induction: the 'strong' double layer to the left induces an opposite charge on the 'weak' double layer to the right, 'strong' means: having a high charge and low regulation capacity, 'weak' means the opposite. The classical case of induction is that initially the right hand plate is uncharged. Absence of a minimum implies attraction. In such cases, between two particles of the same sign of y d . upon diminishing h the repulsion first increases, then passes through a maximum, after which the interaction becomes attractive. The phenomenon also has interesting dynamic implications: if the regulation has a time constant that is comparable to the rate of approach of the two surfaces, and if the rates of adsorption and desorption differ, there may be a hysteresis in the interaction force between snap in and out. In passing it is noted that something similar can also happen between two double layers that are identical with respect to their charges and potentials at large distance, but which have surfaces of different regulation capacities. Anticipating quantitative analysis, the statement can be made that for interaction between dissimilar double layers the potentials or charges of the lower-charged surface are more critical than those of the higher-charged one. This is a consequence of the tendency of diffuse double layers to accommodate most of their charges in the part where the potentials are high, i.e. close to the oHp's. Semi-quantitatively, suppose we have interaction at constant diffuse potential. Let [3.3.27] describe the disjoining pressure after replacing the square of the hyperbolic functions by the product of the two:
and let yf » y%. It is typical for hyperbolic tangents that tanh x -> 1 for high x
PAIR INTERACTION
3.53
whereas tanhx —> x for low x . Hence,
/7el(h) = - ^ p ^
lyf»y$)
[3.6.2]
Consequently, the diffuse potential of the higher charged surface disappears. 3.6b Theory. Diffuse layers only The necessity of considering hetero-interaction was recognized long ago by Smoluchowski1'. Deryagin has paid much attention to it. Some of his papers were published in the Russian literature and during the Second World War, so they did not receive the attention they deserved. The Deryagin-Landau paper, which led to the acronm DLVO, also contained a section on hetero-interaction. More accessible, and better known, is his contribution to one of the Faraday Discussions on colloid stability21. Since then, several elaborations and extensions have become available. As for homo-interaction, in principle two ways are at our disposal to quantify the interaction: (i) Solve the Poisson-Boltzmann (PB) equation, find the Gibbs energy by an appropriate charging procedure, subtract the Gibbs energies for the two double layers far apart, and find Ga el (h). Differentiation with respect to h gives I7el(h) • (ii) Start with the disjoining pressure, i.e., formulate the equivalent of [3.3.21 or 25] and use the PB equation to obtain ym{h), which is now unsymmetrical. Integration provides G ael (h). Let us elaborate these approaches somewhat. Route (i) was followed in subsecs. 3.3a, b. For hetero-interaction the PB equation [3.3.3] remains valid: (^) 2 ^cosh( Z y ) + C]
[3.6.3,
where y = y(h). Establishing C is now slightly more complicated than before. For y(x) curves with a minimum, as in fig. 3.20b and c, we have, as before, C = -cosh(zy m ), so that - ^ - = +i,/2rcosh(zy)-coshfzym)l d(x-x)
zvL
\
[3.6.4]
'-1
As in [3.3.5] the + sign is needed to the right of the minimum and the minus sign to its left. The difference with [3.3.5] is that xm does not coincide with hi2 . For cases without a minimum, as in fig. 3.21, C can be evaluated from the fact that y = 0 at x = x° :
11
M. v. Smoluchowski, Z. Phys. Chem. 9 2 (1912) 129. (As part of his study of the kinetics of coagulation.) 21 B.V. Derjaguin, Discuss. Faraday Soc. 18 (1954) 85.
3.54
PAIR INTERACTION
2 {d(Kx))y=0 from which
£ i f _ ^ _ f =£if_dy_f 2 {d{Kx)J
+ cosh(zy)-2
[3.6.6]
2 [d(icx))y=0
Elaboration is laborious. As for homo-interaction, elliptic integrals and/or numerical analyses are needed. Devereux and de Bruyn11 produced detailed tables for yf[h), y%[h] and Gael{h), to which we return later. Yet another way of finding C is by relating the field strengths at x = 0 and x = h to the corresponding diffuse charges, of{h) and a^W , using [3.3.1], which are now written as {dx)x=0
{eoeRTJ
Kdx)x=h
{£o£RT)
This route was, among others, chosen by Bierman21 and by Chan et al.3) who elaborated it for the case of low potentials. Some scrutiny is needed in deciding the signs if square roots have to be taken in the later steps: in [3.3.1], (3d and (dy/dx)x=0 have opposite signs (because for a positive o^ the potential decreases with distance) but in the situation of fig. 3.20, {dy/dx)x=h and o% have the same sign because x is now counted from the left to the right. All C 's in [3.6.4 and 5] are compatible: at given h the potential between the two oHp's distributes itself in such a way that Ga is minimized (i.e., we are minimizing a functional). From [3.6.7] and [3.6.3], using [3.4.16], we obtain the third pair of solutions C=
-£{
K
J -coshH)
= ^[
K
j -cosh(Zyd)
[3.6.8]
This equation offers, at the same time, a relationship between yf , of , y% and a^ • As for homo-interaction, we also can follow the 'disjoining pressure route', i.e., starting directly from [3.3.23]. The sum of the osmotic contribution and the electrical one (i.e., the Maxwell stress) must of course be the same at any place across the overlapping double layers. The osmotic one is given by 2cRT[cosh(zy)-l] (recall [3.3.24], where we specialized this to y = ym in the symmetrical case), the Maxwell stress is as in [3.3.23], so that
1 O.F. Devereux, P.L. de Bruyn, Interaction of Plane-Parallel Double Layers, M.I.T. Press (1963). The book considers interaction between diffuse double layers at constant yd . 21 A. Bierman, J. Colloid Interface Sci. 10 (1955) 231. 31 D. Chan, T.W. Healy, and L.R. White, J. Chem. Soc. Faraday Trans. 72 (1976) 2844.
PAIR INTERACTION
3.55
/7el(h) = 2 c R T [ c o s h ( Z y ( h ) ) - l ] - M ^ j 2 ^ j 2
[3.6.9]
Combining this with [3.6.3] yields 77el(h) = -2cRT[C(h) + l]
[3.6.10]
Phenomenologically speaking, this final equation is simple and general, but the elaboration is not. We see that the sign of /7el depends on C ; it may change as a function of h . /7el(h) is repulsive if C < - 1 , it is attractive for C > - 1 . On the basis of diffuse double layer theory only, it is virtually impossible to discriminate between the various options of repulsion and attraction, because the sign of 77el is sensitive to the extent of constancy of of, a^ > yf a n d 1/2 u P o n interaction. The resilience of these four crucial parameters against the action of the double layer of the second particle is determined by the two primary (spatial and planar) regulation capacities. For these, no simple general rules can be given, although several advanced partial solutions can be found in the literature.
Figure 3.22. Hetero-interaction between different oxides; schematic. Drawn curves, h = 00 and y d ca C, ; dashed, one of the many options discussed in the text for finite h. Left, curves for surface 1; right, curve for surface 2.
To get some feeling it is worthwhile to consider the interaction between two different oxide layers, i.e., oxides of different ApK and different p.z.c. (see [3.5.20]). Let us assume that for both the £" -potentials are available as functions of pH and c . So, yf (pH,c ,h = 00 ) and y% (pH,c ,h = °° ) are known. Let i.e.p.f < i.e.p.EJ. The two curves have a similar sigmoid shape, as sketched in fig. 3.22. As long as the particles are far apart, the sign of the interaction is simple: at pH < i.e.p.f and pH > i.e.p.° it is repulsive and at i.e.p.° < pH < i.e.p.° it is attractive. This is the primary observation in heterocoagulation. However, upon closer approach a spectrum of possibilities develops, depending on the absolute values of the various potentials and charges and on the regulation capacities of the two layers. For instance, the dashed curve in fig. 3.22 may apply to i/j for low h: due to the close approach of the positively charged second particle y : becomes more negative so that the attractive window is enlarged. As similar things may happen to double layer 2 even on this level a variety of options become available. Let us give a few elaborations.
3.56
PAIR INTERACTION
First, recall that the Devereux-de Bruyn tables11 gave exact numerical results for constant y d 's. So the quality of the various analytical results may be judged by comparison with these tables. Chan et al. (loc. cit.) analyzed some aspects for planar surface charge regulation only. Carnie and Chan2) made this quantitative but only in the DH approximation. This work was extended by McCormack et al.3). In their paper solutions of the PB equation for different values of the integration constant C are given; situations with different regulation mechanisms for the two surfaces are also considered and a variety of <jd{h) and yd{h) curves and equations for Gel were predicted. Bell and Peterson41 developed a graphical method from which the various regulation cases can be read. This work also predicts the conditions under which the disjoining pressure exhibits a maximum as a function of distance. Genxiang et al.5) elaborated the theory for constant yf and yd at such high potentials that the hyperbolic sine may be replaced by only one exponential. Ohshima61 derived (complicated) equations for hetero-interaction at constant o^ , between two plates of finite thickness. See app. 2 for a selection of equations. As shown in sec. 3.3d the DH approximation often gives a rapid approximate result. For hetero-interaction between two plates the following equations, derived by Hogg et al.71, has become popular. G
ffl =£°£K2{pI)2 [{{yff +(^) 2 }(l-coth (y h)) + 2t / f^cose C h(^)]
[3.6.11]
The derivation starts from [3.3.46 and 49], using the appropriate DH equations of sec. 3.3d. The practicality of [3.6.11] stems from the fact that only the two f-potentials (~ y?> yjj ' a r e n e e ded. It should describe weak overlap fairly well, but of course does not suffice to predict the low Kh behaviour. Later, Ohshima et al.8' improved this equation to higher potentials, approaching Devereux and de Bruyn's numerical data. For yf = yA , G
w i = £°£K^T]2 = -2—^2
(yd f [1 - cothfrfc) + cosecOrfi)] = (y d ) [l-tanh(/rfi/2)]
[3.6.12]
Note the typical difference between homo-interaction [3.6.12] and all equations met in 11
O.F. Devereux, P.L. de Bruyn, loc. cit. S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci. 161 (1993) 260. 31 D. McCormack, S.L. Carnie, and D.Y.C. Chan, J. Colloid Interface Sci. 169 (1995) 177. 41 G.M. Bell, G.C. Peterson, J. Colloid Interface Sci. 60 (1977) 376. 51 L. Genxiang, W.H. Ping, and J. Jun, Langmuir 17 (2001) 2167; L. Genxiang, F. Ruijang, J. Jun, and W.H. Ping, J. Colloid Interface Sci. 241 (2001) 81. 6) H. Ohshima, Coll. Polym. Sci. 252 (1974) 257; ibid. 253 (1975) 150. 71 R. Hogg, T.W. Healy, and D.W. Fuerstenau, Trans. Faraday Soc. 62 (1966) 1638. 81 H. Ohshima, T.W. Healy, and L.R. White, J. Colloid Interface Sci. 89 (1982) 484. 2)
PAIR INTERACTION
3.57
sees. 3.3 and 3.4, and hetero-interaction ([3.6.11]). In the former case the interaction can always be written as a product /(y d ) x g{>ch), in the latter this is no longer the case, except at large Kh . This new behaviour is directly correlated with the propensity of sign reversal. The constant charge equivalent of [3.6.13] has been given by Usui11 and reads
Offl = £ ° £ 2 ^ T ' 2 [{(yf)2 + (y2d)2}(coth(yh)-l) + 2yd y d c o s e c ( y h ) j
[3 . 6 . 13]
The difference with the constant potential equivalent is only in the sign of the (coth(K-h) -1) factor. In fig. 3.23 [3.6.11] is illustrated for an attractive and a repulsive long-distance interaction. Here, y d is fixed and positive. For y d < 0 attraction prevails; for y d > 0 repulsion is found at large distance, but for short distance this reverses into attraction. We note that in this approximation the constant charge interaction is just symmetrical to that at constant potential, i.e. it also exhibits sign reversal. As explained before, this is impossible. The origin is in the using of the DH approximation which fails if the potentials become very high, as demanded by interaction at constant charge and low Kh . Therefore, [3.6.13] had little to add; we have only included it for the sake of completeness.
Figure 3.23. Hetero-interaction between two flat plates. Given is aG^\ according to [3.6.11], a = 2F2/e0£K(RT)2 . y$ is fixed at +0.6. The value of yf is indicated.
1J
S. Usui, J. Colloid Interface Set 44 (1973) 107.
3.58
PAIR INTERACTION
3.6c Elaboration on the Gouy-Stern level As discussed before, treatments on the purely diffuse level are inadequate because the regulation of (3d and y d is dictated by the processes occurring in the Stern layers. In particular, there are no realistic conditions under which cfl or y d would remain constant upon overlap. Therefore, we shall now extend the theory of subsec. 3.6b to a pair of double layers, each carrying a Stern- and a diffuse layer. We shall continue to denote the l.h.s. double layer by a subscript 1, the r.h.s. by subscript 2. So, Cj 2 is the inner layer capacitance of double layer 2, etc. The analysis will be carried out with the DH approximation for the diffuse part, which is good enough to account for all the physical phenomena. The required equations are the following. Consider first the diffuse parts. As before, [3.6.9] is the starting equation for the disjoining pressure. It can be written in the form [3.6.10]. General solution [3.3.30] for the potential of two overlapping diffuse double layers in the DH approximation also remains valid, but for further analyses it is expedient to write the elaboration either in terms of diffuse layer potentials yd(h) or y^h)
or charges, af(h) or ofth). The
results are, in terms of diffuse potentials, cjd(h) = ^ £
-B^Hh)
o$(h) = —2
[3.6.14]
[A(ve>(h)sinh(x-h) + B(ve>(h)cosh(x-h)]
[3.6.15]
AW(h) = y d (h)
[3.6.16]
, , y d (h)-y, dl (h)cosh(x-h) BW(M= 2 —sinh(x-h)
[3.6.17]
and, in terms of diffuse charges y d (h) = A(a)(7i)
[3.6.18]
y d (h) = A(o)(h)cosh(x-h) + B(al(h)sinh(K-h)
[3.6.19]
A(o)(h) =
[3.6.20]
Blo)[h) =
[of (h)cosech(ich) + CT?(h)coth(Kh)~]
eoeRTr
[3.6.21]
From this set of equations expressions can be derived for the relative changes of a^ and yd , with respect to their (measurable) values at h = » : O? [h)-O?2(oo) 12 2 , ^ = coth(rh) of2(oc)
y d .(h) jr—cosec(x-/i)-l y d (h)
[3.6.22]
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y?LZ2(h)~y?2(oo) ,
^
y£2M
3.59
(A Ah)
= coth(K-h) + —^—cosec(x-h)-l
af2{h)
[3.6.23]
The last two equations formulate the coupling between charges/potentials on one diffuse double layer to potentials/charges of the other. Together, [3.6.14-23] describe the diffuse parts of the overlapping double layers. In these equations, the coth(x7i) terms dominate at large h, i.e. at weak interaction. On the other hand, the coefficients of the cosec(K'h) terms account for the extents of induction; they are determined by the relative regulation capacities of the two double layers. This takes us to the introduction of the two Stern layers, essentially by extending [3.5.7 and 8], giving F
F' K"
y\(h) = yf(h)—^-fiW(h)
[3.6.24]
^2,1
y\{h) = y^(h) + -^^[AW(h)sinh(rh) + BW(h)cosh(x-h)] C
af[h) =
o$[h) =
^-[yf(h)-y{(h]] oT"Y~ii 2
_ A[yd ( h ) _ y ^ { h ) j
yo(h)
= y}(h) + ^
yo{h]
= yi2{h] +
[3.6.25]
2,2
^
! ^ l
[3.6.26]
[3.6.27]
[3.6.28]
[3.6.29]
where C 21 is the outer Helmholtz layer differential capacitance of double layer 1, etc. These capacitances are considered to be constant, and equal to the respective integral capacitances. The two surface charges of{h) and a^ih) are regulated according to [3.5.21]. In order to be more general we shall for the numbers of surface sites, write JV°j and JV°2 instead of [ROH] as a function of pH, pH$> and pH^ being the two points of zero charge. However, we continue to assume the two surfaces to be amphoteric. Likewise, for
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PAIR INTERACTION
needed. One approach is by using functionals, as in the Reiner-Radke approach". Another alternative21 is based on the fact that the set of equations can be rewritten in terms of the two unknowns yffh) and yijlh), implying that G[yf{h), y^iri)] has to be minimized as a function of yj^fh) and y2{h). Elaboration is beyond FICS but we shall 2)
give some illustrations of the trends, obtained via a lattice theory in figs. 3.24-28. These figures give the distance x between the two surfaces in terms of the thickness of each lattice layer 5. The circles are computed for each (discrete) number of layers, with the lines between them to guide the eye. Double layer 1 is kept at x = 0 and double layer 2 is moved towards 1 in steps, starting from x = 30S ('infinite'). Distributions are given for many values of x. The charges are in numbers of elementary charges per unit cell, but the corresponding values in |iC cm"2 are also given. We use the parameters
Figure 3.24. Hetero-interaction between two surfaces of different but constant surface charges, of =0.04 and <7§ = 0.02 elementary charges/unit cell. AG,^ ; = AGmi 2 = - 1 kT (anion adsorption). e\ l = e\ 2 = 20 , e21 = £\ 2 = 5 0 •
11
E.S. Reiner, C.J. Radke, Adv. Colloid Interface Sci. 47 (1993) 59. J. Lyklema, J.F.L. Duval, Adv. Colloid Interface Sci. (accepted, 2004), where further details can be found. 21
PAIR INTERACTION
0} = JV>1/JV'1(max)
3.61
ff2
= Nls2 /Nls2lmax)
[3.6.30a,b]
to identify the fraction of the iHp that is available for specific ion adsorption. This is a capacity factor contributing to the (chemical) regulation capacity. The electrolyte concentration is given in terms of volume fractions
and i//2 .
In fig. 3.25, other surface charge combinations are considered. Panels a and b refer to attraction. As surface 2 is now negatively charged, the anions near surface 2 become
Figure 3.25. Panels a and b; as in fig. 3.24 but o§ = -0.02 ; panels c and d, cr§ = 0 .
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PAIR INTERACTION
co-ions; they are negatively adsorbed. However, upon sufficiently close approach to surface 1, this negative adsorption becomes positive. Simultaneously a\ becomes less negative because of the attraction by surface 2. The corresponding potential profiles (panel b) follow suit. Upon approach all ^ ' s decrease, the y/2 's become less negative to change sign eventually. Here, i//^ i s the first to change, followed by y/2 and y/%, in this order, as expected. As before, the changes in y/2 s a r e stronger than those in the y/ 's. Panels c and d give interaction between a positive and an uncharged surface. In the absence of regulation the electric interaction would be zero at any h , but in reality attraction occurs because of induction. For x = 305, a2 is still (slightly) finite because of the specific anion adsorption. This excess increases upon approach. At the same time, the strong countercharge a\ becomes a bit more negative. Eventually, for x —> 8 the two countercharges merge (a\ = o2 = ~°i ) a n d are sandwiched between the two surfaces. Panel d gives the corresponding potentials; y/2 = V2 because erg = 0 . Corresponding Gibbs energies are presented in fig. 3.26. In panel a, the gradual transition from attraction to repulsion with increasing c^ is as expected. The primary source for Gel is the two surface charges, but the strength of the interaction is
Figure 3.26. Gibbs energies of interaction between two surfaces of fixed surface charge, corresponding to figs 3.24 and 25. of = 0.04 and AG,^ j = -1 kT throughout. Panel a, AG mi 2 = - 1 kT for various fixed values for 0§ (indicated). Panel b, o^ = 0.02 various values of AGJJJJ 2 (indicated); panel c, as b, but for a§ = 0 ; panel d, as b, for og = -0.02 .
PAIR INTERACTION
3.63
modulated by the (positive or negative) adsorption of anions. Panels b and c show that maxima or minima can occur, depending on the specific adsorption energies of the anions. Strong specific adsorption promotes attraction, mainly because the sum #2 + cr2 becomes very negative, but weak specific adsorption cannot withstand charge reversal; this induction means overall repulsion (panels b + c). In the middle range, there are AGmi 2 values for which attraction/repulsion at large distance is outweighed by repulsion/attraction at short distances. In panel d specific adsorption does take place at surface 2, enhancing the attraction, but from a certain value of AGmi 2 onwards the potentials y/^ become so negative that further increase of AGml 2 does not sort effect any more. Figure 3.27 illustrates the indifferent electrolyte concentration effect. In panel b, surface 2 is uncharged, but because of the relatively strong specific anion adsorption, its ^-potential must be negative. Anionic surfactant adsorption on uncharged surfaces would be representative. At long distance and low cs attraction prevails (because surface 1 together with its Stern layer is positive), but at shorter distance the interac-
Figure 3.27. Hetero-interaction; influence of the electrolyte concentration 0 = c sa j t / 55.55 . of = 0.04 elementary charges/unit cell. AG,^ 1 = - 1 kT . Panel a, og = 0.02 , A G ^ 2 = -2.83 kT ; panel b, 0 ^ = 0 , AG mi 2 = - 1.08 kT ; panel c, o§ = -0.02, A G ^ 2 = ~ fcT • O t h e r parameters as in fig. 3.25.
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PAIR INTERACTION
Figure 3.28. Hetero-interaction between two flat oxide surfaces, regulated across diffuse and Stern counter layers. pHf = 5 , pHg = 8 , <9f=6*} = <9g = 4 = l . 0 = 10~ 3 . Panel a, pH = 3 ; panel b, pH = pH° = 5 ; panel c, pH = 6 . The various combinations of AGmi j =fcT and AG mi 2=kT a r e indicated.
tion becomes repulsive because of the strong anion adsorption on both surfaces. With increasing c s the minimum deepens and is displaced towards shorter distances, to disappear for 0>2.5xlO~ 3 . When both surfaces carry a charge of the same sign (panel a) the long-distance interaction is repulsive, but at low x/ S attraction sets in, resulting In 'sticking' of the two surfaces by anlons captured between them. The maximum In the curve Is also suppressed by electrolytes. In panel c, the surfaces are of opposite sign. In all panels, at very high salt concentration, the Debye length becomes so low that the interaction Is no longer governed by the spatially confined regulation of the charges and potentials but primarily determined by the dielectrics, formed by the two Stern layers. Figure 3.27 also shows that for purely electrostatic reasons interaction curves can be obtained that are traditionally interpreted in terms of double layer repulsion and Van der Waals attraction, although with oversimplified double layer models. It may well turn out that many such interpretations must be revisited. Another qualitatively new feature is that two double layers of the same a^ can exhibit attraction at
PAIR INTERACTION
3.65
very short distance when the RC's of the two double layers differ. Classical DLVO theory can of course not account for that. Finally, we present (without further discussion) in fig. 3.28 hetero-interaction between oxides, including surface charge regulation. Obviously, this example is very important for practice. The general conclusion is that this framework offers promising and versatile prospects. 3.7 Interactions for non-planar geometryIt is obvious that electric interactions involving spherical, spheroidal, cylindrical, etc., particles are at least as important for practice as they are for flat surfaces. We only have to think of sol stability, AFM and adhesion measurements. At the same time the mathematics become increasingly difficult because the PB equation cannot be solved analytically for more-dimensional geometry. This has resulted in numerous attempts toward numerical solutions. With the growing computational powers and accepting increased computer time, these approaches can be made more accurate. On the other hand, in actual measurements one does not always need exact numerical data, if only because the particles and their surfaces are not sufficiently defined for so much detail. Practice therefore also calls for simple analytical approximations. One of the typical features of double layers around spheres is that, because of the radial nature of the field lines, a relatively large fraction is in the low-potential range, making the DH approximation better than for flat symmetry. Nevertheless, even in this approximation the PB equation cannot be solved rigorously without making further assumptions, although the numerical analysis is simpler than for the full PB equation. In the present section we use the interaction between plates as our point of departure, consider the changes that have to be implemented, present numerical results graphically and treat a number of approximations, indicating their validity ranges. 3.7a Interaction between spheres. Basic issues and definitions The general situation is defined in fig. 3.24. In the most general case the radii cij and a 2 and the oHp potentials yf and y2 , and the corresponding diffuse charges of and &2 f° r the isolated particles are different. So are the two Stern layers, but for the present purpose we shall assume these layers to have the same thickness, d. We also let d « a^,a2, implying that the curvature effect of the two Stern layers is negligible. Hence, we may transplant the available theory for regulation (sec. 3.5) without modification. Obviously, this is no longer acceptable for very small particles (nanoparticles) but then other problems pop up as well, such as failure of mean field theories. Therefore, we shall postpone a discussion of such systems until sec. 3.7g (ii). With this the shortest centre-to-centre distance between the spheres is r = a1 +a2 +2d + h
[3.7.1]
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PAIR INTERACTION
Figure 3.29. Identifying the parameters for interaction between two dissimilar spheres. The distance h' in fig. 3.29 depends on the angles Ol and 62. As before, the potentials y d occurring in the final equation are those at h = ~ , and may be identified with the corresponding £'s. For interaction 'at fixed potential' (meaning in practice 'at fixed diffuse potential') y d retains its value upon overlap, for that 'at fixed charge', the magnitude of y d increases when h is reduced. It is typical for spherical geometry that the extent of overlap depends on 0. Upon interaction, the spheres become polarized. The implication is that special kinds of regulation are needed to render y d or a^ constant, i.e. independent of 6. Although such mechanisms are not obvious we shall assume that they exist; at any rate, the following discussion has to be extended by an analysis of spatial regulation. The quantities we are after are the force, /ei
We add the subscript el to avoid confusion with the general J(h) standing for 'a function of
PAIR INTERACTION
3.67
obtained directly by a charging process, similar to that discussed in sec. 3.3b. For both approaches, we must know the potential distribution, i.e., the PoissonBoltzmann equation has to be integrated. We now need it in three dimensions. Recall from [1.5.1.20a] that for a medium of fixed dielectric permittivity the Poisson equation generally reads V2¥=-p/eoe
[3.7.2]
where V 2 y/ { = div grad yi = V • (V y/)) is the Laplacian of yr. In Cartesian co-ordinates
V
V = ff + ff + ff z
z
z
13-7.3]
dx dy dz For cylindrical co-ordinates (r, 9, z),
and for spherical ones (p, 9,
Y = -2T-\P a + ~2 k ^ + c o t 0 h ^ + ^ 2^^Jr [3.7.3b] p2 3p^ 3pJ pz\d9 )d6 p2sinz9d(pz See Lapp.7 for details about these manipulations. The r.h.s. of [3.7.2] is formally not different from the one-dimensional situation, except that y/ is now a function of three variables. Abbreviating this as y/{r), the PB equation can be written in the condensed form V2 (zy(r)) = x-2 sinh(zy(r)) [3.7.4] Equation [II.3.5.9] is the equivalent of [3.7.4] for flat geometry. This is a non-linear partial differential equation, for which no analytical solutions are known, not even in the DH approximation, V2(zy(r)) = K-2zy(r)
[3.7.5]
3.7b Methods and approximations for isolated particles There are a number of numerical approaches for solving [3.7.4], based on multipole expansions1 2 , expansions in terms of double layer potentials ( y m , y d ) or of their hyperbolic functions31, via some finite element (difference) scheme 4 5 6 7 8 ) or 11
A.B. Glendinning, W.B. Russel, J. Colloid Interface Set 93 (1983) 95. J.W. Krozel, D.A. Savillc, J. Colloid Interface Sci. 150 (1992) 365. 31 H. Ohshima, T. Kondo, J. Colloid Interface Sci. 122 (1988) 591. 4) N.E. Hoskin, S. Levine, Phil. Trans. Roy. Soc. (London) A248 (1956) 433, 449. 51 L.N. McCartney, S. Levine, J. Colloid Interface Sci. 30 (1969) 345. 61 J.E. Ledbetter, T.L. Croxton, and D.A. McQuarric, Can. J. Chem. 59 (1981) 1860. 71 R.W. Bowen, A.O. Sharif, J. Colloid Interface Sci. 187 (1997) 363. 81 P. Warszynski, Z. Adamczyk, J. Colloid Interface Sci. 187 {1997) 283. 2)
3.68
PAIR INTERACTION
still otherwise . We consider these treatments beyond the scope of FICS, but shall present some results in sec. 3.7d. In the present subsection we shall discuss the double layer around isolated particles, and some approximations. Let us first consider the relation between y d and &1 for isolated spheres. We need this relationship for converting £'= y d into (7d(h = °o). To this end, relationship [3.3.14] can no longer be applied. Numerical results for the charge and potential distribution
in double
layers
around
spheres
can be found
in Loeb et al's
tabulations21; they include the relation between cfi and y d . This relation depends on c , z and a : the stronger the curvature, the more charge can be accommodated at a given potential y d ; see fig. II.3.12. The best analytical expression is Loeb's equation
m.3.5.56] a ^ - ^ L i n h f ^ V 4 ^ ' 2 ^ 4 ' ! Ka *" L v2 ) J
[3.7.6]
The accuracy is quite good (see table II.3.2). For large Ka the second term on the r.h.s. drops out, and [3.3.14] is approached. In the DH approximation
.
[II.3.5.51]
eneRTyd ,
ad = — ^
^
eneKRTyd (
^—(l + ica) =—° Fa
v
;
F
\ \
a_ i + _!_ V Ka)
[3.7.7]
which for large Ka reduces to
. eneKRTyd crd=—^ iF
[3.7.7a]
and for small Ka to . ene RTyd a6 =—2 2_ Fa
[3.7.7b]
Equations [3.7.7a and b] have a very limited applicability range. Of course, in [3.7.6] yd = y d (h = °o) . For any formula for cfi the total charge in the diffuse part of the double layer is given by g>d =47r(a + d)2crd
[3.7.8a]
Similarly for the surface charge g»° = 4/ra2t7°
[3.7.8b]
Let us now consider Gibbs energies. Recall that the Gibbs energy of a single sphere or a pair of interacting spheres contains (1) the chemical energies involved in adsorption/desorption of charge-determining ions (2) the same for ions specifically adsorbing at the iHp, (3) the electric energies of the surface, the Stern layer and the
11 2)
A.I. Shestakov, J.L. Milovich, and A. Noy, J. Colloid Interface Set 247 (2002) 62-79. A.L. Loeb, J.Th.G. Overbeek, and P.H. Wiersema, The Electrical Double Layer around a
Spherical Colloidal Particle, M.I.T. Press (1960).
PAIR INTERACTION
3.69
diffuse layer and (4) the entropy contribution caused by the uneven distribution of ions. When the charging process is carried out isothermally and reversibly, this entropic part is automatically included. In the plane-parallel case (sec. 3.3b), we covered all of this, restricting ourselves to purely diffuse double layers. For the Gibbs energy of an isolated Gouy-Stern layer, we have derived expressions in sec. II.3.6f, [3.2.2] in the present chapter being the most important result. In sec. 3.5a,b we added the Stern layer after having dealt with the diffuse part, which, in that case could be treated rigorously. The specific adsorption energy entered through [3.5.9 and 10]. A more general, but necessarily more abstract, approach would be to let the diffuse and Stern layer charge themselves simultaneously, leading to the Gibbs energy of interaction at any h as a functional of the potential distribution. In this way, regulation automatically enters the picture. However, as much research so far has been conducted on the purely diffuse part, and because of the easier mathematics we shall follow the previous route and treat the diffuse part separately. The Gibbs energy of an isolated flat diffuse double layer is given by [3.4.4]. Recall that it is the result of a combination of ad/desorption and electric terms and given in J m" 2 . The equivalent for an isolated sphere is Wd
AG"H= f Q^di^ =
yd
A7la
2
RT
[ t^'dy*
o
[3.7.9]"
o
In the constant (diffuse) charge case there is only electrostatic work and
AG°(°o) = - J ^ ' d Q d ' = -
trd
2
RT
o
J yd'dgd'
[3.7.10]
o
where the primes indicate variable quantities during the charging. Note that cfi and yd have opposite signs, so [3.7.9] is negative, whereas [3.7.10] is positive [Qd = -Q°). The former double layer forms spontaneously; the latter does not. As AG°(<>o) is a function of state it does not depend on the way the final state is reached; so when the final y d , a11 set is the same after completion of charging via [3.7.9] or via [3.7.10] or any other route, the two AG°'s are also identical. We note that in the DH approximation where charge and potential are proportional to each other, we would obtain AG°M = - ~ Q V =-2KalRT^<^
[3.7.11]
and
Note that AG°(H is the interfacial excess Gibbs energy for one isolated particle whereas GJ,°'(h) stands for the electrical Gibbs energy of interaction at constant charge between two particles at shortest distance h , with respect to h = °° , i.e. with respect to 2AG°(°°|.
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AG g M = lg> V 2
= 2Ka2RT p
ydgd
[3.7.12]
for the constant charge and constant potential case, respectively. Having [3.7.6] available, we can carry out the integration in [3.7.9], leading to
AG«M = - ^ ^ ^ ^ c o s h f ^ ] , A l o g c o s h ^ l - l ]
[3.7.13]
For a Gouy-Stern layer with d«a we may simply add the two capacitance terms on the r.h.s. of [3.2.2], after multiplication by Ana2 . The result is
Iq
q J [3.7.14]
327ra2cRT[
,(zyd"|
4,
, f zyd ^ 1
cosh -2— + — l o g cosh -2— - 1
r
[
[ 2 J ra
[ 4JJ
Once we have established the Gibbs energy for the two interacting spheres of fig. 3.24 we have to subtract either [3.7.13 or 14] once for a = cij and once for a = a2 to obtain Ggj (fi). For the 'force method' such a subtraction is not required. 3.7c The Deryagin approximation In the treatment of the interaction between flat surfaces, we have encountered a number of recurrent approximations, including the linear superposition approximation (LSA) and that of Debye and Hiickel (DH). These approximations are also used for spherical geometry. Typical is the Deryagin approximation, which is popular in converting analytical equations for flat surfaces into those for spherical symmetry. We already introduced it in sec. 1.4.6a to obtain the Van der Waals interaction between spherical particles if that between flat plates is known. See fig. 1.4.13. The basic idea is to approximate the total interaction between, say two spheres, as the integral over a set of infinitesimal parallel rings, where the Gibbs energy of interaction for each ring obeys a known equation for flat double layers. Also, see fig. 3.30. The approximation is rigorous when there is no lateral interaction between the rings. For electrostatic interaction, this means that the field lines should run parallel to the OjOg axis. This is exact for h' = h, but the condition becomes increasingly poorer in the direction h' ^hY^>h2, see the dashed curves in fig. 3.30. (Note that the field lines emanate normal to the oHp's.) On the other hand, with increasing h' the absolute contribution to the Gibbs energy decreases, so the inaccuracy is blunted. Briefly, the Deryagin approximation is expected to work well for low Kh and large KOL (big particles at short distance)11.
11
G.M. Bell, G.C. Peterson, J. Colloid Interface Set 41 (1972) 542.
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3.71
Figure 3.30. Illustration of the Deryagin approximation for two identical spheres. h'{$) is the local distance. The dashed lines represent field lines.
White1' has elaborated this for arbitrary shapes and Bhattacharjee and Elimelech2' did so through a surface element integration. Here we give the result for two spheres of radii a^ and a 2 G.,(h) = 2rc
aiCt2 1
f Ga_.(h')dh' 2
[J]
[3.7.15]
h
with, as sub-equations, Gel(h) = 7ca| G ael (h')dh'
[3.7.16]
h
for two identical spheres and Gel(h) = 27ca J G ael (h')dh'
[3.7.17]
h
for a sphere and a plate. How good is the Deryagin approximation? Carnie et al. have systematically compared it for the force against exact results obtained numerically under a number of conditions and other assumptions31 such as that of constant potential or charge, the LSA approximation and the DH assumption. One of their results is redrawn in fig. 3.31. For large distance, the Deryagin approximation overestimates the force, whereas it underestimates at low Kh . Below x7t ~ 2 it remains within 10% of the exact results. For larger Ka (not shown) the approximation becomes better. Carnie et al. also studied the quality of the LSA and DH assumptions. Regarding the latter it is noted
11
L.R. White, J. Colloid Interface Set 95 (1983) 286. Sec also R.J. Hunter, Foundations of Colloid Science, 2nd ed., Oxford Univ. Press (2001), sec 11-5. 21 S. Bhattacharjee, M. Elimelech, J. Colloid Interface Set 193 (1997) 273, S. Bhattacharjee, M. Elimelech, and M. Borkovech, Croat. Chim. Ada 71 (1998) 883. 3) S.L. Carnie, D.Y.C. Chan, and J. Stankovich, J. Colloid Interface Sci. 165 (1994) 116.
3.72
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Figure 3.31. Relative error (in %) of the Deryagin force at different yd(cxj) (indicated). Left, constant y d , right, constant a^ ica = 5 . Redrawn from Carnie et al. (loc. cit).
that, although it is better than for plates, it is less so for constant charge than for constant potential because in the former case the potentials have to increase upon overlap, eventually surpassing the DH-validity range. Finally, it may be noted that in the Deryagin approximation the Gibbs energy and the force can always be written as the product of two functions J(a) x g(h). In the exact limit this is no longer the case. 3.7d Numerical results Over recent decades, numerous attempts have been made to obtain precise numerical results for a variety of systems and conditions. Of these, we select some results of the Australian School1' which appear to be the most comprehensive. Figure 3.32 gives illustrations for two spheres of equal radius (tea = 10) at various y d and y d for attraction and repulsion and both for y d constant and o^ fixed. Note that Gel /(y d ) 2 is plotted, so the absolute values for y d = 2 are four times as high as those plotted for y d = 1. The fact that G el /(y d ) 2 decreases with y d means that the repulsion increases less than quadratically with the potential, as predicted in some of the approximated expressions, for instance in [3.7.20-22]. The pronounced maximum for hetero repulsion (panel a) has been encountered before for flat geometry (fig. 3.26 and 27). Repulsion at constant o*1 (panel b) does not show this feature; it is stronger than that at constant y d . The attractive case also exhibits a maximum for constant o 4 (panel d). In addition, this figure illustrates the (lack of) precision of the DH approximation; it is the limit for y d -> 0. The trends in the force are similar to those for the Gibbs energy.
11
J. Stankovich, S.L. Carnie, Langmuir 12 (1996) 1453.
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3.73
Figure 3.32. Hetero-interaction Gibbs energy between two spheres of equal radius. Given is Ggj/lyf1)2 in units of eeo(RT)21KF2 as a function of distance, ra = 10. Panels (a) and (b), i)2 Iy^ = 3 Irepulsion); panels (c) and (d), y^/yf = ~3 (attraction). The value of y d is given. Drawn curves, DH approximation. (Redrawn from Stankovich and Carnie, loc. cit.).
Figures 3.33 and 34 illustrate the radius effects. Figure 3.33 shows the repulsion between relatively small ( KCL = 1) equal-sized spheres. The Gibbs energies are smaller than for large tax and roughly proportional to a, as called for in the Deryagin approximation, see [3.7.16]. Figure 3.34 gives results for spheres of differing radius; panel (b) is the limit for a sphere and a plate (adhesion limit). The Stankovich-Carnie work contains various other comparisons, including that with the Deryagin approximation. We suffice by referring to fig. 3.31. The quality of the DH approximation (drawn curves in figs. 3.32-34) is paradigmatic for the HHF theory ([3.6.11 and 12] for two plates, and [3.7.23-25] to follow for spheres. As the last Illustration of numerical analyses, we reproduce an obvious elaboration, viz., by combining Verwey and Overbeek's exact results for plates with the Deryagin approximation. Recall figs. 3.5-7 and, essentially, our [3.3.16]. Also see 1 '. Such an
V-O, their fig. 35, p. 141. Also reproduced by Overbeek in Colloid Science, Vol. I, Irreversible Systems (H.R. Kruyt, Ed.), Elscvier (1952) 259.
3.74
PAIR INTERACTION
Figure 3.33. Hetero-interaction Gibbs energy for two small spheres ( r a = 1). Repulsive case. Otherwise symbols as in fig. 3.27.
Figure 3.34. As in fig. 3.28 but now for unequal radii.
elaboration was given long ago by Honig and Mul11, who extended the V-O model to interaction at fixed cr*1, essentially as in sec. 3.4a. They write Gel as Gei(h) =
^ ^
,3.7.18]
S ( h )
and tabulate S(h) at constant potential and constant charge for various values of yd (oo). For aqueous solutions at 25 ° z2Gp, (h) .. s — = 4.751xlO n S(h) a
. [Jm- 1 ]
[3.7.18a]
which is represented in fig. 3.35 for interaction at constant y d and constant a^ . The curves are plotted for various yd {h = °°) values which for the constant potential case remains fixed upon overlap whereas it increases for interaction at constant charge. All
" E.P. Honig, P.M. Mul, J. Colloid Interface Set 3 6 (1971) 258.
PAIR INTERACTION
3.75
Figure 3.35. Electric contribution to the homo-interaction Gibbs energy between two spheres for different yd(oo). Exact curves for flat plates converted, using the Deryagin approximation G^f', GJj' . For two spheres of different radius replace a by 2aja 2 l{ax + a 2 ) and for a sphere and a plate by la; hi 2 is the half distance between the two outer Helmholtz layers and a is the particle radius plus the Stern layer thickness. Dashed curve, [3.7.19]. Inset: the low potential case.
curves can be described by the product of functions /(y d ) and g(Kh), as is typical for homo-interaction in the Deryagin approximation. The results are as are as good as the Deryagin approximation and may be compared with their 'flat' counterparts, figs. 3.57 (for constant potential), fig. 3.10 (for constant charge) and with fig. 3.32 (for both). Note that the scalings of these figures are different. The present elaboration again predicts that repulsion at constant charge is stronger than at constant potential, but that the difference is not large and sometimes virtually absent, viz. at large Kh and at high y d . The curve for y d = 10 , where no difference between G(vl and G(o) is observable any more, is only of academic value, because diffuse potentials of about 250 mV are rare; they are only measurable in very dilute solutions, say of 10~4 M (1-
3.76
PAIR INTERACTION
1) electrolyte, where x"~30nm" 1 so that only the region beyond the drawn abscissa axis range is accessible. In practice, therefore, in all experiments for which x"fa>.l the difference between interaction at constant potential and at constant charge is insignificant. In fact, the limitation of fig. 3.31 is not only determined by the quality of the Deryagin approximation but also by the processes which occur in the two Stern layers. It is not easy to imagine regulation processes which, upon interaction, lead to either y d or cfi exactly independent of 9 in fig. 3.24. Note that under coagulation conditions, c — lO^M for (1-1) electrolytes, r—lnm" 1 this regulation is relatively important and interesting, see the inset of fig. 3.30. The function S(h) in [3.17.18 or 18a] differs between interaction at constant potential and constant charge but is independent of a . So, it is concluded that G^' and G^' are both proportional to the radius. This is also the case for the Van der Waals attraction at short h, see [1.4.6.10]. All of this provided the Deryagin approximation is valid. So we see that at those distances where the sum of Gel and G vdw is attractive/ repulsive the absolute value of this sum increases with radius. We shall come back to this feature in fig. 3.44. Finally, the inverse proportionality to z 2 is noted. In fact, this also is the case for interaction between flat plates because in [3.3.17] J[ym,yd) is independent of z. The consequence is that in (2-2) electrolytes the repulsion is 4 times as low as for (1-1). Together with the strong decrease of y d with increasing z this accounts for the very strong dependence of the critical coagulation concentration on the valency of the counterion, the so-called SchulzeHardy rule. 3.7e Analytical expressions A plethora of partially valid formulas has enriched our scientific journals. Notwithstanding their limited range of applicability, they may serve well because they assist us to assess trends quickly, at least semi-quantitatively". Moreover, few practical systems are so well-defined that they deserve sophisticated numerical interpretations. In the present subsection, we shall briefly review a few important equations, referring to Appendix 2 for other examples. Perhaps the best expression for constant yd is obtained from [3.3.28a] and [3.7.15]. For two different a's, el
K
(aj+a 2 )
= —=—J-3 2
K (al+a2)
{ 4 ) rJ
\ + e-Kh
tanh 2 \=2— lnri + e - ^ l
{ 4 J
L
J
[3.7.19]
which for al = a2 = a reduces to 11
See inter alia. J.Th.G. Overbeck, J. Chem. Soc. Faraday Trans. (I) 84 (1988) 3079.
PAIR INTERACTION
GW
3.77
=64nCJTa t a n h ^ ^ l ln[l + e-^]
[3.7.19a]
For sake of illustration, in fig. 3.35 G ^ according to [3.7.19] is included for two values of y d . We see that for low y d the approximation is excellent, but that it overestimates systematically at high y d . The low-potential variant of [3.7.19] is G
,)
=
8,ZSa2cRT icz{al+a2)
2 tar1 + e H r i l ] x
'
L
[3 y 1 9 b ]
J
which for a t = a 2 can also be written as G^) = 2neoea\^-\
mfl + e"^]
[3.7.20]
or as G^} =27if o fa((/ 1 ) 2 ln[l + e-' rh ]
[3.7.20a]
or, introducing the distance r between the centres of the spheres, G|f = 2 7 i £ o f a ( ^ f l n [ l + e-K-[r-2(a+d»]]
[3.7.20b]
where d is the thickness of the Stern layer. For a : * a 2 and dj * d2 these expressions have to be modified accordingly. Equations similar to our [3.7.20] have been known for a long time1'2 . In passing, Wiese and Healy41 showed that in the low potential regime G^1 =G^] -2neo£a(y/dfln[l-e-2Kh]
[3.7.21]
Combination with [3.7.20] gives G^f =-27te o fa((/ 1 ) 2 ln[l-e-' r ' 1 ]
[3.7.22]
Comparable analytical expressions have been proposed by Ohshima . Hetero-interaction between two spheres has already been addressed by Derjaguin at a Faraday Discussion61. The unwieldy presentation was not conducive to much subsequent work. The present primus inter pares equation is that by Hogg, Healy and Fuerstenau (HHF)7), derived using the Deryagin approximation in the DH-limit. We already presented the flat plate case in [3.6.11 ]. Integration according to [3.7.10] gives
11
B.V. Deryagin, Izv. Aka., Nauk SSR, Ser. Khlm. 5 (1937) 1153. B.V. Derjaguin, Ada Physlcochim, URRS 10 (1939) 333. 31 V-O, chapter IX. 41 G. Wiese, T.W. Healy, Trans. Faraday Soc. 66 (1970) 490. 51 H. Ohshima, J. Colloid Interface Sci. 225 (2000) 204. 61 B.V. Derjaguin, Discuss. Faraday Soc. 18 (1954) 85. 71 R. Hogg, T.W. Healy, and D.W. Fuerstenau, Trans. Faraday Soc. 62 (1966) 1638. 2)
3.78
PAIR INTERACTION
61
" 4F2(a,+a2) ^ 1
'
+(y
2» } x
2
[( yi d ) 2 + (yd)2
[3.7.23]
{\-e-«h)
l
;
J
For yf1 = yJj a n d Qj = c^ t m s equation reduces to [3.7.20]. The corresponding Gibbs energy at constant charge was derived by Usui11 following a procedure similar to that leading to [3.7.21]. His results are GW = G W
- £ °!,"; a 2 ( R T ) 2 {tyf f + (yd)2}ln(l-e-^)
1
2
L( yi d ) 2 +(y^) 2
[3.7.24]
[3.7.25]
(l-e-**)
l
'J
The difference between G ^ and G^ is only in the sign of the last logarithmic term. Given our experience with [3.6.13], there are reasons for distrusting the validity of [3.7.25] at low Kh . The trend G(^{h) is similar to that for plates, fig. 3.23. The HHF equation was improved by Ohshima et al.2 by adding correction terms of order (ra)" 1 . These corrections are therefore small at large KCL , where the unmodified HHF equation is a good approximation. Three other half-empirical improvements are refs. 3451 . The last one of these will be included in appendix 2, sec. d.
3.7/ Intermezzo. Comparison of decay functions So far we have met a variety of distance-dependencies in analytical expressions for Gel and /7 el . For a rapid assessment of the steepness of the decays, and other properties, we have collected some of them in fig. 3.36. In table 3.1, the various functions are made explicit and are compared. To highlight the trends /(0) was set unity where possible. For curve 2, a prefactor was needed to achieve this; it equals (In2)~1. From the figure it is seen that the most common function 1 is the steepest. Slightly improved curves yield a less rapid decay. Functions 5 and 6 diverge at low Kh . This is typical for interaction at fixed charge, which becomes infinitely high for h —> 0. Curve 4 is the only one with a bending point. Such a bending point was also observed for the midway potential at low yd, see fig. 3.3. Function 4 'exaggerates' this trend. We note
11
S. Usui, J. Colloid Interface Set 44 (1973) 107. H. Ohshima, D.Y.C Chan, T.W. Healy, and L.R. White, J. Colloid Interface Sci. 92 (1983) 232. 31 G.X. Luo, H.P. Wang, J.Z. Guo, and J. Jin, J. Colloid Interface Sci. 201 (1998) 244. 41 A.V. Nguyen, G.M. Evans, and G.J. Jameson, J. Colloid Interface Sci. 230 (2000) 205. 51 J.E. Sader, L. Carnie, and D.Y.C. Chan, J. Colloid Interface Sci. 171 (1995) 46. 2)
PAIR INTERACTION
3.79
Figure 3.36. Comparison between J(KH) functions. For key, see table 3.1.
that [3.3.41] was derived for the DH approximation. For hetero-interaction and spherical geometry, the distance-function becomes radius-dependent, and such simple decay functions cannot be used. 3.7g Miscellaneous Approaching the end of the treatment of electric pair interactions, let us take stock and flag some remaining problems that deserve attention but for which the available space is too limited for a detailed treatment. (i) Other geometries; cylindrical particles. The principles and methods of numerical and analytical elaborations are similar to those for spheres. The new feature is
3.80
PAIR INTERACTION
the orientation-dependency, which invites a number of intriguing questions: Do charged cylinders align? Form strings? What is the influence of end segments and of chain rigidity? In sec. II.3.5f the diffuse part of the electric double layer around long rigid isolated cylinders was treated. In the DH limit, cJ^iy^) can be written in terms of Bessel functions; for higher potentials numerical procedures are required. Probably the oldest elaboration goes back to Sparnaay
who considered interaction
at constant potential between crossed and parallel cylinders (of given length £). He used the Deryagin approximation which, for geometrical reasons, is better for cylinders than for spheres. His starting equation is [3.3.18]. For crossed cylinders the resulting G^ is independent of £ and reads
G(T)(h)=12^Tatanh2^je_rh
(J)
[3?26a]
(J)
[3.7.26b]
whereas for the parallel orientation GWW
=
?^
t a l
tf^)e-*
Equation [3.7.26a] may be compared with [3.7.19a]; there is a great similarity (also see fig. 3.31, curves 1 and 2), the main difference being that the repulsion between two crossed cylinders is about twice as large as that between two spheres of the same a . After Sparnaay, several authors attacked the same problem. Interaction between a set of identical parallel cylinders has been numerically elaborated by Brenner and McQuarrie21 in the DH approximation, accounting for charge regulation. James and Williams3' extended this, also numerically, to cylinders of different radii, integrating the aforementioned Bessel functions, thus improving upon the DH approximations and Ohshima4' gave explicit analytic expressions for the DH limit. A further generalization was presented by Harries51 who solved the full PB equation numerically for constant y d , cfi , or in-between boundary conditions. By letting the radius of one of the cylinders go to infinity, the interaction between a cylinder and a plate could be considered. Ohshima6' gave analytical expressions for the same systems in the DH approximation, to which we shall return in app. 2 sec. e. He did not need the Deryagin approximation. Gu7' computed the interaction between a cylinder and a sphere in the DH-Deryagin approximation, presenting numerical results for the two radii-depend-
11
M.J. Sparnaay, Rec. Trav. Chem. Pays Bas 78 (19591 680. S.L. Brenner, D.A. McQuarrie, J. Colloid Interface Sci. 44 (1973) 298. 3) A.E. James, D.J.A. Williams, J. Colloid Interface Sci. 79 (1981) 33. 41 H. Ohshima, Colloid Polym. Sci. 274 (1996) 1176. 51 D. Harries, Langmuir 14 (1998) 3149. 61 H. Ohshima, Colloid Polym. Sci. 277 (1999) 563. 7) Y.G. Gu, J. Colloid Interface Sci. 231 (2000) 199. 21
PAIR INTERACTION
3.81
encies. So did Ohshima11. Adamczyk and Weronski2 analyzed the interaction between oblate and prolate spheroids and between such bodies and a flat plate. Interaction between spheroids has also been described by Hsu et al.3). Phillips' work on moreparticle interactions includes an analysis of sphere-cylinder interaction for constant charge and constant potential and a simplified regulation model41. We shall not discuss still other geometries (hollow surfaces, cavities, discs, spheroids, etc.) but briefly return to cylinders in our chapter on polyelectrolytes (V.2). In all studies mentioned, end (or 'cap') effects were neglected. (ii) Nanoparticles. Extension of the previous numerical work to systems with strong curvature offers no technical problems; in fact, we already included an illustration in fig. 3.28. With respect to analytical solutions, the drawback that the Deryagin approximation is invalid for low ra is offset by the improved validity of the DH-linearization. Verwey and Overbeek51 already elaborated this. They presented Gibbs energies for the homo-interaction between two spheres of identical radius at constant y d and at constant cf1 in tabular form, down to Kh = O.\. The limit for a small particle, interacting with a plate takes us to the adhesion limit. There is, however, a point of principle, and that is that in the low Kh limit the assumption of smeared-out electric fields starts to fail. Very small particles carry only a few elementary charges and when the distances between these charges become comparable to h we need more advanced theories in which ion correlations have to be accounted for (in the line of sec. II.3.6b). We can argue that this lowers the repulsion at given h, reasoning that on top of the PB mean field there are now density fluctuations. These fluctuations are attractive, because attractive configurations are favoured over repulsive ones. The decay is steeper than for the mean field part. (We encountered a similar issue with the interaction energy between two freely rotating dipoles according to Keesom: although the electric field of a dipole decays as r~3 , the attraction scales as r~6 , see sec. 1.4.4c.) Fluctuating charge attractions have been statistically analysed long ago in a seminal paper by Kirkwood and Schuhmacher61. (iii) Further discussion of the surface condition. Following common usage, we have distinguished between interaction at fixed potential (y d , that is), fixed charge (a^ ) and (spatial + planar) regulation. Generally speaking, the physico-chemical processes underlying regulation, including their dynamics, deserve more attention. Particularly for low Kh the details of these processes certainly have their influence on the magnitude and, for hetero-interaction, on the sign of the force. The implication is that 11
H. Ohshima, J. Colloid Interface Set 231 (2000) 199. Z. Adamczyk, P. Weronski, Adv. Colloid Interface Set 83 (1999) 137. 31 J.P. Hsu, B.T. Liu, Langmuir 14 (1998) 5383. 41 R.J. Phillips, J. Colloid Interface Sci. 175 (1995) 386. 51 V-O, their tables XV-XX, pages 152-155. 61 J.G. Kirkwood, J.B. Schuhmacher, Proc. Natl. Acad. Sci. USA 38 (1952) 863. 21
3.82
PAIR INTERACTION
the extrapolation of most interaction equations to very short distance is risky. Compounding with features like surface roughness is likely. It is here that much further experimental research is waiting and challenging. It is perhaps appropriate to repeat the caveat that our treatment is restricted to double layers in which charges and potentials may be considered as smeared out. Our approach to regulation (sec. 3.5a-c) is based upon a simple model and requires only a few parameters to cover a wide range of interaction modes. Other authors, facing experiments that were at variance with any of the usual diffuse layer overlap models have proposed alternatives, including mosaic-patterned charges, liquid structure-mediated interactions, or surfaces covered by charge-penetrable layers, contacting the bulk through a kind of Donnan-type equilibrium. The problem with all of these Interpretations is that it is difficult to establish whether a certain proposed mechanism is unique. All of this adds to the challenge. (iv) Surface roughness. Surface roughness remains one of the most important and impervious features of pair interactions. It is important because surfaces are rarely ideally smooth (Fresnel surfaces are rare) but the problem defies easy solution because the nature of the roughness Is rather esoteric. Probably for these reasons, classical texts on colloids pay little attention to it, and neither do we! By surface roughness, we mean all deviations from the Fresnel nature, including asperities, ridges, cavities, and dislocations. More than once, we have had to deal with this phenomenon, for instance in gas adsorption (sec. II. 1.6) and wetting (sec. III.5.5), where it led, respectively, to hysteresis in the adsorption isotherm and in the contact angle. In pair interaction the problem is that when two rough particles approach each other the distance h is not uniform. Only if the morphology is known exactly, can we in principle compute the overall interaction Gibbs energy. If we do not know the morphology we can only model the surface and take some average, which is easier said than carried out. What is a good model? A collection of cones, or spherical caps, spikes, ridges or other protrusions? How are these distributed over the surface? , and how should we average? The extremities of asperities contribute more than their bases, so would one big one blank all the other, smaller ones? Or should we generalize the issue by postulating the surface as having a fractal nature, so that we can use part of the corresponding formalism? What physical reason would a surface have for being fractal, given its synthesis and/or history? From the experimental side, there is not much concrete information either. In various interaction studies, even in those carried out with 'well-defined' surfaces, deviations from DLVO behaviour are often observed at short h that cannot be accounted for by choosing a more advanced interaction formula or by adjusting the regulation parameters. For lack of more information, these deviations are then sometimes attributed to roughness. However, it is mostly not easy to quantify this because the interaction Gibbs energy (or force) and h are both affected. (What is the
PAIR INTERACTION
3.83
zero h reference in AFM measurements for rough surfaces?) Perhaps the only general remark that can be made is that the extent to which roughness has to be accounted for is a matter of scale: when the characteristic size of a surface regularity is (, roughness can be ignored for C «h. For (»h and t»K""1 we simply have to consider the local curvature for which in principle numerical solutions (as in sec. 3.7b and d) or general approximations (as in sec. 3.7c) are available. In all in-between situations, further model studies and interpretations are wanted. For references, see 1! where several older studies are mentioned. In the same breath, chemical surface heterogeneity may be mentioned. In practice, this usually means that the two surfaces contain patches of different y d and/or different regulation capacity. Particle anisotropy may lead to a torque between them. Such patches have also been invoked sometimes to account for deviations from DLVObehaviour at short distance of approach. One of the new features is the extent to which, upon close approach, lateral charge transport in the Stern layer can take place. Problems in evaluating this are similar as for morphological heterogeneity. The two issues may be compounded. For example, the tips of asperities are, chemically speaking, high-energy loci, where counterions may be more strongly bound, leading to a concomitant local decrease of regulation capacity (i.e. a reduced capacity to absorb more counterions, originating from the diffuse part). For a recent elaboration on the HHF level, see ref.2). (v) Influence of the dielectric permeability of the particle. So far, we have tacitly assumed that fP = 0 , so that the Poisson-Boltzmann equation inside the particle reduces to V 2 yP=O
[3.7.26]
This does not mean that the potential inside the solid is zero, but the field is (gradyP=O). Non-zero fP means that the solid particle can be polarized. Semiconductors and gel particles may behave in that way, and so do W/O emulsions and many biological systems. There is a certain connection with issue (iii) in that (part of) the polarization by an incoming particle can be absorbed by the Stern layer, or a layer replacing it. Carnie and Chan31 verified that fP = 0 is often a good approximation for interaction in the ich range that is relevant to stability studies. (vi) Surface 'softness'. Even if the particle surface does not allow penetration of ions, a complication may be that it is mechanically soft, i.e., it has a low Young's modulus. The implication is that in that case interaction also contains elastic work, 'soft' materials may include polymer colloids and proteins.
11 21 3)
S. Bhattacharjec, C-H. Ko, and M. Elimelech, Langmuir 14 (1998) 3365. D. Velegol, P.K. Thwar, Langmuir 17 (2001) 7687. S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci. 155 (1993) 297.
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PAIR INTERACTION
(vli) Chemical forces. At the conclusion of sec. 3.7 it is good to reconsider the basic thermodynamic equation [3.2.6]: any change in the surface excess of any component in the double layer (ions, surfactants, etc.) upon interaction contributes to Gel and /7 el . When the statement is made that 'a layer of adsorbed fatty acids acts as a barrier against aggregation', there is not necessarily a reason for thinking in terms of a mechanical barrier. Resilience against desorption, and hence keeping their charges in place is equally well possible. 3.8 Other contributions to the pair interaction In this section we address ubiquitous non-electric contributions to pair interactions. This category includes Van der Waals and solvent structure-mediated forces. Interactions that occur only for specific systems will be treated elsewhere; in particular, magnetic interactions in sec. 3.10c, and steric ones in chapter V.I. As to the notion of 'non-electric', we repeat that this term is for convenience's sake and in line with common use. It is not implied that the interpretation of Van der Waals forces does not have an electrical origin: polarization of individual atoms or molecules (in the microscopic theory) or polarization of condensed media (in the macroscopic approach). What we have denoted as 'electric' refers essentially to the part caused by the electric double layers, i.e. by free ions, as defined in sec. 1.5. la. Van der Waals interactions have already been discussed in sec. 1.4.6-8, solvent structure-mediated ones in sec. 1.5.4 and, in passing, in sec. III.5.3b. Now we shall briefly summarize and update this matter, anticipating combining of these interactions with Gel in sec. 3.9. 3.8a Van der Waals (dispersion) interactions There are two ways of describing Van der Waals interactions between colloids and macrobodies, the microscopic and the macroscopic approach. The former dates back to Kallmann and Willstatter, who were the first to realize that London-Van der Waals forces, acting between isolated molecules, could be summed to obtain Van der Waals forces between colloids. The basic assumption was the additivity of London pair interaction energies. We now know that this is a fair approximation, predicting the final results within a margin of about 10-20%. This idea lends itself very well to elaboration, as shown in the nineteen thirties by Hamaker and de Boer. Their work, together with the DLVO double layer interaction analysis has boosted the development of our insight into the stability of lyophobic colloids. The macroscopic alternative was developed in the nineteen fifties by Lifshits et al. It is based on the correlation between electric fluctuations of two macroscopic phases. As compared to the microscopic theory it is academically more satisfying because it has a higher level of abstraction. However, implementation is not so easy because for quantification dielectric dispersion data are required, which are not always available.
PAIR INTERACTION
3.85
Because of the macroscopic interpretation, people sometimes talk of dispersion forces or Lifshits-Van der Waals forces. We shall generally call them Van der Waals forces, irrespective of their interpretation. Next, we have to quantify all of this. It follows from the foregoing that we have, in principle, three approaches at our disposal: computation via the microscopic theory, computation via the macroscopic theory and experimentally. All of this has been elaborated in chapter 1.4 and appendix 1.9. Now we shall give an update. Just as is the case for the electric interaction between colloids, the treatment of Van der Waals forces can be carried out on different levels. The simpler approach starts with the finding of the microscopic theory that the (Gibbs) energy of interaction and the disjoining pressure can generally be written as a product of a material constant (the Hamaker constant AJ.-J. for the interaction between particles or macrobodies, i and j , across the intervening medium k) and a function of geometry and distance h , i.e., G vdw (h) =-A l j ( k)/( a ' h '
[3.8.1a]
^VdW^1' = -Aj(k)/'( a ' h '
[3.8. lb]
where the functions /
and / ' stand for the geometric factor for two spheres; for
other geometries these functions can become multi-parametral, although the product with the appropriate Hamaker constant remains valid. Implementation of this approach requires confirmation of the functions /
and / '
by the macroscopic theory and knowledge of Ai((k) for which table I.A9 is available; extensions and updates appear regularly in the literature, see sec. 3.8b. Computation of f
or f
is basically not difficult, although the mathematics becomes more involved
for other than simple geometries. It follows from the macroscopic theory that often [3.8.1a and b] remain good approximations. When that is the case, Hamaker constants can also be computed macroscopically. Macroscopic theory also helps to establish the range of validity of [3.1.8a and b]. So, strictly speaking, AJ.^J can assume different values for different
h so that it is more correct to speak of a
Hamaker function; only over a range of not too large h is it a constant. A typical deviation occurs at large h , where Gvdw{h) decays more rapidly than at intermediate distances. Microscopic theory cannot account for this deviation, but Casimir has predicted the long distance trend, the so-called retarded Van der Waals forces. All of this has been discussed in some length in sees. 1.4.6 and 7. While continuing with the microscopic approach, we must remain conscious of all of these alternatives and restrictions. In the microscopic theory, as derived by Hamaker and de Boer, the interaction is purely energetic, but there is some entropy 'hidden' in the displacement of molecules and in the influence atoms exert on the polarizability of other atoms. In the macro-
3.86
PAIR INTERACTION
scopic theory, the entropy is more explicit in the distribution of fluctuations over the various frequencies. In practice, the temperature dependence is not dramatic because in aqueous solutions mostly only a limited T-window is investigated. We shall use the symbol G for the interaction (Gibbs) energy, without further considering entropic contribution. Hamaker constants have the dimensions of an energy. Generally, we write Aij(k). For two identical particles of nature 1, interacting across water, this becomes A 11(w) . Sometimes, when the details of the nature of the system do not matter, we simply write A . Mostly (see sec. 3.8b) Hamaker constants are positive, so G v d w and ^7 v d w are as a rule attractive. The dimensions of the distance functions depend on the geometry of the system. For instance, for two semi-infinite plates J(a,h) = J(h) is proportional to h" 2 , hence Gmw(h) is in J m" 2 . However, for spheres, f{a,h) is dimensionless so that Gv<;m(h) is in J. The function / ' is such that /7 vdw (h) is in N m~2 for flat plates and in N for two spheres. In sec. 1.4.6 the distance h was counted between the particle surfaces, but as we must now compare with the electric interaction for which h refers to the diffuse parts of the double layers only, we must renormalize all h's in sec. 1.4.6 into h + 2d, see fig. 3.37. For short distances the difference is significant. Let us now recall and modify some important cases.
Figure 3.37. Definition of parameters in the equations for G V ( j w .
For two plane parallel flat layers, per unit area, we have from [1.4.6.2] Gvdw{h) = Vdw
T
\2n(h + 2d)2
[3.8.2]
where the meanings of the symbols are explained in fig. 3.37. It is implicit, in adding 2d in the denominator, that the two Stern layers are assumed to have the same Hamaker constant as the bulk solution. In sec. 3.8b we shall come back to this.
PAIR INTERACTION
3.87
For two spheres of radii a, and a2 , from [1.4.6.13], A[ 2a, a 02 GVr = J- + V d,ww{h) = - — 6 [ (h + 2d) 2 + 2 a , (h + 2d) + 2 a 2 (h + 2d) 2a, a 9 = *—^ + (h + 2d) 2 + 2a, (h + 2d) + 2 a 2 (h + 2d) + 4 a , a 2
+
f
+ ia<
(h + 2d) 2 + 2a,(h + 2d) + 2a 9 (h + 2d) „• 2
11 \\
[3.8.3]
[ (h + 2d) + 2a, (h + 2d) + 2a 2 (h + 2d) + 4a,a 2 J J
which can be derived from [3.8.2] using the Deryagin approximation. When we are not in the first place interested in the shortest distance h but in the distance r = a, + a 2 + 2d + h between the two centres we can also write [1.4.6.12] A\
G
2a,a0
( \
1
vdw ( r ) = ~~^" \~9—;
\r2 + (a, +a0)21]
2a,a0
Z
.
1
Z
7T+~y—;
6 [r 2 -(a,+a 2 ) 2
. ]„
I
^~+mi~9—;
r1-(a^-a^2
1
Z
1
^"M
ro Q A 1
[.3.8.4]
[r2 -(a, - a 2 ) 2 j J
11
Thennadil and Garcia-Rubio have compared this equation with a number of other expressions for the attraction between two spheres of different radii. The limit of Gvdw(r) for long distance (r » a1,a2) is 16Aa,3a| G
vdwf) = — ^ t ^
i3-8-5'
The inverse 6th power is identical to that in the London interaction energy between two atoms (sec. I.4d). For very short distances, (h + 2d) « ava2 • [3.8.3] reduces to G
Vdw'h) = - gh f aiCl2 .
I3-8-6]
6h(a,+a 2 ) All these equations become simpler for two spheres of equal radius (a = al = a 2 ) . From [3.8.3 and 4] we find (see [1.4.6.17 and 16]) G (h)--—\ — + — + vdw " 6 [(h + 2d) 2 +4a(h + 2d) (h + 2d) 2 +4a(h + 2d) +4a 2 + +
ln{ (h+ 2d)» + 4a(h + 2d) 11 [[h + 2d)2 + 4a(h + 2d) + 4a 2 J J
and
11
S.N. Thennadil, L.H. Garcia-Rubio, J. Colloid Interface Set 243 (2001) 136.
3.88
PAIR INTERACTION
respectively. As this last equation only contains the ratio r I a = s it can be converted into A\ 1 2 s 2 — 4-1 G
vdw(s' = - ^ ~2—,T + ^ 6[s - 4 sz
+l n
[3 8 91
^ H sz J
- '
implying that the interaction does not change if we simultaneously multiply distance and radii by the same factor. Two colloidal ivory spheres of 3n radius and IOJJ. distance between the centres attract each other equally strongly as two billiard balls of 3 cm, 10 cm apart. The short-distance variant is Aa G
™{h) = -J^W
[3 8 101
--
Figure 3.38 illustrates the applicability range of [3.8.10] by comparing it with the complete expression [3.8.7]. The parameter choice is more or less representative for two paraffin oil drops in water. It is seen that for very short distance the two are indistinguishable. Differences do show up at larger distances, but then the absolute value of G v d w is so low that the attraction is negligible, unless A11(3) is much higher than in the figure.
Figure 3.38. Van der Waals attraction between two identical spheres according to the microscopic theory. Aj1(3) = 1.25 kT , a = ^=0.2 = 100 nm ; full expression; approximation for h <
For a sphere and a plate, we can start with [3.8.3], setting a1 = a and letting a 2 -> oo . The result is [1.4.6.19]: G
(h) =
vdwl
_ 4 _ £ _+
5
+lnf
&[h + 2d
h + 2d + 2a
h +2d
)}
[3.8.! 1]
{h + 2d + 2a)\
with GVHW(h) = vdw
— 6(h + 2d)
for the very short distance approximation.
[3.8.12]
PAIR INTERACTION
3.89
We also mention an equation, derived by French 11 for the geometry of fig. 3.39 which is representative for a cone + plate configuration, mimicking that in AFM measurements 2 1 . It reads GVv rdi ww(h) =
A,,o,(h + 2d)2(r + h + 2d)sin 2 6+ [h + 2d)2rsin0+ ^ 6(h + 2d) 2 (r + h + 2d) 2
r2{r + h + 2d) 2
[3.8.13]
A variety of equations for the Van der Waals attraction between other geometries [cylinders, rods, laths, etc.) have been presented before, see [1.4.6.21-28], We mention an additional study by Gu and Li 3) on the interaction between a sphere and a cylinder based on the microscopic theory plus a correction to account for retardation. The general result requires numerical analysis but for certain limiting cases there are useful approximations. For instance, for (h + 2 d ) « a [3.8.12] is approached and for very large {h + 2d) the flat sphere-plate approximation works well if the ratio between the cylinder and sphere radius is large. Once the cylinder is long enough, further extension of the cylinder has no further effect on G v d w , i.e., the result applies for that of an infinitesimally long cylinder. For details the reader is referred to the original. All of this establishes a blueprint for treating Van der Waals interactions at intermediate distances. What happens outside this range? For large h retardation sets in, and rigorous analysis requires Lifshits theory. No general rules can be given, because the extent of this phenomenon depends on the natures of the phases, i.e. on their dielectric dispersion spectra. An approximate, relatively rapid numerical method
Figure 3.39. AFM-type interaction underlying [3.8.13]. Both surfaces carry a Stern layer of thickness d. 1!
R.H. French, J. Am. Ceram. Soc. 83 (2000) 2117. In practice, the geometry of the tip in AFM measurements is often not so well defined; because of that, a colloidal particle of defined geometry can be glued to the tip. 31 Y.G. Gu, D.Q. Li, J. Colloid Interface Sci. 217 (1999) 60-69; see also B. Gauthier-Manuel, Europhys. Lett. 17 (1992) 195. 21
3.90
PAIR INTERACTION
is to replace the pair interaction energy between two molecules, on the sum of which the Hamaker approach is based, by Overbeek's approximation [1.4.4.23], An improvement was suggested by Anandarajah and Chen11, and, for practical purposes the correction factor by Zhang et al.2) may be recommended. For very short distances, down to atomic scale, the theory also breaks down. Obviously, G v d w does not go to minus infinity for (h + 2d) -> 0 , as all equations predict. Even Fresnel surfaces cannot approach each other more closely than the interatomic distance in the solid. For crystals this distance is determined by the minimum in the Lennard-Jones interaction, recall fig. 1.4.1. We can correct for that by not only carrying out the integration over all pair interactions over the attractive part (~ r~6) (as in the Hamaker-de Boer approach) but also over the repulsive part (~ r~ 12 ). This leads to a very steep shortrange repulsion between macrobodies31 but the exercise is rather academic because surfaces are rarely that perfect. Surface roughness and other imperfections will play a role at such distances. Lifshits theory also becomes less obvious because it is questionable whether, on that scale, the phases may still be considered 'macroscopic'. For the daily practice of colloid stability studies at such short distances it has to be realized that Van der Waals interactions have to compete with other forces, in particular those mediated by the intervening liquid. These, and other, very short-range interactions may mask the very short distance repulsion. For these reasons, we shall not discuss this limit any further. As a final caveat, all the above theory applies to 'hard' bodies of which the shape does not change upon interaction. See for instance a paper by Vakarelski et al.4), where other references can be found. In AFM measurements sometimes elastic deformation occurs, giving rise to an additional force of interaction. For fluids (emulsion droplets), flattening can substantially increase the Van der Waals interaction51. We shall not discuss interactions between composite materials (layers of different nature61, as in liquid crystals, core-shell latices, adsorbed layers on solids, etc.), although the practical interest is substantial. 3.8b Hamaker constants. Update and extension Appendix 9 of Volume I contains a tabulation of Hamaker constants obtained from
11 A. Anandarajah, J. Chen, J. Colloid Interface Set 176 (1995) 293; J. Chen, A. Anandaraya, J. Colloid Interface Set 180 (1996) 519. 21 B. Zhang, J. Jin, and H.P. Wang, J. Dispersion Set Technol. 20 (1999) 1485. 31 D. Henderson, D.-M. Duh, X. Chu, and D. Wasan, J. Colloid Interface Set 185 (1997) 265. 41 I.U. Vakarelski, A. Toritani, M. Nakayama, and K. Higashitani, Langmuir 19 (2003) 110. 51 J.K. Klahn, W.G.M. Agterof, F. van Voorst Vadcr, R.D. Groot, and F. Grocneweg, Colloids Surf. 65 (1992) 151; S.J. Miklavcic, R.G. Horn, and D.J. Bachmann, J. Phys. Chem. 99 (1995) 16357; D. Bhatt, J. Newman, and C.J. Radke, Langmuir 17 (2001) 116; P.G. Hartley, F. Grieser, P. Mulvaney, and G.W. Stevens, Langmuir 15 (1999) 7282. 61 For an example, see D.Y.C. Chan, D. Henderson, IBM J. Res. Developm. 29 (1985) 11, addressing this for large spherical shells.
PAIR INTERACTION
3.91
theory and experiments. The computed data were obtained by the microscopic and/or macroscopic theory; a large part of the latter stemming from a review by Hough and White . Experimental data stemmed mostly from colloid stability studies, from thin film measurements or using the surface force apparatus. Since the appearance of Volume I (revised print, 1993) theoretical and experimental progress has been made. The theoretical news mostly stems from the better availability of dispersion data (e(i£) spectra over long ranges). There also are better Lifshits-type approximations, in which, based on the seminal work by Ninham and Parsegian , the roles of specific frequency ranges in the spectrum are highlighted. Experimental information has greatly profited from the advent of AFM and so-called optical tweezer techniques. We shall discuss this matter in sec. 3.9, but anticipating this the caveat has to be made that in most systems the final word has not yet been said. As compared with the older colloid stability studies (rate of coagulation, critical coagulation concentrations), AFM offers the advantage of requiring only one particle but, as before, Van der Waals forces rarely come on their own, so that other force contributions must be assessed before G vdw (h) or 17{h) can be obtained by subtraction. For AFM the geometry is usually that of a sphere and a plate, for which we have the required equations available [3.8.11-12]. However, in practice there can be problems with the definition of the surface, and, hence, that of d and h . When the surface is rough subtraction becomes difficult because different types of forces may have a different effective zero point. Sometimes attraction is not seen at all. Some reasons for this observation will be discussed in sec. 3.9 (DLVOE models). Usually, AFM produces hetero-interaction data, implying that Hamaker constants of the Aj2(3) type are obtained. Recall from sec. 1.4.6b, that according to the microscopic theory (and grosso modo confirmed by the macroscopic approach) we may write A2(3)=( A 12- A 13- A 23
+ A
33)
[3.8.14]
Applying the Berthelot principle [1.4.6.5]
Aj = V V S
[3 8 151
--
this may be written as A
12(3) = W A 1 1 ~V A 33)(v A 22 ~\IA33)
reducing to
for homo-interaction.
11 21
D.B. Hough, L.R. White, Adv. Colloid Interface Set 14 (1980) 3. B.W. Ninham, V.A. Parsegian, J. Chem. Phys. 52 (1970) 4578.
[3.8.16]
3.92
PAIR INTERACTION
Although Ajjjgj is always positive, A]2(3) can be negative if A33 is between Ax j and A22 meaning that the Van der Waals forces can be repulsive. Such situations can be met with wetting films (sec. III.5.3) but also for hetero-interaction between a polymer sphere and an inorganic plate in certain organic solvents. Repulsion between two phases 1 and 2 across a medium 3 does not mean that intrinsically 1 and 2 repel each other; across a vacuum they do attract (A12 is always positive). However, large affinities for the intervening medium can drive the particles apart. Essentially this is Archimedes' principle: objects floating on the surface of water are not repelled by gravity, but gravity pulls harder on the denser water. We meet the same issue in the interaction of molecules across a solvent: even in a good solvent the molecules attract each other by Van der Waals forces, but they prefer contacts with the water over those with each other. All of this is found in Flory's ^-parameter to quantify solvent quality, see [1.3.8.9] and sec. II.5.5ff. Another issue is that of the Hamaker constant of the Stern layer. Quantitatively, this is an unsolved, if not insoluble, problem. One difficulty is that the geometry and composition of this layer are not known exactly, even if they are defined at all. The other is that application of Lifshits theory on a thin layer of less-than-macroscopic thickness is questionable and even if this is allowed, how can one get the required spectral information? The phenomenological approach is to treat the colloidal particle as having a core-shell structure with the shell having a thickness d and Hamaker constant A s t . In our analyses, we shall set A st = A solvent , reasoning that Stern layers consist to a large fraction of solvent molecules, even though their density and packing may differ somewhat from those in the bulk. At any rate, this decision makes physically more sense than equating A st = A partlcle which is the tacit assumption if d is neglected in the equations of subsec. 3.8a. As a corollary, we must address the influence of electrolytes on the Hamaker constant of water, ^ = A 33 . In Volume I, and in most of the literature, this effect is neglected, the argument being that even concentrated electrolyte solutions consists predominantly of water. (The mole fraction of 1 molar electrolyte is only 1/55.5). Basically, this is correct, and most of the literature studies do not go beyond that. Nevertheless it should be remembered that: (i) because of the combinational nature of formulas of the type (3.8.16] the precise value of A TO may become very important when this value comes close to Aj t or A 22 and (ii) the screening of the various wavelengths of the dispersion spectrum can be different. The high frequencies are more or less unaffected by electrolytes whereas the zero frequency contribution is. (In the microscopic theory, this is the only term.) Mahanty and Ninham11 concluded the zero frequency screening factor f to be of the
11
J. Mahanty, B.W. Ninham, Dispersion Forces, Acad. Press (1976).
PAIR INTERACTION
3.93
Figure 3.40. Hamaker function for the quartz-water-air system, computed by slightly simplified Lifshits' theory. The influence of the (1-1) electrolyte is indicated. (Redrawn from Nguyen et al., loc. cit.) DLVO-type, viz. / s c =(l + 2x-h)e-2'rfl
[3.8.18]
implying that screening is different between the unretarded and retarded range. Figure 3.40 gives an illustration for a system which is important for flotation, because it involves the attachment of bubbles to mineral surfaces. The first conclusion is that even the sign changes with distance. In pure water, the Van der Waals force is repulsive below h + 2 d ~ 5 0 n m , above that it becomes negative, with -Ai2(w) maximally 0.78 kT . Positive Hamaker constants imply a negative G v d w , meaning that the water film is unstable, and tends to disproportionate into droplets, unless other forces prevent that. See sec. III.5.3a. The second conclusion is that according to this computation, there also is an electrolyte effect which is mostly relatively minor, although at large distances it can even affect the sign. This point demands more precise calculations. All of this may require adjustcment when the particles become very small. The trend is that for nanoparticles the Hamaker constant becomes lower than it is for 'macroscopic' particles. In sec. 3.7g under (ii) we already concluded that this is also the case for G e ] . This reduction of the two DLVO components comes on top of the sheer radius effect, as embodied in, for instance, [3.7.19a, 20] and [3.8.10]. An update of Hamaker constants and other information can be found in appendix 3. Mostly these data complement and better specify those of table Lapp.9. Differences between computed values are usually caused by the availability of, and approximations in, the underlying e(i£) spectra. For these differences, we refer to the original 11
A.V. Nguyen, G.M. Evans, and H.-J. Schulze, Int. J. Mineral Processing 61 (2001) 155.
3.94
PAIR INTERACTION
literature.
3.8c
Solvent structure-mediated
interactions
We have discussed these before, see sec. III.5.3a. Briefly, the structure of a liquid adjacent to a hard wall differs from that in the bulk. Layering takes place, extending over a very few molecular layers, resulting in a density distribution normal to the surface pN (z) displaying oscillations which decay rapidly with z. This is a general phenomenon, observed for hydrophilic and hydrophobic surfaces. The origin is the short-distance molecular repulsion (the +r~12 term in the Lennard Jones interaction). Its occurrence is now generally confirmed by Molecular Dynamics simulations and by experiments (see figs. II.2.4 and 3). This has several important results for colloid science. One is the general finding that in electrokinetics always thin stagnant layers are found: the tangential viscosity of the adjacent layer is much higher than that in the bulk. For the purpose of colloid stability the relevance is that, when two such surfaces approach to such short distance that these structured zones overlap, work has to be done to change them, giving rise to a Gibbs energy Gsolv s t r (h + 2d): solvent structuremediated
interactions.
At extremely short distances these interactions are oscillating, the maxima and minima being determined by the matching of the strong oscillations, i.e., they are alternately repulsive and attractive. For somewhat longer distances, empiry has shown that the decay is of an exponential nature; G
solv.str.
[3.8.19]
in which A is a measure of the structure decay as a function of distance. It is typically short-range: 1 = O (nm) . We had a similar equation before for the disjoining pressure, see [II.5.3.14]11, where the 2d was ignored. From [3.8.19] we now write ^solvstr (h + 2d) = ^ ^ e - W » ' *
[3.8.20]
A
Origin and quantification of these forces is a typical theme of colloid stability. The greatest problem is that, because of their short range, they fall in the category where other contributions to the total interaction become disputable: electrical forces because of regulation, Van der Waals forces because of non-macroscopic behaviour, and where experiments are involved, this requires surfaces that are extremely flat Fresnel surfaces. However, the origin of the phenomenon is well established. All of the discussion above is restricted to hard surfaces. For soft surfaces (LL interfaces) there is no layering of adjacent molecules but rather a gradual change of pN{z)
11
across the phase boundary, obeying a hyperbolic tangent profile, see fig. III.2.6.
Erroneously printed as [5.3.13] on page II.5.35.
PAIR INTERACTION
3.95
The occurrence of stagnant layers has not been confirmed, mostly because the fluidity of such interfaces makes electrokinetic measurements extremely difficult. When one tries to overcome this problem by adsorption of, even minute, amounts of surfactants, this interfaeial mobility can be reduced but then the molecular organization is also affected. Similar factors apply to colloid interactions in thin films and in wetting layers. We conclude that, for soft interfaces, the situation is still open and challenging. 3.9 Extended DLVO theory: DLVOE 3.9a Updating the DLVO model Traditional DLVO theory considers electrostatic repulsion and Van der Waals attraction as the sole, and additive, contributions to pair interaction. The theory is elaborated for flat and spherical symmetries, mostly assuming purely diffuse double layers at fixed potential, and non-retarded Van der Waals forces. With this model, a number of important observations could be accounted for, at least semiquantitatively. These include the very strong influence of the valency of the counterion (the SchulzeHardy rule), the relationship between stability and 'surface potential' (later identified as the f -potential), the rate of coagulation, and weak secondary minimum coagulation for big particles, leading to shear thinning and thixotropy. In addition, DLVO theory provided Ansatzes for the understanding of equilibria in thin films and polyelectrolytes. All of this has supported the essential correctness of the model. However, over the half century after its publication, it transpired that a number of (mostly quantitative) defects required systematic consideration. Several of these have been discussed in the previous sections. The most important ones are: (i) Double layers are not purely diffuse. Only a (small, but very relevant) fraction of the countercharge, depending on the nature of the indifferent electrolyte and its concentration, resides in the diffuse part. (ii) It follows from (i) that the 'surface potential' occurring in the DLVO model must be replaced by the potential of the diffuse double layer. So, when DLVO equations for Ggj are used, the (constant) surface potential in them must be replaced by yfi . Likewise, equations for interaction at constant surface charge, giving G ^ \ require cfi . It is now known that, because of regulation upon interaction, neither \fA nor a^ remains constant (see fig. 3.16). However, experience has shown that for many practical situations equations for Gef work well with i//* ~ £. (iii) Counterion specificity is caused by specific adsorption in the Stern layer. The phenomenon of lyotropic series is essentially absent in DLVO theory. (iv) Van der Waals forces have to be corrected for retardation, except at short distances.
3.96
PAIR INTERACTION
(v) At very short distances, solvent structure-mediated forces have to be added. (vi) The presence of a Stern layer results in Van der Waals forces operating over a longer interaction distance compared to those in the diffuse layer: i.e., (h + 2d) instead of h , where h is the distance between the two outer Helmholtz planes and d the Stern layer thickness. It may be added that in our approach the effects of multivalent counterions and ion correlations are subsumed in the Stern layer. (vii) In the DLVO theory the double layers are assumed to be continually equilibrated. Of these points, the last will be addressed in chapter 4. The others will be discussed now. An analysis like this runs the risk of drowning in the abundance of equations and adjustable parameters. On the other hand, the academically most perfect approach (Gel via charge regulation of Gouy-Stern double layers, and G v d w via Lifshits including retardation) can only be described graphically, whereas in practice there is a demand for simpler analytical approximations that capture the physically essential elements. As a compromise, we shall first treat some general trends and then study the influences of a number of important variables analytically, using some representative equations. The caveat is that, if applied to experiments, the fitting parameters should be used with some reservation. The theory presented here has its roots in the DLVO model but includes a variety of improvements. For the sake of distinction we shall, where needed, call it extended DLVO theory, DLVOE. This name has been used before by van Oss1 but for a much more restricted extension, that we shall not follow: he added so-called acid-base interactions to classical DLVO. Numerous other authors have also extended DLVO theory using these, or other terms. Throughout, G el , G v d w and Gsolv str are considered to be additive. The surfaces will be considered as molecularly flat, which is of course an academic presumption, and free of surfactants except where stated explicitly. 3.9b General features Figure 3.41 gives the general shape of DLVOE interaction curves. It is similar to figs. 1.4.2 and III.5.12. In the top picture, the Gibbs energy is given. For two semiinfinite parallel plates this is expressed in J m~2, for two spheres and other finite objects it is in J. The lower picture gives the derivative, which for two plates is the disjoining pressure, 77, in N m~2, and for two spheres etc. is the force J , in N. The extremes in G(h) correspond to zero values in 77(h) or J(h); the extremes in the latter are found at the bending points in the Gibbs energy.
1
C.J. van Oss, Interfacial Forces in Aqueous Media, Marcel Dekker (1994).
PAIR INTERACTION
3.97
Figure 3.41. General nature of DLVOE homo-interaction curves. Top, Gibbs energy; bottom, disjoining pressure or force, h is the distance between the two outer Helmholtz planes. The distance between the surfaces is h + 2d . The steep ascent for (h + 2d) —> 0 is the Born repulsion between touching surface atoms; its range is less than 0.1 nm. This repulsion also occurs across a vacuum. The increase with decreasing (h + 2d) of 77, or / below h = 0 sketches G solv s j x ; d is about 0.3 nm, depending on the nature of the solvent. The occurrence, heights, and shapes of the various maxima and minima depend on the magnitudes of the three constituting contributions, and hence on c s a l t , the pH, radius etc. Qualitatively, G and 77 exhibit the same extrema, but those for 77 are systematically at larger h . Starting at large (h + Id), minimum
first the shallow secondary
is observed. The reason for its existence is that for large distance, Gel
decays as e~Kh, which for large K is steeper than the decay of G v d w . For low K , GeI extends so far that the secondary minimum becomes invisible. For big particles where h« a
G v d w decays as (h + 2d)" 1 , see [3.8.10]. However, for smaller particles at
large distance the decay of G v d w is according to [3.8.8], which is steeper. Moreover, retardation sets in which also weakens G v d W at large distance. Together with the fact that Gej and G v d w both increase with a (as a first approximation both linearly, see for instance [3.7.19a and 3.8.10]), we arrive at the conclusion that the depth of
3.98
PAIR INTERACTION
Gmin(sec) decreases with decreasing particle size. In fact, for low a it effectively disappears. It is virtually impossible to coagulate nanoparticles in the secondary minimum. Here, the historical fact may be recalled that, when Overbeek found experiments that did not exhibit secondary minimum coagulation under conditions where his theory predicted it, he conceived the concept of retardation. This concept was later elaborated by Casimir, and generalized by Lifshits. The repulsive maximum, Gmax , is of course caused by diffuse double layer overlap. Commonly It is called the energy barrier. This term is sloppy, because to a large extent it is entropically determined, viz., by the diffuseness and regulation of charges upon overlap. Nevertheless we shall continue to use this term because of its common usage. When the barrier is high enough, it can stabilize a sol. Semiquantitatively, the probability that an encounter between two colloids leads to coagulation decays as exp( - Gmax / kT), where G max acts as the activation (Gibbs) energy. Barriers with G max = kT offer no protection against coagulation, whereas those for Gmax> 10-15 kT ensure stability for most practical purposes. In between a range from rapid to increasingly slower coagulation with decreasing salt concentration is found. Although there is no sharp transition between, 'stable' and, 'unstable' sols there are methods of determining rather reproducible critical coagulation concentrations (c.c.c.'s). The c.c.c. is defined as the indifferent electrolyte concentration below which a sol is stable and above which it is unstable; see sec. 3.12. To create an energy barrier there must be a distance range over which Gej exceeds G v d w . This is only possible for low c s a l t , because only then does Gel decay slowly enough to outweigh G v d w . It must be stressed that the range of action of Gel is at least as relevant as its absolute magnitude. Typically, for h —> 0 in equations such as [3.7.19] Gej ~ c salt /K 2 , which is independent of c s a l t , and In [3.7.21 -25] the pre-exponential factor does not contain c
sait
a t
a11-
When two particles can overcome the energy barrier, they can approach each other until they reach the primary minimum, giving rise to primary minimum coagulation. Depending on d , this minimum can be deep and narrow, and is mainly determined by the counteraction between G v d w and G so j vstr . According to DLVO, for very short distances G v d w goes to -•» (see [3.8.10]), whereas Gej remains finite, the former always outweighing the latter. However, it is questionable whether Lifshits theory remains valid for distances having molecular dimensions. Experimentally, it is impossible to establish whether a one or two water interlayer remains between the primary minimum and the surface proper, supposing that the distinction makes any sense at all, given the usual Imperfections in surface structure. In AFM measurements the 'zero distance' more or less coincides with h = 0, provided such a minimum is observed at all. On the other hand, distinction between primary and secondary coagulation is mostly easy. By way of illustration, bacterial adhesion on slides can be followed microscopically. Sometimes It is observed that adhesion first takes place in the secondary minimum, where the bacterium still undergoes lateral Brownian
PAIR INTERACTION
3.99
motion, to later snap into the primary minimum upon which this motion suddenly stops. To the eye it looks like a 'sudden death'. An Interesting issue is whether primary minimum coagulation can be reversed, say by removing the electrolyte. (Secondary minimum coagulation usually is; sols, coagulated in the secondary minimum are usually shear thinning, if not thixotropic). Stated otherwise, can such coagulated systems repeptize by removing the electrolyte? The reply to this question depends to a large extent on the possibility of leaching the electrolyte from the gap, and the ensuing inner-layer regulation. The distance
where the primary minimum is located will be close to h = 0,
implying that it is equal to about twice the Stern layer thickness. For thin liquid films, this thickness can be measured very well; It is the thickness of Newton black films, which is virtually Independent of c galt and equal to twice the surfactant monolayer thickness (chapter V.6). In concentrated systems this minimum acts as the distance of closed approach, with which it is identified in so-called hard sphere models. In sec. 1.4.2 we used the symbol a for it, and this will also be done In chapter 5. So, for spheres a = a + d . We shall now systematically consider the influence of some variables. The purpose is to obtain a quantitative feeling for the effect of some relevant system parameters. To restrict the number of variables we shall, as a rule: (i) not consider the Influence of the temperature, considering that A and Gel are little T -sensitive In the ambient window (note that Gel ~ £l/2T,
see for example
[3.7.13]: and that e decreases with increasing T ). (ii) discuss G only, and not 77. (iii) only consider spherical particles, emphasizing the (relatively) best analytical expressions, viz. [3.7.19a] and [3.8.7]. For pragmatic reasons we shall start with the approximation using constant (yd . 3.9c Influence of the Hamaker constant and retardation For this case, trends are illustrated in fig. 3.42. The value of A = 3 kT is more or less representative for poly(styrene) across water; that of 30 kT, for a number of hard ceramic materials (see app. 2) and heavy metals (app. 1.9). Retardation is estimated following Dagastine et al.1 . For A - 3 kT , Gel prevails everywhere except at short distance. For A = 30 kT, attraction prevails everywhere. To stabilize such sols electrostatically, about twice as high diffuse potentials are needed (Gel is about proportional to (y d ) 2 ). Retardation becomes significant only at larger distances. However, in these pairs of graphs the overall effect is not visible, in panels a and b, because Gei dominates; and in panels c and d, because coagulation in the primary minimum takes place anyway. Retardation becomes significant when the value of Gtot is more critical. In the remaining part of this section, retardation Is ignored.
11
R.R. Dagastine, D. Prieve, and L.R. White, J. Colloid Interface Set 231 (2000) 351.
3.100
PAIR INTERACTION
Figure 3.42. Homo-interaction according to DLVO theory. Two spheres, a = 1 0 0 n m , d = 0 nm , y d (constant) = 1.5 , c salt (l-l) = 10"2 M . Equations [3.7.19a] and [3.8.7]. Panel a, A = 3 kT non-retarded; panel b, A = 3 kT, retarded (almost identical to panel a); panel c, A = 30 kT non-retarded; panel d, A = 30 kT , retarded.
3.9d Influence of the Stern layer thickness (d) The effect of d depends on the conditions. Those in fig. 3.43 are chosen in such a way that d dominates the fate of the sol. The reason for the strength of this effect is that the position of the energy barrier is such that G v d w is in its steeper part. In the example, Gmax increases by a factor of more than two between d= 0 and d = 0.3 nm . It is unlikely that d is much larger than 0.3 nm (that would require very thick hydrated layers, which have never been proven for hydrophobic surfaces without steric interaction). It is inherent in classical DLVO theory that d = 0 . The implication is that DLVO theory underestimates Gtot because at the position of the maximum G v d w is lower than was thought. For this reason, Hamaker constants derived from coagulation data, treated in terms of classical DLVO theory, are underestimations. In the figure it is seen that the position of the maximum is only slightly affected by the choice of d . It is concluded that d is an important parameter that may not be set to zero. However, it is not a parameter which is readily established independently.
PAIR INTERACTION
3.101
Figure 3.43. DLVOE theory. Homo-interaction between two identical spheres (a = 100 nm), y d = constant = 1.1. Electrolyte, 0.05 M (1-1). Total interaction curves from [3.7.19a] and [3.8.7]. A = 3 kT . Shown is the influence of the Stern layer thickness d .
Figure 3.44. DLVOE theory. Homo-interaction between two identical spheres of variable radius a (indicated). Parameters: A = 3 kT , c s a l t = 0.1 M , d = 0.3 nm , y d = 1 . Computations based on [3.7.19a and 8.7].
3.9e Influence of the particle radius (a) This influence is rather straightforward: upon increasing a the repulsive part becomes more positive, the attractive part more negative; Figure 3.44 illustrates this. Gtot is not exactly proportional to a, as would be predicted by simplified shortdistance approximations. However, the deviation from the common intersection point on the distance axis, predicted by the approximated theory, is negligible.
3.102
PAIR INTERACTION
3.9J Influence of the electrolyte concentration and valency. The SchulzeHardy rule Let us start with the interaction at constant potential, as in DLVO. However, we replace y/° in DLVO (assumed to be fully determined by Nernst's law, and hence being independent of c salt ) by y6-, which decreases with increasing c salt . For that, we use old empirical data, collected by Eilers and Korff1 , which can be fairly well represented by yfi = £ = const. z~2c"1/4 where we have fixed the constant by letting y d = 1 at c s a l t =10 seen in fig. 3.45.
[3.9.1] l
M . The first result is
Figure 3.45. Homo-interaction between two Identical spheres (a = 100nm, A = 3 fcT) according to [3.7.19a and 8.7]. Given is the (1 -1) electrolyte concentration. y d ss y e k = 1 at 100 mM; values at other salt concentrations are according to [3.9.1]; see text. Panel a, d = 0 nm ; panel b, d — 0.3 nm .
The maximum decreases very rapidly with c s a l t . For DLVO (panel a), the critical coagulation concentration is between 50 and 100 mM, for DLVOE it is between 100 and 200 mM. In order to define a c.c.c. value on the basis of such curves we must anticipate sec. 3.12 and briefly discuss two classical methods for determining c.c.c. experimentally. An old-fashioned method is the coagulation series method, which involves a series of vials, containing a fixed amount of sol, and adding increasing amounts of electrolyte to establish the concentration above which sedimentation of aggregated particles becomes clearly visible after a preset time. The method involves stirring and/or shaking steps with the purpose of making coagulation as efficient as
" H. Eilers, J. Korff, Chem. Weekblad 33 (1936) 358; Trans. Faraday Soc. 36 (1940) 229, their table IV.
PAIR INTERACTION
3.103
possible, essentially combining perikinetics with orthokinetics (see ch. 4). In this way, fairly sharp borderlines between 'stable' and 'unstable' sols are obtainable. An alternative is the rate of coagulation method, in which this rate is mostly measured optically as a function of c s a l t . Plots of logW against logc salt consist of a descendingand a horizontal branch, whose intersection is identified as the c.c.c. Here, W is a standardized measure of the probability that a pair encounter leads to aggregation. For the present purpose it is important that both techniques show the same trends (with respect to valency and nature of the electrolyte), although for the rate- method, c.c.c. values are higher by —30-60%. Phenomenologically, the electrolyte behaves as if it is not equally efficient between these methods. There has also been a third method, advocated by the Yugoslav School1' in which in statu. nascendi sols are coagulated, yielding even higher c.c.c. values, but the same trends. This comparison of methods may be summarized by stating that there is a method-specific threshold, say G max (h = hmax) = bkT, discriminating between stability and instability. With this in mind, analytical analyses can be made to find approximate relationships between c.c.c. and z , i.e., to account quantitatively for the Schulze-Hardy rule. It is noted that most of our equations apply to symmetrical (z-z ) electrolytes, but as the co-ions are negatively adsorbed their role is minor, so we may interpret z as the valency of the counterion. To obtain the required relationship we use the fact that at G
max
t h e
derivative 77(fi) = -dG{h)/dh = 0 .
We shall elaborate this for the case where G max is composed of [3.7.19a] and [3.8.10].
However,
for
mathematical
convenience
we
replace
ln[l + e~'rfl]
by
0.693 erKh , which is a good approximation (see fig. 3.36). We further assume that the particles are not too small, i.e., (hmgx +2d) « a , so that [3.8.10] may be used instead of [3.8.7]. Then, _
„
44.36^-cRTa
GtatCW) " — ^ . 44.38.cRT, tot
,0(zud}
Aa
Kh
tanh* ( ^ j < ^ " 1 ^ 7 ^ =bkT
t a n h 2
K
( ^ &_Kh \ 4 J
^
A ^ 12(h + 2d) 2
and, from 77 tot ( h m a x ) = 0 , 44.367rcRTa K
,9(zyd) _Kh tanh2 ^ — e max =
I
4
J
Aa 12
CW+2d>
FT
-or.-, [3.9.4]
which can be combined with [3.9.2] to give
12
11
^ --bkT *-(Vax+ 2 d ) 2
=
^ 12(Vax+ 2 d )
M. Mirnik, Nature 190 (1961) 689.
[3.9.5]
3.104
PAIR INTERACTION
This equation may be called the implicit DLVOE expression for h m a x • It can be compared with DLVO theory, which sets b = 0 and Id = 0 ; whence simply, K-h max =l
(DLVO)
[3.9.6]
The elaboration of [3.9.5] is straightforward, although a little clumsy. We shall take the case with b = 0 kT, considering that the d - effect is more relevant than the beffect; then, simply, ic(hmax + 2d) = 1
(DLVOE)
[3.9.7]
We note from [3.9.7] and fig. 3.43b that the maximum is situated not far beyond twice the Stern layer thickness. Because of the ' 2d effect', hmax
is less dependent on c s a l t
than in the DLVO model. The physical prediction is that, as soon as two particles have overcome the maximum, G solv s t r takes over (not drawn in the figure) so that there is little or no room for a primary minimum. In order to find such a minimum, G m a x should be low; i.e., electrostatic stabilization must be suppressed. Perhaps this is the reason why in the interpretation of scattering experiments with concentrated systems (chapter 5) simplified interaction equations often suffice, such as those consisting of a repulsive exponential decay plus an attractive pit. Equation [3.9.7] can be re-substituted into [3.9.4], yielding, ^9(zyd}
44.36;rcRT
tanh K
2
AK2
0K.rt
2lcd
-2— \e\ 4 )
=
(b = 0)
[3.9.8]
12
where c = c.c.c. Basically, this equation therefore comes down to a definition of the c.c.c. in terms of y d , A and d . At this point it is more than historically interesting to recall the Schulze-Hardy rule, already briefly alluded to in the present chapter. This empirical rule dates to the end of the 19th century1'21, and is still generally valid for hydrophobic colloids: it states that the stability decreases very strongly with the valency of the counterion. One of the main tasks for DLVO to solve was to explain this rule quantitatively. To that end, an equation similar to [3.9.8] was invoked. The difference with DLVOE is that Stern layers were ignored, implying that exp(2/cd) ~ 1 and yA ^> y°. Hence, for the spheres we selected, DLVO would have written31 44.36.cRT K
J ^ ) ^
\ 4 J
12
As K1 ~ cz 2 it follows that
11
H. Schulze, J. Prakt. Chem. 25 (1882) 431; 27 (1883) 320. W.B. Hardy, Proc. Roy. Soc. 66 (1900) 110; Z. Physlk. Chem. 33 (1900) 385. 31 DLVO theory considers flat plates, leading to a somewhat different shape for Gel. 21
PAIR INTERACTION
c.c.c. = const.
3.105
tanh 4 (zy°/4) „" , A2z6
[3.9.10]
and as y° was supposed to be high, tanh 4 {zy° / 4) —> 1, so that c.c.c. ~ z~ 6 . The const, in [3.9.10] is 8xlO~ 2 2 mM . Although Overbeek himself warned that the z" 6 dependence is not generally valid11, for a while, this power law was a success story, even to the extent that non-fitting data were laid on a Procrustes' bed to force agreement21. Our analysis shows that the relationship is much more complicated. We refrain from a detailed discussion, but note that an important improvement of the interpretation of the Schulze-Hardy rule is already obtained by replacing y° by y d , taking only the linear term of the tanh series expansion: c.c.c. = ^
^ , 256 A 2 z 2
[3.9.11,
implying that the relationship between c.c.c. and z depends on the way in which y d changes with these variables, and this will depend on the system conditions such as the pH and the nature of the electrolyte. For example, for systems obeying [3.9.10], c.c.c.— z" 6
but
mostly 2
determined by the z~
[3.9.11] applies, in which case the and the z-dependence of y
d
z-dependence
is
which, in turn, depends on
system conditions such as pH and the nature of the electrolyte. For example, for systems obeying [3.9.1] c.c.c. ~ z~5 . A number of secondary phenomena can now also be accounted for, such as ion specificity: under otherwise fixed conditions, stronger counterion adsorption results in a lower y d , and because of the 4th power this propagates very strongly in the c.c.c. values. We shall return to this matter in subsecs. 3.9i and in sec. 3.12. In [3.9.11] the influence of the radius is not strong; it becomes stronger after [3.9.5], in the transition to [3.9.7] the bkT contribution is retained. We shall not discuss this in more detail, partly because the assumption of interaction at constant y d is fallible. However, the general conclusion is that DLVOE gives a satisfactory interpretation of the Schulze-Hardy rule. 3.9g Interaction at constant o~d This case can be elaborated with the equations at our disposal. For example, for the case with low y d we can obtain G^
from [3.7.22]. The potential in this equation
can be obtained from the fixed charge using [A.2c.l]. The resulting y d is mildly concentration-dependent via the parameter p = {SeoecRT)~l/2, but for ambient values of a6
(say, a few uC cm"2 ) { p < r d ) 2 » l , so that y d becomes concentration-indep-
endent and constant. Hence, GJ,j again assumes a shape containing a concentrationindependent coefficient, then a fixed [zyd )2 and a decay function, all comparable to
11
Sec e.g. J.Th.G. Overbeek in Colloid Science (H.R. Kruyt, Ed.) Vol. I. Elsevier (1952) p. 308. In this connection, some authors identified the (qualitative!) Schulze-Hardy rule with the z~6 power law. 21
3.106
PAIR INTERACTION
[3.9.2]. So, it is questionable whether it is useful to go through all these computations, the more so since neither y d nor cfi remains constant upon interaction. 3.9h Interaction between regulated surfaces The Implementation of regulation on the Gouy-Stern level no longer leads to (deceptively) simple analytical expressions. Instead we must rely on numerical results. Theory for Gel has been derived for flat surfaces in sec. 3.5c, of which fig. 3.19 was our illustration. In fig. 3.46, these curves have been reworked to spherical symmetry, using the Deryagin approximation, and the Kd effect has been Introduced after adding G v d w according to [3.8.9]. The short vertical lines at h = 0, (2d = 0.6 nm) indicate the strongly repulsive short-range Gsolv s t r . The most critical pH range between pH = 4.6 and 5.0 is magnified.
Figure 3.46. Homo-interaction between two oxide surfaces pH° = 4.255 with a° regulated across charge-free Stern layers. Given is the influence of the pH. Parameters, Cj = 120|iFcm~2 , JVS =5xlO 14 cm" 2 , K ^ I C T 1 , Kb = 10" 3 , c s a l t =0.1M. z = l, a = 100nm, A = 3 kT, d = 0.3 nm . Figure 3.47 illustrates the effect of regulation across the Stern layers by varying the specific Gibbs energy of counterion binding ". As before, Kt = exp(-AadsGmi /RT) (see [3.5.10a]), where A ads G mi is the molar Gibbs energy of chemical bonding. A large range of such Gibbs energies is scanned. Values below K{ = 1 correspond with positive (repulsive) values of A ads G inl ; for Ki = 1 the electrolyte is indifferent; for A ads G mi ~ 1 kT (characteristic of simple ions), K( ~ 2.7; and for surfactants (AaH.=Gmi - 1 0 fcT), K, is in the thousands. We see that for K. < 1 there is a dUo
IIll
1
1
For this and the following figure, Gel for flat plates is from J. Lyklema, J. Duval, Adv. Colloid Interface Set (subm. 2004). This paper also contains y d (h), al[h) and a (h) curves.
PAIR INTERACTION
3.107
Figure 3.47. Homo-interaction between two surfaces of fixed positive surface charge (10 |j,Ccm~2) with regulation across the Stern layers. Given is the influence of specific adsorption of anions at the inner Helmholtz plane via K; (indicated). Parameters; Cj =120|jFcm- 2 , C 2 = 2 0 n F c m - 2 , A = 3 kT , d = 0.3 nm . Values of Kj: a = 0.2 , b = 0.5 , c = 5, d = 10, e = 20, f = 100, g = 500 , h = 10 3 , i = 3 x l O 3 , j = 5xlO 3 .
moderately shallow secondary minimum, in inward direction followed by strong repulsion, determined by the screened surface charges. Counterions can only adsorb because they are electrostatically attracted. For already weak specific adsorption (curve c, A ads G mi —1.6 kT), the combined action of chemical and electric attraction reduces Gel substantially, and for curve d (—2.3 kT) it has become virtually invisible. These elaborations demonstrate how sensitive is the specific adsorption under realistic experimental conditions. For very high attractive AG mi 's, superequivalent adsorption (overcharging) takes place, and the system restabilizes again but now with | cr'l >| a°\. This is the basis for the familiar irregular series. For this charge reversal a value of about 8 kT for AaHc.Gmi is needed. See further subsec. 3.9j. id CIS
IH1
J
The last illustration involves hetero-interaction; see fig. 3.48. This is the most general case. The conditions for curves a and c would represent electric repulsion, had there been neither regulation nor specific adsorption, whereas for curves b and d Gej would be zero under these restrictive conditions. However, as in fig. 3.47, specific adsorption does play an important role. This role is so strong because the proximity of the other, also positive, particle makes y1 more positive and hence boosts the specific adsorption by induction. Not many kTs per ion are needed to achieve superequivalent counterion adsorption on particle 2, leading to attraction (see panel a). In panel b the situation is mostly attractive, but for very low AGmi these ions are desorbed (by induction), leading to a weak repulsion for chemical reasons. In the
3.108
PAIR INTERACTION
Figure 3.48. Hetero-interaction between two regulating Gouy-Stern layers. General parameters, aj = a2 = 100 nm , di = d2= 0.3 nm , c s a l t = 0.1 M , of = +7.11 |iC cm 2 , AG mi j = -kT . Panels a and b, A = 0 ; panels c and d, A = 3 kT . Values of A G ^ indicated. Panels a and c, o§ = +3.56 nCcm~ 2 , A G ^ 2 = -2.83 kT ; panels b and d, a§ = 0 .
borderline region between attraction and repulsion, shallow maxima or minima can be found, which are too low to detect experimentally. Comparison of panels a and b, with c and d, respectively, shows that only with the uncharged second surface can G v d w outweigh G el . When both surfaces carry a charge there is some lowering of Gmax , and the combined effect of attraction for h —> 0 deepens the primary minimum. The main quantitative conclusion is that the depths of primary minima can be equally well determined by Van der Waals, as by hetero-electrostatic, interactions, so care has to be taken not to over interpret data in terms of only the former. 3.9i Lyotropic (Hofmeister) series In this section we discuss ion specificity, e.g., the difference in properties between (counter)ions of the same valency. Such phenomena are also known as ion size effects. The result is that certain sequences arise in properties measured for ions of increasing/decreasing size, but at fixed valency. For such sequences we use the name
PAIR INTERACTION
3.109
lyotropic series; the term is considered equivalent to Hqfmeister series1]. So far, in FICS, attention has been paid to such series in several places, including electrolyte solutions (hydration), in sec. 1.5.3; in electric double layers, sec. II.3.6d; table II.3.7; fig. II.3.41 and 3.41, and in ionized monolayers, sec. III.3.8b. Now we shall discuss these series in connection with the stability of hydrophobic colloids. Lyotropic series hardly occur in DLVO theory (Verwey and Overbeek spent only a few pages on them), but form an essential part of DLVOE. In figs. 3.47 and 3.48 it was concluded that, for hydrophobic colloids on the verge of destabilization, small differences in specific adsorption Gibbs energy are reflected magnified in the c.c.c. The reason is that they propagate into t/d which occurs in the c.c.c. to the fourth power (see [3.9.11]). These small ion-specific differences (at c~ c.c.c.) are generally too small to detect electrokinetically, but stability studies are valuable in explaining the non-electrostatic contributions of ion adsorption. In the literature, these phenomena are rather neglected. For example, they are ignored, or almost ignored, in the standard books by Russel et al. and by Hunter, respectively (sec. 3.15). Ignoring the 'chemistry' implies ignoring such important features as hydrophobic bonding and water structure, for which much relevant information is now available. Basically, the reason why specific adsorption contributes to stability is the same as for the adsorption of charge-determining ions. The latter phenomenon is responsible for the spontaneous formation of electric double layers around sol particles. When two such particles meet the double layers cannot develop fully, so that work has to be done to push them together. Depending on the type of regulation, this work can be chemical, electrical, or a mixture of both. The same can be said of specific adsorption, the difference with the adsorption of c.d. ions being merely quantitative. Mostly, the Gibbs energy of simple counterion adsorption is O(kT), whereas that of c.d. ions is O(10fcT), see for example, fig. II.3.61. This difference is related to the way in which these ions adsorb, and where. The adsorption of simple cations takes place at the iHp, and is probably to a large extent entropically determined (via the solvent structure) . Adsorption at the iHp is also the case for ions such as hydrolyzed La3+ or surfactant ions. For phosphate ions, which can make very strong chemical bonds with a number of oxides it is a matter of taste whether one would call them 'specifically adsorbed counterions' or 'charge-determining ions'. For all these ions, the rule is that if they are displaced upon interaction, chemical work has to be done, whereas if they remain in place the potential goes up and the work is electrical, with 'a bit of both' (regulation) being the most general phenomenon. Figures 3.47 and 3.48 gave examples, with A ads G m j as the (sensitive) parameter.
Hofmeister himself (1850-1922) studied specific salt effects on the precipitation of proteins, i.e. on a special type of hydrophilic colloids. 21 J. Lyklema, Mol. Phys. 100 (2002) 3177.
3.110
PAIR INTERACTION
Table 3.2. C.c.c. values In mM for some negatively charged sols' Sol
Electrolyte
As2S3
Au
Agl
LiCl
58
-
-
LlNOg
-
-
165
NaCl
51
24
-
NaNO 3
-
-
140
KC1
49.5
-
-
KNO3
50
25
136
RbNO 3
-
-
126
MgCl2
0.72
-
-
Mg(NO3)2
-
-
2.60
CaCl 2
0.65
0.41
-
Ca(NO 3 ) 2
-
-
2.40
SrCl 2
0.635
-
-
Sr(NO 3 ) 2
-
-
2.38
BaCl 2
0.69
0.35
-
Ba(NO 3 ) 2
-
-
2.26
*' Experiments by Freundlich et al., and Kruyt and Klompe, compiled by Overbeek in Colloid Science I (H.R. Kruyt, Ed., (Elsevier, 1952) p. 307.
We shall now discuss lyotropic series from an experimental point of view, with the aim of establishing the empirical laws behind them. By way of introduction, see table 3.2, which gives a survey of old (but not obsolete) c.c.c. values obtained by the 'coagulation vial' method. More recent data will follow in sec. 3.12, but the table already shows a number of clear trends: (i) c.c.c. decreases from Agl —> As 2 S 3 —> Au , which is at least in line with the fact that A increases in this direction, but differences in y d may also be involved. (il) The Schulze-Hardy rule Is obeyed; the ratio between z = 2 and z = 1 is not far from 2 6 = 64 , but depends significantly on the counterion size. (iil) There are obvious ion specificities. For Agl these are clearer than for the other two. The sequence Is Inverse (c.c.c. is lower for bigger counterions) . (iv) The specificity is relatively more pronounced for monovalent than for bivalent ions. This is a recurring trend, and has two reasons; (1), for monovalent ions the c.c.c.
Recall from sec. II.3.10h that a series is called direct when the measured quantity increases with the radius of the unhydrated ion; it is inverse when it is in the opposite direction.
PAIR INTERACTION
3.111
is much higher, so coagulation takes place in a higher part of the adsorption isotherm and, (2), the electrical contribution for bivalent ions is twice as high as that for monovalent ones. Let us emphasise the negatively charged Agl system for z = 1. For this system, <7°(pAg) curves are available; in 10" 1 M electrolyte the lyotropic series is direct; at a given pAg, a° (RbNO3) > <7° (KNO3) > cr°(LiNO 3 ): see fig. II.3.41. In terms of pure DLVO theory it is incomprehensible that sols with higher surface charges could be more susceptible to coagulation. However, in DLVOE it is obvious that, as a result of specific adsorption, simultaneously y d decreases whereas o° increases. Lyotropic sequences depend on the nature of the surface. By way of illustration, we repeat and extend in table 3.3 some sequences compiled earlier in table II.3.8. Table 3 . 3 . Direction of the lyotropic sequence for the binding of alkali-ions. System
Direction
Agl
direct
Agl (high T)
no specificity
Hg
direct
SiOH (BDH)
direct
TiO 2 (rutile)
inverse
TiO 2 (lowp.z.c.)
direct
a-Fe2O3
inverse
Y-A12O3
inverse
Behenic acid
inverse
Remarks
Very minor effect
Calcined sample
Monolayers, include TMA+, TEA+
References: see table II.3.8. The monolayer example comes from III.sec.8b. direct and inverse mean: stronger adsorption for the largest (smallest) ion, respectively. Direct sequences in binding imply inverse series in coagulation. From these observations it is concluded that lyotropic sequences do not reflect a purely ion size effect (if they did, we ought to always find the same sequence). Rather, they are the result of ion-surface interaction. The nature and, particularly, the size of the surface sites onto which the ions adsorb play a decisive role. On the almost ideal mercury surface, no sites are distinguishable and the remaining specificity is so small that it is experimentally detectable only with the most sensitive techniques. By combining stability studies with surface charge measurements it is possible to compute A a d s G m ; . The results depend somewhat on the available data and the assumptions made.
3.112
PAIR INTERACTION
Most advanced would be the charge regulation model, leading to figures like 3.47 and 3.48, but a simple model on the basis of constant y d can also be invoked. For example, from measured c.c.c values, and using [3.9.11], y d can be found, and hence a^ . With £7° available from titration at c salt = c.c.c, o"' can be deduced, from which ^ads^m c a n b e found by using an appropriate isotherm equation, for example that by Frumkin, Fowler, and Guggenheim (FFG), see sec. II.3.6d. There is some uncertainty about the values to be selected for the c.c.c. and as to the model for the inner double layer. For Agl sols, two independent approaches1 yielded for Li+ , K+ and Rb+ ions the values -3.6 kT , -4.0 kT and -4.2 kT , and -2.1 kT , -2.5 kT , and -2.9 kT , respectively. (The series given on p. II.3.135 is again slightly different.) The differences between these series illustrate the limits of quantification. The absolute values are only a fraction of the corresponding hydration Gibbs energies (compare table 1.5.4). Alkali ions on Agl are not dehydrated. Rather, the water structure of the counterionadsorbed I~ ion pair is affected, and the corresponding contribution to A ^ G ^ is largely of entropic origin . More on lyotropic sequences will follow in sec. 3.12 and, in a different context, in the next subsection. 3.9j Overcharging; charge reversal Overcharging is the phenomenon in which more countercharge adsorbs than is required for compensation of the surface charge. As a result, the f -potential inverts its sign, hence the name 'charge reversal'. However, this term is sloppy because the surface charge does not reverse its sign. On the contrary, its absolute value increases because of the increased screening. The phenomenon has been known for a long time, Freundlich already described it in one of the first editions of his Kaplllarchemie. For colloid science, overcharging is an immensely important phenomenon, and it is mostly achieved by the more powerful adsorption of ionic surfactants and polyelectrolytes. It is a means of controlling the sign of yd or a^ and, in this way, controlling the sign of the counterions to which the c.c.c. is very sensitive. Here, we are interested in the basic principles of colloid interaction and therefore only consider inorganic electrolytes. As long as the surface charge is smeared out, overcharging can only take place if counterions adsorb superequivalently and this can occur only when they have a specific attraction for the surface, i.e., a chemical attraction in addition to the Coulombic one. The charge- and potential distribution in double layers for this situation has been discussed in sec. II.3.6c and d; also see fig. II.3.20. There is much empirical information on superequivalency and overcharging, but only some of it is obtained under sufficiently defined conditions (pH, c salt , effect of the nature of the adsorbent) to be helpful to our understanding. We now summarize the evidence in terms of some general rules. 11
J. Lyklema, Adv. Colloid Interface Set 100-102 (2003) 1.
PAIR INTERACTION
3.113
(i) Very weak specific adsorption. It is probably safe to state that no ion is really one hundred percent indifferent. NaF on mercury approaches this condition. Alkaliions on Agl (previous subsection) and on most oxides, are weakly specifically adsorbing. The adsorption Gibbs energies are, at most, a few kT. Often, the binding only takes place in pairs with surface charge groups ( Na+ - I~S on Agl, Na+ - ~ O - S on oxides, where S stands for the groups on the surface of the solid). Insofar as such ions do not adsorb measurably on the uncharged surface, they cannot affect the p.z.c; but when the surface charge is high, specificity shows up in the double layer capacitance and in the c.c.c. See figs. II.3.41, 43 and 63. Overcharging does not occur. (ii) Moderate specific adsorption. Some alkali ions do adsorb on uncharged surfaces, and hence do shift the p.z.c. We can take as an example, Li+ ions on a-Fe2O3 , probably caused by the possibility of substituting Fe3+-ions in the solid isomorphically. These ions do exhibit overcharging, at least over a short pH-range around the p.z.c. The effect is too minor to detect electrokinetically or by stability experiments. Specific adsorption of this kind is more common for monovalent anions than for monovalent alkali- or alkaline earth cations. This difference is probably mainly caused by their larger radius (and, hence polarizability), but even at the same ionic radius there are substantial differences between the enthalpies and entropies of hydration of cations and anions (table 1.5.4). A typical illustration is given by relatively simple anions such as NO3 which can shift the p.z.c. of Agl; nevertheless it is difficult to stabilize Agl sols that are positively charged by adsorption of the charge-determining Ag+ ions, because of strong co-adsorption of NO3 ions. Differences between monovalent cations and anions also show up with inert 'adsorbents', such as water vapour. It is easier for anions than for cations to enrich the water/vapour interface; consequently anionic lyotropic sequences in the surface pressure are more pronounced (figs. II.3.73, 75, III.4.18). In this connection, it may be repeated that oil drops and other apolar materials tend to become negative with respect to water. The reason is the same: because of their larger polarizability, anions accrue more strongly onto, or into, these materials than do cations. Moderate specific adsorption, as meant here, does lead to overcharging, but for solid surfaces its detectability via electrophoresis or stability studies is hard because it is measurable only under conditions where the sols are unstable. The evidence stems essentially from double layer studies, including the shift of the p.z.c. with c salt which takes place in the opposite direction as the i.e.p., which can be measured by streaming potentials. (iii) Multivalent ions, especially cations, form a class of their own, because overcharging has frequently been observed for them. The problem is that it is not the ion as such which causes the overcharging, but one, or more, hydrolyzed species. Com11
A. Breeuwsma, J. Lyklema, Discuss. Faraday Soc. 52 (1971) 324.
3.114
PAIR INTERACTION
plexes of the type [M£+(OH~) ](xz"y' form in the solution and these can be very surface active1 . An additional factor is that the composition of adsorbed complexes depends on pH and often differs from that in solution2 . The identification of these surface complexes, and their relevance for the interpretation of Schulze-Hardy type phenomena and overcharging, therefore requires much systematic research, first by studying the effect of pH on surface charge, f-potential, and stability. Really comprehensive studies are not known to the present author. Many partial investigations have been carried out with oxides, not least because of their relevance in natural waters and in ore beneficiation, but the problem there is that changes in pH simultaneously influence the surface charge and the composition of the complexes. In this respect, experiments with the classic Agl system are helpful because the surface charge is controlled by pAg (and not by pH) whereas the composition of the complexes is fixed by pH (but not by pAg). For this system Overbeek3 , relying on work by Troelstra and Kruyt , describes the status quo in 1950, which has not changed materially since then. The conclusions are: (1) At low pH, Th 4+ and Al3+ -ions do not form complexes in the solution or in the double layer. They coagulate Agl sols with very low c.c.c.'s (0.013 and 0.067 mM according to ref. 3) ) but cannot overcharge. (2) At higher pH, depending on the nature of the metal ion, complexes are formed. Now overcharging is observed at sufficiently high c s a j t , and the sols are restabilized, having become positively charged. Eventually, a second c.c.c. is attained, but now with NO3 as the counterion. In this way, irregular series develop in the coagulation behaviour. (3) For bivalent metal ions, overcharging is observed if they are hydrolyzable and the pH is not too low. Examples: Cd2+ , Zn2+ and Be2+ as their nitrates show it, but Ba2+ does not. (4) There are indications that at very high pH the adsorbability of the hydroxycomplexes decreases, eventually to disappear completely. Some of these trends are sketched in fig. 3.49. It appears that these observations are 'classical' in the sense that they are still representative of the basic phenomena. The present author is not familiar with more recent studies in which stability and electrokinetics for hydrolyzing counterions are systematically studied as a function of c salt and pH for a variety of systems. Overcharging in electrokinetics on its own is a familiar observation. Excellent
11 Much information on complex formation of metal ions and their adsorption, both as a function of pH can be found in W. Stumm, J.J. Morgan, Aquatic Chemistry; Chemical Equilibria and Rates in Natural Waters, 3 rd ed. Wiley (1996). 21 E. Matijevic, J. Colloid Interface Set, 43 (1973) 217. 31 J.Th.G. Overbeek, Stability of Hydrophobic Colloids and Emulsions, in Colloid Science, Vol. I, H.R. Kruyt, Ed., Elsevier (1952) 314. 41 S.A. Troelstra, H.R. Kruyt, Kolloid-Beihefte 54 (1943) 277, 284.
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3.115
Figure 3.49. Sketch of the stability regions for Aglsols in the presence of hydrolyzable multivalent counterions. — high pH, - - - low pH. The values of f and the two c.c.c.'s depend on the nature and valency of the cation.
illustrations for colloidal silica and rutile (TiO2) are given in a series of papers by James and Healy and by James et al.2). Their electrolytes included Cd(NO3) and Co(NO3)2 and convincing evidence was presented that adsorption of hydrolyzed metal ions is responsible for the overcharging, to which we shall return in sec. 3.13a. A recent striking observation concerns electrophoresis on air bubbles31. In NaCl solution £" becomes more negative with increasing pH, the more so the lower c s a l t . Around pH ~ 3 there is an indistinct i.e.p. Addition of CaCl2 enlarges the window where f > 0, but in A1C13 very high positive £"s are found above pH ~ 3. As with Agl, at pH>10 the Al- effect disappears. The interesting phenomenon is that water vapour does not actively attract ions; rather the preferential exclusion of one type of ions (mostly the cations) leads to enrichment of the other type at the interface and, hence, to the most negative ^-potential. Only hydroxylated Al3+ can compete, but at high pH these complexes prefer to stay in the solution. This last phenomenon must, of course, also be operative with Agl and other systems. 3.9k Coagulation by electrolyte mixtures The influence of mixed counterions is a rather subtle phenomenon: if counterion 1 has a certain coagulation propensity, and counterion 2 a higher one, is a 50-50 mixture then 50/50 effective, or more so, or less? Figure 3.50 shows the various possibilities that have been considered. As the matter is only interesting when c.c.c.-l * c.c.c-2 the fractions of the respective c.c.c.'s are plotted on the two axes, to make the figure symmetrical. So, when salt 1 has monovalent counterions and salt 2 bivalent ones, the salt 2 axis is more stretched. The figure must be read as follows: starting from pure salt 2, experiments are carried out with 90%, 80%, ... etc. of c.c.c2, where the concentration of salt 1 (as a fraction of c.c.c.-l) is established to achieve
1
R.O. James, T.W. Hcaly, J. Colloid Interface Sci. 40 (1972) 42, 53, 65. R.O. James, P.J. Stiglich, and T.W. Healy, Faraday Discuss. Roy. Soc. Chem. 59 (1975) 142. 3) C. Yang, T. Dabros, D.Q. Li, J. Czarnecki, and J.H. Masliyah, J. Colloid Interface Sci. 243 (2001) 128. 21
3.116
PAIR INTERACTION
Figure 3.50. Map of the possible trends for coagulation by salt mixtures. (1) Synergism, (2) additivity, (3) superadditivity, (4) antagonism. The axes give the c.c.c.'s as fractions of the c.c.c.'s of the pure electrolyte.
coagulation. Only when c.c.c.-l = c.c.c-2 may additivity be expected. This is the least interesting situation. More generally, the two c.c.c.'s are different, especially so when the counterions have different valencies, or very different specific adsorption Gibbs energies. Older investigations, particularly by the Russian School", gave ambiguous results, partly because they did not know what to look for, and partly because the surface potential (or charge) was not controlled. In the framework of DLVOE theory an important role is played by the specifically adsorbed countercharge. For a particle of fixed surface charge the charge distribution over the two competing counterions at the iHp will be determined by an ion exchange mechanism. For zx = z2 , ^- = Kl2^°2 ' C2
[3.9.12]
K 1 2 =e- ( A G mi-^ G Jn2)/^
[3.9.13]
According to this rule, the Stern layer is, over the entire concentration range, richer in the more strongly specifically adsorbing ion, leading to a higher coagulation propensity, or to a lower c.c.c, with synergism as the generally expected result. For systems of fixed y° , the situation is more subtle because a° can now vary over the range. In fact, titration experiments can be carried out to establish relative preferences. In unpublished experiments by the present author and Mrs. E. Akkerman, it was indeed found that, on negatively charged Agl at given pAg, a° was more negative in RbNO 3 + LiNO3 mixtures than in proportion to their concentration. However, the trend was barely outside experimental error. More systematic stability studies were carried out by Storer 21 . For Agl-sols in MgSO4 + NaNO3 mixtures he
11 21
Yu.M. Glazman, V.M. Barboi, I.M. Dykman, various papers in Kofi. Zhur. (1956-1965|. C.C. Storer, Ph.D. Thesis, University of Bristol (1968).
PAIR INTERACTION
3.117
found superadditivity (curve 3 in fig. 3.50) but for poly(styrene sulfonate) latices he found synergism. Another experimental illustration in a slightly different vein, stems from the field of clay minerals1'2'31, where the cation exchange capacity, c.c.c. (see sec. II.3.10d) of montmorillonite was measured as a function of pH in the presence of Al3+ salts. Here, the £ ratio plays an important role. The additional aspect is that adsorption of hydrolyzed aluminium species takes place, with a pH-dependent composition. More experimental studies are welcome. 3.91 DLVOE theory. Conclusion. Alternatives? As compared to classical DLVO theory, DLVOE covers a number of phenomena that are quite common in practice, and which go far beyond the Poisson-Boltzmann level. These additional phenomena include ion size effects, non-electrostatic (specific) adsorption, and the effect of ions and surfaces on the structure of water. The shortcut to keep all these features tractable was to account for them only in the Stern layer, that is the layer where they dominate. A further advantage is that diffuse theory is only needed for that part of the double layer where the PB premises hold. For this part we can fall back on DLVO, with the improvement that yd(h) follows from regulation. One may perhaps generalize the difference between DLVO and DLVOE in that the former is essentially electrostatic, whereas the latter also exposes the chemistry in the wider sense (see the discussions in sec. 1.5.1). Given our goal of explaining the fundamentals, a number of issues that may be called 'extensions' remained consciously underexposed. To these belong: surface roughness and (chemical) heterogeneity, covalent binding of ions to surfaces, the very specific interactions met in biochemistry and immunochemistry, surface porosity (as with membranes or bacterial cells), adsorbed dipole layers, the presence of adsorbates, and other geometries beyond flat and spherical. However, much of the basic information for the elaboration of these extensions is already available. Regarding the shapes of the particles, provided h> d there is no difference of principle between objects of different geometries, but for very strong curvature the double layer structure may have to be reconsidered. We have already mentioned nanoparticles (sec. 3.7g), and in ch. V.2 we shall discuss polyelectrolytes, i.e., highly charged thin cylinders. It should be re-emphasized that the present theory considers pair interactions only, i.e., the interaction between two colloids, with their double layers, embedded in a large volume of fixed p, T, c salt , pH, etc., but not containing other particles. This is not what is usually considered. Stability studies mostly involve multiparticle systems whose properties do not necessarily reflect the additive sum of pair interactions. Formally, one could write for the pair interaction Gibbs energy in a multiparticle system 11
J.L. Ragland, N.T. Coleman, Soil Sci. Am. Proc. 24 (19601 457. M.T. Kaddah, N.T. Coleman, Soil Sci. Am. Proc. 31 (1967) 328. 31 M.G.M. Bruggenwert, P. Keizer, and P. Koorevaar, Neth. J. Agricult. Sci. 35 (1987) 259. 21
3.118 G[r) = G[r)paiI + G[r,{ri})
PAIR INTERACTION [3.9.14]
where the second term on the r.h.s. stands for the multiparticle contribution, {r(} representing the (averaged) positions of all particles, i. Here, we only consider the first term on the r.h.s. A test for the reliability of c.c.c. as a characteristic of pair interaction is that the experimental value should be independent of the sol concentration. For many systems this has been verified, for instance for Agl sols in the presence of poly(vmyl-alcohol)11 and for latices2'. In more concentrated systems, the countercharge is always shared by a number of particles, and the Donnan salt exclusion is not necessarily the sum of those per pair. This has its consequences for the osmotic pressure and Maxwell stress in the interparticle space. Even the phase behaviour of concentrated colloids is not linearly correlated to the pair interaction: for entropic reasons, phase separation can occur in systems of homodisperse repelling spheres, whereas upon attraction a porous gel can be formed, which prevents the formation of such phases. Because of these collective properties of concentrated systems, we shall treat them separately (ch. 5). In that chapter generalization [3.9.14] will also be discussed from a more fundamental point of view. The (conditional) correctness of DLVOE theory can, of course, only be assessed experimentally. This will be done in sees. 3.12 and 3.13. However, we shall now mention two alternatives that have attracted attention. (i) In order to explain certain aspects of the phase behaviour of concentrated colloids (charged colloids with very extended double layers). Sogami and Ise 3 ' 4 ' added to the DLVO interaction an attractive contribution of electrostatic origin, acting between pairs of colloids at large distances. This concept has been heavily debated. We shall not review the various aspects, but note that the SI model; (1), does not indicate basic flaws in DLVO or DLVOE theory (the validity of our [3.2.6] is not addressed); (2), is based on multiparticle assemblies and; (3), considers electrolyte concentrations that are so low that the double layer only contains counterions. Under these conditions, pairs of double layers become ill-defined. We shall not discuss this but come back to systems with very little electrolyte in sec. 3.11. (ii) Models have been developed to improve the Poisson-Boltzmann equation by accounting for ion size and ion correlation effects, and by avoiding the mean field assumption. We have considered these already for isolated double layers in sec. II.3.6a and b, where we concluded that deviations from Gouy-Chapman behaviour become significant only under conditions where simple Stern corrections do the job perfectly. So, for practical reasons there is no reason for going into such discrete ion models,
11
G.J. Fleeer, J. Lyklema, J. Colloid Interface Sci. 55 (1976) 238. C.N. Bcnslcy, R.J. Hunter, J. Colloid Interface Sci. 88 (1982) 546. 3) I. Sogami, N. Ise, J. Chem. Phys. 81 (1984) 6320. 4) Ordering and Phase Transitions in Charged Colloids, A.K. Arora, B.V.R. Tata, Editors. VCH Publishers (1996). Also see, E. Ruckenstein, Adv. Colloid Interface Sci. 75 (1998) 169. 21
PAIR INTERACTION
3.119
the more so because they are mathematically involved. However, academic reasons are a different matter. There is no doubt that ion correlations do exist and that they can explain a number of phenomena, such as overcharging. In particular, such discrete charge models are needed under conditions where the surface charges may no longer be considered as smeared out and for multivalent ions. However, the problem is that such phenomena have been predicted only under conditions where traditional chemical interactions also satisfactorily account for them. A recent illustration is given in a review by Quesada-Perez et al.1' on overcharging in colloids. We conclude that there are definite academic reasons for remaining aware of discrete-ion theories with their conesquences for ion correlation phenomena and overcharging. At present, the challenge is to carry out experimental investigations in such a way that specific ion binding is unambiguously excluded. Considering these reservations, we shall base our discussions in sees. 3.11, 12 and 13 on DLVOE-type analyses, but in chapter V.2 we shall briefly return to this matter. 3.10 Forced pair interaction By forced Interaction between pairs of colloids we understand interaction under the influence of external fields. Some of these have already been, or will be, considered, (IV.sec. 4.5, 5.3, V.sec. 2.5), but we shall briefly review them for the sake of systematics. Magnetic fields will be introduced as a new feature. 3.10a Gravity When isolated colloidal particles are subjected to gravity, or when they are (ultra-) centrifuged, they will sediment at a rate that is primarily determined by the density difference between particle and medium, and the particle's radius. At issue is now the situation where one particle is fixed, say, because it has settled onto the bottom of the vial, and the other is pushed down onto it. What happens depends on the height of the energy barrier as compared to the potential energy resulting from gravity. When the barrier is high enough the particles may be compressed, but upon 'switching off the external force (turning the vial) they will redisperse. If the barriers are too low, they may stick together, depending on the depth of the primary minimum. In fig. 3.51 the interaction curve, in the absence and presence of the potential energy G
grav—K-mw)9h
[3-10.11
caused by gravity, is sketched. Here, m and mw are the masses of the particle and
M. Quesada-Perez, E. Gonzales-Tovar, A. Martin-Molina, M. Lozada-Cassou, and R. HildalgoAlvarez, Chemphys Physchem 4 (2003) 234.
3.120
PAIR INTERACTION Figure 3 . 5 1 .
DLVOE
interaction curves with gravity (or centrifugational) contribution superimposed (dashed curve).
Coagulation occurs when the maximum is sufficiently lowered. Just as with common coagulation, there is a transition zone between stability and instability. In the sketched situation the primary minimum deepens slightly, but there may be more subtle cases as in figs 3.46 and 3.47 where gravity just suffices to keep the particles together. In practice, sedimentation is often investigated for concentrated systems. This augments Gffav . A particle in a sediment feels the collective load of the particles above it. Hence, in [3.10.1] the r.h.s. must be multiplied by the number of superstanding particles and a kind of packing factor, indicating the contribution in the vertical direction. In sec. V.8.3d, sedimentation in concentrated emulsions will be treated. Other, early experiments were reported by El-Aasser and Robertson11 (latices), Rohrsetzer et al.2) (Prussian blue) and Melville et al.3) (Agl-sols). The practical interest is enormous; one can think of the deposition of layers, the formation and ageing of slurries, the manufacture of television screens, the shelf stability of paint dispersions, etc. In sec. 2.3d sedimentation as a means of determining particle masses was discussed. The latex example is important as a step towards coalescence and the subsequent formation of a coating. More fundamental are questions regarding reversibility after decompressions (completely or partly? Sintering? If yes, what is its time scale, i.e. how rapidly and by what steps can sediments solidify? ...). Surface roughness will also play its role. As a counterpart, colloid interaction at zero gravity, or under microgravity may be mentioned. We refer to a paper by Folkersma et al.41, where other references can be found.
11 2) 31 4)
M.S. El-Aasser, A.A. Robertson, J. Colloid Interface Sci. 3 6 (1971) 86. S. Rohrsetzer, I. Kerek, and E. Wolfram, Kollold-Z.Z. Polym. 2 4 5 (1971) 529. J.B. Melville, E. Willis, and A.L. Smith, Trans. Faraday Soc. I 6 8 (1972) 450. R. Folkersma, A.J.G. van Diemen, and H.N. Stein, Adv. Colloid Interface Sci. 8 3 (1999) 71.
PAIR INTERACTION
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3.10b Electrical and mechanical forces Because these two types of force often operate together, we describe them in combination. The application of an electric field E to a charged colloid leads to electrophoresis. For two identical particles in each other's neighbourhood, not much changes because the electrophoretic mobility u of the two is the same. This remains so when the two have different radii, provided u is still in the Helmholtz-Smoluchowski regime. When the f -potentials are different, so that one moves faster than the other, the interaction is essentially of a hydrodynamic nature: then, the trajectories have to be analyzed11. Similarly, when the colloid is coagulated, application of E leads to electro-osmosis. All of this has been described extensively in chapter II.4 on electrokinetics. New features arise when E becomes very high (^ 106 V m" 1 ), particularly when the conductivity of the dispersed particles substantially exceeds that of the dispersion medium. Then, the application of the field leads to a significant increase in the viscosity. Switching the field off results in a reduction of the viscosity. This phenomenon is called electrorheology, and the sol becomes an electrorheologicalfluid: it is caused by particle alignment after dipoles are induced. In dilute sols chain-like structures are typically formed, whereas in concentrated sols, rather networks are created. The stronger E , the longer the strings and the larger the aggregates. Theory for forced interaction is much more complicated than for the gravity case because the forces are not additive in an applied electric field. The original homogeneous isotropic field and the superimposed dipole field are both screened by the same countercharges of which the displacement is difficult to compute. Moreover, the dipole induced in one particle of the pair in turn also affects the dipole moment in the other. The electrical repulsion Gibbs energy decays as aln(l - t~Ktx), see [3.7.19a], whereas the interaction energy between unscreened dipoles decays as a6 /he. This follows from the considerations in sec. 1.4.4. Some simplification can be achieved by applying AC fields of high frequency; in that case the relaxation of the diffuse parts of the double layer is suppressed. In this connection the phenomenon of dielectrophoresis may also be mentioned. It was introduced in sec. II.4.5a; recall that it means particle displacement in heterogeneous fields. When the dielectrophoretic mobilities of pairs of particles are different, pattern formation of the particles may result. For an illustration and elaboration see21. Some literature examples include an experimental study by Bezruk et al.3) on TiO2 and a styrene-methacrylate copolymer in butyl alcohol and heptane. Aggregates were formed whose size decreased with frequency and with the LiCl concentration. Adriani 11
For an illustration see S.C. Nichols, M. Loewenberg, and R.H. Davis, J. Colloid Interface Sci. 176 (1995) 342. 21 M. Trau, S. Sankaran, D.A. Savillc, and I.A. Aksay, Langmuir 11 (1995) 4665. 31 V.I. Bezruk, A.N. Lazarev, V.A. Malov, and O.G. Us'yarov, Koll. Zhur. 34 (1972) 321 (transl. 276).
3.122
PAIR INTERACTION
and Gast11 studied dilute, sterically stabilized polylmethyl methacrylate) latices. They determined the chain length distribution; the average length as a function of E increases rapidly above a certain threshold. When chains form, the dynamics also become a determining factor. Anderson21 gave a theoretical approach to the dipolar interaction force and estimated the yield stress. For a review see ref. 3I. Dukhin et al. investigated the hydrodynamics of pairwise interaction in an electric field41. Gong and Marr5) described confined colloids of which the ordering could be controlled by the application of an electric field: as before, the polarization of the particles is responsible for this phenomenon. Marshall et al.61 reported the influence of applied electric fields on concentrated suspensions of poly (methacrylate} in a chlorinated hydrocarbon. These suspensions are shear-thinning, implying that mechanical forces can help to get the particles out of the secondary minimum. At low shear rate application of the field raises the viscosity drastically, upon three orders of magnitude. However, at very high shear rate, where the viscosity is low, the field effect is virtually absent. Then all the particles are, and remain, apart. Regarding shear, distinction can be made between its effect on pair interaction and the reverse phenomenon, the influence of particle interaction on rheology. Application of strong shear can, in principle, deform double layers and hence affect the pair interaction. A more general phenomenon is the influence of shear on the relative motion of liquid with respect to a solid, and on the rate of approach of particles without double layer distortion. The former gives rise to electro-osmosis and related electrokinetic phenomena, already treated in chapter II.4. For unstable sols the latter leads to, orthokinetic coagulation, which will be treated in sec. 4.5b The most familiar reverse phenomena are the primary and secondary electroviscous effects, both accounting for the increase of the viscosity of dilute stable sols by the presence of electric double layers. For very dilute stable systems the viscosity increase of the fluid by the presence of a dispersed phase obeys the Einstein equation
n=1w[l + l
13.10.2]
where r]w is the viscosity of the liquid (~ water) and (p the volume fraction of the dispersed phase. The primary effect accounts for the influence of double layers on the 11
P. Adriani, A.P. Gast, Faraday Discuss. Chem. Soc. 90 (1990) 17. R.A. Anderson, Langmuir 10 (1994) 2917. 31 A.P. Gast, C.F. Zukoshi, Adv. Colloid Interface Set 30 (1989) 153. 41 A.S. Dukhin, Roll. Zhur. 48 (1986) 439; 49 (1987) 858; A.S. Dukhin, V.A. Murtsoukin, Koll. Zhur. 48 (1986) 240; A.S. Dukhin, O.G. Us'yarov, Koll. Zhur. 49 (1987) 1055; N.I. Gamayunov, V.A. Murtsovkin, and A.S. Dukhin, Koll. Zhur. 48 (1986) 233. 51 T. Gong, D.W.M. Marr, Langmuir 17 (2001) 2301. 61 L. Marshall, C.F. Zukoski IV, and J.W. Goodwin, J. Chem. Soc. Faraday Trans. (I) 85 (1989) 2785. 21
PAIR INTERACTION
3.123
factor of (p, whereas, when the solution Is not so dilute that pair interactions play a non-negligible role, a secondary electroviscous term has to be added which, in line with the previous sections, is proportional with tp1. More details and elaboration will follow in sec. 6.9, where more information on the relation between particle interaction and rheology can also be found.
3.10c Magnetic forces Between electric and magnetic interactions in colloidal systems there are a number of analogies and some basic differences. 1. An analogy is that in electromagnetic waves the propagation of the electric and magnetic field obeys similar laws, compare [1.7.1.1b] with [7.1.2]. For static fields the way in which the medium is accounted for is also similar. 2. A basic differences is that, unlike the electrostatic case, the north and south pole of a magnetic dipole always remain paired, whereas in electric double layers free ( + ) and ( - ) charges can be found far apart. 3. A consequence of the permanence of north-south poles is that forces involving magnetic dipoles are directional, whereas electric forces emanating from a point charge are simply radial. Hence, to describe magnetic phenomena vectorial notations are often required. 4. Electric double layers are ubiquitous in aqueous media; special measures have to be taken if one needs a fully uncharged particle. On the other hand, magnetic phenomena are only significant for a very limited number of materials. For this reason, in FICS electrostatics receive much attention whereas magnetics are considered as a speciality, though with interesting intrinsic properties. Let us elaborate this systematically. Regarding electric fields in a medium, recall from sec. I.4.5f that in a medium of permittivity eoe the dielectric displacement D and the field strength E are related through [1.4.5.12]
D = eoeE
[3.10.3]
Here, D (in Cm" 2 ) and E (Vm" 1 ) have different dimensions, this difference being accounted for by the natural constant eo = 8.85 x 10~12 C2 N"1 m 2 = Fm"' (many more decimals in I. app. 1), also known as the permittivity of free space. The extent to which a medium can polarize under the influence of the applied field is expressed by e (dimensionless), the relative dielectric constant. The corresponding magnetic field equation reads B = u0MH
[3.10.4]
where B is the magnetic induction (in T, for tesla; = NA"1 m"1 = JA^m" 2 = Vsirr 2 ,
3.124
PAIR INTERACTION
H the magnetic field strength ( Am"1 = Cs^m" 1 ), ju° the permeability of a vacuum -
4TI xlO" 7 NA"2 = VC"^ 2 !!!" 1 and ju is the dimensionless (relative) permeability
characterizing the magnetization propensity of the medium. In passing it may be noted that there is much confusion in the codification and terminology of electromagnetism11. We largely follow the IUPAC recommendation mentioned in sec. 1.5a, particularly those by Mills et al. These are also in line with the approach by Rosensweig2', whose book is recommended for further reading. The possibility that materials can become magnetized depends on the presence of elementary magnetic dipoles, or their creation by an applied field, and quantitatively, on the extent to which they are aligned. On the basis of the value of ju materials can roughly be classified into three categories31. Media can be: (i) diamagnetic
[JU<1).
Diamagnetism is usually a small effect, opposing the
applied field and caused by a change in the orbital motion of electrons. For our purpose this category does not play a role. (ii) paramagnetic
(//>1), stemming from the alignment of atomic or molecular
magnetic dipoles by the torque resulting from H. Paramagnetism decreases with temperature because Brownian motion of the particles opposes the alignment. Typically, in paramagnetic materials there is no long-range ordering unless a field is applied. (iii) ferromagnetic,
/ / » 1, up to 104 , characterized by domain formation,
i.e.
assemblies of similarly oriented dipoles that are already present in the absence of an applied field, and which can be aligned when the material has been previously exposed to a field. A related typical phenomenon is that of hysteresis; upon increasing and decreasing H the magnetization does not follow a reversible path. The phenomenon that domains remain oriented after the field has been switched off is called remanent magnetism. Ferromagnetism is exhibited by iron, cobalt, nickel and many of their oxides and alloys, some rare earths and a few other compounds. When it occurs, its effects are impressive. Ferromagnetism disappears above a certain temperature, the Curie temperature, which is material-specific. In electrostatics the extent to which a medium is polarized is quantified by the polarization P (in Cm"2 = Cmm" 3 , i.e. the induced moment per unit volume). The magnetic equivalent is the magnetization, volume
or magnetic dipole moment per unit
1
M (in A m " ) . F o r electrostatics, from II.4.5.17a]
D=£OE +P
[3.10.5]
whereas
For a comparative discussion on different units see "Which SI?" by P.C. Scholten, J. Magn. Magn. Mater. 149 (1995) 57. 21 R.E. Rosensweig, Ferrohydrodynamics, Cambridge Univ. Press (1985), Dover reprint (1997). Rosensweig (loc. cit.) discerns more types, including antiferromagnetism and ferrimagnetism.
PAIR INTERACTION
3.125
B = MOH + MOM
[3.10.6]
is the magnetic equivalent. In electricity P has the same dimensions as D whereas in magnetics M dimensionally corresponds to H 1J. Electric polarization is related to the polarizing field through the (electrical) susceptibility xe °f which the defining equation is P = eoXtE = ^ -
[3.10.7]
The magnetic equivalent is
M=x H=Xj
^
[3 10 81
jjr
- -
where ZmaR i s the (dimensionless) magnetic susceptibility. Electrostatically, a particle can be polarized by an applied field according to [1.4.4.5 and 6] p i n d = aE = 47ieoa3 E
[Cm]
[3.10.9]
The magnetic equivalent is relevant for paramagnetic particles. It reads
mi d =/
" ^6 a f m a g B
[Am2
'
[3 10 101
'-
Here, we write ct mag for the radius of the particle to become magnetized in distinction from a for colloidal spheres; in many applications magnetic cores are embedded in larger colloidal particles (see fig. 3.52 below). As in electrostatics, the magnetization of a system equals the magnetic moment per volume unit M = Vm x orientation factor. In electrostatics %t = e - 1 ; likewise Zmag=A-l
[3.10.11]
For diamagnetic materials Zmaa <0, for paramagnetics it is O(10~3) and for ferromagnetics O(103), depending on H. The next step is discussing magnetic colloids or colloidal ferrojluids, i.e., sols in which the particles can be magnetized or already carry a magnetic dipole. The issue is very relevant for practice, production of magnetic data storage devices being one of the current interests. Preparation and stabilization of such sols requires a certain craftsmanship. A specimen has been given in sec. 2.4d. One of the main issues is size control. Simple comminution of ferromagnetic materials by prolonged milling and dispersion in water or oil has its problems: the resulting particles are heterodisperse and mostly the attraction between them is so strong that aggregation and precipitation
11
In [1.7.2.6] M had a different dimension, but this had no further consequences.
3.126
PAIR INTERACTION
ensues. Stabilization can be achieved sterically, or by anchored surfactant layers and this is relatively more easy for smaller particles. Small particle size implies that the magnets in it are mostly single-domain, with a size in the order of 10 nm. This contributes to preparing a better defined system. Nowadays it is possible to prepare colloidal ferrofluids that, depending on conditions and the presence of external fields, can be stable, with liquid flow ability. Such sols are different from the 'magnetic fluids' that are in use for clutches and brakes; these consist of slurries in oil of larger multidomain iron particles; the system solidifies upon application of a magnetic field. As for the types of magnetic colloids that have been made, let us give a few illustrations. Buske et al. L2) prepared dispersions of magnetite (Fe 3 O 4 ) in water or oil by reacting a mixture of ferrous-ferric solution with concentrated ammonia at low temperature. An amorphous precipitate was formed, which was converted to crystalline Fe 3 O 4 by heating. Growth was stopped by adding fatty acids, acting as stabilizers. The resulting particles were ellipsoids with a size ratio below 2:1 and sized between 5 and 15 nm. This recipe was modified by Shen et al.31 and improved by van Ewijk et al.4), who also synthesized iron oxide cores embedded in silica spheres. The idea is that tetra-ethoxysilane is hydrolyzed and polymerized on the surface of the magnetite particles, thus stabilizing them. The resulting particles behave colloidally as silica sols with magnetic hearts. In passing, it is also feasible to bind the magnetic material to the surfaces of SiO2 particles. Magnetic cores can also be incorporated in latex particles. Such latices are even commercially available and find widespread application. Step three is: what happens when magnetic, or magnetisable, particles interact and when an external field is applied? That depends on the nature of the magnetism (paramagnetic or ferromagnetic?), on the action of other interaction forces (repulsive or attractive?) and on the concentration of the system (pair or multipair interactions?). Figure 3.52 sketches what happens when a field H is applied. Diamagnetic particles (top) carry no magnetic dipoles in the absence of a field, but dipoles are induced when the field is applied. Because of the low susceptibility rather high fields are needed to achieve polarization. Single-domain ferromagnetic particles (bottom) already carry dipoles in the absence of a field but these are randomly distributed when the system is dilute. Application of the field leads to polarization by orientation. These dipoles are much stronger than in case (a); the difference is larger than suggested by the lengths of the arrows. Orientation is counteracted by thermal motion. At low H the magnetization M is proportional to H , but with increasing H it bends off to 11
N. Buske, H. Sonntag, and T. Gotze, Colloid Surf. 12 (1984) 195. N. Buske, H. Sonntag, Mater. Set Forum 25-26 (1988) 111. 31 L.F. Shen, A. Stachowlak, S-E.K. Fateen, P.E. Laibinis, and T.H. Hatton, Langmuir 17 (2001) 288. 41 G.A. van Ewijk, G.J. Vroege, and A.P. Philipse, J. Magn. Magn. Matter 201 (1999) 31. 21
PAIR INTERACTION
3.127
Figure 3.52. Schematic presentation of the influence of an applied field on paramagnetic (a) and singledomain ferromagnetic particles (b), leading to superparamagnetic behaviour of the entire sol. The hatched shells suggest protecting layers (polymeric, surfactant, silica ...) within which the iron oxide core is embedded.
strive for a plateau in which all dipoles are parallel in the direction of H . With respect to the terminology, in a system like fig. 3.52 all individual particles are ferromagnetic, but the system as a whole behaves as if it were paramagnetic. Sometimes the term superparamagnetic is used. The consequences for stability depend on the degree of orientation and the interparticle distance, but it is immediately seen that particle strings can be formed. For a quantitative analysis, recall from [1.4.4.3] that the energy of an electric dipole with moment p in an electric field E is given by u = -p-E = -Epcos0
[3.10.12]
where 9 is the angle between p and E . Similarly, for a magnetic dipole the moment m follows from u
= -m-B = -mBcos0= -fiom-H
[3.10.13]
This can be taken as the defining equation for m (IUPAC). The SI units of m are [ Am2 or JT" 1 ]. The electric field resulting from an electric dipole, measured at a distance r from the half-way distance between the two charges is [1.4.4.4] E(r) =
5iP^
£_^
[3.10.14]
3.128
PAIR INTERACTION
where the first term on the r.h.s. is the component of E in the r direction, the second that in the direction of p . For a magnetic dipole, represented by a cylinder of length d, area a d and surface pole density of ±pm , Rosensweig11 derived for the magnetic equivalent of [3.10.14] H(r) ~ P m C t d Q [3cos0r-dl 47y/or3 valid for r»d. and um
[3.10.15]
Here, d is a dimensionless unit vector. As cos0=d-r, pm = //oM
= add , the volume of the magnetic dipole, [3.10.15] can also be written as
H(r)=
Vm g
l [3(d-r)r-d]
[3.10.16]
So, H and M have the same dimension [ Am"1 in SI units], the spatial position is determined by r and the decay is again proportional with r"3 . According to Rosensweig, for two (ideal) dipoles ml and m 2 the interaction is generally U
mag =-M°™S2
[3K-r)(m2-rHml-m2)]
[3.10.17]
where the absolute value is given by the coefficient before the square brackets, the directional part by the term in square brackets, the bold symbols represent dimensionless unit vectors, and the dot products represent cosines. For instance, for two identical parallel dipole moments m^ = m2 = m , m 1 -m 2 =l and, if 6 is the angle between m and r , U
mag=-^[3cos^-l]
[3.10.18]
For ferromagnets m is the permanent dipole moment, whereas for paramagnetic materials m follows from [3.10.10]. In passing it is noted that the factor [3cos2 0-1] also occurs in the (second order orientational) ordering parameter, see for instance [1.6.5.58], [III.3.5.1] and [III.3.7.13] The extent of alignment of ferromagnets (not their absolute values) is of course also determined by the applied field. Their average orientation is determined by the ratio " m a g /kT = /uQmH/kT (see [3.10.13]) between (magnetic) energy and (thermal) entropy. For a dilute system the relation between M and H is given by M(H) = M s a t L | ^ L _ J where L(x) is the so-called Langevin Junction
11
Rosensweig, loc. cit.
[3.10.19]
PAIR INTERACTION
3.129
L(x) = cothx-— [3.10.20] x This equation describes the trend mentioned above in the discussion of fig. 3.52b. The first term of the series expansion is linear:
{dH)H=0
3/cT
where p N is the particle number density [ m~3 ]. Sometimes this slope is called the Initial susceptibility, or Langevin susceptibility. For a dlpole, located at the centre of a spherical core of radius a m , m can be assessed from the bulk saturation magnetization M sat using m = (7r/6)Msata^lag, where M sat is about 4.8xlO~ 5 Am~ 1 for magnetite. All of this completes the framework for computing the magnetic interaction (Gibbs) energy, which has to be added to G el , G v d w and other interaction contributions. It Is beyond this book to elaborate this but we note a few trends. Magnetic forces attain their maximum when the dlpoles are aligned, (i.e. In strong fields) and when the magnetic core is large (high a m ). The maximum attractive energy equals -2// o m 2 /47tr3 , as follows from [3.10.8]. This orientation-averaged Interaction is of O(JcT), which Is not high, but the coupling energy in an external field, which obeys [3.10.13] becomes very high in strong fields. The forces decay as r~3 , which at large r has a longer range than G v d w . For larger particles G mag always prevails over G v d w because the former increases more strongly with particle size than the latter. For instance, with Fe3O4 the transition in the dominance is around a ~5 ran. Further elaborations are needed to establish the conditions where repulsive electric interaction can compete with magnetic attraction. When concentrated magnetic colloids have to be stabilized against aggregation, encapsulation in stabilizing polymers, silica, etc. can be very effective. Such layers essentially cut out the short-distance part of the magnetic attraction, more or less as the d-effect in sec. 3.9.4. In equations such as [3.10.17 or 18] this means that the r"3 has to be replaced by h = [ r - 2 ( a - a )]~3, see fig. 3.52. When the magnetic attraction is not very strong, strongly bound surfactants or terminally anchored polymer brushes may serve the same purpose. Through manipulation, parameter situations can be sought for reversible (as a function of H ) secondary minimum coagulation, with the depth determined by Gm + Gel . In the literature a variety of elaborations and applications can be found. By way of illustration we mention studies on osmotic pressure2' and neutron or visible light scattering31, both resulting from many-body interactions. There are numerous studies
11
For some elaborations, see C. Tsouris, T.C. Scott, J. Colloid Interface Sci. 171 (1995) 319. F. Cousin, V. Cabuil, J. Molec. Liquids 83 (1999) 203. 31 L.F. Shen, A. Stachoviak, S-E.K. Fateen, P.E. Laibinis, and T.A. Hatton, Langmuir 17 (2000) 288. 21
3.130
PAIR INTERACTION
on aggregation kinetics1 23>4) . In many situations strings are formed upon application of the field, as in ref. in which the aggregation/deaggregation kinetics is also studied and in ref.6). Applications in the domain of magnetic tapes were already mentioned. Part of the information can be found in the patent literature, not in the least because of the possibility of imprinting memories. Magnetic coagulation of para- and diamagnetic materials is used in the treatment of fine-grained ores: capturing these by coagulation of the tailings can prevent loss of valuable minerals71. A challenging palaeobiological example is that of chain formation of biogenic magnetic materials in bacteria, which keeps them oriented in the magnetic field of the earth. The colloidal properties were studied by Philipse and Maas8). In the field of chemical physics, the phase behaviour of dipolar fluids has drawn attention, partly stimulated by a seminal paper by de Gennes and Pincus91. For an excellent review, see ref.10'. 3.10d Optical forces Strong laser beams can exert a measurable radiation pressure on colloid particles. This phenomenon has recently been exploited to measure interparticle forces. We shall come back to this in sec. 3.12d. 3.11 Pair interactions in non-aqueous media The emphasis in FICS is on aqueous colloids, not in the least because most life phenomena and many industrial processes proceed on water-basis. Regarding pair interactions, the most typical element is the presence of dissociated electrolytes in solution with the related phenomenon of formation and screening of electric double layers. In other media these phenomena are less self-evident which has its conesquences for the pair interaction. Depending on the dielectric permittivity quantitative or even qualitative differences occur and these are the topic of the present section.
11
M.R. Parker, R.P.A.R. van Kleef, H.W. Myron, and P. Wyder, J. Colloid Interface Sci. 101 (1984) 314. 21 J.B-. Hubbard, P.J. Stiles, J. Chem. Phys. 84 (1986) 6955. 31 M. Ozaki, F. Egami, N. Sugiyama, and E. Matijevic, J. Colloid Interface Sci. 126 (1988) 212. 41 J.H.E. Promislow, A.P. Gast, and M. Fcrmigier, J. Chem. Phys. 102 (1995) 5492. 51 M. Fermigier, A.P. Gast, J. Colloid Interface Sci. 154 (1992) 522. 61 K. Butter, P.H.H. Bomans, P.M. Fredcrik, G.J. Vrocge, and A.P. Philipse, Nature (Materials) 2 (2003) 88. 71 J.Svoboda, J. Zofka, J. Colloid Interface Sci. 94 (1983) 37, 81 A.P. Philipse, D. Maas, Langmuir 18 (2002) 9977. 91 P.G. de Gennes, P. Pincus, Phys. Kondens. Mater. 11 (1970) 189. 101 P.I.C. Texeira, J.M. Tavares, and M.M. Telo da Gama, J. Phys. Condens. Matter 12 (2000) R411.
PAIR INTERACTION
3.131
Table 3.4. Pair interactions in various media: Rough classification in terms of dielectric constants. Category
I
Range of dielectric const. £5
Non-polar (= apolar)
II
(25°C) n-Hexane n-Dodecane Benzene p-xylene Dioxane Diethyl ether Dimethyl ether
1.89 2.02 2.27 2.3 2.2 4.3
Very low ion concentration, low o~°, substantial y/° ~ g, very extended double layers. Difficult to stabilize electrostatically.
5.0
Some electrolyte dissociation, may be enough to increase the electrostatic interparticle repulsion.
>11
1 -Hexanol 1-Butanol 1-Ethanol Methanol Nitrobenzene DMF1' Water FA2)
Electrostatic stabilization is supported by electrolyte addition. Degree of dissociation depends on e and the nature of salts and liquids.
-25-78.5
Water + miscible compounds of III
Polar mixtures
1
Typical properties
Liquid non-ionic surfactants such as: C 1 4 EgP 3 (plurafac) C4E2 5.7 C 2 E 3 (trioxitol) 10.1 11.1
Polar
IV
£
-5-11
Low polar
III
Examples
Dimethylformamide;
21
13.3 17.8 24.3 32.6 34.8 36.7 78.5 109.5
As category III with the possibility of preferential adsorption of one of the liquid constituents.
Formamide.
3.11a Classification By way of introduction, it is helpful to classify solvents on the basis of their dielectric constant e. Roughly speaking, four categories can be recognized, as in table 3.4. The borderlines are not sharp; the lower e, the more the specific physicochemical properties of the solvent show up. The physical distinction between the various categories is essentially a matter of screening. Screening of surface charges and of ions in solution, that is. Screening proceeds through two mechanisms:
3.132
PAIR INTERACTION
(i) by the solvent. Quantitatively it is determined by the susceptibility %e defined in [3.10.7] and which equals e-1 . This quantity governs the extent to which the solvent can polarize (= displace charges in it so as to compensate the electric field, created by the ionic charges that are introduced). The issue of situations in the absence of electrolyte is somewhat academic. In practice, such systems are not easily prepared: aqueous systems always contain H+ and OH" ions, often also (hydro-Jcarbonates from the CO2 in the air, and spurious ions leaching from the vessels. The introduced particles may also contain adhered electrolytes that dissolve upon dispersion. Salt-free non-aqueous solutions are easier to make, but then the sols can hardly be stabilized electrostatically. From the academic side the issue has drawn some interest. One of the problems is that the double layer becomes ill-defined because there is no reference potential far away from the surface. Rather the screening is determined by the counterions of the other particles, so that it becomes sol concentration-dependent. The consequence is that pair interactions lose their meaning. Instead, multiparticle interactions have to be considered, requiring different approaches, for instance by using cell models. In the domain of polyelectrolytes, see sec. V.2.2b, this problem has attracted much attention, mostly from the theoretical side. Conformational adjustments as a function of the architecture and line charge of polyelectrolytes are interesting issues. Here, we shall not discuss this any further, but refer to studies by Fuoss et al.11, Alfrey et al.2), Oosawa3' and Ohshima4' on the double layer structure in salt-free media. (ii) by the electrolyte, quantitatively determined by the Debye length K~1 . For categories III and IV the screening is strong and dominated by electrolytes, whereas for category I it is weak and mainly achieved by the solvent. Consequently, in III and IV the development of double layers, the creation of electric interaction between double layers, the effect of electrolytes and the c.c.c. as a stability criterion all remain qualitatively valid, though with significant quantitative differences. The equations of the previous sections remain applicable, mutatis mutandis. However, in non-polar media the situation is vastly different: ions are so poorly screened that they form only to a minor extent, if the electrolyte dissolves at all. Critical coagulation concentrations cannot be defined and the formation of double layers obeys different rules. As this category differs the most markedly from the usual aqueous ones we shall discuss it first. 3.11b Apolar media (group I)*1 Our task is to consider quantitatively the main contributions to G(r) and find 11
R.M. Fuoss, A. Katchalsky, and S. Lifson, Proc. Natl. Acad. Set USA 37 (1951) 579. T. Alfrey, P.W. Berg, and H. Morawetz, J. Polym. Set 7 (1951) 543. 31 F. Oosawa, Polyelectrolytes, Dekker (1971). 41 H. Ohshima, J. Colloid Interface Set 247 (2002) 18. *' In this section the terms 'non-polar' and 'apolar' are considered synonymous. 21
PAIR INTERACTION
3.133
stability criteria. What are the differences compared to aqueous systems? It is expedient to start with G el . It is very difficult to charge dispersed particles in apolar liquids; in fact, it is difficult to disperse them at all. Mostly, stabilizers are needed; low- or high-molecular mass surfactants, of which we now emphasize the former category. Electrophoresis has shown that, under appropriate conditions and with the right chemistry, some charges do develop on the particles. In fact, the wellknown fire hazard by sparking upon pumping of fuel through pipes is another dramatic illustration. This charge accumulation is the result of streaming potentials that can build up to high values in the absence of back-conduction. Two mechanisms can be responsible for that: spontaneous charge transfer between surface and liquid and adsorption of ionic surfactants. Regarding the former mechanism, in the absence of surfactants the ion that is most generally responsible for the charge transfer is the proton. It does not occur freely but only as bound to (molecules or atoms in) the solid (SH) and solution (HB): SHj + B" +± SH + HB <=± S~ + H 2 B +
[3.11.1]
The resulting sign of the surface charge depends on the relative acidity/basicity in the Br0nsted sense of solid and liquid. By comparing the signs of the charges (electrokinetically) for a variety of solid-liquid systems the relative acidities/basicities can be established. A historical example stems from Verwey1' who established electroosmotically that SiO2 is more electronegative than TiO2, which is more negative than ZrO2 in water, acetone, ethanol, methanol, in which each oxide is decreasingly negative in this direction. This trend is in line with the p.z.c.'s in water (II, app. 3). Although the example refers to group III the principle persists in group I. More examples can be found in the classical reviews by Lyklema21 and Parfitt and Peacock31 and in the more recent one by Morrison41. The second mechanism, adsorption of ionic surfactants, follows paths that differ substantially from those in aqueous solvents. A surfactant can become ionic only upon dissociation, and dissociation only occurs when the newly formed ions are screened. Screening, in turn, has to be achieved by the solvent only. In apolar media this goes better with bigger ions, because these create weaker electric fields around them. When a surfactant dissociates into, say a small cation and a big anion, as with NaDS or Aerosol AOT51, it is the bigger anion that remains in solution, the smaller cation being driven out to the surface, rendering it positive. As compared to aqueous systems, this is just the other way around. Similarly, surfactants such as tetraalkyl-ammonium bromide tend to charge the particle negatively. 11
E.J.W. Verwey, Rec. Trav. Chim. 6 0 (1941) 625.
21
J. Lyklema, Adv. Colloid Interface Sci. 2 (1968) 67. G.D. Parfitt, J. Peacock, in Surface and Colloid Science, E. Matijevic, Ed., 10 (1978) 163. I.D. Morrison, Colloids Surf. A71 (1993) 1. Sodium di-(2-ethylhexyl) sulfosuccinate.
31 41 51
3.134
PAIR INTERACTION
These are the two basic mechanisms, but in practice11 these trends are often obscured by additional processes, of which some are difficult to control. One Is the surface composition of the particle; depending on Its way of preparation, the surface may contain inadvertent chemicals Interfering with the adsorption mechanism or donor-acceptor mechanism. As a result, it is sometimes hard to achieve reproducibility between investigations with the same material, but of different history. Another factor is the presence of water. Trace amounts can have a drastic influence on the fpotential even in amounts that are below the detectability of classical analytical techniques, like the Karl Fischer titration. The effect depends on the nature of the liquid, but the trend is that water adsorbs onto the particles, overtaking the chargeforming mechanism. Results are ambiguous: water can improve the charge formation but also screen it because it allows more counterions to adsorb. As the former trend prevails at low water concentration cw the latter at high c w , f(cw) graphs with maxima are sometimes observed. For more information, see the reviews mentioned and refs. 23) . Measuring (-potentials in non-polar media requires special precautions. The low conductivity makes it difficult to concentrate the applied field (in electrophoresis, electro-osmosis) entirely to the system and avoid stray currents. Moreover, suppression of electrode polarization is an issue, and so is the presence of water. Solutions to such problems were already elaborated by van der Minne and Hermanie41. For the conversion of mobilities into £-potentials, tea must be known, which poses two more problems. The first is that colloids in apolar media are often partly aggregated (see below), so that the aggregate is the moving entity. The second is the identification and counting of the ions present. Sometimes elemental analysis is needed but conductivity measurements are often helpful because the conductivity K is related to the concentration ci of the ionic species that are needed to compute K , see [1.6.6.8]: K =
Xi;lici
[3.11.2]
where A( is the molar conductivity of ion i. For a mixture of ions this sum may still be unsolvable, but in non-polar media there are often not that many dissociated species. In particular, for solutions of only one surfactant the dominant species is the surfactant ion. The molar conductivity can be assessed from the corresponding quantity in water, using Walden's rule,
Aif^ /lj(w) 1
=
^ L
,3.11.3,
?7(solv.)
' See previous page, footnotes 2-4. A. Kitahara, Non-aqueous Systems, in Electrical Phenomena at Interfaces, A. Kitahara, and A. Watanabe, Eds., Marcel Dekker (1984) chapter 5. 31 M.E. Labib, Colloids Surf. 29 (1988) 293. 4) J.L. van der Minne, P.H.J. Hermanie, J. Colloid Sci. 7 (1952) 600; ibid 8 (1953) 38.
PAIR INTERACTION
3.135
Figure 3.53. Comparison between the potential distribution in an aqueous (w) and a non-polar (np) double layer. Schematic.
Admittedly, Walden's rule is not good for very small ions, but in the present context only bigger ions count. Kitahara (loc. cit.) has given some illustrative conductivity data. Still another idea for assessing K , is by measuring the double layer relaxation time t, which can nowadays be carried out electroacoustically. Setting r~(Dk 2 )"' see [1.6.6.31], K can be computed if D is known. Having mastered these problems and obtained reliable f-potentials, the next step is much easier than in aqueous media: the virtual absence of screening electrolyte ( K —> 0 ) makes the equations for the potential distribution and the pair interaction simpler in that unscreened Coulomb interactions are acceptable replacements for the Gouy-Stern interpretation. See fig. 3.53. Here, the assumption is made that in both cases the surface charge sits on the solid proper. Had, in the np case the charge originated from ion adsorption, it would have been located at the i.H.p. For the potential distribution this would not have made a significant difference. In fact, for the non-polar case y/° ~ £. The surface charge ( a°, or a{, depending again on the chargedetermining mechanisms) equals -crek . For a spherical particle we may use [3.7.8a] for the total surface charge. Furthermore,
{a + d) For K —> 0
{a + d)
Gel(r) is well-represented by Coulomb's law.
Gp](r) = l/ H (r) = —2
4neoer
=
2
L_
r
[3.11.5]
3.136
PAIR INTERACTION
and for the decay \f/(r) in a single double layer we may use [II.3.5.50], W)
=
^ - i E l * e-^r-(a+d»]
[3.11.6]
for K —> 0 this reduces to ¥(r)
= i;(^L
[3.H.7]
r which again is just the Coulomb decay as Is easily verified (Coulomb: y/(r) = g/4%eo£r , use [3.7.8a] and [3.11.4]). The physical meaning of these equations is twofold: (i) The low e in the numerator of [3.11.4] implies that even low values of a° (or CT* ) lead to relatively high potentials; double layers in apolar media have low capacitances. Experiments confirm this; f-potentials in hexane, toluene, etc., are of the same order of magnitude as those in water. (ii) The slow decay of [3.11.7] means that the interaction force (low disjoining pressure if measured between parallel plates) is low, but that it has a long range. Unless the sols are extremely dilute the particles 'sit on each other's double layers'. Otherwise stated; from a stability point of view such sols behave as if they are concentrated. In turn, this insight has two consequences: (a) It can be entropically favourable if the particles spontaneously aggregate to a certain extent, the agglomerates playing the role of kinetic units; they are far enough apart to have their thermal degrees of freedom not restricted by their colleagues. In fact, such agglomerates are often observed. For electrophoresis, aggregates can act as the 'kinetic units'. Note that such aggregation spontaneously makes the sols heterodisperse. (b) Stability-wise, in concentrated systems G(r) has to be extended by a multi-pair contribution, as in [3.9.14]. Overall stability and phase behaviour now require a more general treatment, as will be given in chapter 5. Only when the sol may be represented as a dilute collection of aggregates will the pair interactions dominate. For concentrated systems there is mostly no unambiguous route to define stability in terms of pair interactions only. We shall come back to this in sec. 3.12. and in chapter 5, but state already that techniques like particle counting, rheology, sedimentation and light scattering may be used as a replacement of (or in addition to) the determination of c.c.c.-values. Correlation of stability with C, -potentials proves the role of electrostatics in interaction; its absence indicates the prevalence of other contributions; often that of stabilizing adsorbed surfactant or polymer layers. This takes us to the other contributions to G(r). With respect to Van der Waals forces the situation does not differ much from that in water. Of course, the Hamaker constant A must be adjusted and often the influence of protective layers adsorbed on A
PAIR INTERACTION
3.137
has to be considered. New vista can be opened by adjusting the compositions of solvent, particle and protective layer in such a way that A of one of the combinations is zero {contrast matching). Solvent structure forces are ubiquitous for fluids against a hard wall, so they also exist in non-polar liquids. In fact, as such fluids often consist of big molecules, the phenomenon can even become more prominent. Recall that in fig. II.2.2, we already gave a prototype, measured in octamethylcyclotetrasiloxane (OMCTS) between mica surfaces in the surface force apparatus. These seminal experiments have since been confirmed and extended, for instance by the Australian group1'21. We repeat that hard surfaces are required. Adsorption of surfactants may render the surface softer and hence reduce Gsolv s t r , but of course the adsorbate now does provoke other interaction contributions. For the long-range interaction Gsolv s t r has, of course, no consequences. Finally it is noted that c.c.c.-values in apolar media lose their meaning. Instead, other stability criteria have to be invoked, including sedimentation, rheology (is there a yield value?), optical (scattering) and particle counting. We shall return to these issues in sec. 3.12. 3.11c Low polar media (group II) This group constitutes an intermediate between groups I and III/1V and has a number of idiosyncrasies justifying the status of a special category. The distinctive characteristic is that, although the solvents are far from polar, enough electrolytes dissolve and dissociate to create ionic screening, resulting in a steeper potential decay than in the previous category and hence in a substantial electrostatic pair repulsion (the force, proportional to the slope of the Gibbs energy-distance curve, increases). The quantitative argument is that K1 = const. c s / e; if, with respect to group I, e is say doubled, but c s increases thousandfold K2 increases with a factor of 500. An unusual phenomenon ensues: higher electrolyte concentrations lead to increased electric repulsion. So far systematic data only refer to liquid non-ionic surfactants as the solvent, but it is likely that later more representatives of this group will be found. Experimental evidence stems to a large extent from a detailed and systematic study by van der Hoeven31. A variety of solids dispersed in liquids, as mentioned in table 3.4 under group II, were studied by elemental analysis, dielectric and conductivity measurements, electrophoresis, sedimentation, confocal scanning laser microscopy, /-ray absorption and rheology (compression modulus, viscosity). (The dielectric permittivity is still too low to measure the stability in terms of c.c.c.-values.) The
11
H.K. Christenson, R.G. Horn, and J.N. Israelachvili, J. Colloid Interface Sci. 88 (1982) 79. H.K. Christenson, D.W.R. Gruen, R.G. Horn, and J.N. Israelachvili, J. Chem. Phys., 87 (1987) 1834. 31 Ph.C. van der Hoeven, J. Lyklema, Adv. Colloid Interface Sci. 42 (1992) 205.
21
3.138
PAIR INTERACTION
Figure 3.54. Influence of the ^-potential on the shear thinning of suspensions of anhydrous Na 2 CO 3 in C n E 6 . (Courtesy Ph.C. van der Hoeven.) wealth of material allowed to establish the role of electrostatics in stability and particle Interaction. Figure 3.54 is an illustration. Here, the solvent was Imbentin, a commercial non-ionic of the C J J E 6
type. The particles were positively charged by the
surfactant dodecylbenzenesulphonic acid (HDBS). In the absence of surfactant the f potential is about -4 mV, the system is unstable and strongly shear thinning. A concentration as low as 0.02 M suffices to increase £ to above +60 mV, with an accompanying reduction of the low shear rate (y) by almost a factor of 100. Addition of HDBS up to 0.12 M reduces the absolute value of C, somewhat; all of this is reflected in rj{y) . At very high y the particles are thermodynamically kept apart, irrespective of the pair interaction. This is one of the many illustrations of the electric stabilization in such systems.
3.1 Id Polar liquids To a large extent stability of these classes III and IV is similar to that in water. The solubility of electrolytes is mostly high enough to measure and interpret c.c.c's. It is readily verified that in equations like [3.9.10 and 11] the constant is proportional to e3 . So, making this explicit, c.c.c. = ^
^
[3.11.81
where e, f and A all vary with the nature of the liquid, or with the composition of the mixture. Additional features can be incomplete dissociation of the electrolyte and, in mixtures, preferential adsorption of one of the components in the Stern layer. Incomplete dissociation of electrolytes that are strong in water is mostly no problem
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for e^35; below that value the most notable effect is that of the reluctance of the solution to carry multivalent ions. For example, in water-ethylene glycol mixtures with a high fraction of the latter Ba(NO3)2 does not completely dissociate into Ba 2+ and NO3 ions but forms Ba(NO3)+ ions. In investigations of the valency influence on c.c.c. this phenomenon has to be taken into account (although for the stability of Agl sols the required Ba(NO3)2 concentration is so low that association is still negligible1'). Preferential adsorption from a mixture obeys the rules set forth in sec. II.2.3, see for instance21. When the preferential adsorption is strong, the adsorptive acquires the status of surfactant. An example is butanol (BuOH), which is only partly soluble in water and hence adsorbs preferentially on Agl. Although BuOH adsorption reduces the (negative) surface charge, the stability is initially increased; apparently counterions are displaced from the iHp to the diffuse layer31. As the BuOH molecules adsorb with the butylgroup towards the surface, and the OH's sticking out, the particle is 'hydrophilized' with as a typical accompanying result, inversion of the lyotropic sequence for alkali ions4'. Experiments of this kind are helpful to extract information on the inner double layer part. It is recalled that subtle differences (particularly referring to ion specificity) are easier observed in c.c.c. measurements than electrokinetically. In a similar vein, Vincent5' reported on the stability of poly(styrene) latices in mixtures of water with urea, methanol, ethanol and propanol. De Rooy et al.6) systematically studied the stability of Ag, Agl, a-FeOOH and Cu-phthalocyanine sols in mixtures of water with methanol, ethanol, isopropanol and acetone. The intention was to find out whether changes in stabilization by the admixed component had to be attributed to either the diffuse or the Stern layer, or to both. For monovalent electrolytes and for water-rich mixtures for all cations changes in the diffuse layer prevail, i.e. the e3 effect of [3.11.8]. An exception was observed for Ag sols in 95% ethanol, where in bivalent and trivalent cation nitrates, irregular series were obtained, as in fig. 3.49, suggesting overcharging. Systematic electrophoresis measurements should confirm that (data not available). Obviously this is a specific feature of chemical origin, perhaps caused by the presence of citrate on the surfaces of Ag particles. 3.12 Measuring pair interactions Having developed detailed theory for the pair interaction G(h) or G{r) between macrobodies or colloidal particles, the logical next step is to verify these functions experimentally. Attempts of doing that date back to the nineteen twenties and thirties, when the nature of these forces was only embryonically understood. A first 11
J. Lyklema, J.N. de Wit, Colloid Polym. Sci. 256 (1978) 1110. G. Machula, I. Dekany, Coll. Surf. 61 (1991) 331. 31 J. Lyklema, Pure Appl. Chem. 48 (1976) 449. 41 B. Vincent, B.H. Bijsterbosch, and J. Lyklema, J. Colloid Interface Sci. 37 (1971) 171. 51 B. Vincent, Adv. Colloid Interface Sci. 42 (1992) 279. 61 N. de Rooy, P.L. de Bruyn, and J.Th.G. Overbeek, J. Colloid Interface Sci. 75 (1980) 542. 21
3.140
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classification of such experiments can be made between (i) particles in a sol and (ii) macrobodies or isolated particles. The first category only yields indirect results: it is impossible to sequester two particles out of the numerous colleagues and identify their interaction force. Instead, some collective property, reflecting the multi-pair interaction is measured, like the critical coagulation concentration (c.c.c), phase behaviour, sedimentation, rheology or light scattering. Nevertheless, such methods have greatly contributed (and still do contribute) to fostering our insight into the factors controlling the interaction. Category (ii) methods are direct but mostly demand more delicate instrumentation and manipulation; for some techniques, like those of the surface force apparatus, the choice of the material (mostly mica) is limited. The more recent scanning force microscopy (SFM) is in this respect less restrictive. In both categories the definition of the system is a recurrent requirement. In this respect category (i) experiments have two advantages, both resulting from their large interfacial area: it is possible to determine surface charges by colloid titration and the presence of surface active impurities is less of a problem because of the better buffering capacity. For both categories £"-potentials can in principle be obtained, (for mica plates in a flat cell) by one of the methods described in chapter II.4a, but if one only has f -potentials one cannot distinguish between, say different modes of interaction (constant potential, constant charge or regulation). We shall classify thin liquid film studies as belonging to group (ii). Most of this will be treated in chapter V.6. In line with the chosen set-up, we shall emphasize solid particles without adsorbates, deferring steric stabilization to chapter V.I and the stability of emulsions to sec. V.8.3. 3.12a Obtaining pair interactions in multiparticle assemblies The two classical methods of measuring colloid stability (in terms of c.c.c. values or rates of coagulation) upon addition of electrolyte do not yield G{r) curves, but they have proven their usefulness in establishing a number of important features. Moreover, they do not require fancy instrumentation. The achievements, persisting till today, include (i) Detection whether stabilization has an electric or non-electric origin; in the former case it is salt-sensitive. (ii) Verification that the rate is obeying Smoluchowski, or Fuchs/Reerink type of kinetics (chapter 4). In dilute systems the initial rate is quadratic in the particle number concentration, with the height of the energy barrier, G max in figures such as 3.41 etc., determining the probability of permanent sticking. (iii) Checking the counterion valency (z) effect. For hydrophobic sols the qualitative Schulze-Hardy (SH) rule is usually observed, although the strict z~6 dependence, predicted by DLVO must not be expected (sec. 3.9.6). When for certain systems deviations from the SH rule are observed, one has to consider interaction mechanisms
PAIR INTERACTION
3.141
beyond DLVO theory. Example: some latices exhibit only a weak z-effect, probably because part of the stabilization is of a steric nature. (iv) Estimating Hamaker constants, using [3.9.11] or alternatives. The constant in that equation can be determined (see the theory in sec. 3.9.6), although the equation to be used depends on particle size and shape, which for heterodisperse systems are not usually known. As long as most of the interaction is dominated by the outer part of the diffuse double layer different modes of interaction (constant potential, constant charge or regulation) cannot be discriminated, and perhaps this is not necessary for the present purpose. The result most sensitively depends on the value of yfi , which can be measured in situ as f. This method is semiquantitatively correct but not very precise: the spread in the Hamaker constants in Lapp.4 (obtained by this approach) is larger than that in app. 3 of the present volume, although there is no systematic difference between the absolute values. (v) Particle size and shape remain problematic. Sophisticated theory can account for the radius or shape-influence on the c.c.c, but the experimental evidence is not unambiguous. For instance, theory predicts linearity with a in logW-logc salt plots11 whereas experimentally the slopes are rather independent of a. Particle surface roughness and dynamic origins have been proposed as explanations. (vi) Measuring lyotropic sequences, indicating the necessity of including Stern layers with ion- and material-specific properties (sec. 3.9.9). From an experimental point these methods have in common that they are not very prone to spurious additives (because of the large surface area, acting as a buffer). The required particle size and £-potential measurements can be carried out in situ. However, as in electrophoresis measurements for better visual or instrumental observation, dilution is needed and care should be taken that the dilution-liquid is contaminant-free and has exactly the same composition as the dialysate of the sol. In passing it is noted that stability ratio measurements have a wider applicability range than just for hydrophobic sols. For instance, W[t) dependencies have been invoked as a stability measure for non-aqueous dispersions, just after having the solid mechanically dispersed in the liquid carrier. All the above-mentioned experimental features were already established long before it was possible to prepare homodisperse sols with well-defined shapes and surface properties. An important step ahead, but still in the domain of multiparticle systems, was a series of experiments by Barclay and Ottewill , aimed at obtaining the filtration pressure as a function of the sol volume, i.e. basically as a function of particle distance. Sols of montmorillonite or latices were compressed, the spacing between the surfaces 11
W is the stability ratio, see sec. 4.3. L.M. Barclay, R.H. Ottewill, Special Discuss. Faraday Soc, I (1970) 138, 164; L. Barclay, A. Harrington and R.H. Ottewill, Kolloid Z.-Z. Polymere 250 (1972) 655; R.H. Ottewill, Progr. Colloid Polum. Set 67 (1980) 71. 21
3.142
PAIR INTERACTION
was measured (from X-ray or from the volume and surface area), so that a primitive /7(h) curve could be obtained. Upon reversal of the compression there is an initial hysteresis, the first pressure always being higher, but after two cycles the process became reversible. Possibly the clay platelets aligned in a parallel fashion, although domain formation could not be excluded. An illustration is given in fig. 3.55. Curves for 10"1 M NaCl were also measured (not shown); they are steeper. There is no indication of a primary minimum, supporting the action of G s o ] v s t r , as in [3.8.9]. No independent titration or electrokinetic data were reported. The montmorrilonite plate surface carries a constant a°, but as the counterion adsorption varies upon compression cf1 and yfi will regulate. The authors fitted the curves with constant values of y/d or a^ and were able to adjust these parameters in such a way that the experimental data were reasonably well reproduced, although the value o^ =-18|iC cm"2 seems to the lower side given the cation exchange capacity which would rather correspond to -10 \xC cm" 2 . So, these experiments cannot discriminate between different modes of interaction. 100
Figure 3.55. /7(h) curves for Na-montmorrilonite particles in 10- 4 MNaCl. - - experiment, theory for constant (Z1 = 250 mV , ibid, for const, cfi = -18 |iC cm" 2 . 1 atm = 1.013 x 105 Nm~ 2 . (Redrawn from Barclay and Ottewill, loc. cit.)
10
0.1 10
15
20
25
30 n
h
Still another route to obtain G[r) runs via radial distribution Junctions g(r) , the central idea being that this quantity is under certain conditions measurable and related to G(r). Academically speaking this is an appealing approach because it addresses, on an advanced level of abstraction, the multiparticle system as such. However, the practical elaboration has so far defied the unambiguous establishment of G{r). The procedure only works for dilute sols of stable homodisperse sols. Let us collect the required equations, starting with the definition [1.3.9.22] g(r) = pN(r)/{pN) where (pN) is the average particle number density. For sufficiently low (pN),
[3.12.1] g(r) is
PAIR INTERACTION
3.143
the only distribution function to be considered; it is a pair distribution function. After [1.3.9.23] g{r) was related to the pair energy via g{r) = exp[-u(r)/kT]. Here u{r) is usually called the 'potential of the mean force', but in reality this is not a good term because u{r) has the dimensions of an energy and not of a potential. To be precise, it is the isothermal reversible work of bringing the two particles from infinity to distance r , including all the work needed to adjust the positions of all the other particles and of ions in the double layers11. Hence, it is a Gibbs, or Helmholtz energy or a grand potential (depending on conditions). For this Gibbs energy we have [3.9.14], which for low pN leads to g(r) = e"~ G(r W /RT
[3.12.2]
if G{r) is expressed per mole of particles. There are three options for establishing g{r) experimentally for homodisperse sols, viz. from light scattering (or scattering of other types of radiation), Theologically and via the osmotic pressure. Basically all three rest on the relation between g[r) and the isothermal compressibility (see Ornstein-Zernike equation, [1.3.9.32]). For the interpretation of light scattering the link is in the structure factor S(q) where q is the scattering vector (see chapter 1.7) qU) = ^
^
l
[3.12.3]
A
and 9 is the scattering angle and X the wavelength of the incident light. Measurement of the scattered intensity as a function of q gives the product of two q- dependent factors, the form factor P{q) and the required structure factor S(q) . For homodisperse sols of spherical particles of radius a , P(q) is known21 =
[3(sinqa-qacosqa)f
[
(qa)3
J
The form factor accounts for the non-interacting part of the scattering, whereas S(q) is a measure of the time-averaged correlation caused by interaction. It is not surprising that S(q) is related to g(r). In appendix I.I 1, see [I.A11.2], we showed that S(q) and g{r) are each other's Fourier transforms31 S(q) = l + 4^7wf [g(r)-l] S m ( g r ) r 2 dr J qr
[3.12.5]
o
For the difference (in electrostatics) between the mean potential and the potential of mean force, see sec. 1.4.3c. 21 A. Guinier, G. Fournet, Small Angle Scattering ofX-Rags, Wiley (1955). For more details and elaborations, see D.J. Cebula, J.W. Goodwin, C.C. Jeffrey, R.H. Ottewill, A. Parentich, and R.A. Richardson, Faraday Discuss. Chem. Soc. 76 (1983) 37.
3.144
PAIR INTERACTION
g(r) = l + — | f [S(q)-l]^Elq2dq 2 27l pN J qr
[3.12.6]
Rajagopalan and Rao discussed the application of these Fourier inversions in practice . For the Theological interpretation the measurable linking quantity is the storage modulus at high frequency G'(co —> °») = G'(°°) . Again for stable dilute sols of homodisperse particles of radius a , m=PNkT
+ ^p2N] g{r]±\!^\dr
o
"-
,3.12.7,23) J
This equation contains dG(r)/dr , that is: the force required to displace particles with respect to each other by mechanical means. As g{r) obeys [3.12.2], G'(°°) is implicitly related to G[r) so that in principle the latter quantity is obtainable. The (osmotic) pressure of a colloid can, in dilute sols, also be related to g(r), using [1.3.9.26]
77 = < p N k T > - ^ j r ^ V ) . 4 ; n - 2 d r 6 J o
dr
[3.12.8]
This equation contains, as [3.12.7] does, an ideal entropic term and a contribution due to interaction, the second virial term, accounting for pair interactions. Equation [3.12.8] may be looked at as the equation of state of the sol. In experiments where the particles are platelets, and where h can be assessed, g(r) is directly accessible; when that is not the case, as for dilute homodisperse sols, the g{r) route via [3.12.8] is needed. All these three approaches in terms of g[r) require extremely accurate experiments on very well defined and characterized particles because via [3.12.2] g{r) depends sensitively on G{r) . As a result, mostly the inverse route is preferred: for G{r) an assumption is made and then the appropriate macroscopic variable predicted, and the result compared with experiment. The significance of such procedures for determining G(r) functions is limited, the more so as the solution is not necessarily unique. In many cases oversimplified interaction functions serve well to 'explain' S(q) (see chapter 5), etc. Accurate light scattering data are e.g. available for ionic micelles. From S(q) spectra G(r) can be computed, from which at best it can be decided that only a fraction 11
R. Rajagopalan, Langmuir 8 (1992) 2898: R. Rajagopalan, K.S. Rao, Phys. Rev. E. 55 (1997)
4423. 21 R.W. Zwanzig, R.D. Mountain, J. Chem. Phys. 43 (1965) 4465. 3) J.W. Goodwin, R.W. Hughes, S.J. Partridge, and C.F. Zukoski, J. Chem. Phys. 85 (1986) 559.
PAIR INTERACTION
3.145
of the countercharge participates actively in the interaction. This information can be obtained in a more direct way, though. Goodwin et al. (toe. cit.) verified [3.12.7] for weakly flocculated latices (high c s a l t , but a C 12 E 6 stabilizing adsorbate) and fitted C, to account for their G'[pN) curves. In addition to these 'systematic' approaches there are a number of more incidental phenomena and approaches from which information on G{r), or some properties of it, is obtainable, including yield values in rheology, electroviscous effect (sec. 6.9), Schiller layers", sediment volumes and rates of (ultra)centrifugation21. Hydrodynamics and electrokinetics have also been proposed. For instance, van de Ven et al.3) inferred interparticle forces from the change in trajectories of two interacting particles in shear flow while Velegol et al.4i5) estimated the attractive force holding a pair of latex particles together by differential electrophoresis. Interesting as such procedures may be, none of these has so far developed into an everyman's tool. 3.12b Interaction between macrobodies, SFA In a number of cases it is possible to measure the force between macroscopic objects (plates, plate and sphere, crossed cylinder and filaments61, etc.). When these macroscopic objects have the same Hamaker constant and the same surface properties as systems of colloidal size, these forces may be converted to those between the corresponding pairs of particles. Measurements of this type are known as 'surface force measurements' (although van der Waals interactions are body forces), and the apparatus are called surface force apparatus, SFA. We shall also use this term. In sec. 1.4.8 the topic was briefly introduced, emphasizing dispersion forces. Since then, SFA have been perfected and extended and comprehensive data on DLVOE and other interactions are now available. For a detailed description of the various variants of SFA see the literature; reviews of this matter7'8'91 help to find the pertinent references and the historical development. Many familiar names contributed to this history, including Deryagin and Overbeek, indicating that the founders of DLVO theory also participated in the quest for experimental verification. The present-day SFA versions all go back to pioneering work by Tabor and Winterton101 and Israelachvili and Adams1'21. The technical difficulties 11
K. Furusawa, S. Hachisu, Sci. of Light 17 (1968 1; J. Colloid Interface Sci. 28 (1968) 167. J.B. Melville, E. Willis, and A.L. Smith, J. Chem. Soc. Faraday Trans I 68 (1972) 450. 31 T.G.M. van de Ven, P. Warszynski, X. Wu, and T. Dabros, Langmuir 10 (1994) 3046. 41 D. Velegol, J.L. Anderson, and S. Garoff, Langmuir 12 (1996) 4103. 51 G.L. Holtzer, D. Velegol, Langmuir 19 (2003) 4090. For a more recent illustration of Deryagin's crossed filament method, see Ya.I. Rabinovich, B.V. Deryagin, Roll. Zhur. 49 (1987) 682; Coll. Surf. 30 (1988) 243. 71 P.F. Luckham, B.A. de L. Costello, Adv. Colloid Interface Sci. 44 (1993) 183. 81 P.M. Claesson, T. Ederth, V. Bergeron, and M.W. Rutland, Adv. Colloid Interface Sci. 67 (1996) 119. 91 V.S.J. Craig, Colloids Surf A129-130 (1997) 75. 101 D. Tabor, R.H.S. Winterton, Proc. Roy. Soc. A312 (1969) 435. 21
3.146
PAIR INTERACTION
that had to be negotiated included at least the following three elements. (i) Preparing surfaces of well-defined geometry. Regarding the latter, as it is extremely difficult to arrange, and keep under close approach, two surfaces exactly parallel, experiments are usually carried out with a sphere and a plate, two spheres (see sec. 12c) or with crossed cylinders, of which the idea goes back to Tomllnson31, long before Hamaker and de Boer derived their equations for the Van der Waals attraction between macrobodies (see chapter 1.4). It was this symmetry that induced Deryagin, who together with Abrikosova pioneered this technique for gold fibers, to conceive the conversion of equations for plates into those for non-flat geometry. This approximation now carries his name, see sec. 1.4.6a. According to this approximation for the interaction force /(hKcrossed cylinders) = /(h)(sphere and plate) = 27KzG(h)(flat plates)
[3.12.9]
where the radius of the sphere is the same as that for the cylinders. The approximation holds for a » h . Because of [3.12.9] measured forces are usually given as J{h)/a . (ii) Preparing molecularly smooth, meticulously clean surfaces. Polishing turned out not to lead to surfaces flatter than a few nm. The solution was to work with in situ cleaved mica sheets. These can be bent and glued onto, say glass rods of desired radius. When the mica is prior to that silvered on the rear side it becomes lightreflecting, which is needed if light interference is used for measuring h . (iii) Measuring f(h). For measurements in a liquid J(h) can be attractive and repulsive at different h 's. Usually / is measured as a force required to deflect a cantilever or compress or extend a spring. Calibration can be done with samples of known weight. However, some balancing (feedback) is needed to prevent the two macrobodies from jumping into contact after an attractive range is reached. This balancing has the problem that different springs are needed for strong forces at shorter displacements, and for weaker forces at larger displacements. In the IsraelachviliAdams apparatus this was solved by a three-stage device; one for coarse control, allowing the surfaces to be positioned within about 1 |im over a range of about 2 cm, a medium-range control (1 nm over about |im) and a third precision-control (0.1 nm over about 200 nm). Distances were measured by multiple beam interferometry4'5'61. This solves the positioning but not the problem with the spring system becoming unstable, to (partly) overcome that problem a device has been developed in which the spring constant can be varied by changing the length of the cantilever. White light is " J.N. Israelachvili, G.E. Adams, Nature 262 (1976) 774. J.N. Israelachvili, G.E. Adams, J. Chem. Soc. Faraday Trans. I 74 (1978) 975. 31 G.A. Tomlinson, Phil. Mag. 6 (1928) 695. 41 S. Tolansky, Multiple Beam Interferometry of Surfaces and Films, Oxford University Press (1949). 51 J.N. Israelachvili, J. Colloid Interface Scl. 44 (1973) 259. 61 R.A. Quon, J.M. Levins, and T.K. Vanderlick, Colloid Interface Scl. 171 (1995) 474. 21
PAIR INTERACTION
3.147
used, which is led through a monochromator, leading to so-called FECOS (fringes of equal chromatic order). In most cases the fringes are curved; from the shapes of the fringes the local curvature can be computed. The position of the tip is used for calculating the separation. Upon contact, under the influence of strong attractive forces, the surfaces (or the glue) can be flattened. There are of course alternatives; for instance, h can also be measured capacitatively11. All told the apparatus allows the measurement of forces between 10"7 and 10"4 N over distances of 0.2-200 nm. In later versions further improvements were realized2'3'4'51. Special reference should be made to systems that employ a bimorph force sensor61. The most familiar apparatus is known as MASIF, for Measurement and Analysis of Surface Interaction Forces. A bimorph is a sandwich of two piezoelectric plates bound together with their polarization directions facing each other. It acts as a cantilever and has a fixed relation between the deflection and the voltage output, allowing the direct detection of the (particle or surface-) displacement with respect to the spring to which it is attached. So virtually it is a force (sensor). The MASIF apparatus is smaller than the SFA and can measure interactions between, say glass beads that were originally molten to render them spherical and smooth. A limitation of the SFA is that it is restricted to mica and a very few other materials (sapphire and glass). Although mica is hydrophilic and molecularly smooth, for the purpose of checking DLVOE theory it would be desirable to also have more materials available. Lacking these, attempts have been made to modify the mica surface, e.g. by plasma treatment or by adsorption of surfactants or polymers. However, such adsorptions give rise to additional interactions, such as steric, mosaic-like (for patchwise adsorption) and dynamical ones (for limited rates of desorption upon approach). Moreover, they make the short-distance Gsolv s t r contribution softer. Given our present aim, we shall not discuss these. Neither shall we analyze the (disputed) long-distance hydrophobic attraction. Even in the absence of intentionally added adsorbates the definition of the double layer on mica poses problems. Mica stems from the mineral muscovite. Muscovites from different sources exhibit small differences in chemical composition. Upon cleaving, thin aluminosilicate layers are formed, kept together by essentially K+ ions. These intracrystalline ions can, to some extent, be exchanged against other ions, e.g. by soaking with (NH 4 ) 2 SO 4 , CaCl2 or BaCl2 7) but mostly under rather extreme conditions (e.g. high temperatures), that are not usually encountered under ambient
11
P. Franz, N. Agrait, and M. Salmeron, Langmuir 12 (1996) 3289. J.N. Israelachvili, Proc. Nail. Acad. Set USA 84 (1987) 4722. 31 J.N. Israelachvili, P.M. McGuiggan J. Mater. Res. 5 (1990) 2223. 41 J. Klein, J. Chem. Soc. Faraday Trans. 179 (1983) 99. 51 J.L. Parker, H.K. Christenson, and B.W. Ninham, Rev. Set Instr. 60 (1989) 3135. 61 J.L. Parker, Langmuir 8 (1992) 551; Progr. Surface Sci. 47 (1994) 205. 71 K. Jasmund, G. Lagaly, Tonminerale und Tone, Steinkopf (1993) chapter 3. 21
3.148
PAIR INTERACTION
SFA conditions. So, in air cleaved mica surfaces consist of a fixed negative surface charge a°, entirely compensated by K+ ions at the iHp:
11
R.P. Mitter, K.S. Rajagopalan, J. Indian Chem. Soc. 25 (1948) 591. G.L. Gaines, J. Phys. Chem. 61 (1957) 1408. 31 J.N. Israelachvili, G.E. Adams, J. Chem. Soc. Faraday Trans (I) 74 (1978) 975. 41 R.M. Pashley (a), J. Colloid Interface Set 80 (1981) 153; (b) ibid, 83 (1981) 531; (c) R.M. Pashley, J.N. Israelachvili, ibid 97 (1984) 446; (d) R.M. Pashley, ibid 102 (1984) 1. 51 J.S. Lyons, D.N. Furlong, and T.W. Healy, Austr. J. Chem. 34 (1981) 1177. These measurements have later been essentially confirmed by P.J. Scales, F. Grieser, and T.W. Healy, Langmuir 6 (1990) 582 and by N. Debacher and R.H. Ottewill, Colloids Surf. 65 (1992) 51. 21
PAIR INTERACTION
-40
3.149
_ mV
-60
Figure 3.56. f -potentials of freshly cleaved mica plates D green mica (three samples), o ruby mica, curve A fused SiO2 , pH = 5.8. (Redrawn from Lyons et al.)
-80
-100
-120
-4
-3
-2 C
KC1
Zeta potentials are more negative than on fused silica,
[3.12.10]
i.e., it is direct in the sense of table 3.3, and therefore inverse in ( and
3.150
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Clearly this phenomenon results from specific adsorption of Al-hydroxides. Once the double layer properties of isolated mica surfaces are established the interaction force between two such layers can be computed, using the regulation strategy of sec. 3.5. Although a° is fixed, there is of course no reason to postulate that either / or i/ 1 would remain constant upon interaction. Accepting that double layer compositions on isolated mica surfaces deserve better characterization, it is interesting to summarize what the force measurements did contribute so far. We refer to the results by Israelachvili, Adams and Pashley, mentioned before. There are minor differences between different mica samples and sometimes cycli have to be passed before the force is hysteresis-free, provided the measurements are reproducible. Figure 3.57 gives a typical illustration. In this case the solution contains only K+ as the counterion, so o^fh) is unaffected by exchange, unless there area also protons in the system. The semilogarithmic J{h) curves have a linear part, becoming steeper with increasing c s a l t , corresponding to the long-distance exp(-xTi) decay, predicted by most theories (sec. 3.7f). The crossing of low c s with high c s curves is qualitatively in line with fig. 3.8 (not necessarily quantitatively because fig. 3.8 applies to the constant ifA case). When the solution contains other cations the long-distance part becomes more curved on a semi-logarithmic scale; exchange upon approach can also contribute to this curving. This exchange affects y/1 and &1. For h 5-2.5 nm often a short-range repulsion is observed, somewhat depending on the mica sample. Its visibility depends on the type and concentration of the electrolyte.
Figure 3.57. Semi-logarithmic plot of the SFA interaction force between two crossed mica cylinders. Given is the force per unit of radius a . Mica in the K+ -form, in solutions of KNO3 (concentration in M indicated). In 1 M KNO3 the part above 4 nm was attractive. (Redrawn from Israelachvili and Adams (1978) loc. cit.)
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In some cases this repulsive force is so strong that no attractive part can be observed, in line with figs. 3.45b, 3.46 and 3.48. On the other hand, the attractive part can be seen in cases where c salt » c.c.c.; from such curves the Hamaker constant could be established at 4.7 kT . For short range repulsion that is water structure-mediated, a thickness of about 2.5 nm seems a bit large (compare fig. 3.43). This thickness should be independent of the nature and concentration of the electrolyte, but Pashley's experiments do not support this. Moreover, he finds irregularities in the lyotropic sequence and identified binding of hydrated cations as the cause for these 'hydration layers'. As long as no further insight into the double layer structure on mica and a more advanced regulation model are available these ideas wait for confirmation. In this connection, the binding of H3O+ and the group of other cations probably obeys different laws, as it is the case for oxides. Auger electron spectroscopy (AES) and ESCA studies may help to discriminate11. Measurements in MgCl2, CaCl2, SrCl2 and BaCl2 solutions confirmed the DLVOE picture, that at concentrations above about 10~3 M (above the 'c.c.c.') attraction prevails. An advantage of SFA measurements over traditional coagulation studies is that interactions can also be studied far above the c.c.c. It was found that for c s ^ l M (about thousand times the c.c.c.) bivalent ions could overcharge the mica surface; after that the curves are similar to those of the monovalent electrolytes, but now with Cl~ as the counterion. Measurements in LaCl3 indicated overcharging at very low concentration. Formation of hydroxides could be the origin but no pHdependency studies were carried out to support that, see sec. 3.9.10. So, at this stage for mica the occurrence of a number of DLVOE features is at least semiquantitatively confirmed: exponential long-distance decay, the existence of short-range water structure-mediate repulsion, Schulze-Hardy rule, and overcharging by superequivalent adsorption of hydrolyzed ionic species. However, there is room for further studies in which the properties of the inner part of the double layer on mica are better quantified. An exemplary example of such a study has been provided by Shubin and Kekicheff21, who for LiNO3 as the electrolyte also investigated the pH effect and found the force to increase strongly with pH from 3.8-10. The primary minimum not only disappears at high cLiNO but also at elevated pH. With increasing pH the Stern charge gradually changes from H+-originated to Li+-originated. This finding called for a more systematic investigation of the ion exchange properties, which these authors carried out theoretically with the triple layer model. The analysis went along the lines of sec. 3.5c with if/* and d as important parameters. For the many details see the original paper. In summary, SFA results in combination with systematic studies of the inner part of the double layer do support and refine DLVOE theory.
11 P.M. Claesson, P. Herder, P. Stenius, J.C. Eriksson, and R.M. Pashley, J. Colloid Interface Sci. 109 (1986) 31. 21 V.E. Shubin, P. Kekicheff, J. Colloid Interface Sci. 155 (1993) 108 (102 references).
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SFA has also been used for measuring interactions with a sol between the surfaces (for instance, silica sols11) and for experiments under shear21. 3.12c Individual particles near surfaces, SFM In the previous volumes a number of surface-imaging technique results have been given, which had in common that the surface is scanned by a sharp tip. Depending on the mode of operation the tip is kept at a given height above the surface or allowed to follow its contours. See sec. 1.7.lib, figs. II. 1.3 and 4 and figs. III.3.66-68. These techniques come under the general name scanning probe microscopy (SPM) or more specialized names as scanning force microscopy (SFM), scanning tunnelling microscopy (STM) and atomic force microscopy (AFM). The names SFM and AFM are synonymous; the former term is often used because the latter is not entirely correct. Since the pioneering work of Ducker et al.3) the SFM is not only a versatile tool for surface imaging, but it has also grown into an important apparatus for the measurement of surface forces. Basically, a particle is glued to the lower side of a cantilever spring, the deflection of which can be measured, giving the force (after calibration, from Hooke's law). The particle size is of O(|i's), i.e. it is to the larger side of colloids. It is also possible to measure the force between two particles; one of these must the be fixed onto the solid. SFM techniques are under development; certain 'dedicated' variants are commercially available. Simple as such a device may seem, there are many technicalities to negotiate before reliable and reproducible experiments can be carried out. Attaching the probe to the tip requires a glue that after setting becomes rigid, so that it does not interfere with the measurements. Nor may molecules dissolve from it, because those could adsorb on the particle, inducing spurious steric and other interactions. The deflection of the cantilever is commonly measured by monitoring the position of a laser spot, reflected from the back of the cantilever to fall evenly on a pair of diodes. Displacement of the cantilever leads to an off-balance current from the diodes that can be translated into a deflection. From the deflection the force can be deduced. Probe and surface are piezoelectrically moved towards or away from each other. The distance h follows from the displacement of the piezo and the cantilever deflection. In this way no absolute value for the distance h can be found; to that end a 'hard wall contact' is usually selected. The assumption is that under that condition expansion of the piezo only leads to deflection of the cantilever, meaning that the surfaces are assumed incompressible (the 'constant compliance region')41. Soft materials or the presence of soft adsorbates may pose problems. For technical details see the review by Claesson et al. cited in
11
D. Atkins, P. Kekicheff, and O. Spalla, J. Colloid Interface Set 188 (1997) 234. J.N. Israelachvili, P.M. McGuiggan, and A.M. Homola, Science 240 (1988) 189. 31 W.A. Ducker, T.J. Sendcn, and R.M. Pashley, Nature 353 (1991) 239; Langmuir 8 (1992) 1831. 41 I. Larson, C.J. Drummond, D.Y.C. Chan, and F. Grieser, Langmuir, 13 (1997) 2109. 21
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subsec. 3.12b and the IUPAC recommendation, mentioned in sec. 3.15a. For our present purpose the significance is that we have a technique available for the measurement of pair interactions between, say two spheres, or between a sphere and a plate. Conversion into the corresponding interaction Gibbs energy for flat plates follows from ^—^- (sphere-plate) = 2nG{h) (flat plates) a
[3.12.11]
(compare [3.12.9]). An advantage over SFA is that a variety of materials can be investigated. Nowadays forces of O(pN) can be measured at distances down to 0.01 nm1J. As technical improvements continue to be implemented more and better data may be expected in the near future. An intrinsic problem is that the surfaces of materials other than mica are rarely smooth on an atomic level. The implication is that there is no well-defined plane in which the surface charge resides. Moreover, because of their short range, Van der Waals forces are much lower for rough materials in contact than for smooth ones. Because of these reasons, measurements at very short distances are difficult to interpret. In table 3.5 some technical aspects of SFA, MASIF and AFM are compared. The list is not complete; soft surfaces and/or surfaces carrying adsorbates are excluded. Regarding the time-dependence, in SFA and MASIF the surfaces approach each other so slowly that the double layers may be continually at equilibrium. Whether this remains the case upon jump-in may deserve study. In the literature many examples of good SFM force measurements can be found. In view of our purpose of checking DLVOE theory only those are useful where no surfactants or other adsorbed layers are present and where the surfaces of particle and substrate are smooth and electrochemically characterized, preferably by both titration and electrokinetics. By way of illustration we refer to the work by Larson et al.2' on the interaction between Stober silica spheres and flat monocrystal surfaces of sapphire (a-Al2O3). These surfaces were in isolation characterized by micro-electrophoresis and by streaming potentials, respectively, (on the Helmholtz-Smoluchowski level interpreted in terms of C, -potentials). The surface roughness of the A12O3 (^0.5 nm over 5 |im2 ) was determined. The electrolyte was 10"4 or 10~3M KNO3 . The i.e.p. of SiO2 was between 2 and 2.5, but that of the alumina posed a (familiar) problem: it was 4.2 whereas the p.z.c. of powdered A12O3 is rather 8.5-9, see table II. app. 3b. Most likely this difference is caused by small amounts of acid groups (silica?) on the A12O3 surface. No surface conduction data were available. Force-distance curves (fig. 3.58) were at least qualitatively in line with the electrokinetic charge sign. For the hetero-interaction case of opposite f -potentials (pH = 3.9) the authors did not try
11 21
J.Y. Walz, Adv. Colloid Interface Sci. 74 (1998) 119. I. Larson, C.J. Drummond, D.Y.C. Chan, and F. Grieser, Langmuir, 13 (1997) 2109.
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Table 3.5. Comparison between SFA, MASIF and colloidal probe SFM measuring techniques. Issue
SFA
MASIF
Probe SFM
Directly measured
h as a function of piezo position
Deflection of bimorph as a function of piezo position
Deflection of cantilever spring as a function of piezo position
Data derived
Deflection of leaf spring —> force
Time scale of measurements
Relatively slow; mostly step-wise approach and retraction
Nature of surface
Transparent(bendable) surfaces
Spherical surfaces
Any type of surface, radius 2-20 |im
(dis)advantages
(With mica) atomically flat over ~10 2 nm 2
(With molten spheres) atomically flat over ~10 2 nm 2
Surface roughness offer problems
Determination of radius of curvature
From interferometry
Microscopic, or from micrometer screw
From (SEM) particle size
Miscellaneous
Small A/V ration, hence susceptible to contamination. Relatively laborious.
h ; h = 0 from constant compliance region
Relatively fast1)
Lower interaction area resolution similar to that in SFA or MASIF.
*' In some devices the surfaces can be kept at a fixed distance for awhile. a regulation model (as in sec. 3.12.6c) but simply considered interaction at constant o^'s or i/^'s , finding their result between these two. No systematic studies are available yet to verify the borderline cases, predicted in the mentioned sec, where attraction at short distance is possible between surfaces of the same diffuse charge sign at long distance. The same authors also reported on the SiO2-TiO2 system11. It is interesting to compare the SiO2-Al2O3 study with an independent one on the same materials by Veeramasuneni et al.21. These authors' A12O3 had an i.e.p. of 9.1 (p.z.c. not measured), the results were, mutatis mutandes similar, and so was the interpretation. Systematic studies of specificities deserve more attention. Trends may become complex when with hetero-interaction the directions of the lyotropic sequences are
11
I. Larson, C.J. Drummond, D.Y.C. Chan, and F. Grieser, J. Phys. Chem. 99 (1995) 2114. S. Veeranasuneni, M.R. Yalamanchili, and J.D. Miller, J. Colloid Interface Set 184 (1996) 594. 21
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Figure 3.58. Force between a S1O2 sphere and an 01-AI2O3 -plate, measured by SFM. Electrolyte, 10~ 3 M KNO3 . (Redrawn from Larson et al., loc. cit.)
different between the two surfaces. Compare table 3.3. See for Instance ref.11 for the SiOg-mica system. Extending work by Meagher on the same system in CaCl22), Fielden et al.3) reported indications of oscillatory forces at very high salt concentrations (5 M, three orders of magnitude above the critical coagulation concentration). This may be in line with our previous conclusion that ion correlation forces do occur, but under conditions that are not met in classical stability studies. Short-distance repulsion are more or less routinely reported, but oscillatory forces appear restricted to solvents of big organic molecules41. An interesting new development is that of SFM measurements as a function of an applied electric potential difference, thus creating a forced interaction, as in sec. 3.10. Such studies help to understand the charge formation process. Illustrations include refs567) 3.126. Springless measurements The title may sound somewhat derogatory but this subsection deals with a number of sophisticated techniques, by which (mostly repulsive) interactions between particles
11
I.U. Vakarelski, K. Ishimura, and K. Higashitani, J. Colloid Interface Set 227 (2000) 111. L. Meagher, J. Colloid Interface Sci. 152 (1992) 293. 31 M.L. Fielden, R.A. Hayes, and J. Ralston, Phys. Chem. Chem. Phys. 2 (2000) 2623. 41 S.J. Oshca, M.E. Welland, Langmuir 14 (1998) 4186. 51 R. Raiteri, M. Grattarola, and H.J. Butt, J. Phys. Chem. 100 (19961 16700. 6) A.C. Hillicr, S. Kim, and A.J. Bard, J. Phys. Chem. 100 (1996) 18808; K. Hu, F.R.F. Fan, A.J. Bard, and A.C. Hillier, J. Phys. Chem. B101 (1997) 8289. 71 D. Barten, J.M. Kleijn, J. Duval, H.P. van Leeuwen, J. Lyklema, and M.A. Cohen Stuart, Langmuir 19 (2003) 1133. 21
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Figure 3.59. Tangential translation of a trapped sphere near a planar wall. and surfaces are measured without requiring mechanical devices. Instead, G(h) is monitored hydrodynamically or optically. A hydrodynamic technique for the measurement of pair interactions has been elaborated by Alexander and Prleve11. The principle on which it rests is as follows. Consider an isolated particle that has sedimented under gravity towards an electrically repelling horizontal surface. It will come to rest at an equilibrium shortest distance h , where Gel and Ggrav J u s t balance, s e e ng- 3.59. Due to Brownian motion the height will fluctuate around h . The probability of finding it between h and h + dh will be P{h)Ah , where P{h) is related to G(h) according to a Boltzmann relation. Thus, P(/i) = Aexp[-G(h)/kT]
[3.12.12]
with A determined from the normalization oo
Jp(h)dh = l o
The height is determined from the horizontal displacement under shear, as sketched in the figure. When the shear rate is y the tangential velocity of the centre of the particle equals v [h] = y(h + a) minus a lag by the hydrodynamic interaction with the wall. For this last effect, theory is available21, so that from measured displacements h can be derived. In this case this was carried out stroboscopically against a dark field. P(h) is obtained from the histogram of displacements. In this way G{h) is obtained, after subtraction of the gravity contribution [An13)a3Apgh . Results were analyzed for latices, measuring the displacement of two particles of different size every two seconds using a stroboscope against a dark field. The method is interesting, but has a size window: for a measurable profile P[h) the particles should neither be too large nor too small. Moreover, the interaction should be repulsive. 11 21
B.M. Alexander, D.C. Prieve, Langmuir 3 (1987) 738. A.J. Goldman, R.G. Cox, and H. Brenner, Chem. Eng. Sci. 22 (1967) 653.
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3.157
Figure 3.60. Explanation of the principles of TIRM. The substrate must be transparent, with refractive index ng > nw and the angle of incidence di should be large enough to attain total reflection. The evanescent wave in the water /e(z) induces a height-dependent scattering 7(h) from the particle from which fi can be determined. This seminal experiment has triggered a number of developments in which the application of a shear field could be avoided by invoking optical techniques. The most important are the following.
(i) Total internal reflection microscopy [TIRM) is a technique for measuring P(h) based on the scattering of particles in an evanescent field. Figure 3.60 sketches the principles. We recall the principles of evanescent waves, set forth in sec. 1.7.10a, repeating that the substrate must be transparent. The intensity Ie(z) of the evanescent field decays exponentially with z, with a decay length of O(100 nm). The scattered intensity I(h) can be measured with an accuracy of about 1%, so h can be established down to ~1 nm. The technique has been reviewed by Prieve11 and Walz21; also see ref 3.4.5)
(ii) Optical levitation and trapping. That light can exert a force was already known by Kepler, who explained by this mechanism why the tails of comets always point away from the sun. By the same principle a (laser) beam exerts a tuneable radiation pressure on a colloidal particle. For a particle above a surface this force comes on top of the colloidal interactions we want to measure and gravity. When, by applying radiation pressure, a particle is captured at a certain position, one speaks of optical trapping. The radiation force is very weak and the technique became possible only 11
D.C. Prieve, Adv. Colloid Interface Sci. 82 (1999) 93. J.Y. Walz, Curr. Opln. Colloid Sci. 2 (1997) 600. 31 M.A. Brown, E.J. Staples, Faraday Discuss. Chem. Soc. 90 (1990) 193. 41 D.C. Prieve, N.A. Frej, Langmuir 6 (1990) 396. 51 S.G. Flicker, S.G. Bike, Langmuir 9 (1993) 257; S.G. Flicker, J.L. Tipa, and S.G. Bike, J. Colloid Interface Sci. 158 (1993) 317. 21
3.158
PAIR INTERACTION
after the development of powerful lasers to produce a photon density that was high enough to levitate particles against gravity. A typical illustration, taken from ref.2) is given in fig. 3.61. The left, descending part is dominated by electrostatic repulsion, and can be well represented by DLVO theory at constant \fA 's. The right, ascending branch is the gravity part (proportional to h ), augmented by the radiation pressure. For illustrations from other research groups, see3'41. (iii) Optical tweezers. Ashkin has also shown51 that it is possible to use a single laser beam to confine a dielectric particle in three dimensions. Use is made of the force that a particle of higher refractive index than its surrounding medium experiences towards the region of high field intensity in a radial field. In this way single particles can be optically trapped in space. When two of such trapped particles are manipulated to come close together one speaks of optical tweezers. The term is also used when only one particle of the pair is trapped. The technique has been pioneered by, among others, Crocker and Grier61. For a review, see71. (iv) Evanescent Wave Light Scattering (EWLS) stands for a variant of TIRM in
Figure 3.61. Gibbs energy of a 10 \i polystyrene sphere in a 0.5 M NaCl solution, levitated above mica on glass at three levels of radiation pressure (indicated). For better visibility the profiles have been offset by 1 kT. (Redrawn from Walz and Prieve, loc. cit.) 11
A. Ashkin, Phys. Rev. Lett. 24 (1970) 156. J.Y. Walz, D.C. Prieve, Langmuir 8 (1992) 8073. 31 T. Sugimoto, T. Takahashi, H. Itoh, S. Sato, and A. Muramatsu, Langmuir 13 (1997) 5528. 41 A.R. Clapp, R.B. Dickinson, Langmuir 17 (2001) 2182. 51 A. Ashkin, Opt. Lett. 11 (1986) 288. [check xx radiation pressure only?) 61 J.C. Crocker, D.G. Grier, Phys. Rev. Lett. 73 (1994) 352. 71 D.G. Grier, Curr. Opin. Colloid Sci. 2 (1997) 264. 21
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3.159
which P(h) is not obtained by following the fluctuations of one given particle as a function of time but by simultaneously measuring the positions of a large collection of identical particles. The technique has been implemented by Polverari and van de Ven11, using an impinging jet flow, imposed over the surface. 3.12e Measuring techniques, and conclusion The general conclusion of this section is that we are contemplating a fascinating development in instrumentation, from which in the near future more interesting news may be expected. So far the trends are; (i) The DLVO part of the DLVOE works well unless the distances are short (smaller than a few nm). Often one can get away with constant a^ or constant y/d pictures, although there is no theoretical justification for such models. For long distance the difference between the two is minor. (ii) Short-range solvent structure-repulsion is often observed. It may mask Van der Waals attraction. (iii) Progress in the understanding of the complex of interaction forces demands additional characterization of the inner double layer part (titration, electrokinetics and possibly surface conduction measurements). (iv) Dealing with surface roughness will remain a challenge. (v) There are no indications for the occurrence of ion correlation effects in the usual range of potentials and concentrations studied, but they may become relevant beyond this range, in particular at electrolyte concentrations far exceeding the c.c.c. Searching for such features remains a challenge. Generally, the most challenging task ahead appears to be homing in on the E of DLVOE. 3.13 Case studies: Oxides and latices The purpose of this section is to select, from the vast literature on lyophobic sol stability and pair interactions, a few systems that are typical because of their practical interest and the availability of double layer data in conjunction with those on interaction. The main issue is establishing the validity range of DLVO theory and, where its application limits are exceeded, to find out whether DLVOE suffices and/or whether still other improvements have to be considered. Many of these additional features are system-specific. Our choice of systems covers titanium oxide, silica and polymer colloids (latices). TiO2 is an amphoteric, highly insoluble oxide of which the double layer has been extensively studied. SiO2 is an acidic oxide with pH-dependent solubility; it has a stability that appears poorly correlated with the surface charge. Latices are often used as model colloids because they are relatively easily made homodisperse. Their surfaces can be positive, negative or amphoteric, depending on the 11
M. Polverari, T.G.M. van de Ven, Langmuir 11 (1995) 1870.
3.160
PAIR INTERACTION
synthesis, but pair interaction may involve a steric component. So, these three systems show a variety of features which make them suitable paradigms for systematic stability studies. Obviously, pair interactions often also recur in Volumes IV and V, for instance in interaction dynamics (chapter 4), the behaviour of concentrated systems (chapter 5), electrovlscous effects (sec. 6.9b), polyelectrolytes (chapter V.2), ionic micelles (chapter V.4), and thin films (chapter V.6). In all these cases and in the three examples to be studied in this section, pair interaction is just one of the many features considered, with the implicit consequence that interpretations of the pair interaction often do not go beyond the most simple equations. Otherwise stated, the interaction models of the previous sections and of chapter 4 are mostly ahead of the experiments. For a rigorous analysis the composition of the double layer and stability information must be simultaneously available. Certainly as a function of important variables, such as pH and c salt , that is rarely the case. Many studies address only one, or a limited number of, variables; of course these can be interesting in their own right. With all of this In mind, the selection of the illustrations becomes somewhat arbitrary. However, in order to focus on the basic mechanisms, we will keep the discussion focused by avoiding studies where added components, such as surfactants and polymers, give rise to additional phenomena. 3.13a Titanium oxides Titanium oxides are widely applied, for instance in paper, toothpaste, ceramics, pharmaceutical products and paints. One of the reasons Is their high refractive index, i.e. they can act as 'whiteners.' In the pigment industry these are known as white pigments. A further application is as a photocatalyst. Photocatalysis on the surface of TiO2 involves electron transfer and this transfer depends on the composition of the double layer11. The process has for instance been studied in the light-induced reduction of cytochrome-C by colloidal TiO2 21 and In the electron transfer of methyl viologen31. (The methylviologen cation is the most frequently used electron relay In light-induced water photolysis.) For a review see41. Specific applications sometimes require intentional modification of the material. For instance, in photocatalysis doping is applied to modify the Fermi levels (sec. II.3.10e). A more colloidal-like modification is that of covering TiO2 surfaces with a thin SiO2 layer. This has a dual purpose. The first stems from the complication of TiO2 being sensitive to ultraviolet light, which is strongly absorbed by the particles. As a result, white paints, particularly when exposed to sunlight at elevated temperatures, degrade by catalysis; they obtain grey-blue colours. SiO2 layers on the surface act as 11
J.R. Darwcnt, A. Lepre, J. Chem. Soc. 82 (1986) 2323. P. Cuendet, M. Graetzel, Bioelectrochem. Bioenerg. 16 (1986) 125. 31 G.T. Brown, J.R. Darwent, J. Chem. Soc, Chem. Commun. 2 (1985) 98. 41 A.L. Linsebigler, C.g. Lu, and J.T. Yates, Chem. Revs. 95 (1995) 735. 21
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3.161
UV filters, so they reduce the problem. Secondly, as the Hamaker constant of SiO2 is between that of water and TiO2 (table in app. 3), such layers attenuate the Van de Waals attraction between the particles. Colloidally, such modified particles behave as silica particles with a heart of titanium oxide. The presence of a protective SiO2 sheet is electrokinetically readily diagnosed through the lowering of the i.e.p. (around 2 for SiO2 , around 5.5 for TiO2 ). In this section we emphasize uncovered particles. Titanium oxide comes in two crystal modifications, anatase and rutile. Both are tetragonal. Rutile is thermodynamically stable, anatase is metastable. Hence, spontaneous transformation from anatase to rutile always takes place; however, it only proceeds readily at temperatures far above the ambient slow conditions of aqueous sols. So, for practical reasons we can interpret TiO2 as exhibiting two stable modifications. Between these, the double layer properties hardly differ; differences with respect to the pristine p.z.c. (Table II.A3) are virtually within experimental error. At most, pH° (anatase) is a few tenths above pH° (rutile). It must be kept in mind that anatase crystals may be covered by a thin surface layer of rutile. In this connection, it must be noted that there are several ways to prepare colloidal TiO 2 , but these procedures do not necessarily yield identical particles. Surface properties generally depend on the mode of preparation; this feature will not be discussed here, although it plays a background role in the characterization. Milling of natural TiO2 mineral is of industrial relevance, but for the present purpose we are more interested in better defined particles. Matijevic et al.11 obtained sols containing narrow, spherical rutile particles from hydrolysis of acidified TiCl4 solutions, also containing Na2SO4 . In this group, TiO2 sols were also synthesized by hydrolysis of aerosols consisting of TiCl4 or Ti-ethoxyde2). After dispersion in water, sol properties such as the i.e.p. and size distribution can be monitored by adjusting the process conditions. These sols are mostly dilute. Titanium tetraethoxide has also been used as the precursor to obtain homodisperse TiO2 sols31. Illustrations 3.64 and 3.65 (below) have been obtained with these sols. Chhabra et al.4) reported the synthesis of colloidal rutile or anatase sols via the micro-emulsion method. The resulting particles are rather small (20-25 nm), but have a poorly defined surface chemistry because of surfactant remnants stabilizing the micro-emulsions. Besides sols, TiO2 electrodes can also be prepared, for instance, using anatase films by oxidative anodic hydrolysis of TiClg 51. In Volume II a great deal of double layer information on TiO2 can be found, which we shall not repeat in extenso. The pH° (table II.A3) at 25°C is about 5.5, decreasing to about 5.1 at 70°, see fig. II.3.37. From the pH° and its temperature coefficient, it is possible to obtain the standard Gibbs energy, enthalpy and entropy of charge 11
E. Matijevic , M. Budnik, and L. Meites, J. Colloid interface Set 61 (1977) 302. M. Visca, E. Matijevic , J. Colloid Interface Set 68 (1979) 308. 31 E.A. Barringer, H.K. Bowen, Langmuir 1 (1985) 414. 41 V. Chhabra, V. Pillai, B.K. Mishra, A. Morrone, and D.O. Shah, Langmuir 11 (1995) 3307. 5) L. Kavan, B. O'Regan, A. Kay, and M. Gratzel, J. Electroanal. Chem. 346 (1993) 291. 21
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formation,
as defined in [II.3.8.16]. At 25°C,
TA c f S^ 1 =l.l
AcfG^ = 8.76 ; A cf H^ = 9.9
and
k J m o l e 1 . So, the double layer formation is mainly enthalpically
determined, although the entropy increase also contributes. Using a (site-binding) model, it is possible to obtain intrinsic pKa and pKb values for the proton uptake and release. The results depend on the model for the basic reactions; our results can be found in table II.3.5. A typical titration curve, also including f -potentials ( f = yrA ), was given in fig. II.3.63. In that figure, the i.e.p. and p.z.c. coincide for
c s
meaning that at least up to these concentrations, the electrolyte (KNO3 ) is indifferent. These curves exhibit the characteristic salt (screening) effect, whereupon increasing c s at fixed pH, | a°\ increases but | y/d\ decreases. When the electrokinetic charge density | oe]i\ is plotted as a function of | a°\, the characteristic behaviour is obtained, starting with | o*k| = |
Specific Gibbs binding energies of monovalent ions on TiO2 (rutile or
anatase) in units of kT per ion.
Cations on negative TiO9 Li + -0.5 Na+ -0.6 -0.45 K+ NH+ -0.7 Cs + -0.2
Anions on positive TiO 9
cr NOi
-0.2 -0.14
ClOg
-0.06
r
-0.05
Data computed from the compilation of Bourikas et al, loc. clt. Regarding lyotropic sequences, for alkali ions the basic binding trend is Li+ > Na+ > K+ > Cs + , i.e. it is inverse and in line with our previous predictions, see tables II.3.8 and 3.3 in the present chapter. The interesting feature is that calcination of the sample leads to the reverse order. Changes in the surface hydration are responsible for this inversion. After publication of Vol. II, the various site-binding models have been applied to a large number of titration curves for rutile and anatase, and reviewed by Bourikas et al.11. These authors support and extend our lyotropic sequence. They tabulate their best estimates (average from about 20 references) for log Kc
and
log K a , where Kc and Ka stand for the cation and anion pair formation constants, respectively, A
9mi
= 2 3 kT
on K
P c
negative o r
2 3 kT
and
positive
K
t h e lonlc s
P a•
anatase ec
and
rutile
surfaces.
Using
c
P ifi binding Gibbs energies of table
3.6 are found. The values are very low, meaning that these ions are virtually indifferent. Perhaps at very elevated salt concentrations specificity can show up . The relatively
11
K. Bourikas, T. Hiemstra, and W.H. van Riemsdijk, Langmuir 17 (2001) 749. M. Kosmulski, A.S. Dukhin, T. Priester, and J.B. Rosenholm, J. Colloid Interface Sci. 263 (2003) 152. 21
PAIR INTERACTION
3.163
1.67xlO"3 M •P-oZ
# 0.33xl0~ 4
3
1.67xlO" —•"" 0.33xl0" 4
-A-A-1.67X10"5
Figure 3.62. pH-dependence of the electrophoretic mobility of rutile in the presence of several concentrations of Ba(NO3)2 (indicated). (Redrawn from Fuerstenau et al., loc. cit.) high value for Li+ may be related to the closeness of its crystal ionic radius (0.060 nm) to that of Ti 4+ ions (0.068 nm), which would allow superficial isomorphic uptake by the solid. For haematite (radius Fe3+ Ion 0.064 nm), this phenomenon was observed before; with this system, Ll+ ions can even shift the p.z.c.11. In connection to this, it has been verified that Li+ ions can be incorporated in anatase electrodes21. Fuerstenau et al.3' reviewed the adsorption of alkaline- and alkaline earth ions for a variety of oxides. For the alkaline earth ions in rutile and anatase, the direct order is mostly found: Ba2+ > Sr 2+ > Ca2+ > Mg2+ . The fact that mono- and bivalent cations exhibit different lyotropic orders must mean that their adsorption mechanisms have a different, valency-determined, origin. Actually, all alkaline earth ions can even cause overcharging in a given pH window. Figure 3.62 illustrates this for Ba(NO3)2 . The rutile sample was synthesized from TiCl4, and the remaining HC1 was removed. The curve in 0.33xlO~6M Ba(NO3)2 was about identical to that in dilute NaNO3 , which was virtually inert. The pristine i.e.p. was about 6.6, about one pH unit above the pristine p.z.c. The main message communicated is that, at sufficiently high pH, Ba2+ ions bind so strongly that the sign of £ is inverted. In more concentrated Ba(NO3)2 solutions it even becomes impossible to make the particles electrokinetically negative. On the basis of this figure alone, one cannot determine the origin of this superequivalent adsorption. In this respect the results in the presence of Co(NO3)2,
11
A. Breeuwsma, J. Lyklema, Discuss. Faraday Soc. 52 (1971) 324. D. Fattakhova, L. Kavan, and P. Krtil, J. Solid State Electrochem. 5 (2001) 196. D.W. Fuerstenau, D. Manmohan, and S. Raghavan. in Adsorption from Aqueous Solutions, P.H. Tewari, Ed., Plenum (1981) 93. 21
3.164
PAIR INTERACTION
Figure 3.63. Like fig. 3.61, but now for Co(NO3)2 . Supporting indifferent electrolyte, 10~3 M KNO3 . (Redrawn from James and Healy, loc. cit.)
obtained by James and Healy11 , give more evidence. See fig. 3.63. In this example, the pristine i.e.p. ~ p.z.c. and the measurements in the presence of Co(NO3)2 are extended to higher pH, where overcharging is again inhibited. The authors present evidence for the adsorption of hydroxycomplexes as the origin of the charge reversal, in line with the trend in fig. 3.49. Two of the items of evidence for this mechanism are (i) the mobility at high coverage approximates that of Co(OH)2 and (ii) the trend of u(pH) is very similar to that on SiO2 , i.e. determined by the chemistry of the Stern layer rather than by the electrostatics of the surface. So, although the adsorption of the first Co2+ ions may be related to the properties of the surface, upon maximum coverage the particle behaves electrokinetically as if encapsulated by cobalt hydroxides. For the trivalent Al3+ counterion, the situation is represented in fig. 3.642). In this case, f -potentials are plotted, computed from mobilities using the Wiersema-LoebOverbeek tables31. This figure demonstrates strong adsorption of alumino-hydroxy complexes. It is noteworthy that the trend is very similar to that at the air-water interface, mentioned at the end of sec. 3.9j. In both cases, the adsorption starts at low pH, passes through a maximum, to disappear above pH = 10. Again, this supports that £ is more controlled by the chemical composition of the Stern layer rather than by the nature of the 'particle.' The electrokinetic behaviour of the sol in the presence of multivalent cations is strongly contrasted to that of anions, which hardly adsorb and
11
R.O. James, T.W. Healy, J. Colloid Interface Scl. 40 (1972) 53. G.R. Wiese, T.W. Healy, J. Colloid Interface Sci. 51 (1975) 434. 31 P.H. Wiersema, A.L. Loeb, and J.Th.G. Overbeek, J. Colloid Interface Sci. 22 (1966) 78. 21
PAIR INTERACTION
3.165
Figure 3.64. Electrokinetic potential on rutile in the presence of 10 **M KNO3 and various concentrations of added AlINC^^ (indicated). (Redrawn from Wiese and Healy, foe. cit.)
do not exhibit overcharging . All of this proves that for the present system overcharging has a chemical origin. So far there is no need to account for this phenomenon as a result of ion discreteness. The next question automatically becomes how all of this makes itself felt in the stability of TiO2 sols. Starting with the classical rate of coagulation measurements, figs. 3.65 and 66 give typical illustrations. These data have been obtained for homodisperse TiO2 sols, in which the particle surface consisted of anatase21 and was electrochemically well characterized; the surface charge and f -potential were both measured, data similar to fig. II.3.63, and cr°(pH) curves were analyzed with a site-binding model. Ionic leftovers of the synthesis were removed by washing procedures and ageing. Eventually, the pristine p.z.c. and i.e.p. became 5.2 and 5.5, respectively, within a few tenths identical to the usual values (Vol. II, app. 3). The Stern layer capacity was very high, indicating a porous surface. Qualitatively, fig. 3.65 exhibits expected trends. The critical coagulation concentration (c.c.c.) is much lower for Ba2+ than for K+ and increasing the pH till = 9 (further away from the p.z.c.) makes the sol more stable. The latter effect is much smaller since upon increasing pH | £] rises much less than | <j°|. Figure 3.66 gives the counterpart; logW as a function of pH at fixed KC1 concentrations. There is a window 11 21
A. Fernandez-Nieves, F.J. de las Nieves, Colloids Surf. A148 (1999) 231. E.A. Barringer, H.K. Bowen, Langmuir 1 (1985) 420.
3.166
PAIR INTERACTION
Figure 3.65. Stability ratios obtained by photon correlation spectroscopy for TiO2 sols, prepared by hydrolysis of titanium tetraethoxide + KC1, pH = 7.5; • , KC1, pH = 8.9; A , BaCl 2 , pH = 7.5. Temperature, 25°C ± 0.2. (Redrawn from Barringer and Bowen, loc. cit.)
Figure 3.66. As previous figure, but now presented as logW(pH) for KC1 at different concentrations (indicated). around the i.e.p. where coagulation is fast and W = 1; in fact, logW is slightly negative as compared with the rapid coagulation at pH = 8.8 in fig. 3.65. The window is not entirely symmetric. On the r.h.s., where cations are the counterions, the slope dlogW/dpH is much more concentration-dependent than on the l.h.s., where anions play this role. Weaker specific adsorption of the anions, as found before, may be responsible for this difference. Quantitatively, according to fig. 3.65, c.c.c. for Ba(NO3)2 is about a factor of 100 lower than c.c.c. for KNO3. When [3.9.8] is applied in the simplified form for d = 0 and low (/* => f [3.9.11], at most a factor of 4 between z = l and z = 2 can be accounted for, so the remaining factor of 25 must be attributed to differences with
PAIR INTERACTION
3.167
respect to t, amounting to [f(Ba2+)/£(K+)]4 = 0.04 or f(Ba2+)/f(K+) = 044 at the pertinent concentrations, which cannot be read from the paper but which is a reasonable ratio considering that Ba 2+ probably adsorbs specifically. The authors computed the Hamaker constant A11(w) according to Reerink and Overbeek". For KC1 as the electrolyte, the result differed between the c.c.c, above the p.z.c. (3.3 kT) and below it (7.0 kT). Apart from the fact that there should be no difference between positive and negative surfaces, the absolute value is too low as compared with presentday information (table A3.2) by a factor of 2-4. The conclusion is that these data deserve further scrutiny, for which sec. 3.9e may be helpful. The inverse lyotropic sequence in the affinity of alkali ions (table 3.7) should, according to our model, lead to a direct sequence in £ and, hence, in the c.c.c.values. This was confirmed experimentally by Kallay et al.2). For a rutile sol they found at pH = 10, 0.039, 0.060 and 0.136 mmole dm" 3 for LiCl, KC1 and CsCl, respectively, just the reverse from Agl-sols. With hydrolyzing ions the results are, of course, much more striking. Figure 3.67 illustrates this for rutile sols in the presence of complexing and adsorbing Alcomplexes, of which the electro kinetic potentials were given in fig. 3.64. Addition of A1(NO3)3 leads to displacement of the minimum (logWsl) to higher pH values. Unlike the situation in pure KC1, where on the high pH side the surface becomes negative due to adsorption of OH~ ions, in the presence of A1(NO3)3 the surface + Stern layer becomes negative due to adsorption of Al-hydroxycomplexes. This difference leads to an asymmetry in the shape of the logW(pH) curve, even if only because the value of d in [3.9.8] becomes pH-dependent. Chemical and kinetic issues now enter the discussion: what is the composition of the hydroxycomplex, and is this composition the same for the bulk and adsorbate and how fast do the complexes adsorb? Speciation of Al-hydrocomplexes as a function of pH is quite complicated and beyond this section. The composition of adsorbed complexes most likely differs from those in the dissolved state, as extensive studies by Matijevic and co-workers have shown31. This conclusion may also be drawn from the collected work in Stumm's group4'. The dynamics of the processes also received attention in the experiments described in fig. 3.67. It is not the absolute value of the observation time that is relevant, but the characteristic time of complex formation, (adsorption and change in the adsorbed state, if any) with respect to the particle interaction time (difference between the Deborah numbers). From this figure (and a few other arguments) it may be inferred 11
H. Reerink, J.Th.G. Ovcrbcck, Discuss. Faraday Soc. 18 (1954) 74. N. Kallay, M. Colic, D.W. Fucrstenau, H.M. Jang, and E. Matijevic, Colloid Polymer Sci. 272 (1994) 554. 31 E. Matijevic, J. Colloid Interface Sci. 43 (1973) 217. 41 W. Stumm, J.J. Morgan, Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters, Wiley, 3 rd ed. (1991). 21
3.168
PAIR INTERACTION
Figure 3.67. Stability ratio of the TiO2 sol of fig. 3.64, 1CT4M KNO3, curve A 7 6 6 4xlO~ M; curve B, 1.2xlO~ M; curve C, 5.3xlO~ M; curve D, 1. xlO~ 5 M; curve E, 2.1 x 10~^MAl(NO3)3 added. Arrows denote pH values for which ( = 0. Ordinate axes staggered for ease of presentation. (Same reference as fig. 3.64.)
that the adsorption process is completed within seconds. However, for a further understanding of the dynamics, studies on the much shorter time scales of particle interaction are needed (sees. 4.4 and 5). (Rates of) coagulation studies are of course not the sole means for studying particle interactions. First, we recall the work by Larson et al. on direct force measurements among TiO2 surfaces, included in table A3.2. Rheological measurements, see for example2'3'41, are extremely relevant for practice, for instance in the formulation of paints. However, the translation of rheological characteristics, such as yield values and the various moduli into pair interactions, is not straightforward (see sec. 6.13) and, moreover, the experiments are often carried out with ill-defined systems. As a consequence, the line of interpretation is mostly the
11
I. Larson, C.J. Drummond, D.Y.C. Chan, and F. Griescr, J. Am. Chem. Soc. 115 (1993) 11885. 21 F. Mange, P. Couchot, A. Foissy, and A. Pierre, J. Colloid Interface Set 159 (1993) 58. 31 G.E. Morris, W.A. Skinner, P.G. Self, and R.St.C. Smart, Colloids Surf. A155 (1999) 27. 41 J. Gustafson, E. Nordenswan, and J.B. Rosenholm, Colloids Surf. A212 (2003) 235.
PAIR INTERACTION
3.169
other way around, in that certain observations are explained in terms of (changes in) interaction. For example, this is the case with respect to parameters such as the influence of pH, added Ca 2+ , the relation of the yield value to the (distance from) the i.e.p,. etc. Sedimentation is even more difficult to interpret in terms of pair interactions. 3.23b Silica Silicon Is one of the most abundant elements In the earth's crust, of which it comprises 28%. In that crust, It is a major component of most minerals, sands and clays. The silicium atoms are always bound to oxygens. The purest natural form of silica (sillcum oxide, SiO2 ), is quartz. Silicates are highly insoluble in water, unless they are (strongly) hydrolyzed. Even the human body contains some silica, approximately seven grams per adult. In tissues such as cartilage, it Is usually complexed to glycoproteins, whereas in blood it is mostly found as hydrolyzed silicic acid. Silicon (or silica) is a minor constituent of bones and also occurs in joints. In industry, an abundance of technical products for specific purposes are made of these materials. Our present interest is focused on colloidal and suspended silica11. Colloidal, and related forms of silica, come in a variety of modifications: crystalline or amorphous, fumed or fused, precipitated or gel-like, porous or non-porous. Numerous industrial silicas are available under a variety of trade names. They are prepared by different techniques and consequently their sizes and surface properties may vary; sometimes they are tailor-made for specific applications, which may mean that their surfaces are modified in a specific way. When commercial samples are subjected to stability studies it is always recommendable to verify the surface properties: is the i.e.p. In the right bracket (around 2)? is the sample soluble in water? are titrations reversible?, etc. Well-defined silicas have been made and characterized in various research laboratories. Water-dispersable silicas are invariably hydrolyzed. For our present purpose, it Is relevant that after the pioneering work by Her, the Inventor of the ludox sol, various homodisperse SiO2 sols have been synthesized. The affinity for water and the electric double layer of silica and its variants have been studied by a range of authors. Illustrations can also be found in FICS. In fig. II.1.26 the influence of surface treatment (outgassing, hydrophobing, ...) on the adsorption of water vapour is given for Cab-O-Sil and fig. II. 1.28 gives the statistical thickness of adsorbed water layers on native and modified Aerosil 200 at three temperatures. Table II. 1.3 (In sec. II.3f) gives enthalpies of wetting (immersion) for a variety of solids and includes quartz, amorphous and pyrogenic S1O2, from which it can be deduced that this enthalpy is much higher for quartz than for the other two modifications. The case studies of the wetting by water in sec. III.5c include water on 11
See for instance R.K. Her, Colloidal Silica in Surface and Colloid Science, E. Matijevic, Ed., Vol. VI (1973) 1-100.
3.170
PAIR INTERACTION
quartz (figs. III.5.17 and 18), indicating the occurrence of so-called a- and (3-films, related to the primary and secondary minimum in /7(h) isotherms for the films. Our tabulation of contact angles (Table III.A4) does not contain an entry on oxides, but it is a fact that various silicas are easily wet with water, i.e. they are hydrophilic, provided they are clean and fresh. Stripping the surface-bound water and heating makes the surfaces more hydrophobic. The extent of reversibility upon exposing such dehydrated surfaces to water depends on the type of silica. Double layers on SiO2 , in the sense of cr°(pH) curves, are given in figs. II.3.64 and 65. In the first of these figures, the typical behaviour of precipitated SiO2 is shown at four KC1 concentrations. The p.z.c. is between 2 and 3; with increasing pH the surface charge initially hardly increases, but at higher pH values, depending on cKC1, it becomes progressively more negative. However, this rise does not reflect a simple, physical, double layer picture; a° may become more than twice as high as can be accommodated by dissociating all hydroxyl groups on the surface, whereas the corresponding ^-potentials are not exceptionally high. Figure II.3.65 shows that this exceptionally high titration charge is reserved for the rather porous precipitated silica; quartz and pyrogenic silica do not exhibit such behaviour. For most types of SiO2, the pristine point of zero charge is between 2 and 2.5 (see table II.A3), meaning that under ambient conditions all silicas are negatively charged. Extremely low pH values are needed to make their surfaces positive; in fact, sometimes it has been debated whether the surface can be charged positively at all. When a p.z.c. clearly above 2.0-2.5 is found, this may imply the presence of (trace-) amounts of impurities on the surface. For example, for the classical Ludox pH° = 3.5 probably by the presence of minor admixtures of Al3+ . Regarding the proton binding enthalpy, recall fig. II.3.61 where this quantity is given as a function of pH° for a variety of oxides. This graph only covers oxides with zero points above 5, but when the data are extrapolated to pH° = 2 a value for this enthalpy of about zero, or even positive, is found. The implication is that the formation of the double layer on silica must be entropically driven. For most silicas, direct lyotropic orders are found in specific binding, (Rb+ > K+ > Na+ > Li+ ), in line with the Pearson rule. One could say that silica and Li+ are both strongly hydrated and therefore avoid contact. All of this is background for the idiosyncratic stability behaviour of SiO2 sols. As anticipated in sec. II.3.10c, this is not in a direct way correlated to the pH. This peculiar trend came to light in classical studies by Allen and Matijevic11 and Depasse and Watillon21. Allen and Matijevic investigated two commercial Ludox samples, called AM and HS. Both have non-porous spherical particles of amorphous silica with a BET area of about 210m 2 g~ 1 . The size distribution is relatively narrow and the 11 21
L.H. Allen, E. Matijevic, J. Colloid Interface Sci. 31 (1969) 287; 33 (1970) 420. J. Depasse, A. Watillon, J. Colloid Interface Sci. 33 (1970) 430.
PAIR INTERACTION
3.171
Figure 3.68. Electrophoretic mobility of silica sols, a = 120 |j.m , c salt =0.1M. (Redrawn from Depasse and Watillon, loc. cit.)
particle radius a is about 15 \im. The difference between AM and HS is that the former contains 0.2% A12O3 . Depasse and Watillon used a homemade silica prepared by alkaline polymerization + dehydration of silicic acid. During syntheses, sizes were controlled by the slow addition of monosilicic acid to obtain fractions with a = 32, 54, 62 and 120 urn , with standard deviations below 0.10. Both pairs of authors measured the stability from a variant of light scattering to obtain stability ratios and c.c.c. values. For a full understanding of the double layer composition (and its peculiarities!) sets of <7°(pH,csalt) and f(pH,csalt) ought to be available, but in refs.1'2' only sparse electrokinetic data are available (in ref. ' some titrations in Na+ + Ca+ mixtures are included. Figure 3.68 gives mobilities in NaCl and KC1 according to21. The data for Ludox HS are similar (in these studies organic buffers had to be added and non-zero mobilities could be observed down to a (non-pristine?) i.e.p. of 1.6). The data of fig. 3.68 show no particularities at all, except for the fact that substantial mobilities could be observed at such high salt concentrations where many other sols are already unstable. Stability diagrams from the same sources are reproduced in figs. 3.69-71. Although the two groups of researchers use different silicas and employ different stability criteria, the results agree at least qualitatively in three respects: (i) At low pH the sols are very stable, even in concentrated electrolyte. (ii) With increasing pH (i.e. with increasing negative surface charge and increasing negative £ -potential), the sols become less stable and coagulate at pH's of at about 7-8. (iii) At still higher pH, restabilization is found with K+ or Cs+ as the counterion, but not with Li+ or Na+ .
11 21
L.H. Allen, E. Matijevic, loc. cit. J. Depasse, A. Watillon, loc. cit.
3.172
PAIR INTERACTION
Figure 3.69. Extinction coefficient, measured 6 s after mixing the silica sol with 1.5 M electrolyte as a function of pH; A, a = 62 fim ; o and • , a = 54 |j.m . (Redrawn from Depasse and Watillon, loc. clt.)
Figure 3.70. As fig. 3.69 but now for NaCl and LiCl. • , a = 62 |im ; for o and A , a = 54 nm .
Figure 3.71. Critical coagulation concentration of Ludox AM for the indicated electrolytes as a function of pH, measured 1 hr. after salt addition. Data for Ludox HS are similar. (Redrawn from Allen and Matijevic, loc. cit.)
PAIR INTERACTION
3.173
It Is very likely that in broad lines this behaviour must be explained in terms of strong hydration under near-uncharged conditions. At this point, SiO2 sols behave as if they are hydrophilic. In terms of DLVOE theory, this can be visualized by interaction curves like those in fig. 3.46 where d Is so large that beyond that distance no significant Van der Waals attraction remains. Note from table A3 that the Hamaker constant of silica is much lower than that of other oxides, so it is relatively easier to stabilize SiO2 sols electrostatically. The strong hydration layer is in line with the resilience of the surface against giving off protons at pH values up to —5-6. Recall that the adsorption enthalpy of protons is very low, so the uptake and release of protons Is entropically controlled. At these low pH values DLVO does not apply. At more elevated pH the hydration layer breaks down; at the same time the negative surface charge starts to increase. The strong hydration layer can no longer protect the particles against coagulation. We are discussing the rise in the extinction in figs. 3.69 and 70 and the decrease of the c.c.c. at a given pH In fig. 3.71, both in the pH ~ 7-8 range. Both pairs of investigators agree that the effectivlty to coagulate the sol increases in the direction Li+ < Na+ < K+ < Cs+ . This is in line with the direct lyotropic order in the binding of these ions1'. In line with this, c.c.c. values obey the inverse order, i.e. L1+ > Na+ > K+ > Cs+ . All of this is perfectly in line with the DLVOE model. However, upon a further pH rise, restabilization for electrostatic reasons may in some cases take over. The evidence from figs. 3.69-71 shows that its occurrence depends on the nature of the counterion: no restabilization with the (weakly binding) Li+ or Na+ ions or stabilization in the presence of the (stronger binding) K+ or Cs+ ions. The problem is that titration data do not exhibit a clear distinction between the pair Li+ , Na+ and the pair K+ , Cs+ . The origin of this restabilization at very high pH remains subject to dispute. It may be mentioned that a similar trend has been found indirectly via yield stress measurements by Franks21, reproduced in figs. 3.72 and 73. The so-called Geltech silicas, used in these studies, contain amorphous spherical and approximately homodisperse particles. The electrokinetlc potentials of fig. 3.72 were obtained using an acoustosizer, converting mobilities Into f s via the Smoluchowski equation (analyses using the O'Brien-White approach are less complete but exhibit the same trend). The yield stresses in fig. 3.73 depend on the volume fraction, particle size and structure of the particle network (which are kept constant) and on the attraction between the particles, which must be overcome to induce flow. This study confirms the trends observed previously. There is a destabilization range around pH 7-8, with increasing particle attraction from Na+ to Cs+ and restabilization at very high pH in at least KC1 and CsCl.
11
Th.F. Tadros, J. Lyklcma, J. Electroanal. Chem. 17 (1968) 267; (measured for precipitated silica, but also found for other S1O2 modifications). 21 G.V. Franks, J. Colloid Interface Sci. 249 (2002) 44.
3.174
PAIR INTERACTION
Figure 3.72. Zeta potentials of Geltech silicas in 0.4 M solutions of the indicated chlorides. (Redrawn from Franks, loc. cit.)
Figure 3.73. Yield stresses of 40 vol% dispersions of the same silica under the same conditions as in fig. 3.71. It may be concluded that the restabilizatlon in some electrolytes is well-established, but it cannot be a DLVO feature. Several mechanisms have been proposed, most of them involving the interference of cations with the hydration layers on the silica surface. Depasse 11 reviewed some of these models and suggested that the fact that Na + and Li + ions bind strongly to OH~ ions, whereas K+ , Rb + and Cs + do not associate at all with OFT ions, would explain the distinction in high pH restabilization behaviour between these two groups of ions. Tschapek and Torres Sanchez 21 looked at the influence of heat treatments (calcinations at high temperatures) on the c.c.c. for NaCl. This study underlines the relevance of hydration layers but does not shed light on the peculiarities at high pH. The present author believes that some dissolution of the silica 11 21
J. Depasse, J. Colloid Interface Sci. 194 (1997) 260. M. Tschapek, R.M. Torres Sanchez, J. Colloid Interface Scl. 54 (1976) 460.
PAIR INTERACTION
3.175
at elevated pH may play a role. The phenomenon is well established and, among other things, it leads to some irreversibility in the titrations; moreover, the presence of silicic acids in the solution has been analytically detected. It can be imagined that terminally anchored oligomeric silicic acid chains act as steric stabilizers; chemical affinities of counterions for such chains may then account for the observed specificities. Perhaps this suggestion deserves more systematic study. It may be added that Yaminsky et al.11 also reported similar 'chemical' influences in direct force measurements. Returning to the lower pH range, the interaction between SiO2 particles has also been studied by other techniques than classical coagulation studies. Penfold and Ramsay21 measured the structure factor S(q), see [3.12.3-6], by small angle neutron scattering, interpreting the obtained radial distribution function g{r) in terms of interparticle interactions. As is mostly the case in such studies, an oversimplified expression for G{r) was used for the fitting, viz. [3.7.20b], with yA (~ £) substituted from y°. The virtue of this study is that this technique is shown to work; the pH influence could also be established. However, the resulting potentials differ significantly from those measured directly. The interpretation deserves a better interaction model. In a similar vein, Chang et al.3) studied SiO2 sols by light scattering, small angle X-ray scattering and osmotic pressure measurements. They could show these techniques to be internally consistent, including the question whether the osmotic pressure from light scattering (sec. I.7.8f) was identical to that measured directly? However, the interpretation of the derived 'effective surface charge' remained obscure. Zerrouk et al.4) used the same scattering technique for studying Ca2+ -induced coagulation at pH 7.59. They did not try to obtain interaction Gibbs energies but focused on the fractal properties of the resulting coagulates; obviously, these structures are related to G(r). The double layer in the presence of Ca2+ (and Ba2+ ) ions has been investigated long ago5) and this can also be said about the stability61. As for the alkali ions, the lyotropic order in the affinity is direct: Ca2+ < Sr 2+ < Ba 2+ . Meagher71 measured the heterointeraction between mica and silica spheres by AFM and found the same trends as Depasse-Watillon and Allen-Matijevic reported for alkali counterion: short range repulsion caused by hydration around the p.z.c. but attraction at higher pH.
11
V.V. Yaminsky, B.W. Ninham, and R.M. Pashlcy, Langmuir 14 (1998) 3223. J. Penfold, D.F. Ramsay, J. Chem. Soc. Faraday Trans. (I) 81 (1985) 117. 31 J. Chang, P. Lesieur, M. Delsanti, L. Belloni, C. Bonnetgonnet, and B. Cabane, J. Phys. Chem. 99 (1995) 15, 993. R. Zerrouk, A. Folssy, R. Mercier, Y. Chevallier, and J.C. Morawski, J. Colloid Interface Sci. 139 (1990) 20. 5) Th.F. Tadros, J. Lyklema, J. Electroanal. Chem. 22 (1969) 1; M.A. Malati, S.F. Estefan, J. Colloid Interface Sci. 22 (1966) 307. 61 R.K. Her, J. Colloid Interface Sci. 53 (1975) 476. 71 L. Meagher, J. Colloid Interface Sci. 152 (1992) 293. 21
3.176
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Figure 3.74. Electrophoretic mobility of colloidal SiO2 (0.1 g dm- 3 ) in 1CT3M (HNO3+ KNO3) in the presence of various added concentrations of Co(NO3) (indicated) as a function of pH. (Redrawn from James and Healy, loc. cit.)
Several studies report on the adsorption of Al3+ and other high-valency hydrolyzable cations on the electrokinetic properties and stability of SiO2 sols. By way of illustration, fig. 3.74 gives electrophoretic mobilities in the presence of Co(NO3)2, taken from the work by James and Healy11. It is interesting to compare this figure with its counterpart for TiO2 , fig. 3.63. At pH below 6 the SiO2 surface is negative, but this does not lead to substantial Co2+ adsorption, which seems unexpected, the more so as on the positively, or slightly negatively charged TiO2 much adsorption takes place. Apparently it is again the hydration layer on the silica which makes it resilient against the uptake of Co2+ or one of its hydroxides. On the other hand, at higher pH, when this layer is absent, the behaviour on silica is similar to that on rutile. Eventually at high pH and high coverage of Co(OH)2, the particle coated this way behaves electrokinetically as a Co(OH)2 particle with a heart of silica. 3.13c Latices Latices are polymer colloids, colloidal spheres of which the particles consist of polymer. Historically, the name latex (singular) came from the milky sap, which can be tapped from Hevea Braziliensis trees, and which contain 36% hydrocarbon from which natural rubber can be made. Nowadays latices can be made synthetically and because it is not so difficult to prepare them homodispersely, they have been, and still are, popular model colloids. Besides this, they find wide applications, for instance as 11
R.O. James, T.W. Healy, J. Colloid Interface Set. 40 (1972) 53.
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coating material, in paints and a variety of other household products, in medical science, and immunochemistry (as carriers for proteins and immunoglobulins; illustrations to follow in chapter V.3), and they also act as a link in immunological tests such as ELISA. For our present purpose, our attention will be focused on its colloid stability. Before doing that, let us recall that in earlier volumes of FICS several experiments with latices have already been reported. Fig. II.3.29 describes the measurement of the number of sulphate groups on the particle surface by eonductometric and potentiometric titration, figs. II.4.34 and 35 give the resistance and streaming potentials, respectively, of polystyrene latex plugs, fig. II.4.29 shows
J. Hearn, M.C. Wilkinson, and A.R. Goodall, Adv. Colloid Interface Sci. 14 (1981) 173-236. R. Arshady, Colloid Polym. Sci. 270 (1992) 717-32. 31 Q. Wang, S.K. Fu, and T.Y. Yu, Progr. Polym. Sci. 19 (1994) 703-753. 41 Colloidal Polymers, Synthesis and Characterization, A. Elaissari, Ed., Marcel Dekker (2003). 51 J.W. Vanderhoff, H.J. van den Hul, R.J.M. Tausk, and J.Th.G. Overbeek, in Clean Surfaces: Their Preparation and Characterization for Interfacial Studies, G. Goldfinger, Ed., Marcel Dekker (1970). 61 H.J. van den Hul, J.W. Vanderhoff, Brit. Polym. J. 2 (1970) 121. 71 D.H. Everett, M.E. Giiltepe, and M.C. Wilkinson, J. Colloid Interface Sci. 71 (1979) 336. 81 J.N. Shaw, J. Polym. Sci. C27 (1969) 237. 91 A. Kotera, K. Furusawa, Y. Takeda, and K. Kudo, Kolloid-Z.Z. Polymere 239 (1970) 677. 101 A. Kotera, K. Furusawa, and K. Kudo, Kolloid-Z. Z. Polymere 240 (1970) 837. III K. Furusawa, W. Norde, and J. Lyklema, Kolloid Z.Z. Polymere 250 (1970) 908. 21
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face charge and elaborated more systematically by Goodwin et al. ll2) , see also31. The trend is for surfactant-free latlces having larger particles than those made in the presence of emulsifiers. For some purposes this may be a drawback, but the advantage Is that after preparation no surfactants have to be removed. In passing, it is not always straightforward to ascertain completeness of removal; however, when the surfactants are so strongly bound that they cannot be detectably stripped, they do perhaps not interfere with later studies. In interaction studies, large particles can lead to secondary minimum coagulation (fig. 3.44). Nowadays latices can be synthesized with a variety of bulk and surface properties, sometimes tailor-made. Regarding the bulk of the particles, copolymerizatlon can increase the hardness (e.g. copolymerlzation of dlvinylbenzene with styrene) and the particles can be made to conduct (e.g. with poly(pyrrol)) and show magnetism (to carry out studies as described in sec. 3.10c). Fluorescent labels can be embedded or refractive index matching can be achieved: (1) striving for identity of the refractive index with that of the medium, so that in scattering studies the contribution of the bulk Is bleached out and only the surface layer is seen and (ii) reducing the Hamaker constant across the liquid. Such model systems have proved useful in the study of non-aqueous latices, sterically stabilized by oligomerlc brushes, acting as a parapet. Core-shell latices can also be prepared, for example with a core providing for mechanical strength or providing a specific refractive Index with a (thin) shell to impart specific stability improving properties. The Hamaker constant depends on the matching of the (complex) dielectric permittivities of the polymer and the solvent (sec. 1.4.7), but the surface layer contribution must not be Ignored. Some typical values can be found in table A3.4. The charge-determining groups on the surface can be negative or positive, strong or weak, or the surfaces can be made amphoteric. The most familiar negative group is the sulphate group, originating from K 2 S 2 O 8 (potassium persulphate), which is a popular initiator. Sulphate groups are strong (as an electrolyte) but (chemically) somewhat liable to hydrolysis, which may yield the weaker -COOH and non-dissociating -OH groups. Carboxylic groups can appear on the surface with H2O2 (hydrogen peroxide) or certain organic substances as the initiator. Positive groups Include the tetramethyl ammonium and amldlne, -(NH2)2 ; the latter may hydrolyze at high pH and/or high temperature to give amides. For a polystyrene latex with a pH-independent positive surface charge, see . It may be added that y -radiation has also been used to initiate
11 J.W. Goodwin, J. Hearn, C.C. Ho, and R.H. Ottewill, Brit. Polym. J. 5 (1973) 347; Colloid Polym. Sci. 252 (1974) 464. 2) J.W. Goodwin, R.H. Ottewill, R. Pelton, G. Vianello, and D.E. Yates, Brit. Polym. J. 10 (1978) 173. 31 J.H. Kom, M. Chainey, M.S. El Aasser, and J.W. Vanderhoff, J. Polym. Sci. A, Polymer Chem. 30 (1992) 171. 41 J. Blaakmeer, G.J. Fleer, Colloids Surf. 36 (1989) 439.
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the polymerization process. (ii) Surface characterization. It follows from the previous paragraphs that latices are versatile model systems, and as such they find wide application. However, for really quantitative studies in terms of DLVO or DLVOE theory, a lot of water has to flow under the bridge. To that end, the composition of the surface has to be established. The issues to be solved can be summarized as follows: (1) What is the nature of the surface charges and what is their surface density? (2) Which fraction of the countercharge resides in the Stern layer? (3) Is the surface hairy? The answers to these three questions are to a certain extent correlated. When there is only one type of charged groups the determination of the surface charge o° is relatively straightforward. To that end, conductometric and/or potentiometric titrations can be carried out. Surfaces containing only strong groups are the easiest to characterize. They give rise to interaction at constant a°, and by titration only one number is to be measured. Figure II.3.29 gives an illustration11. For weak groups, the charge is pH-dependent and (potentiometric) titrations have to be carried out and analyzed in the same way as for oxides. Such latices exhibit charge regulation upon particle interaction. The problems become more challenging when the surface carries more than one group, say carboxyls, next to sulphates. In those cases, mostly conductometry2'31 and potentiometry have to be carried out and analyzed, preferably at more than one indifferent electrolyte concentration. Figure 3.75 gives an illustration of a o° (pH) graph for an amphoteric latex; it resembles the a° (pAg) graphs for silver iodide41 more than those for oxides (fig. II.3.59 and 63). Questions (2) and (3) above are coupled because hairiness and the position of the effective slip plane are related. Moreover, the conversion of electrophoretic mobilities or streaming potentials (for latex plugs) into £ -potentials depends on the hairiness. Addressing this last issue first, let us repeat that in sec. II.4.6e,f, reference has been made to a number of mobility- tea curves for latices, displaying a maximum. When these curves are unwittingly converted into £ -potentials, using Helmholtz-Smoluchowski or O'Brien-White theory, values of £ (csalt) curves are found passing through a maximum, which is physically unrealistic. It was shown that in this conversion ion mobility in the stagnant layer has to be accounted for. If that is properly done, £" becomes a decently decreasing function of c s a l t . See figs. II.4.29 and 30 and ref.51' where the spurious maximum in £" is an artefact. Regrettably, incorrectly computed
11 More illustrations in P. Bagchi, B.V. Gray, and S.M. Birnbaum, J. Colloid Interface Sci. 69 (1979) 502. 21 W.T. McCarvill, R.M. Fitch, J. Colloid Interface Sci. 66 (1978) 204. 31 M.E. Labib, A. Robertson, J. Colloid Interface Sci. 77 (1980) 157. 41 B.H. Bijsterbosch, J. Lyklema, Adv. Colloid Interface Sci. 9 (1978) 147. See also figs. II.3.41 (lyotropy) and 42 (capacitances). 51 M. Minor, A.J. van der Linde, and J. Lyklema, J. Colloid Interface Sci. 203 (1998) 33.
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Figure 3.75. Surface charge as a function of pH for an amphoteric latex carrying sulphate, carboxyl and amine groups. The curves are individual and were shifted to merge at the i.e.p. = pH° . Electrolyte, KNOg , concentration indicated. (Redrawn from I.H. Harding, T.W. Healy, J. Colloid Interface Set 107 (1985) 382.)
£ 'csait) curves continue to appear in the literature; based on such data, no convincing DLVO/DLVOE interaction curves can be constructed1'21. The relation between the hairiness and surface conductivity is that the distribution of polymer segments dictates the tangential mobility of counterions in the stagnant layer. As this flow pattern is not easily predicted theoretically, the best solution is to carry out additional conductivity measurements (in plugs), splitting the total conductivity in a bulk and interfacial part. See3' for an illustration. With this information available, one can at least compute a f -potential and arrive at a zeroth order G(h) assuming £" to remain constant upon interaction. One step better is to establish the charge o"1 in the stagnant layer from charge balance, (a° +al +o~ek = 0), obtaining a° from titration and crek from the C,potential, using PB theory. With a model for the adsorption isotherm in the stagnant layer, a f(h) or tr^th) relation can be set up, including regulation and, hence, a better expression can be found for G(h). There is much circumstantial evidence for the unintentional presence of 'hairs,' which are remnants of the polymerization process and difficult to control. Why would, upon polymerization, a chain end, carrying a sulphate group submerged into the Jl
M. Elimelich, C.R. O'Melia, Colloids Surf. 44 (1990) 165. D. Bastos, F.J. de las Nieves, Colloid Polym. Set 271 (1993) 860. Conduction in the stagnant layer was considered by A.F. Barbero, R.M. Garcia, M.A.C. Vilchez, and R. Hidalgo-Alvarez, Colloids Surf. A92 (1994) 121. 21
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growing droplet move out till this group Is exactly at the Interface? There are arguments for this group to stick out into the solution. Most likely the hairiness (number and lengths of the hairs) varies from one batch to the other. Suspicion about the presence of hairs comes from various sources. One of these is that the counterion valency effect is usually less pronounced than with 'bald' hydrophobic colloids. Information is mostly indirect, i.e. via a combination of electrokinetic characterizations (with the problem indicated above) with stability studies and determination of the particle radius. There are indications that heat treatment of latices reduces the hairiness, but it remains uncertain how much, or how little, hairiness is left after the treatment because there is no option for measuring it. We refrain from a systematic discussion. Some literature examples are refs.1'2'3'41. {Ill) Stability. Although the precise characterization of the surfaces of the latices will remain a problem for a while, a variety of studies have been undertaken to investigate the collective stability and phase behaviour of latices, exploiting their beautiful spherical shape and homodispersity. Everybody who has made latices in the lab and has dialyzed them to get rid of the excess electrolyte observes the development of opalescent colours inside the bag. The explanation is that with decreasing salt concentration the range of the electric repulsion increases till eventually all particles 'see' each other. In that sense, the sols effectively become concentrated. The system responds by ordering itself. Diffraction of light ensues and, given the sizes of the particles, the results are in the visible part of the spectrum. This typical illustration of collective stability has been applied to make artificial opals and polymer colloid crystals51. Extensive discussions of concentrated sols and their phase behaviour will follow in chapter 5. Techniques for studying or utilizing such systems include light scattering (3.12a). (a) Light scattering and other optical techniques® ''8'9) , which all have in common that rather advanced optical and mathematical techniques are coupled to a poor description of interparticle interaction. The fact that one can get away with simplified G(r) curves stems from the measuring conditions. The ordering and, hence, g(r) are mainly determined by the outer, repulsive part of G(r), which is rather independent of the structure of the inner double layer part. When measurements are carried out under
11
J.B. Smitham, D.V. Gibson, and D.H. Nappcr, J. Colloid Interface Scl. 45 (1973) 211. W.M. Brouwer, R.L.J. Zsom, Colloids Surf. 24 (1987) 195. 31 J.E. Seebergh, J.C. Berg, Colloids Surf. A100 (1995) 139. 41 X. Wu, T.G.M. van de Ven, Langmuir 12 (1996) 3859. 51 T. Okubo, Progr. Polym. Set 18 (1993) 481. 61 D.J. Cebula, J.W. Goodwin, G.C. Jeffrey, and R.H. Ottewill, Discuss. Faraday Soc. 76 (1983) 37. 71 U. Apfel, R. Grunder, and M. Ballauff, Colloid Polym. Set 272 (1994) 820. 81 C. Johner, H. Kramer, S. Batzill, C. Graf, M. Hagenbuckle, C. Martin, and R. Weber, J. Physique 114 (T994) 1571. 91 R.Y. Ofoli, D.C. Prieve, Langmuir, 13 (1997) 4837. 21
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attraction one must know the composition of the double layer in more detail. Of the many studies, we refer to Okubo who found a counterion specificity in the formation of iridescent crystals in the direction H+ < Na+ < TMA+ < TBuA+, which is in line with the direct lyotropic order expected for specific binding to hydrophobic surfaces. (b) Small angle neutron scattering (SANS) and X-ray scattering. We refer to an extensive review by a Japanese group21, mainly intended as a pabulum for a dispute on the nature of the pair interactions. The technique also works for the study of crystallized latices via Bragg reflections. (c) Measurement of osmotic pressures or pressure-induced coagulation31. Rymden41 concluded from osmotic pressure studies that there must be substantial counterion binding, although it was difficult to quantify. Pressure-induced compaction is mostly carried out in (ultra-)centrifuges5 6'71. (d) Rheology. Concentrated latices are suitable model systems for the study of volume fraction-dependent viscosity, rj((p). See table 6.4 and, for the electroviscous effects, sec. 6.9b. The relation with G(r) is not very prominent. (e) Coagulation by electrolytes, which is more in line with the previous subsections. The main difference with approaches (i) - (iv) is that now mainly dilute sols are investigated. Consequently, the relation with G{r) is more direct. Critical coagulation values are usually obtained from logW-logc studies. Immediately a problem arises: the rates should be radius (a)-dependent but that is mostly not observed, whereas the c.c.c. ought to be independent of a, although it is often found to be dependent. The former feature probably has a dynamic origin, see sec. 4.4, whereas the latter may be of a chemical nature (prolonged particle growth may lead to different surface structures). By way of illustration, Ottewill and Ranee81 reported c.c.c. values for dialyzed poly(tetrafluoroethylene) (PTFE) latices at pH 3, where multivalent ions do not hydrolyze. The results are 47, 74 and 0.16 mM for NaCl, Ba(NO3)2 and A1(NO3)3 , respectively. These values do not obey a clear power law, as demanded by [3.9.10 or 11], and, as £"-potentials were not reported, one cannot make statements about the double layer composition. However, in a subsequent paper91, the same authors found, in line with sec. 3.9j, that upon increase of pH, the stability is governed by specific adsorption of Al-hydroxy complexes having charges below +3. A study more in line with the demands for a profound analysis of the interaction 11
21
T. Okubo, J. Chem. Soc. Faraday
Trans. 8 7 (1991) 1361.
S. Dosho, N. Ise, K. Ito, S. Iwai, H. Kitano, H. Matsuoko, H. Nakamura, H. Okumura, T. Ono, I.S. Sogami, Y. Ueno, H. Yoshida, and T. Yoshiyama, Langmulr 9 (1993) 394. 31 A.E.J. Meijcr, W.J. van Megen, and J. Lyklema, J. Colloid Interface Sci. 66 (1978) 99. 41 R. Rymden, J. Colloid Interface Sci. 124 (1988) 396. 51 T.G. Lanyi, G. Horvath-Szabo, and E. Wolfram, J. Colloid Interface Sci. 98 (1984) 72. 61 S. Rohrsetzcr, P. Kovacs, and M. Nagy, Colloid Polym. Sci. 264 (1984) 812. 71 M.W. El-Aasscr, A.A. Robertson, KolloidZ. Z. Polymere 251 (1973) 241. 8) R.H. Ottewill, D.G. Ranee, Croat. Chem. Ada 50 (1977) 65. 91 R.H. Ottewill, D.G. Ranee, Croat. Chem. Acta 52 (1979) 1.
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has been conducted by Behrens et al.11. These authors worked with homodisperse carboxylated latices of which the double layer was systematically studied by potentiometric titration and electrophoresis. The dependence of o° on pH and c salt was modeled by a site-binding model. It is known that good fitting of the titration curves does not yet guarantee a good prediction of C, -potentials. In this case, mobilities were converted into
S.H. Behrens, D.I. Christl, R. Emmcrzael, P. Schurtenbergcr, and M. Borkovcc, Langmuir 16 (2000) 2566. 21 R.H. Ottewill, in NATO-ASI Series 303, Ser. C. (1990) Scientific Methods to Study Polymer Colloids and their Applications, 129. R. Hidalgo-Alvarez, A. Martin, A. Fernandez, D. Bastos, F. Martinez, and F.J. de las Nieves, Electrokinetic Properties, Colloidal Stability and Aggregation, Kinetics of Polymer Colloids, Adv. Colloid Interface Sci. 67 (1996) 1 (443 refs.).
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is that in many applications only particle repulsion at long distance has to be considered, which is not particularly sensitive to the static and regulation properties of the inner layer. See, for example, curves 1, 2 and 3 in fig. 3.7f. On the other hand, this inner layer plays a very important role in attractive situations, as in adhesion or electrodeposition. In the present volume of FICS, pair interactions also recur in chapters 4 and 6; in volume V this is the case in chapters 2 (interactions between polyelectrolytes), 3 (interaction between proteins and surfaces), and 6 (disjoining pressures across thin liquid films). The formation of electrical double layers, sometimes including interaction, plays a role in ionic micelles (chapter V.4), micro-emulsions (chapter 5), and, to a lesser extent, foams and emulsions (chapters V.7 and 8). Even if in these chapters pair interactions do not recur as such, several features encountered in the present chapter (such as the influences of particle radius, counterion valency, and specific ion effects) do play their roles. As an entry to applications beyond FICS, we can return to some of the ten phenomena mentioned on p. 1 of chapter 1, reconsidering the interpretations offered on the pages thereafter. The influence of salinity on delta formation (example 1) is an obvious and direct consequence of DLVO-rype salt influence on pair interaction. Extrapolating the DLVO rule, one may expect stronger sedimentation in harder water, carrying more Ca2+ salts. (Invariably such particles are negatively charged so that cations are the counterions.) Sterically stabilized particles are insensitive (or at least much less sensitive) to saline. In rivers, estuaries and especially oceans, such steric barriers often have a biogenic origin; 'in oceans all particles have at least once been eaten.' Knowledge on stability in natural waters is useful in the treatment of waste water. Getting rid of small particles is mostly achieved by adding minor amounts of electrolytes with high cation valency in combination with polymeric flocculants. Besides this colloidal intervention, it is appropriate to let nature do the purification on its own devices by letting suitable bacteria break down matter that they consider 'edible.' Here again, colloid interaction plays a role in the forming of bacterial colonies. As expected, this interaction has DLVOE-type plus steric aspects. Similar phenomena can be recognized in the structure of soils. Soils consist of a mixture of minerals (mostly silicates), quartz, clays and some biogenic material. For agricultural purposes, soils must have an open structure (channels needed for the transport of nutrients) and at the same time have sufficient mechanical strength. In colloidal terms, this means that soils should be in the coagulated state. In practice this is mostly automatically achieved because of the presence of sufficient Ca2+ . In 1953 the Netherlands suffered from serious floods during which large areas of fertile arable land were covered by seawater. Reclaiming the land by repairing the dikes and pumping the water out was not enough to restore the desirable soil structure because the salt water gave rise to exchange of the Ca2+ against Na2+ ions, leading to stability
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in the colloidal sense, and hence to a very compact, inaccessible soil. The Dutch, with their common sense of colloid stability, prevented this problem by scattering gypsum before ploughing. As to gold sols, phenomenon 2, as long as these are electrostatically stabilized they are readily coagulated by minor amounts of electrolyte because gold has a high Hamaker constant (table A3.2). Coagulation studies carried out with this system belong to the oldest available; stability changes are easily optically monitored and the critical coagulation concentrations exhibit clear Schulze-Hardy trends. It is of historical interest that Zsigmundy's ultramicroscopic monitoring of the coagulation of gold sols, mentioned in sec. 1.4, led Smoluchowski to develop his famous theory for the kinetics of fast coagulation (sec. 4.3a), long before particle interactions were understood. Paints (phenomenon 4) and other pigment dispersions are multicomponent mixtures of a variety of materials, including one or more pigments, builders, polymers, surfactants, corrosion inhibitors, etc., dispersed in a fluid that may be aqueous ('water borne') or non-aqueous ('solvent'). Such multicomponent mixtures are subject to a variety of practical demands, dictated by the intended application: hiding power, gloss, ultraviolet light- and high temperature resistance, etc. These demands lead back to Theological demands, in particular as to the effective viscosity as a function of the rate of shear, (rate of shear thinning), sometimes thixotropy and other rheological phenomena, see chapter 6. On top of this, for purposes requiring atomizing and spray coating at elevated temperatures, as in automotive painting, defect-free, heat-resistant films must be formed. It is needless to state that this branch of technology is subject to continuing research. The point is to recognize the pair hetero-interactions in it. In this respect let us note that, because of environmental reasons, there is a trend of replacing solvent-borne by water-borne systems, which automatically implies a growing interest in the electrical contribution to the interaction. As a result, interest is growing on the double layer properties of the solids involved, on the possibilities of modifying them and on the dynamics of hetero-interaction. Shelf stability is at least as important as the instantaneous interaction. Van der Waals attractions are dictated by the Hamaker constants of the materials involved. These cannot be changed, but application of coatings may help to reduce the attraction; a thin layer of SiO2 on TiO2 particles reduces the effective Hamaker constant because the Hamaker constant of SiO2 is between that for TiO2 and that for water1'2'31. In sec. 3.13a it was stated that such layers also have an optical screening effect, so in practice the coating has a dual function. Now addressing item 8 of the introductory phenomena, we come to the domain of ceramics. Such materials have been known since antiquity; see also the introduction to chapter 2. Besides objects of art and household goods, like china, there is now a
11
M.J. Void, J. Colloid Interface Sci. 16 (1961) 1. D.W. Osmond, B. Vincent, and F. Waite, J. Colloid Interface Sci. 42 (1973) 262. 31 B. Vincent, J. Colloid Interface Sci. 42 (1973) 270. 21
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demand for ceramics for technical purposes. The aim is to make materials of high mechanical strength that are resistant against temperatures as high as 2000 K, so that they can compete with, and replace, metals and alloys. Basically there are two ways of manufacturing, the dry and the wet route. Both have in common that a raw material has to be selected. Historically such clays have always played an important role, next to a number of minerals. Today a variety of synthetic, inorganic raw materials are used, sometimes tailor-made for special application, for example silicium carbide and nitride for manufacturing hard, strong, temperature-resistant ceramics. Via the dry route the powder is, after intermediate steps, fired as such, whereas via the wet route the material is dispersed in a medium (mostly aqueous), which is then baked. It is here that colloid stability enters, and as far as this stability is not achieved by surfactants or polymers, it is of an electric origin and determined by the laws described in this chapter. The colloidal system acts as a precursor, which has to be densified to attain the appropriate Theological consistency. These slurries are slip-cast in a mold, having the shape wanted for the final product. This molded shape is called the green, the name stemming from the often slightly greenish colour of pastes for making porcelain. Eventually the green is fired to obtain the desired product. Colloidally, the greens must be very dense and therefore stable. Upon firing, no cracks may form, calling for strict ordering on the colloidal scale. This, in turn, requires homodisperse systems or spacefilling mixtures. Stability can be attained electrostatically or sterically. Sometimes the latter is easier in the 'pre-green' state, but upon firing the added organic must burn completely. For further reading see the relevant literature1'21. This does by no means exhaust the wide range of applications of pair interactions. Key words such as slurry compaction, photographic films, magnetic tapes, adhesion, electrodeposition and flotation, point to the richness of the topic. 3.15 General references There are no books or reviews treating the contents of the present chapter in more detail than presented here. However, there are many references (a) treating interactions as part of more extended texts on colloids, (b) dealing with special aspects in more detail than here, and (c) emphasizing applied aspects of colloid stability. These references are organized accordingly. 3.15a IUPAC recommendations In addition to those mentioned in sec. 1.5a and 6.15a, we mention J. Ralston, I. Larson, M.W. Rutland, A.A. Feiber and J.M. Kleijn, Atomic Force Microscopy and Direct Surface Force Measurements in Pure Appl. Chem., in course of publication (2004).
2)
T.A. Ring, Fundamentals of Ceramic Powder Processing and Synthesis, Acad. Press (1996). W.M. Sigmund, N.S. Bell, and L. Bergstrbm, J. Am. Ceram. Soc. 83 (2000) 1557.
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3.15b References containing parts on pair interactions Authors Div. Adv. Colloid Interface Set 100-102, (2003) a special volume, in honour of Prof. Overbeek, Th.F. Tadros, Ed. (Includes contributions related to pair interactions: lyotropic sequences (Lyklema), coagulation of clay minerals (Lagaly and Ziesmer), stagnant layers (Hunter), black films (Platikanov, Nedyalkov and Petkova), particle adhesion (Adamczyk), coagulation by hydrolyzing metal salts, (Duan and Gregory), and sedimentaiton of charged colloids (Philipse and Koenderink). A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th ed. John Wiley (1997). (Familiar textbook on interface science; contains rather condensed chapters on double layers and Van der Waals interactions). Handbook of Surface and Colloid Chemistry, K.S. Birdi, Ed., CRC-Press, 2nd print (2003). (Multi-authored volume, contains topics related to the themes of our chapters III. 1,2 and volume V.) H.J. Butt, K. Graf and M. Kappl, Physics and Chemistry of Interfaces, Wiley (2003). (Contains, besides chapters on double layers and interactions, contributions on interfacial thermodynamics, adsorption, wetting and some themes of FICS volume V.) Electrical Phenomena at Interfaces; Fundamentals, Measurements and Applications, A. Kitahara, A. Watanabe, Eds., Marcel Dekker (1984). (Volume 15 of the 'Surfactant Science' series; contains chapters on double layers, electrokinetics, and stability.) P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, 3 rd ed., Marcel Dekker (1997). (Well-known textbook on colloid science, contains several chapters dealing with the themes of the present chapter.) R.J. Hunter, Foundations of Colloid Science, Oxford University Press, 2nd ed. (2001). (This well-known textbook on colloid science contains chapters on double layers, electrokinetics, dispersion forces, particle interactions.) J.N. Israelachvili, Intermolecular Forces, with Application to Colloidal and Biological Systems, Academic Press, 2nd ed. (1991). (Discusses in some detail, several aspects of this chapter, in particlar the Van der Waals part.) G. Lagaly, O. Schulz and R. Zimehl, Dispersionen und Emulsionen; Eine Einfiihrung in die Kolloidik feinverteilter Stoffe einschliesslich der Tonminerale, Steinkopf (1997). (A modern general text in the German language, covering a wide spectrum of colloidal phenomena. Also contains biographies of well-known colloid scientists (by K. Beneke).)
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W.B. Russel, D.A. Savllle and W.R. Schowalter, Colloidal Dispersions, Cambridge University Press (1989). (Emphasis on the more physical aspects; written in a condensed style. Contains chapters on single and interacting double layers, dispersion forces, electrokinetics and some applications.) See also the references to chapter 1. 3.15c References addressing special aspects of pair interactions Authors Div. Discussions of the Faraday Society-Faraday Discussions of the Chemical Society, as far as they are devoted to colloid stability and related topics; starting from 1939: (1939) Electrical Double Layers, abandoned owing to the outbreak of the 2nd World War, but published In the Transactions; (1950) Electrical Double Layers, no. 18 (1954) Coagulation and Flocculation; no. 42 (1966) Colloid Stability in Aqueous and Non-aqueous Media; no. 65 (1978) Colloid Stability; no. 76 (1983) Concentrated Colloidal Dispersions; no. 90 (1990) Colloidal Dispersions; no. 123 (2002) Non-equilibrium Behaviour of Colloidal Dispersions. Z. Adamczyk, P. Warszynski, Role of Electrostatic Interactions in Particle Adsorption, in Adv. Colloid Interface Sci. 63 (1996) 41, and Z. Adamczyk, P. Weronski, Application of the DLVO theory for particle deposition problems, ibid. 83 (1999) 137. (Two reviews with some overlap, 148 and 228 references, respectively. Both treat pair Interaction in some detail, including hetero-interaction and particles of non-planar shapes. The theory is applied to particle deposition (adhesion), including the inherent fluid dynamics.) L. Bergstrom, Hamaker Constants of Inorganic Materials, in Adv. Colloid Interface Sci. 70 (1997) 125-169. (Lifshits-type calculations, discussions of the contributions of the various ranges in the dispersion spectrum, tabulation for inorganic materials.) W.R. Bowen, F. Jenner, The Calculation of Dispersion Forces for Engineering Applications, in Adv. Colloid Interace Sci. 56 (1995) 201-243. (Review of principles and elaborations; comparison between different approximations for the Hamaker function.) B. Cappella, G. Dietler, Force-distance Curves by Atomic Force Microscopy, in Surf. Sci. Rept. 34 (1999) 1-104. (Detailed review, 236 refs., about instrumental principles and handling; results obtained up to June 1999.) P.M. Claesson, T. Ederth, V. Bergeron and M.W. Rutland, Techniques for Measuring Surface Forces, in Adv. Colloid Interface Sci. 67 (1996) 119-183. (Review, 179 refs., focus on technicalities, describes a wide range of techniques.)
PAIR INTERACTION
3.189
V.S.J. Craig, A Historical Review of Surface Force Measurements, in Colloids Surf. A129-130 (1997) 75-94. (109 refs., also contains technicalities and results.) R. Ettelaie, R. Buscall, Electrical Double Layer Interactions for Spherical ChargeRegulating Colloidal Particles in Adv. Colloid Interface Set 61 (1995) 131. (Review, 30 refs., introducing an improvement to the DH-approximation and site bindingregulation between flat surfaces.) R.H. French, Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics in J. Am. Ceramic Soc. 83 (2000) 2117-46. (Review, 235 references, mostly on Lifshits-type computation of Hamaker constants with a few applications to colloid and interface science.) Ph.C. van der Hoeven, J. Lyklema, Electrostatic Stabilization in Non-aqueous Media in Adv. Colloid Interface Sci. 42 (1992) 205-277. (Detailed review, 107 refs. on the stability of colloids In media of low polarity.) P.F. Luckham, B.A. de L. Costello, Recent Advances in the Measurement of Interparticle Forces in Adv. Colloid Interface Sci. 44 (1993) 183-240. (Review, 165 refs., describes the principles of surface-surface, particle-surface, and particle-particle interaction methods.) J. Lyklema, Principles of the Stability of Lyophobic Colloidal Dispersions in Nonaqueous Media, in Adv. Colloid Interface Sci. 2 (1968) 66. (Classical review, 191 refs.) I.D. Morrison, Electrical Charges in Non-aqueous Media, in Colloids Surf. A71 (1993) 1-37. (Review, 182 refs., on the origin, measurement and interpretation of charges on particles in apolar and low-polar media.) H. Ohshima, Electrostatic Interactions Between Two Spherical Colloidal Particles in Adv. Colloid Interface Sci. 53 (1994) 77-102. (Review, 20 refs., pays attention to 'soft' surfaces and particles with non-zero dielectric permittivity.) Ordering and Phase Transitions in Charged Colloids, A.K. Arora, B.V.R. Tata, Eds., V.C.H. Publishers (1996). (Much information on various aspects of phase formation in concentrated, mostly homodisperse, colloids. Interpretation has some emphasis on the so-called Sogami-Ise interaction.) G.D. Parfitt, J. Peacock, Stability of Colloidal Dispersions in Non-aqueous Media in Surface and Colloid Set, E. Matijevic, Ed., 10 (1978) ch. 4. (Classical review, emphasis on media of low dielectric constant.)
3.190
PAIR INTERACTION
D.C. Prieve, Measurement of Colloidal Forces with TIRM in Adv. Colloid Interface Sci. 82 (1999) 93-125. (Review, 53 refs. of optical technical techniques for measuring particle interactions, emphasizing levitation and related procedures.) E.S. Reiner, C.J. Radke, Double Layer Interactions between Charge-regulated Colloidal Surfaces: Pair Potentials for Spherical Particles bearing Ionogenic Surface Groups in Adv. Colloid Interface Sci. 47 (1993) 59. (Review and generalization of charge regulation models, Gibbs energy functionals, various elaborations in the DH approximation; comparison between various approaches.) 3.15c! Pair interactions: applied aspects Many specific applications can also be found in the references of subsecs. 15a and 15b. for instance, the second half of the book by Kitahara and Watanabe (15a) is devoted to applications and Lagaly et al.'s book treats the application of pair interactions to clay minerals. Aut. Div. Nanoparticle Assemblies, in Faraday Discuss, Roy. Soc. Chem. 125 (2004). (Various papers on the lower border of colloids.) Colloid Chemistry in Mineral Processing, J.S. Laskowski, J. Ralston, Eds., Elsevier (1992). (Collection of papers dedicated to J.A. Kitchener, treating the interfacial and colloidal aspects of flotation. Contains applications of particle-particle and particle-bubble interactions.) M. Elimelech, J. Gregory, X. Jia and R.A. Williams, Particle Deposition and Aggregation: Measurement, Modelling and Simulation, Buttterworth, Oxford (1995). Handbook of Applied Surface and Colloid Chemistry, K. Holmberg, Ed., John Wiley (2001). (Extensive, emphasizes the role of surfactants.) A.V. Nguyen, H.-J. Schulze, Colloidal Science of Flotation, Marcel Dekker (2004). (Colloids, bubbles, interactions; statics and dynamics, with applications.) H. van Olphen, An Introduction to Clay Colloid Chemistry, 2nd ed. John Wiley (1977). (Application of DLVO theory to clay minerals, old, but not dated.) T.A. Ring, Fundamentals of Ceramic Powder Processing and Synthesis, Academic Press (1991). (Contains much information on colloid stability aspects.) Nanostructure Science, A World-Wide study, R.W. Siegel, Ed., Kluwer Academic Publishers (1999). (Review by a panel.)
4
DYNAMICS AND KINETICS
Marcel Minor and Herman P. van Leeuwen 4.1 Introduction 4.2 Diffusion of colloidal particles 4.2a Nature of particle motion; various timescales and force field effects 4.2b Probing Brownian motion of particles by dynamic light scattering (DLS) 4.3 Coagulation kinetics 4.3a Diffusion-controlled particle-particle approach; rapid coagulation 4.3b Particle-particle approach with repulsive forces; slow coagulation 4.4 Electrodynamics of particle-particle interaction 4.4a Double layer relaxation routes 4.4b Double layer relaxation in particle-particle interaction 4.5 Electrodynamic relaxation of colloids in external fields 4.5a A qualitative comparison of relaxation times 4.5b Particle-particle interaction in a hydrodynamic field; orthokinetic coagulation 4.6 Aggregation and fractals 4.7 Applications 4.8 General references
4.1 4.3 4.3 4.9 4.11 4.11 4.15 4.18 4.18 4.26 4.33 4.33 4.37 4.44 4.51 4.53
This Page is Intentionally Left Blank
4 DYNAMICS AND KINETICS MARCEL MINOR AND HERMAN P. VAN LEEUWEN
4.1 Introduction Many of the preceding chapters, in this volume as well as in the previous ones, focus on equilibrium properties of colloids and interfaces which can be interpreted on the basis of elements of reversible thermodynamics. In non-equilibrium situations where processes come into play (see e.g. chapter 1.6 on transport phenomena and chapter II.4 on electrokinetics), we inevitably have to leave the trusted routes of equilibrium thermodynamics to enter the field of irreversible thermodynamics. Gradients in thermodynamic and mechanic quantities, i.e., generalized forces, then lead us to processes such as mass transport, governed by the elementary principles of conservation (in space and time) of mass and momentum, under a variety of initial and boundary conditions. The force fields involved include those of chemical, electrical, hydrodynamic and gravitational nature. As we have outlined before (Volume I, chapter 6), thermal energy, leading to Brownian motion, is essential for the very existence of many stable colloidal dispersions. It is related to the frequent stochastic encounters between the colloidal particle and the molecules of the medium in which it is dispersed. The thermal energies of medium molecules and dispersed particles, i.e. the kinetically most elementary components of the system, obey the principle of dynamic equipartition. The rigorous details of their motion occur on tiniescales that vary with the mass of the particle considered (cf. [1.6.3.10]). For colloidal particles this typically comes to O(lCT8)s , whereas for molecules in the medium it is smaller by some four orders of magnitude. This distinct separation of tiniescales makes the theoretical description of Brownian motion of colloidal particles less involved. So, for this purpose the medium can be seen as a continuum, characterized by macroscopic hydrodynamic parameters such as its viscosity. When we take a closer look at colloidal dispersions and try to understand their dynamic behaviour, the picture of an isolated hard sphere is far too primitive. Particles dispersed in electrolyte are generally carrying electric charge and are surrounded by a more or less diffuse cloud of countercharge in the solvent. In such a real dispersion we recognize several typical dimensions, viz. the particle radius, the average distance between individual particles, and the double layer thickness (see fig. 4.1). FurtherFundamentals of Interface and Colloid Science, Volume IV J. Lyklema (Editor)
© 2005 Elsevier Ltd. All rights reserved
4.2
DYNAMICS AND KINETICS
Figure 4.1. Characteristic elementary dimensions in an electrocratic colloid: particle radius a, average particle separation (h), and diffuse double layer thickness 1 / K .
more, all the elements are in continuous diffusive movement characterized by specific mobilities. Since diffusion over a certain distance can be associated with a corresponding time, we may identify a number of characteristic relaxation times for the system pictured In fig. 4.1. Let us first define the typical tlmescales of elementary processes in a colloid. Consider the pair of particles In fig. 4.1 which is assumed to be fairly unstable, that is subject to a certain rate of coagulation. For aggregation to materialize, the particles have to approach each other. Under conditions where diffusion is the leading transport process, the characteristic tlmescale will be of the order of h 2 / D , where h Is the mean particle separation and D is the particle diffusion coefficient. For a standard colloid with a of O(102) nm, K'X ofO(10)nmand h of O(103) nm (that Is, a volume fraction of order 1%), and a D of O(10~12) m 2 s" 1 , this timescale of particle-particle approach comes to the order of I s . The final portion of the particle-particle approaching process will be retarded due to repulsive double layer overlap and accelerated by Van der Waals attraction. With increasing stability of the colloid the retardation gains Importance and typical times for diffusion across the double layer region may grow far above O (K~2 ID), that is O(l 0"4) s for the above standard colloid. During the period of double layer overlap the diffuse double layer will generally respond quickly. This Is so since Ions are fast, and the typical dimension of the region of perturbation, O([a//f] 1/2 ), is small, i.e. the relevant double layer response time is O(a/K\Dion), which comes to O(10~6) s for the aforementioned colloid. The surface charge distribution may relax via various mechanisms. For example, re-equllibratlon via diffusion along the surface will typically take place on a timescale of O{a /KD?oa). Double layers around colloidal particles are also dlsequllibrated under the influence of external forces such as an applied electric field. The difference with the situation of particle-particle Interaction is not In the nature of the relaxation processes but In the simpler nature of the perturbation. Thus the application of well-established techniques such as electrokinetics, electroacoustics and dielectric spectroscopy (see II.4) Is extremely helpful In elucidating the nature and features of double layer relaxation processes in a given frequency range or Its equivalent reciprocal time domain. When particles of comparable size encounter and stick together, the process is
DYNAMICS AND KINETICS
4.3
called 'coagulation' or, when steric interactions are involved, 'flocculation'. On the other hand when a particle is attached to another much larger particle, or to a surface, this is termed deposition. The differences between the physics of interaction in coagulation and deposition are quantitative, rather than qualitative. Deposition theories usually involve properties of fields around large bodies, which are assumed to be unperturbed by the small particles. Such a separation of the typical length scales simplifies theoretical work significantly. Deposition by external fields is of major significance for industry. Colloidal films are made under the influence of electric fields (electrophoretic deposition in car painting, paper printing, herbicide deposition and the manufacture of television screens) or liquid flow (deposition in packed filter beds). In this chapter we shall analyze in detail the precise nature of the motion of colloids in terms of fundamental steps and trajectories. Such information is of basic importance for the understanding of elementary colloidal properties such as the frequency of particle encounters, rates of diffusion-controlled coagulation, reduction of aggregation rates as a result of repulsive energy barriers, extent of electric double layer relaxation during particle-particle interaction, etc. We shall discuss the physicochemical dynamics of colloids and analyze some of the kinetics involved with typical processes in colloidal dispersions. The term dynamics is used here in the sense that it refers to the capability of the colloid to restore equilibrium after an external perturbation of, say, chemical or electrical nature. For a specific process in the colloid, for instance coagulation or ion adsorption at the particle surface, we shall use the term kinetics in relation to the description of its time course and rate. The Brownian motion of particles, molecules and ions will be taken as the starting point, before considering diffusion across potential barriers1'. We shall invoke various types of interaction, e.g. chemical, electrostatic and Van der Waals. The nature and mathematical description of the ensuing relaxation behaviour will be discussed at some length. The kinetics of the primary coagulation steps will be given ample attention, as will further aggregation and gel formation. 4.2 Diffusion of colloidal particles 4.2a Nature of particle motion; various timescales and force field effects Stochastic processes and Brownian motion of free particles have been discussed extensively in sec. 1.6. Here we briefly review Langevin's force balance and the Chandrasekhar equation. The latter represents the generalization of the Fokker-Planck equation to phase space (see sec. 1.6.3c and [6.3.20]). It describes the displacement of colloids due to Brownian motion down to timescales as small as those on which the velocity of the particle is unrelaxed. As explained in sec. 1.6.3b, the Langevin21 equation 1) S. Chandrasekhar, Stochastic Problems in Physics and Astronomy, in Rev. Mod. Phys. 15 (1943) 1. 21 P. Langevin, C.R. Acad. Set. Paris, 146 (1908) 530.
4.4
DYNAMICS AND KINETICS
|I.6.3.4] essentially represents the balance of the particle acceleration, friction and fluctuating random forces. The solution of the Langevin equation demonstrates that for very short times the movement of the particle is uniform, and proportional to t , whereas for longer times the displacement develops a stochastic nature and becomes proportional to t 1 / 2 .
Figure 4.2. Different trajectories for particles 1 and 2 with identical radii but different masses m. The particle in (a) is the heavier one (rrtj > m 2 ) and has the larger correlation length, as indicated by arrowed bars. Figure 4.2 gives an example of a diffusion path for two particles of the same radius but with different masses (rrij > m 2 ). Both particles are assumed to be released from the origin at t = 0 . On timescales where the particles have already made many uncorrelated movements the root-mean-square displacement (compare [1.6.3.2a]) is given by
(r2)I/2=(6Dtf2=f^r
[4.2.1]
which is identical for both particles. Nevertheless, the average rate of displacement of the lighter particle 2 is larger than that of particle 1 which is the result of the equipartition principle (U2) = 3 £ I
[4.2.2]
where m is the particle mass. In reaching the positions at time t , the lighter particle has made many more random steps than the heavy one. This is so since the velocity component relaxes on a characteristic timescale r , defined as T = m/J
[4.2.3]
where / is the friction coefficient; compare [1.6.3.5]. Equation [4.2.3] shows that r is proportional to m, whereas the velocity is proportional to l / m 1 / 2 . The correlation length in the diffusion path is defined by the distance travelled during the time r , and
DYNAMICS AND KINETICS
4.5
is also denoted as the velocity persistence length in the diffusion process. Since the velocity is given by (kT/m)l/2, the correlation length is of the order of 1/2 (m//)(/cT/m) , which is larger for the heavier particle (see fig. 4.2). For a given period of time t, the particle has passed through {t/r) = tf/m of those elementary steps. Hence, the r.m.s. displacement is of the order of the product of the length of one step and the square root of the number of steps,
K)-.2(«Lr.(jrr.of(i»fl which is in agreement with [4.2.1]. The correlation length scales as a 1 / 2 . Thus, large and heavy particles make bigger steps. For a 1 p.m particle with density 103 kg/m3 dispersed in water, this correlation length is of the order of a few tenths of a nm. Now let us suppose that a diffusing particle is confronted with an energy barrier. In the case of a DLVO-type repulsive barrier (see further on in this chapter), the width of the barrier is typically K'1 which, for stable electrocratic colloids, is larger than 10 nm. Because the correlation length for diffusion of the particle is much smaller than the barrier width, the barrier is crossed via a large number of steps. It is then allowed to use the diffusion equation instead of the Chandrasekhar equation (see below). On the other hand, certain physical barriers can be very localized. As an example we mention concentrated emulsions, stabilized by emulsifier or adsorbed protein layers. These layers are usually a few nm thick and may give rise to very stable systems due to steric repulsion. Hence, the energy barrier is high and relatively narrow. Another example is that of tethered oligomers, so-called brushes. Theory for that is presented in chapter V.I and several illustrations of the stability of concentrated sols, stabilized by brushes, are given in chapter 5 of this volume. Furthermore, ageing processes in coagulated sols can also be related to high and narrow energy barriers. In order to quantify the particle transport outlined above, we will derive expressions for transport of Brownian particles across an energy barrier. We will not restrict ourselves to wide barriers but start with the general equation, largely following the treatment given by Chandrasekhar1'. Recalling the Chandrasekhar equation in one dimension ([16.3.20]) we write for the probability P{x,vx,t) of finding a Brownian particle with respect to phase space and time, that is, at a given position and a given time with a given velocity
£ dt
+u
f ^ dx
=A ,fl + *P+q dv
dv
|£
dvz
[4.2.5,
where F is the external conservative force per unit mass of the particle and
1 S. Chandrasekhar, Stochastic Problems in Physics and Astronomy, in Rev. Mod. Phys. 15 (1943) 1.
4.6
DYNAMICS AND KINETICS
P=Jlm
[4.2.6]
q = /3kT/m
[4.2.7]
Note that Z?"1 is the pertaining relaxation time constant r . It is easily verified that [4.2.5] satisfies the Maxwell-Boltzmann distribution (comp. sec. 1.6.3c)
[
jmvl+m
I
J
P(x,ux)~exp - ^
[4.2.8]
where
Figure 4.3. DLVO-type potential energy field.
Let us now consider a one-dimensional potential field as sketched in fig. 4.3. For colloidal double layers at equilibrium,
[4.2.9]
where Q represents the maximum value of q> and w is a measure of the steepness of
p
[4.2.10]
DYNAMICS AND KINETICS
4.7
where the function W{x,vx) satisfies the boundary conditions x=0
:
x -> »o :
W=0 ] ... . W —> 1 J
[4.2.11]
Starting from a given initial condition on P(x, vx), the profile around the barrier will develop quickly to the quasi-stationary state. As particles travel across the barrier, a diffusion layer (depletion layer) is being built up. For the one-dimensional semi-infinite situation considered here, the depletion layer continues to grow since diffusion is the sole transport mechanism under consideration. For a narrow barrier at short distance from the surface (in electrocratic colloids this is at Otx""1) ) and timescales well beyond {DK2)~1 , the particle concentration would vary significantly only over distances much larger than the barrier width. Thus, the depletion layer is essentially covering distances beyond xb, for which (p and W are zero and the problem reduces to that of the ordinary diffusion equation. With the aid of this latter equation, pN at x = xb can be related to p^ , the bulk concentration, and the non-stationary nature as arising from the growth of the diffusion layer can be taken into account. We will treat the procedure in some detail for spherical geometry (see sees. 4.3a and b). However, in practice onedimensional diffusion on longer timescales develops a convective character due to temperature and pressure fluctuations ('natural convection'). Here, by setting 3P/3t = O, we describe the situation for times large compared to the time of developing the distribution in the barrier region, but short compared to the time for developing a significant diffusion layer beyond the barrier (hence pN ~ p*N beyond xb ). Substitution of the expression for the potential ([4.2.9]) into the Fokker-Planck equation
? =Z ^ K ) 8t
m dvxy
x
>
+
^^
m2
14.2-12]
dv2
as introduced in detail in 1.6, yields an equation for W which can be solved. Resubstituting the solution into the expression for P gives (2nkTY1/2(nt-fi\1/2 P = p*\±^L\ £_£
I m )
{ 2nq )
-m(v2-w2x'2]\°°r [ exp — '-
2kT
exp
J
P
[
[
2c
[4.2.13]
? J
where cc = /J/2 + [(/? 2 /4) + ii> 2 ] 1/2
[4.2.14]
x' = x-xm
[4.2.15]
£=ux-ax'
[4.2.16]
and ^ is a dummy integration parameter. The flux J of particles crossing the barrier can be found by integration
DYNAMICS AND KINETICS
4.8
[4.2.17] in which we consider the two limiting situations P»w:
J = p'N{2nkTyi/2
Dpml/2w
P«w.
J = ptN(2nkT)-1/2-^--exp(-9*/kT)
exp(-g>* IkT)
[4.2.18] [4.2.19]
where we have used Stokes' law ([1.6.3.31]), i.e. D = kT/Gnrja, and defined 0* = Qm as the height of the barrier in units of energy. The physical meaning of the two limits given by eqs. [4.2.18] and [4.2.19] is illustrated in fig. 4.4. In the limit of p» w, the energy barrier is relatively extended, such that the velocity component of the particle in the x -direction is able to relax during the process of crossing the barrier. During this passage there will be many uncorrelated diffusion steps and the velocity distribution in the barrier will be close to the Boltzmann distribution. The mobility of the particle, rather than its mass, is of importance. In the other limiting situation with P«w the particle is unable to relax its velocity component while crossing the barrier. Those particles that have enough energy at the foot of the barrier, will pass. The velocity distribution around the barrier is far from Boltzmann-like and here the mass of the particle is the relevant parameter, rather than its mobility.
Figure 4.4. Different conditions for particles crossing an energy barrier: a) the case where the correlation length is small compared to the barrier width, and b) the opposite case. Notice that p and ix> represent the reciprocals of characteristic time constants for travelling over the correlation length and the barrier width, respectively.
DYNAMICS AND KINETICS
4.9
Table 4.1. Values of /? and w estimated for different particle radii a. Parameter values: 77= 10~3 Pas; p=10 3 kg m" 3 ; K~l = 10 nm ; Q* = 10 kT ; T = 300 K . a/m 10" 8 10- 7 10" 6 105
pis-1 4.5xlO 10 4.5 xlO 8 4.5 xlO 6 4.5 xlO 4
w/s~l 4.5xlO 8 1.4xlO7 4.5 xlO 5 1.4xlO4
Table 4.1 shows some values for /? and w calculated for particles with different radii and a barrier with fixed height and width. Clearly, /? is much larger than w for particles below the micron scale. This means that for most particle colloids the situation of fig. 4.4a prevails. Only for relatively heavy particles and narrow barriers may it be necessary to apply the Fokker-Planck equation instead of the mere diffusion equation. For time intervals very large compared to the time of relaxation of the velocity component (/7"1) and sufficiently wide barriers (/?/u)»l), the Fokker-Planck equation reduces to
| i = divr f Dgradrp - j : gradr0* j
[4.2.20]
which is the diffusion equation extended with an additional term to account for migration in the force field (compare [1.6.7.2]). 4.2b Probing Brownian motion of particles by dynamic light scattering (DLS) A popular method to investigate Brownian motion of colloid particles experimentally, is dynamic light scattering (DLS). We refer to chapter 1.7 for the general background of the technique and ways of evaluating experimental data and to chapters 2 and 5 of this volume for more illustrations. The conventional DLS setup follows movement of Brownian particles when displaced over distances of the order of the wavelength A and beyond. As shown in sec. 4.2a, the velocity autocorrelation function of an individual 100 nm Brownian particle decays on a time-scale of the order of 1 ns during which it moves over a distance of the order of some tenths of a nm, i.e. much smaller than X . Hence, particle displacements as monitored by DLS are composed of very many independent small displacements. The extension of DLS to multiple scattering dispersions, socalled diffusing wave spectroscopy (DWS), calls for concentrated systems and allows for registration of particle motion on sub nm length scales. As a result of multiple scattering, DWS may track particle movements over distances down to the correlation length. From the Fokker-Planck equation, or the time-dependent particle velocity equation [1.6.3.10], we have
4.10
DYNAMICS AND KINETICS
(Ar2(t)} = 6tD 1-—(l-exp(-/t/m))
[4.2.21]
which reduces to the known limits of velocity relaxation t»m/J:
t«m/J:
[4.2.22] t2
[4.2.23]
m for extreme values of the time constant ml J. The problem with this approximation is that it ignores the effect of particle motion on the flow of the surrounding fluid. A friction term of the nature -Jv{t) implies that friction is supposed to work instantaneously. By including inertia and a memory term in the Langevin equation, the transient effect in fluid motion can be taken into account. The resulting equation has been solved by Laplace transform techniques to a generalized equation for (Ar2(t)) which retains the limit given by [4.2.22] and differentiates that of [4.2.23]1'2'. Figure 4.5 shows the time evolution of the normalized displacement (Ar(t)2)/6t of |im sized particles for different volume fractions and the comparison with theoretical prediction on the basis of [4.2.21], as well as the effect of the inclusion of the timedependent friction term in the Langevin equation. Clearly the agreement between experiment and theory is very good, provided the inertia in the friction term is taken
Figure 4.5. Time evolution of Dp(t) = (Ar2{t)>/6t for 1.53nm poly(styrcnc) particles in aqueous electrolyte for different volume fractions tj>. Marks on the right indicate self-diffusion coefficients as predicted by Batchelor1 . The dashed line through the 2.1 per cent data is the theoretical prediction for (Ar2(f))/6f of Hinch21. The dash-dot line represents 14.2.20] with a time-independent friction coefficient. (Redrawn from D.A. Weitz, D.J. Pine, Dynamic Light Scattering, W. Brown, Ed., Clarendon Press (1993) pp. 652.) 11 21
G.K. Batchelor, J. Fluid Mech. 74 (1976) 1. E.J. Hinch, J. Fluid Mech. 72 (1975) 499.
DYNAMICS AND KINETICS
4.11
Into account. Another noteworthy aspect is that in the short time limit, (Ar(t) 2 )/6t is independent of the volume fraction. This is understandable since in that region we have (Ar(t) 2 )/6t = — kT t/m
(see [4.2.23]); even if there is hydrodynamic interaction,
we should still have an average kinetic energy of —kT.
4.3 Coagulation kinetics 4.3a Diffusion-controlled particle-particle approach; rapid coagulation As early as 1917, and unaware of the origins of attractive and repulsive forces between particles but having seen Zsigmondy's11 data for the rate of coagulation of gold sols, Smoluchowski2' recognized the need to distinguish between what he called 'rapid' and 'slow' coagulation of colloids. The notion 'rapid' relates to colloids where all of the encounters between individual particles lead to a permanent contact, whereas the qualification 'slow' denotes colloids in which only a relatively small fraction of the collisions results in coagulation. The upper limit of coagulation rates is solely determined by the diffusive properties of the particles. In order to estimate the magnitude of the 'rapid' coagulation rate, Smoluchowski adopted a simple scheme in which one central particle acts as a perfect sink for other particles implying that every encounter leads to permanent contact. The diffusive flux of particles towards the central one is then a measure of the frequency of particle-particle encounters. The simple Smoluchowski approach is conceptually helpful and offers a good first-order approximation for the initial rates of rapid coagulation. The apparent limitations will be briefly outlined later on in this section. We consider the simple case of a homodisperse colloid with spherical particles in the absence of any external force field. In the 'rapid' coagulation limit, the time-
Figure 4.6. Definitions in connection with the Smoluchowski scheme for 'rapid' coagulation of spherical colloids.
" R. Zsigmondy, Zur Erkenntniss der Kolloide, Jena (1905); English translation: J. Alexander, Colloids and the Ultramicroscope, New York (1909). R. Zsigmondy, Z. Phys. Chem. 9 2 (1917) 600. 2) M. von Smoluchowski, Z. Physlk. Chem. 92 (1917) 129.
4.12
DYNAMICS AND KINETICS
dependent diffusive flux J of particles towards some designated central particle is easily found from Fick's laws, together with the appropriate boundary conditions. Details on derivations of J are given in sec. I.6.5f. A sketch of the situation with coagulating particles is given in fig. 4.6. For spherical colloids with radius a, the effective coagulation distance is la. Thus the boundary conditions of the 'rapid' coagulation problem are given by r = 2a:
/OM=0
[4.3.1a]
r->°°:
PN=PN
[4.3.1b]
where r is the distance from the center of the central particle, pN is the particle number concentration and p^ the value of p N in the bulk. The resulting equation for the incoming particle flux J 2 a (given as the number of particles that arrive at r = 2a per unit time) is given by [1.6.5.41 ] which we write here in the form11 J2a =167ia2DpyO;{(7iDptr1/2 + (2ar 1 }
[4.3.2]
where D is the particle diffusion coefficient and t is time, counted from the pristine situation of a completely random particle distribution. Equation [4.3.2] shows that for times well beyond a 2 ID the transient term (TID t)~ 1/2 becomes negligible in the flux as well as in the time integral of the flux, that is the total amount of particle encounters. The flux then reaches its steady-state value SnaD p*^ . It should be noted that the flux stands for the number of particles coming in at r = 2a towards the central particle per unit time. The magnitude of the flux is therefore immediately identical to the number of collisions that one particle experiences per unit time, i.e. the collision frequency v v[= J2a(steady state)) = 8-naD p^
[4.3.3]
For aqueous dispersions of particles in the size range from 1CT7 to lCT6m the characteristic time for reaching steady-state {a2 /Dp), to which we come back in more detail in sec. 4.3b, ranges from 1CT1 to 1 s. For a given particle radius a (and related diffusion coefficient D ) the collision frequency is simply proportional to the particle concentration, that is, to the volume fraction (j> of the colloid. Table 4.2 collects some typical figures. From [4.3.3] it is immediately found that v=6
[4.3.4a]
Combining this with D =kT/6nr]a ([1.6.3.32]), we finally obtain v=(kT/n^a 3 )^ 1
[4.3.4b]
' Just for convenience, fluxes from the medium towards the central particle at r = 0 will be counted as positive here, though the formal sign of such fluxes is negative (comp. 1.6).
DYNAMICS AND KINETICS
4.13
Table 4.2. Orders of magnitude of the collision frequency ( v) and the time constant for rapid coagulation (r rap ) for colloids with particle radius a and volume fraction <j>. a (m)
O(D p )(m 2 s-!)
O(a 2 /D p ) (s)
3xlO7
10-12
io-i
3xlO" 6
IO-1 3
102
O(V)(S-!)
O(r rap )(s)
lO- 4
1 lO- 2
1 102
lO- 2
lO- 3
103
lO- 4
10"5
105
lO- 2
which illustrates, again, that at constant volume fraction the collision frequency increases with decreasing particle radius according to a cubic relationship. With the explicit expression [4.3.3] for the collision frequency
v at hand, it is
straightforward to compute the ensuing coagulation rate dpjj / dt for the case where all particle encounters give rise to aggregation. The total number of collisions in a dispersion is ^ vp^ per unit volume and unit time; the factor ^ corrects for two particles being involved in one collision. Proper counting further includes: (i) a factor of 2 because one collision corresponds with the elimination of 2 primary particles, and (ii) another factor of 2 to take into account that the central particle in the Smoluchowski model is itself also subject to Brownian motion. The effective D be shown to be equal to 2D
can
. The coagulation rate follows as
d / ^,/dt = -167iD p a^ I 2
[4.3.5]
which has the characteristic form of a second-order process with a diffusion-controlled rate. The time dependence of the concentration of remaining primary particles follows immediately from integration of [4.3.5] PNW/po=(l with T
+ t/TTap)~1
[4.3.6]
being the time constant for rapid coagulation
TTap=(l6naDpPo)~1
[4.3.7]
where pQ is p^ at t = 0 . Figure 4.7 sketches the course of p^ with time a n d compares the curve with some comparable decay functions. For t/r
well below unity, the
dependence approaches a simple linear function, whereas for large t/r is m u c h more tailing. The magnitude of r
11
the behaviour
varies with particle size and particle
J.Th.G. Overbeek, in Colloid Science, H.R. Kruyt, Ed., Kinetics of Flocculation, chapter VII, Elsevier, Amsterdam (1952) p. 278.
4.14
DYNAMICS AND KINETICS
Figure 4.7. The course of some functions J as a function of time t : ( ) 1/(1+ t / r ) with / / / O = I for t/r = 1 , i.e. [4.3.6]; ( ) exp(-t/r); ( ) exp(t / T) erfcft /r)l/2; (—•)
n-l/^t/r)- 1 / 2 .
concentration. Typical values are in the range between fractions of seconds for small particles at high concentration, and thousands of seconds for large particles at lower concentrations. Comparing the expression for r with those for v, [4.3.3] and [4.3.4], it is easily seen that r rap is proportional to (a2 /D)/ and that r r a p /(a 2 /D) grows above unity for 0 decreasing below approximately 0.1. Hence, for such volume fractions neglection of the transient term in the pertaining diffusion equation [4.3.2] is justified. Obviously, the simple Smoluchowski approach has its shortcomings and limitations. The process of coagulation is described as the annihilation of particles by diffusion-controlled collision with other particles. The practical reason for this approach was that at that time Zsigmondy's data11 only contained numbers of particles, without a discrimination with respect to size, because his study was based on the Tyndall effect (ultramicroscope). The transient term in the pertaining diffusion equation [4.3.2] is not rigorously described by (TiDt)"1^2 because the diffusing particle and the target surface are identical spherical objects. A consequence is that viscous corrections are significant (see sec. 4.5b). Another problem is that only doublet formation is counted and that these doublets are considered incapable of forming triplets and multiplets. Smoluchowski recognized this and later formulated a more general treatment of evolving multiplet formation. Generally, the net formation of doublet, triplet ... n-fold particles is given by the coagulation of p-fold and q-fold particles with
11
R. Zsigmondy, Z. Phys. Chem. 9 2 (1917) 600.
DYNAMICS AND KINETICS
4.15
p + q = n , and the elimination of n-fold particles by further coagulation with any other particle. In terms of a rate expression this comes to n-1
.
~df = X ^.n VVn-P " P" X KqPqPn-p p=l
I43'8'
q=l
where pn is the short-hand notation for p^ of particle type n, and the fc's represent the pertaining coagulation rate constants, which contain the mutual diffusion coefficients and the radii of the particular combination of multiplet particles (comp. [4.3.5]). It is easily shown11 that for not too different radii of the particles involved this leads to an expression for T that differs from [4.3.7] by a factor of 2. The essence is that one collision now corresponds with the removal of one particle, the multiplet being practically as effective in further coagulation as a primary particle. Thus the time evolution of the decrease of the total particle concentration pT is similar to that for the monomers ([4.3.7]), with the time constant r doubled p T (t)/p 0 =(l + t / 2 r r a p r 1
[4.3.9]
For the sake of completeness we also give the equations for aggregates of type j , where j is the number of constituting primary particles21 Pi{t)/p0
=(t/2r rap )J- 1 (l + t/2r r a p rJ- 1
[4.3.10]
As illustrated in fig. 4.8, for any aggregation number j > 1, the time evolution of p. passes through a maximum. Except for a slight increase of k from fcjj to Jc,2 to kl3, experimental data on poly(styrene) latices agree satisfactorily well with theoretical prediction3'. 4.3b Particle-particle approach with repulsive force; slow coagulation The first and most obvious extension of the simple 'central sink particle' approach by Smoluchowski is to incorporate particle-particle interaction forces into the diffusion equation. Since a force F modifies the velocity by an amount FIJ where J is the friction coefficient (see sees. 4.2 and 1.6.3b for details), the particle flux expression has to be extended with a term FpN / J . Under steady-state conditions, as explained in sec. 4.3a, the expression for the flux J loses the transient term and we may write it in the form
11
J.Th.G. Overbeek, Colloid and Surface Chemistry. A Self-study Course, Part 2, Lyophobic Colloids, MIT, Cambridge/Massachusetts (19721. 2 M. Elimclech, J. Gregory, X. Jia, and R. Williams, Particle Deposition and Aggregation: Measurement, Modelling and Simulation, Buttcrworth-Heinemann, Oxford (1995). 31 E.G.M. Pelsscrs, M.A. Cohen Stuart, and G.J. Fleer, J. Colloid Interface Set, 137 (1990) 362.
4.16
DYNAMICS AND KINETICS
Figure 4.8. Normalized particle concentration pjpo as a function of time ( . a) Theoretical dependence of pjpo on r / 2 r according to [4.3.9]. b) Experimental data for pjp0 versus t for 600 nm poly(styrene) latex particles. (Courtesy E.G.M. Pelssers.)
where g(r) = G{r) I kT , G(r) being the Gibbs energy of interaction and dG(r)/dr the force. Note that this amounts to writing the Nernst-Planck equation [1.6.7.1] with the terms for diffusion and flow, which is applicable for not too short timescales and not too narrow energy barriers (see sec. 4.2a). Combining [4.3.11 ] with the Stokes-Einstein relation between the friction coefficient /
and the diffusion coefficient D for spher-
ical particles, [16.3.31 and 32], and taking into account the movement and the velocity modification F / / of the central particle, we arrive at
^^[^^'T]
[4 3131
-
and this equation is to be solved under the boundary conditions defined by [4.3.1] extended with the conditions on g[r) r = 2a:
pN=0
(g(r) = -°°)
[4.3.13a]
r = <*>••
PN=PN
(g(r) = 0)
[4.3.13b]
DYNAMICS AND KINETICS
4.17
The solution11 provides us with the steady-state particle concentration profile for r>2a
pN(r) = p; e xpl- g (r))- J 7 ( D g ( r ) ) J ^ & P
d x
[4.3.14]
CO
where ^ is a dummy integration parameter. Note that [4.3.141 reduces to a Boltzmann profile for J = 0 (equilibrium). Applying condition [4.3.13a] we obtain the flux as l6nD ap'N ^—™
J =
[4.3.15] 2
la J exp(+g(%))dZ/x la
which for g —> 0 approaches \6nDap"N, viz. the same as expression [4.3.3] with a factor of 2 accounting for the movement of the central particle. For rapid coagulation, J follows from [4.3.15] in the limit g = 0 and hence we obtain the retardation factor or stability ratio W as 2a
J W=
rap d
' = la \ exp{g)dz/x2
[4.3.16]
For a large variety of cases, the extension of the force field is small compared to the particle radius. Furthermore, only in the vicinity of the maximum does the potential profile contribute significantly to the integral in [4.3.13]. If we follow the formulation of the curvature of the barrier as given by [4.2.8], i.e. if kTg = Q*—w*2 (r-r m ) 2 , then [4.3.16] develops to W = — exp{Q*/kT) [ exp -——x 2
d
X
[4.3.17]
from which 1 (271/cTl1^ W = —exp(0*/JcT)la w*
[4.3.18]
We note that this result can also be derived from expression [4.2.16] for J3»w by multiplying with the effective surface area of the central collecting particle (47i(2a)2) and accounting for its diffusion (replace D by 2D ). The non-dynamic properties of G{r) and its dependence on the electrolyte concentration have been extensively discussed in chapter 3 (Pair Interactions). For details we refer to sees. 3.2-3.9 and app. 2.
11 J.Th.G. Overbeek, Kinetics of Flocculation in Colloid Science, Vol. I, ch. VII, H.R. Kruyt, Ed., Elsevicr (1952); B. Chu, Molecular Forces. Based on the Baker Lectures of P.J.W. Debye, John Wiley, New York (1967).
4.18
DYNAMICS AND KINETICS
Numerous attempts have been made to test the validity of the basic findings of the Smoluchowski analysis of colloidal coagulation rates or, for that matter, colloidal stabilities. The stability factor W is experimentally accessible and has indeed played a prominent role in the development of the notion of colloidal stability. The log W-log c plot is of a bilinear nature (see sec. 4.4b) and allows for determination of a kinetic c.c.c. (critical coagulation concentration). Examples are given in chapter 3 of this volume, sec. 3 and figs. 3.65 to 3.67. There is a vast amount of literature on the interpretation of W for a variety of colloids11. As a general trend, the theoretical prediction of a pronounced dependence of W on particle size is practically not confirmed for a variety of types of colloids. In sec. 4.5b we show some data referring to the stability of latex particles in a flowing dispersion and the picture is the same. Here we briefly refer to some kinetic findings on the initial stage of coagulation of polystyrene latex particles. Lichtenbelt et al.2) used a stopped-flow technique to measure the rate constant of rapid coagulation by electrolyte. For the initial process of doublet formation average rate constants fcn of about exlO'^m-^s" 1 were found, approximately half the Smoluchowski value for perikinetic coagulation. The difference is fully due to the effects of Van der Waals attraction and hydrodynamic interaction341, taking reasonable values for the Hamaker constant. There is no clear dependence of fcj j on the particle radius a, in line with the Smoluchowski theory. This contrasts with the stability factor W for which the predicted variation with a is not experimentally found and for which a number of explanations have been suggested including those based on double layer dynamics51. In the sees. 4.4 and 4.5 we shall return to various aspects of the issue. 4.4 Electrodynamics of particle-particle interaction 4.4a Double layer relcucation routes The electric double layer around an individual charged colloidal particle has been extensively discussed in chapter II.3 and pair interaction is considered in great detail in chapter 3 of this volume. Conductive relaxation and diffusive relaxation have been briefly indicated as potential routes of re-equilibration of a perturbed double layer. Here we shall analyze the dynamic properties of the double layer in more detail by considering the various basic elements of possible relaxation processes for individual particles. More specifically we shall discuss:
11
W.B. Russel, D.A. Savillc, and W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge (1989). 21 J.W.T. Lichtenbelt, C. Pathmamanoharan, and P.H. Wierscma, J. Colloid Interface Sci. 49 (1974)281. 31 L.A. Spielman, J. Colloid Interface Sci. 33 (1970) 562. 41 E.P. Honig, G.J. Roebersen, and P.H., Wierscma, J. Colloid Interface Sci. 36 (1971) 97. 51 J. Lyklema, H.P. van Leeuwen, and M. Minor. Adv. Colloid Interface Sci. 83 (1999) 33.
DYNAMICS AND KINETICS
4.19
(i) interfacial transfer of ions that exchange between medium and colloid phase, (ii) diffusive transport of exchangeable ions into/from the bulk medium, from/ towards the colloid/medium interface, (iii) conductive and diffusive transport of non-exchangeable ions in the diffuse layer(s) at the colloid/medium interface, and (iv) lateral diffusion of ionic species along the colloid/medium interface. Each of these basic transport processes will be characterized at such a level that we shall be able to estimate the extent of double layer relaxation of colloids in interaction with each other, with macroscopic surfaces and with externally applied electric or hydrodynamic fields.
Figure 4.9. Elements of double layer relaxation processes in a colloidal dispersion: (i) interfacial transfer of exchangeable ions; (ii) diffusion of exchangeable ions in the medium; (iii) ion migration in the diffuse layers; (iv) lateral diffusion of surface ions.
Figure 4.9 gives a schematic picture of these elementary processes. Certain colloids, e.g. semiconductor colloids, may exhibit all of these processes simultaneously, others only one or two of them. Diffuse double layer relaxation in the medium, (iii), is always present in electrocratic colloids. (I) Interfacial ion transfer For certain types of electrocratic colloids, ion exchange between the dispersed phase and the medium is the charge-regulating mechanism. The classical example is silver iodide. It has a Frenkel defect structure with Ag+ ions at regular lattice positions in dynamic equilibrium with interstitial Ag+ ions and Ag+ vacancies i A gXgi]°^ A g^ + i v A g + r
i4-4-1!
4.20
DYNAMICS AND KINETICS
where [Ag^j] represents the regular lattice occupation, Agt an interstitial and [V
+
]~ a vacancy. In contact with an aqueous medium, Ag+ can be transferred
between the two phases and net charge separation can be generated. The Gibbs energy of activation for the interfacial Ag+ transfer reaction is reflected by the exchange current^ density jQ Agt c
>
which is defined by the forward and backward rates of
Aggq under equilibrium conditions
jn=Fk./ti c , =FkA, c , J A ° <*eq AgJq A<*eq Ag+
[4.4.2]
w h e r e k a n d k a r e t h e rate c o n s t a n t s for t h e interfacial transfer of A g + from Agl t o solution and back, respectively, (in ins" 1 ), and the products Fk., c + and Fk., c + represent the equal but opposite Ag+ fluxes (mol m~z s~l). For small disequilibrations, where the relation between the ion transfer rate and the potential is (essentially) linear, j o is coupled to the ion transfer resistance RtT per unit area (Q. m 2 ) according to Jo~
i K
14-4.3]
' tr
The characteristic time constant r^ for the ion transfer can then be formulated as r tt = R^C
[4.4.4]
where C is the differential double layer capacitance of the interface (F m~ 2 ). Values of r
for different colloids may vary over many orders of magnitude, as illustrated by the
electrochemical literature1'. For silver iodide r
is less than O(10~5)s , which implies
that the actual transfer of Ag+ across the interface generally is not rate-limiting. It will be shown below that diffusion of Ag+ is the slower process (sec. (il)). For oxides, where the double layer relaxation is coupled to (fast) chemisorption of H+ /OH~ , r
is
generally very small, i.e. below O(10~6)s 2) . (ii) Diffusion of exchangeable ions in the medium In an electrocratic colloid under coagulation conditions, the medium usually has a certain concentration of exchangeable, charge-determining (cd) ions and an excess of some indifferent electrolyte. The transport of the cd ions is then similar to that in electrochemical cells with an excess of 'supporting' electrolyte over 'electroactive' species. In this situation, the conductive contribution to the transport of the cd ions may be neglected with respect to the diffusive term. Thus the dynamic flux J of species i towards/from the supposedly spherical surface of a colloidal particle is given by the diffusion equation [1.6.5.41 ] 1 A.J. Bard, L.R. Faulkner, Electrochemical Methods. Fundamentals and Applications. 2'"' ed., John Wiley (2001). 21 S. Trasatti, Electrodes of Conductive Metallic Oxides, Part B, Elsevier, Amsterdam (1981).
DYNAMICS AND KINETICS
Ji=DiAcil[(nDityl/2+a-1^
4.21
[4.4.5]
where D; is the diffusion coefficient of i, Acf the driving concentration difference (which, in the derivation of [4.4.5], is supposed to remain constant; see sec. 1.6 for more detail), t is time and a is the particle radius. The (7tD1t)~1/2 term represents the linear diffusion component and a"1 accounts for the convergent nature of the diffusion. The physical meaning of the (7iD(t)~1/2 term is that of the reciprocal thickness S~l of the diffusion layer as it develops with time. For example, for t of the order of 10~ 5 s, i.e. comparable to the effective timescale of particle encounters (\/D K2), and with D{ being O(10~9)m2s"1 (small ions in water), S is OdCr^M-m. Thus, for such a timescale and a particle radius of O(l)nm , Si a remains well below unity and the ionic diffusion is essentially linear. Such conclusions are highly relevant in analyses of particle-particle interactions: linear diffusion is often dominant but in the region of double layer overlap the concentration profiles are very complicated, the particle separation, O(K~X) , being again smaller than S. The item will recur later in this section. We note that for nanoparticles the interaction time is typically 10~8 s (for a K~1 of 1 nm), and the a"1 term is of the same order of magnitude as the (7iDit)~1/2 term in [4.4.5]. The characteristic time constant for an ion transfer process, controlled by linear diffusion in the medium, appears as the argument in the relaxation of the Galvani potential difference between the two phases (A0-A0 ) after an instantaneous charge injection (A(*-A<*eq) = (A^-A ( * eq ) 0 experfc(t/r ld ) 1/2
[4.4.6]
where (A0-A0 )O represents the initial overpotential generated by the charge injection, i.e. the value at t = +0 . Reinmuth11 has shown that rld is defined by a diffusion parameter aw and the differential interfacial capacitance C rld = 2
[4.4.7]
where
^ F M ( 2 ^
[4 4 81
- '
which is known to electrochemists as the Warburg coefficient, with c* the bulk concentration and Di the diffusion coefficient of the exchangeable species. This coefficient also appears in expressions for the complex impedance for linear diffusion towards an interface: Z = (1 -i)<mr 1/2 where i is the imaginary unit and a> the angular frequency. See sec. II.3.7c for more detail. 1!
W.H. Reinmuth, Anal Chem. 34 (1962) 1272.
4.22
DYNAMICS AND KINETICS
(Hi) Ionic conduction in diffuse layers The process of charging or re-equilibrating an electric double layer at the interface between a dispersed particle and a surrounding medium involves such phenomena as mass transfer polarization and relaxation of ionic atmospheres. Let us first consider the medium side of the interface and the relatively simple case of a symmetrical electrolyte (z+ = |z_| = z; c+ = c_ = c). The thickness of the diffuse layer is related to the Debye length K~1 which is given by [1.5.2.10]. Near coagulation conditions, the value of r" 1 is O(l)nm which means that double layer overlap between particles at average separation is not significant in a not too concentrated colloidal dispersion. Given this restriction on the particle concentration we may utilize the derivation by Buck11 of the diffuse layer relaxation function for spherical symmetry. The starting point is the conservation equation for the time-dependent concentrations of the ionic components of the medium
dcL=D d\ 3t
' 3r 2
|
2 3ct : z . F q p y RT ( 3 r 2
r 3f|
2 3y] z,F 3^pc, | 2c,A r dr)
(a)
RT dr { dr
r J
[4.4.9]
(b)
where, by comparing with e.g. [1.6.6.3], we recognize the diffusive part in (a) and the conductive part in (b). The latter is the gradient of the flux, with c4 and y/ as the variables in a spherical symmetry. If we further assume D+ = D_ = D and realize that the space charge density p is given by zF[c+ -c_), [4.4.9] can be rewritten as dp Jd2p 2 dp 4z2F2c*3(/| 2 z — = D —£- + — - - K p + —\
dt
[dr2
r 3r
F
RTr
4.4.10
3r J
where 9 2 p/9r 2 is the linear diffusion term, (2/r)/(3p/3r) the spherical diffusion term, K2p the linear conduction term and (4z2F2c*)/(3^/3r)RTr the spherical conduction term. For colloids with xa » 1 it is reasonable to neglect the spherical contribution to the conduction: the typical migration distance is O(l/x") and this is then much less than the radius a (cf. similar argument in the analysis of diffusion processes under (ii)). Thus, Buck's analysis for a spherical electrode in a large volume of medium '^electrode << ^medium' Yie^s a n analytical solution if only the leading terms in r are retained. For equal Di 's the resulting relaxation function for the time evolution of the potential difference across the interface follows as
11
R.P. Buck, J. Electroanal. Chem. 23 (1969) 219.
DYNAMICS AND KINETICS
{A
4.23
exp(-t/rj
\£o£
1
[4.4.11]
+
Ve~e [e^^e) 172 - 271 " 172 ^/^' 172 ^^/^) 172 )]
where (A
where K is the specific conductivity of the medium (fi^'m^l. This expression is identical to the formulation of the Maxwell relaxation time in [II.3.13.22] and [II.3.13.25]. In expression [4.4.11] the coefficients a I ee0 and l/K££o represent the geometrical capacitance of the medium with characteristic dimensions a and permittivity eeo , and the capacitance of the diffuse layer with thickness 1/r, respectively. The first term on the r.h.s. of [4.4.11] represents the primary conductive relaxation, whereas the terms containing t 1/2 stand for the diffusive contribution. Note that for large t » re the term erf(t/r e ) 1/2 approaches unity and all other terms cancel. Thus we obtain A
Figure 4.10. Potential profile after instantaneous injection of a charge density Aa° . (1) The situation at t = +0 , with no regions of net space charge in the medium, i.e. only dielectric polarization. (21 The situation for t/ Te >> 1 , i.e. after development of interfacial double layers.
4.24
DYNAMICS AND KINETICS
displaced yet. So the linear potential profile simply reflects the dielectric behaviour of the medium with permittivity eoe and capacitance Eoe/a difference Aaa/eoe;
and hence a potential
situation (1) in fig. 4.10. Over the Maxwell relaxation time regime
(see [4.4.12]) the migration of ions annihilates the initial potential profile and transforms it into the known profile with only finite potential gradients in the double layer region at the interface. For large Ka the potential jump is typically much smaller than Aaeoela
and amounts to
Aol£o£tc.
Hunter 11 solved [4.4.10] for one-dimensional transport conditions and equal diffusion coefficients of all ions
^
= 4^-*V*.«l dt
[ dxA
[4.4.13,
J
under the initial and boundary conditions t =0
: p(x,t) = O
[4.4.14a]
x->oo : p(x,t)->0
[4.4.14b]
x = 0 : dp(x,t)/dx = -K2A<j
[4.4.14c]
Using Laplace transformation, this set of equations can be solved to give p{x,t) = ^-\expHot)erfc 2 I
— ^ = -JK2DI \24Dt
)
-
( x VI exp(x-x)erfc — = = + *Jic2Dt }
UVDt
I4 4 151
- -
)\
which was also obtained by Buck. Profiles of p(x), for various times, indeed show a constant slope of KACF/2 , thus obeying the boundary conditions. Figure 4.11 illustrates this 2 '. In the case of unequal diffusion coefficients of the cations and anions in the electrolyte, the situation is much more involved. Concentration polarization comes into play, comparable to that described in some detail for the so-called far-field effect in electrokinetics. See sec. II.4.6c for details. A major message contained in [4.4.11] is that in colloidal dispersions the diffuse double layer in the medium is able to relax very rapidly on the timescale of particleparticle encounters. For example, for a 1CT1 M 1-1 electrolyte solution K~X is O(10~9)m and re is O(10~9)s . This means that diffuse double layer relaxation is practically complete within some 10"8 s after perturbation.
11 R.J. Hunter, Foundations of Colloid Science, Vol. II, 1989, ch. 13 (based on material provided by R.W. O' Brien). 2 We note that the corresponding equation in Hunter's book, [13.5.9], and the ensuing behaviour of p{x), as given by his fig. 13.5.1, suffer from a minor incorrectness. R.J. Hunter, loc. cit.
DYNAMICS AND KINETICS
4.25
Figure 4.11. Calculated values for the normalized space charge density p[x)l~KAa° as a function of normalized distance KX for various values of time (given as Dic2t ) after application of an instantaneous charge injection corresponding to a change in surface charge density Ao° . K2Dt = 0.01 (A), 0.1 ( • ), 0.5 (A), 1 (X), 2 (O), 100 (•). Depending on its conducting properties, the dispersed phase may also possess
diffuse double layer features as outlined above for the medium. For the example of Agl, which has a conductivity on the level of a semiconductor, the particle charge is realized by an excess of Agt or [V^+l in a space charge layer adjacent to the interface. The equilibrium fraction of interstitials at room temperature is O (10"7). From this it is readily found that the thickness of the space charge layer is about 50 nm. The relaxation time r sc is given by an expression analogous to [4.4.12] r
s c
=^-
[4.4.16]
where e and K are the relative permittivity and the specific conductivity of the particle phase, respectively. With K for dispersed Agl not less then 10" 2 fl"'m" 1 v, we find that rsc is at least O(10"8)s . With decreasing conductivity of the dispersed material, the value of TSC increases. In the limit of essentially insulating particles, as e.g. with poly(styrene-sulphate) latex, there is no charge carrier transport inside the dispersed phase. The particle charge resides at localized sites on the surface, and this may give rise to different double layer properties and hence to different stability against coagulation. (iv) Lateral diffusion of surface ions More often than not, specific adsorption of ions is appreciable under coagulation conditions (about 10~' M 1-1 electrolyte). Then a significant fraction of the double 11
T. Takahashi, K. Kuwabara, O. Yamamoto, J. Electrochem. Soc. 116 (1969) 357.
4.26
DYNAMICS AND KINETICS
layer charge is located in the Stern layer. Exchange of ions like Li + , Na+ , Mg 2+ , Ca2 + between the Stern layer and the diffuse layer is generally fast11 with time constants well below 1CT6 s . Down to this timescale the Stern layer and the diffuse layer are purely capacitative, without any ion transfer resistance in between. This finding is confirmed by MD simulations21 indicating that lateral transport of monovalent ions in the Stern layer is almost as fast as in the diffuse layer because these two layers are virtually short-circuited. Electrokinetic and dielectric spectroscopic data suggest that lateral mobilities of Stern layer ions is lower than but of the same order of magnitude as their bulk mobilities (see II.4.6e/f for details). This implies that ionic migration inside the stagnant layer is a route of importance in such double layer perturbations as provoked in particle-particle interaction. The fundamental time constant for surface diffusion TS is (cf. [4.4.12]) T =-^— s 4Df
[4.4.17]
where D? is the surface ion diffusion coefficient and d is the typical distance over which the ions are migrating. For example, for double layer polarization in an external electric field the magnitude of d would be of the same order as that of the particle radius a . Then for a of O(10~7)m and D° of O(10~9)m2 s" 1 , we get a value for r s of the order of 10 5 s , corresponding to a relaxation in the 0.1 MHz regime. For (spherical) particles in interaction, the typical migration distance d is (a/zc)1^2 which generally is at least an order of magnitude smaller than a. Relaxation via lateral diffusion of ions in the Stern layer is then correspondingly faster and may indeed be of significance in stability analyses (see sec. 4.5). 4.4b Double layer relaxation in particle-particle interaction Slow perikinetic coagulation of hydrophobic colloids is a dynamic process. The resulting rate of aggregation is limited by the extent to which the particles are able to pass the energy barrier between them. Obviously, the rate of coagulation depends on the height (and shape) of the energy barrier, which on the DLVO level is described as the net result of attractive Van der Waals and repulsive electrostatic forces. Computation of the electrostatic forces on the basis of equilibrium thermodynamics, as assumed in chapter 3, is justified only when the relaxation of the double layer is fast compared to the rate of approach. See sees. 4.4a and 4.5 for details. The electric double layer associated with dispersed particles is the result of specific attractions between ions and the particle surface. The extent of charge accumulation
" A . J . Bard, L.R. Faulkner, Electrochemical Methods. Fundamentals and Applications, 2nd ed., John Wiley, New York (2001). 21 J. Lyklcma, S. Rovillard, and J. De Coninck, Langmuir 14 (1998) 5659.
DYNAMICS AND KINETICS
4.27
onto the surface and the equilibrium structure of the double layer are determined by a balance of chemical and electrostatic interactions (see sec. II.3.2). If overlap of double layers occurs, for example during a collision, this balance will be affected, leading to a certain degree of adjustment or complete relaxation of the double layer. The first 'dynamic' theory of particle-particle interaction has been presented by Dukhin and Lyklema11. They developed a perturbation theory which accounts for small transient deviations from the equilibrium double layer structure due to retarded desorption of charge-determining ions. In a method similar to that of Spielman21 and Honig et al.3), who incorporated hydrodynamic interactions into the SmoluchowskiFuchs theory4', they calculated a modified diffusion coefficient. This coefficient has been used to calculate the colloid stability ratio W , which is a measurable quantity. However, this method is valid only in the region of small deviations from equilibrium. Preferably, in further studies this restriction should be removed because the perturbations are generally not small, as will be seen. A stability theory is 'dynamic' if it accounts for the extent of double layer relaxation during interaction. The two extremes of such a theory are well-known: either the double layer is in full equilibrium at any instant during the encounter, i.e. interaction takes place at constant (chemical) potential (cp), or no surface charge relaxation takes place during interaction, leading to interaction at constant (surface) charge (cc), see sec. 3.2. In both limiting cases the surface charge density is independent of the timescale of interaction. Calculations have shown that the nature of the interaction, i.e. at constant potential or at constant charge (see sees. 3.3, 3.4 and 3.5), may have a substantial impact on the stability of sols. On going from cp to cc, the Gibbs energy of interaction increases (see sec. 3.4) and hence the extent of double layer relaxation decreases, rendering the colloid more stable. Since transient deviations from equilibrium cannot be generally excluded, the availability of a 'dynamic' theory is indispensable. Some experimental motivation for the development of such a theory exists as well. The theory as presented in sees. 4.3a and b, predicts a pronounced particle size dependence of the colloid stability. Up to now, experiments on a variety of systems, i.e. latices51, Agl6) and haematite71, do not seem to confirm this prediction. Anyway, measurements of the 'rapid' coagulation time constant r for well-defined homodisperse colloids over a range of values of the particle radius a, are extremely scarce. If the timescale of an encounter would depend on the particle size, and hence, " s . S . Dukhin and J. Lyklema, Langmuir 3 (1987) 94; Faraday Discuss. Chem. Soc. 90 (1990) 261. 21 L.A. Spielman, J. Colloid Interface Set 33 (1970) 562. 31 E.P. Honig, G.J. Roebcrsen, and P.H. Wiersema, J. Colloid Interface Sci. 36 (1971) 97. 41 M. von Smolucbowski, Z. Physik. Chem. 17 (1917) 129; N. Fuchs, Z. Physik 89 (1934) 736. 51 M. Elimelich, C.R. O'Melia, Langmuir 6 (1990) 1153: R.H. Ottewil, J.N. Shaw, Discuss. Faraday Soc. 42 (1966) 154. 61 H. Rcerink, J.Th.G. Overbeek, Discuss. Faraday Soc. 18 (1954) 74. 71 N.H.G. Penners, L.K. Koopal, Colloids Surf. 28 (1987) 67.
4.28
DYNAMICS AND KINETICS
on the extent of disequilibration of the double layer, then a 'dynamic' theory could provide a clue for solving this problem. As outlined in sec. 4.3c, double layers in colloids can relax by a variety of mechanisms, depending on the nature of the charges involved. The basic relaxation time of the diffuse double layer is relatively fast, although for certain surface charges and mobilities of the species involved, coupled diffusive processes may take much longer to reach completion. Adjustments of surface charges, generated by specific adsorption of ionic species in the medium, may be much slower, certainly if the very adsorption/desorption step takes place at a limited rate. Therefore, we shall treat this particular type of transient double layer disequilibration in some detail. More specifically, we will consider the case where, on the timescale of r int , the rates of desorption/adsorption of charge-determining ions are limited (compare category (i) in sec. 4.3c).
Figure 4.12. Definition of the geometrical parameters, characterizing interaction between two spherical particles with radius a at separation h(G), h being the shortest separation at e = 0 .
Let us consider two identical spherical particles with radius a; see fig. 4.12 for the definition of the geometrical details. It is supposed that the medium contains two types of electrolyte, one of them with an adsorbing charge-determining ion A and the other being indifferent. The indifferent electrolyte is present in large excess over the adsorptive ion, similar to conditions for exchangeable ions as outlined in sec. 4.3c and in line with the usual situation. Consequently the concentration of A is immaterial for diffuse double layer relaxation rates and, for that matter, conduction is negligible in the mass transfer of A . The binding of A is assumed to follow a Henry type adsorption isotherm implying that under equilibrium conditions we have ^q=ZAFrA=ZA^CA^A
14-4.18]
where
11 G.A. Martynov, Elektrokhimiya istry, p. 418).
15 (1979) 474 (in English translation, Soviet
Electrochem.-
DYNAMICS AND KINETICS
4.29
AadguA the depth of the potential energy well for A , yAq the normalized equilibrium potential in the plane of adsorbed A (yA = e \//A I kT). Under dynamic conditions, the change of FA with time is generally given by d/
A/dt =
fc
adC£[/T5X-/A]-fcdes^A
I 4 - 4 - 19 !
where kad and kdes are the adsorption and desorption rate constants, respectively, and P^3* is the limiting value of F . In the case of identical particles, the disequilibration will always provoke desorption of A as the particles approach each other. Let us consider, by way of example, that the two particles under consideration have acquired a positive charge by adsorption of Az+ ions from the medium. Their surface charge density <xA is positive and so is the dimensionless potential y in the double layer region. When the double layers of these two particles start to overlap, the electrostatic equilibrium situation is perturbed, resulting in an increased potential. The potential increment (y - y eq ) is the primary driving force for perturbation that initiates the relaxation process. In our example with the desorption/ adsorption of A as the rate-limiting steps, the concentration just outside the adsorbed layer (cA) follows the potential perturbation in accordance with Boltzmann's equilibrium conditions. So, in the transient situation we have c° = c°' eq exp(-AyA)
[4.4.20]
where AyA = (yA - yAq) and c A eq is the equilibrium value of cA for AyA = 0. The surface charge density <JA, or for that matter FA, lags behind yA and cA and will tend to re-establish equilibrium by decreasing its magnitude via desorption. Initially, it is the decrease of cA (by a term exp(-AyA) which gives rise to a finite net desorption rate. This is easily seen in the rate equation [4.4.19] where the first term in the r.h.s. is lowered, so that the second term dominates. By combining [4.4.20] with [4.4.19] and using the equilibrium condition dr A / dt = 0 , we find that d
^ A / d t = KdcTq[nr
-^A>xp(-AyA)-kdesFA[l-exp(-AyA)]
[4.4.21 ]
which holds under conditions where FA remains close to F^ for the individual particle, i.e. either at short times or at small perturbations. For AyA « 1, that is for potential perturbations well below 25/z A mV, exp(-AyA] may be linearized and [4.4.21 ] reduces to dr
A / d t = kdesrAAyA
14-4.22]
Since under these conditions FA ~ F^ , the time evolution of oA follows as d f 7 A /dt = -)cdestTA<'AyA
[4.4.23]
4.30
DYNAMICS AND KINETICS
Figure 4.13. Sketch of the variation of the net desorption/ adsorption rate dFA/dt, as composed of the individual adsorption and desorption terms, with the overpotential Ay A . The arrow indicates the lowering effect of the initial AyA on the adsorption rate. The possible variation of the rate constants fc ad a n d ^des w i t h ^A i s s e lected.
Figure 4.13 pictures the effect of a change of yA (and cA ) on the adsorption/ desorption rate balance. We might add that for larger perturbations, the rate constants Jcad and fcdes will probably not be constant over the pertaining range of potentials. Equations [4.4.21-23] look fairly simple but the problem is that Ay varies with time in a complicated manner as related to the time course of the particle encounter. Attempts have been made to estimate the average velocity (i>) at which the two particles approach each other under steady-state flux conditions11. From [4.4.3] and [4.4.14] we may obtain (u) from the transport coefficient, i.e. the ratio between flux and local particle density n {v)
= Jln=
r/ 2Pa«p[g(r)]
r
2
J exp[g(r)]d^/^
[4 4 24] 2
2
Subsequently, the average velocity can be used in defining a characteristic average time of interaction (fint) , that is, the typical time of overlap of two double layers, each with thickness K~]
'<•- J % 2K-1
It is assumed that r lnt is dominated by the time required for the particles to move from h = 1K~1 to the maximum of the barrier since, after crossing that, the velocity sharply increases due to the progressive increase of the Van der Waals attraction (see sec. 1.4.6). Returning to the geometry of the problem, as sketched in fig. 4.12, we may quite generally formulate the charge density
S.Yu. Shulepov, S.S. Dukhin, and J. Lyklema, J. Colloid Interface Set. 171 (1995) 340.
DYNAMICS AND KINETICS
aA(h,O) = a1+Kj
*2&**
4.31
[4.4.26]
where
a
kdes
dx
K(V)
[4427]
where 0^(9) represents the local equilibrium surface charge density in the situation of particle-particle interaction. On the basis of this elementary relaxation equation, the implicitly coupled11 problem can be solved by an iterative procedure. Both the interaction time constant r int and the resulting stability ratio W, as defined by [4.3.16] can be computed. A number of results are given in fig. 4.14. The results confirm that for a given electrolyte concentration the particle size influences the velocity (v) and hence the interaction time r ^ . As a consequence, the extent of double layer relaxation during a particle encounter varies with the particle radius a in the sense that relaxation is more complete for bigger particles. Over a certain range of values of fcdes this feature may practically eliminate the sensitivity of W to the particle size. Although it seems tempting to generalize this finding, the model used and the approximation adopted are too simplified to warrant that. Rather these fairly qualitative findings should stimulate the application of sophisticated numerical
Figure 4.14. The interaction time r int (a) and the dynamic stability ratio W (b) as a function of the (l-l)-electrolyte concentration as computed on the basis of [4.4.23]-[4.4.27]. Particle radii (nm): 100 (O), 150 (A), 200 (A); fcdes = 10" 1 s" 1 ; <7^q = 1 ; Hamaker constant =12kT . (Redrawn from J. Kijlstra and H.P. van Leeuwen, J. Colloid Interface Scl. 160 (1993) 424.)
1
Note that AyA and (u) are interdependent.
4.32
DYNAMICS AND KINETICS
computation techniques thus more rigorously resolving the electrodynamics of colloidal interaction. The discussion presented so far applies to homodisperse colloids and perfectly spherical particles. Important deviations from such ideality are those due to (i) heterodispersity and (ii) surface roughness. (i) Particle size/heterodispersity effects. Some of the double layer relaxation processes outlined in sec. 4.4a depend on the particle radius a, others do not. The scaling of some of these with a is collected in table 4.3 and set against the interaction time r int • ^ o r m o s t relaxation routes, t/Tint scales with a" 1 , implying that the extent of double layer relaxation increases with increasing particle size. Thus, in the dynamic regime, colloids would lose some stability with increasing a and in heterodisperse systems the smaller particles would be the ones with the highest stability against coagulation. (ii) Surface roughness. It is easily understood that small protrusions on the particle surface, with dimensions far below a but of the order of /r"1, may impact heavily on the DLVO interaction energy1'. Shulepov2' tackled the problem by summing all interaction energies at the level of a refined Deryagin approximation. Calculation of the effective pair interaction implies averaging over all possible configurations. In such an approach, the essential dynamic argument is the possibility of rotation of particles in order to find the (strongly preferred) pair configurations with minimized repulsion. Such reasoning would lead to an elementary dynamic explanation for the relatively low stability of colloids with rough particle surfaces. Table 4.3. Scaling of various relaxation time constants r with the particle radius a in particle-particle interaction. Type of relaxation process
r
Scaling with a
(Eq.) T
7/7
int
constant
a" 1
constant to
a"1 to a~1'2
(i)
Interfaclal transfer of exchangeable ions
R^C
[4.4.4]
(ii)
Diffusion of exchangeable ions in dispersion medium
2 ^
[4.4.7]
(ill)
Space charge transport
eeo/K
[4.4.12]
constant '
a" 1
(iv)
Lateral diffusion of surface
a/2DaK
[4.4.17]
a
constant
a l/2
a,
ions Due to the complicated geometry, in between linear (independent of a) and spherical (proportional to (a//r} 1/ ' 2 ), the effective r will vary with a to a power between 0 and 1/2. For Ka»\ essentially tangential relaxation. 11
S. Bhattacharjee, C.-H. Ko, and M. Elimelech, Langmuir 14 (1988) 3365. S.Y. Shulepov, Surface Roughness and Particle Size Effect in Brownian Coagulation, PhD Thesis, Delft University, The Netherlands (1997). 21
DYNAMICS AND KINETICS
4.33
Much of the implementation of dynamic aspects of colloidal properties still awaits due attention. 4.5 Electrodynamic relaxation of colloids in external fields 4.5a A qualitative comparison of relaxation times The preceding section made it clear that rate-limited charge relaxation can have a significant impact on colloid stability. The most obvious method to experimentally study charge relaxation mechanisms is by observing the response of sols to an externally applied electric field. Such an applied field is likely to cause an impact on the charges similar to that of the perturbing fields in particle-particle interaction but the geometry and the transient features are of a much simpler nature. Over several decades, dielectric spectroscopy and particle electrophoresis (see II.4) have been exploited to retrieve equilibrium and non-equilibrium double layer characteristics. These studies mainly focused on the diffuse part of the double layer. Only more recently have attempts been made to relate the electrodynamic properties of colloids to relaxation of charge in the compact part of the double layer (the Stern layer)12). Below we shall try to implement the available knowledge on electrodynamics of Stern layers, as outlined in sec. 4.4, into a global overview of double layer relaxation processes in colloids and their responses to externally applied fields. The following considerations are generally relevant: (i). The effective timescale of observation r obs must be tuned to the charge relaxation process of interest. For instance, the slow mechanisms responsible for sintering of aggregates cannot be studied by dielectric spectroscopy or electrophoresis. Generally, processes for which the ratio between the characteristic time constant and r obs is much larger or much smaller than unity ( De » 1 or « 1), cannot be investigated. (ii). The relaxation process must have its impact on some measurable quantity: charge relaxation gives rise to significant modification of the ohmic and capacitive components of the complex currents in dielectric spectroscopy, or the particle mobility in electrophoresis. Electrodynamic measuring methods cannot always identify the relaxation mechanisms. A well known example is the insensitivity of the d.c. particle mobility to the polarized Stern layer31 (as long as Stern ions are unable to desorb/ adsorb but polarize the Stern layer via lateral redistribution). In this section we will discuss some other examples of such (in)sensitivities. (iii). In order to experimentally characterize the dynamics, which may involve different processes on different timescales, it is often necessary to apply more than one technique. For example, when Stern layer conduction takes place, £ cannot be obtained 11
J. Lyklcma, H.P. van Lccuwcn, and M. Minor, Adv. Colloid Interface Set 83 (1999) 33. ^ M. Minor, Electrodynamics of Colloids, PhD Thesis, Wageningcn University, The Netherlands (1998) Chapter 7. 31 R.W. O'Brien, L.R. White, J. Chem. Soc, Faraday Trans. 74 (1978) 1607.
4.34
DYNAMICS AND KINETICS
Figure 4.15. Meaning of the governing time constants for lateral and normal relaxation processes, rji and t^, respectively. (From J. Lyklema, H.P. van Leeuwen and M. Minor, Adv. Colloid Interface Set 83 (1999) 33.)
from the electrophoretic mobility only; an additional technique is required, for instance dielectric spectroscopy or conductivity studies. Then, from the two techniques not only £ is obtained but also the surface conductivity Ka. Likewise, if the conductivity of the Stern layer Kcs Is obtained, the average lateral mobility of ions in that layer can be inferred provided the Stern charge a' is known from an additional technique, e.g. tltration in combination with electrokinetics. For a discussion of the high-frequency limits, see below. A number of relaxation mechanisms were discussed in section 4.3c. Now we consider the situation of a spherical colloidal particle, surrounded by a (partly) occupied Stern layer and a diffuse layer. To keep the analysis phenomenological, we distinguish between two types of relaxation processes, one parallel to the surface (II) and one normal to it (1). Ions in the Stern layer are considered to be able to move along the surface and adsorb/desorb from the surface towards/from the bulk (see fig. 4.15). In this framework we phenomenologically introduce two characteristic times, viz. r.|, needed for tangential redistribution of ions in the Stern layer and r__ | needed for establishing equilibrium between the Stern layer and the adjacent diffuse layer. The nature and magnitudes of t and T± derive from the governing elementary relaxation processes for a given system (see 4.3c). Depending on the magnitudes of ry and rL as compared to r obs we can classify a number of typical regimes, including; »' W * i l a n d T o b s / r i « 1 This would be the case for an a.c. field with a high frequency (1/ r o b s ). Stern ions would then be neither able to exchange with the diffuse layer, nor would they redistribute within the Stern layer, which would thus remain unpolarized. This corresponds
DYNAMICS AND KINETICS
4.35
Figure 4.16. Theoretical frequency dispersion of the permittivity As (b) and conductivity AK (c) for a double layer (a) with y^ — 4 in which half the immobile surface charge is compensated by mobile counterions in the Stern layer. Different curves hold for different values of the mobility of the Stern ions (given as D^^^/Da). (Data taken from J. Lyklema, H.P. van Leeuwen, and M. Minor, Adv. Colloid Interface Sci. 83 (1999) 33.)
with the Schwarz model of double layer dynamics11. (") Tobs >h « ! a n d robs lTi » l (implying TJ, » r± ). In this case the lateral redistribution of Stern layer ions is too slow to be significant. On the other hand, they do exchange with ions in the diffuse layer and thus generate polarization of the Stern layer. This situation applies to several electrostatic phenomena, comp. chapter II.5. Since relaxation of the Stern layer may follow two simultaneous routes (_L and ||), the overall time constant rs is given by l/r s = l/ty + l / r ± , in which the smaller one of Tj, and Tj_ dominates. In the case of an external field, rj, derives from surface diffusion of Stern ions over 7ia/2 {tea » 1) with ^ ~ a2 /2D° (comp. [4.4.15] in sec. 4.3c). As outlined in sec. 4.3d, zL is related to the limited rate of desorption of ions from the Stern layer: r± ~ l/fc^es • ^ v w a v °^ e x a m p l e w e consider a given double layer composition in some detail. Figure 4.16a pictures the case of an outer Helmholtz plane potential yd of 4. Half the fixed surface charge is assumed to be compensated by mobile 11
G. Schwarz, J. Phys. Chem. 66 (1962) 2636.
4.36
DYNAMICS AND KINETICS
Figure 4.17. Theoretical frequency dispersion of the permittivity for three different types of Stern layers. In all cases TM —> oo (no lateral mobility). Case 1: r ± ^ 0 , i.e. the Stern layer is in (local) equilibrium with the diffuse layer. Case 2: free co-ion exchange in the absence of surface charge. Case 3: retarded co-ion exchange ( r ± = 10 3 a 2 / 2D b u l k ). (Data taken from M. Minor, PhD Thesis, Wageningen University (1998).)
counterions in the Stern layer, the other half being located in the diffuse layer. For the case where tL —> °° (no exchange between Stern layer and diffuse layer) the frequency dispersions of the dielectric increment Ae and the conductivity increment AK are shown in fig. 4.16b. The full curves are the responses of the diffuse layer only, i.e. the situation for rs -> °» . Changes in the dispersion caused by the relaxation of the Stern layer are also shown. The mobilities of the Stern ions were decreased in steps of a factor of 10 starting with the bulk mobility. The relaxation frequency of the additional dispersion shifts downwards by the same factor since t « l / D ° . The conductivity increment spectrum shows a similar effect of Stern layer polarization. Hence, low frequency dielectric dispersion is able to quantify relaxation characteristics of a Stern layer11. Another interesting case is given In fig. 4.17, displaying the permittivity spectrum for various cases where the lateral mobility of Stern ions is completely Insignificant. For a purely diffuse layer (complete absence of Stern ions), the results almost coincide with those for the 'free counterion exchange' results (case 1, in fig. 4.17). Therefore low frequency dielectric dispersion (LFDD) cannot discriminate between these two situations. The reason is that if a counterion adsorbs from the diffuse layer into the Stern layer, the non-linear diffuse layer (we assume y d >> 1) tends to respond by locally losing a counterion. However, this occurs by the counterion skipping position, i.e. leaving the far fields (concentration and potential beyond the diffuse layer) unaffected. On the other hand, when co-Ions are able to exchange (case 2) the response is affected. The reason for this is that when a co-ion travels from the diffuse layer into the Stern layer to increase the Stern layer charge locally by ze, the non-linear diffuse layer prefers to respond by locally gaining a counterion instead of losing a co-ion. Via tangential transport in the diffuse double layer, together with normal transport just 11
R.W. O'Brien. J. Colloid Interface Set 92 (1983) 204.
DYNAMICS AND KINETICS
4.37
beyond this layer, both a co-ion and a counterion are delivered locally. Hence, the far fields are affected, and so is the dielectric response. In this connection, also see figs. III.4.24 and 25. In the case of retarded co-ion exchange and at very high frequencies, the Stern layer is unable to respond via normal exchange, so the response is due to the diffuse layer only. However, at much lower frequencies the Stern layer is able to relax via exchange and the permittivity increment approaches that for the case of free co-ion exchange. Hence, LFDD is sensitive to this relaxation of the Stern layer and yields the pertaining relaxation time t±. We conclude this subsection by noting that all the theories relate to macroscopically observable phenomena, such as the a.c. conductivity of a colloidal dispersion and the particle mobility to microscopic dynamic and static characteristics of the Stern layer, such as the Stern layer ion mobility, D°, and charge density
11 21
C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 94 (1998) 2441, 2583. M. Smoluchowski, Z. Phys. Chem. 92 (1917) 129.
4.38
DYNAMICS AND KINETICS
radius a( that, by centre, are located within the infinitely long cylinder with radius R = cr + a{, will eventually collide with the central particle j . The rate at which this occurs is'' J
i =4 M j o R ' J z ^ - z 2 d z = | M«ij
I 4 - 5 •! 1
where z represents an arbitrarily chosen axis, perpendicular to the direction of the flow through the cylinder. Hence, the number of collisions between particles i and j per unit time and volume, i.e. the collision frequency Vj., reads v
y=-^MPj(ai+aj)
I4-5-2'
and the limiting rate constant for rapid orthokinetic coagulation follows as k
ij=f*(ai+aj)
[4.5.3a]
or on
ka=— ya3
[4.5.3b]
for initially monodisperse sols. The major difference with perikinetic coagulation is the cubic dependence on particle radius. The rate constant for fast perikinetic coagulation is given by
gjgrK + ^j)3 or
for monodisperse sols. On the basis of [4.5.3] and [4.5.4] we can compare the orthokinetic and the perikinetic rate constants for a monodisperse colloid with particle radius a kjortho) _ 4r?ia3 kiperi) kT
[4 5 5]
which is of order unity for 1 |im particles in aqueous medium, with ^=10~ 3 Pa s, under mild agitation (stirring in a coffee cup). These simple derivations are far from exact, but confirm the enhanced coagulation rate in stirred dispersions of big particles. The trend is that Brownian motion is the major coagulation mechanism for submicron particles, while the orthokinetic mode prevails for larger, supermicron particles.
11 M. Elimelech, J. Gregory, X. Jia, and R. Williams, Particle Deposition and Aggregation: Measurement, Modeling and Simulation, Butterworth-Heinemann, Oxford (1995).
DYNAMICS AND KINETICS
4.39
Differential velocities Collision rates can be increased whenever particles obtain different velocities when subjected to an external force. For instance, larger particles will cream or sediment faster than the smaller ones, causing collisions on their way, a 'broom effect'. The same arguments hold if the gravitational field is replaced by an electrical field (electrophoresis) or a magnetic field (magnetophoresis). Consider two different spherical particle species i and j , which attain different velocities u( and u, in a force field. This can occur because the particle radii are different or because the forces acting on the particles are different. We assume that the particles follow straight lines and consider a central particle with radius a . . Then within the infinitely long cylinder with radius Rr = a. + ai, again 50% of all particles will eventually collide with particle a,. The rate at which this occurs is J^Pi^juj-UjI
[4.5.6]
The hydrodynamic forces and torques acting on an isolated pair of spheres in shear flow are well known. Differences in behaviour of spheres of macroscopic and microscopic dimensions are believed to be due to the effects of interaction forces (see sec. IV.3.) between the spheres and to rotatory and translational Brownian motion. Curtis and Hocking11 calculated the doublet formation from the real non-rectilinear trajectories of two identical spheres in shear flow, taking hydrodynamic interactions, Van der Waals attraction and electrostatic repulsion into account. Van de Ven and Mason21 presented a rigorous analysis and distinguished three different regimes: (i) Brownian motion is negligible but the interaction forces are considerable; i.e. orthokinetic coagulation (ii) Brownian motion is appreciable, both Brownian motion and interaction forces (= Van der Waals, electric and hydrodynamic) must be taken into account (iii) Both Brownian and interaction forces are negligible. Under such conditions particle capture can never occur but orbital pairs may exist. As an extension of the Smoluchowski collision rate, [4.5.2], Van de Ven and Mason wrote for the capture frequency per particle 32 , v = —« o r/M
[4-5.7]
where aQ is a dimensionless capture efficiency, accounting for the deformation of the streamlines around the central particle. The equations governing the trajectories were solved numerically and the boundaries of the capture cross section (from which the orthokinetic capture efficiency was calcul11 21
A.S.G. Curtis, L.M. Hocking, Trans Far. Soc. 66 (1970) 1381. T.G.M. van de Ven, S.G. Mason. Colloid & Polymer Sci. 255 (1977) 468.
4.40
DYNAMICS AND KINETICS
Figure 4.18. Orthokinetic capture efficiency a0 as a function of CA for various values of the
ratio CR/CA. ated) were determined by trial and error. « 0 was determined as a function of shear rate y, double layer potential, y/d , Hamaker constant, ionic strength and particle size. For the situation where electrostatic repulsion is absent, Van de Ven and Mason11 derived the following approximate semi-empirical expression for aQ
where CA is the ratio between the attractive force and the hydrodynamic force, and / is a function of the (normalized) London wavelength A . Assuming A = 100 nm, / equals 0.79, 0.87 and 0.95 for particles of radius 2, 1 and 0.5 urn, respectively. It is interesting to note that in this situation the capture frequency is not proportional to y but J — y°-82 . Figure 4.18 shows the capture efficiency for a situation in which electrostatic repulsion cannot be neglected. Curves 1-7 correspond to increasing potential iffi (or decreasing Hamaker constant A ) for particles with a radius of 2 um suspended in 1 mM (1-1) electrolyte. The parameter CR is defined as the ratio between the repulsive force and the hydrodynamic force CR= K
y J 3rjya2
and the ratio C R / C A follows as 11
T.G.M. van de Ven, S.G. Mason, loc. cit
[4.5.9a]
DYNAMICS AND KINETICS
4.41
Figure 4.19. Histogram of doublet formation efficiencies for an aqueous dispersion of 2 urn poly(styrene) latex spheres in 0.001 M KC1 (with i/A = —41 mV . The dashed curve is an experimental fit with the sudden drop in efficiency drawn at the critical shear rate y cr j t (15.7 s~' ) for transition from secondary to primary doublets. Below / c r j t all doublets are secondary and above yCTit all are primary.
[4.5.9b] Calculations show that a suspension can be unstable at low and high shear rates, but stable at intermediate rates. At low shear rates coagulation takes place in the secondary minimum, at high shear rates this happens in the primary minimum. The effect of decreasing the ionic strength is similar to that of increasing the surface potential.11 Figure 4.19 shows experimentally determined doublet formation efficiencies21 showing the critical shear rate yc (15.7 s"1) where transition from secondary to primary doublets occurs. Figure 4.20 illustratively depicts the trajectories of shear-induced encounters of latex spheres in 50% aqueous glycerol. These data are taken from Takamura et al.3) and show how the particle trajectories are affected by the attractive or repulsive nature of the interaction. The impact of shear on predominantly perikinetic doublet formation (regime ii) was studied by Van de Ven and Mason41. For sufficiently small Peclet numbers, they derived that the flux is given by j =
' t o V ' % (l + P e l r /2 a ; / r) + o(Pe tr ) 2aj2Qexp(g)^
[4.5.10]
with ft =0.5136,
11 21 31 41
T.G.M. van de T.G.M. van de K. Takamura, T.G.M. van de
Vcn, Ven, H.L. Ven.
Colloidal Hydrodynamics, Academic Press, London (1989). S.G. Mason, loc. cit. Goldsmith, and S.G. Mason, J. Colloid Interface Sci. 82 (1981) 175. S.G. Mason, Colloid & Polym. Sci. 255 (1977) 794.
4.42
DYNAMICS AND KINETICS
repulsion
Figure 4.20. Trajectories of shearinduced collisions of 2.6 \xm PS latex spheres in 50/50 glycerol/water showing the projection on the xy-plane of the paths of the centres of spheres from the midpoint between them. At the centre is the exclusion sphere which cannot be penetrated if the collision occurs in the xy-plane. (Redrawn from Takamura et al., 1981).
«P=
2a exp
jr-
[4.5.11]
L ^#
and where P
6 t r
- ^
,4.5.12,
is the translation^ Peclet number. It is generally assumed that the capture frequency for purely Brownian aggregation and that for aggregation induced by shear only ( J B and J s , respectively) are simply additive J = Js + JS=
2a
!
M P
ex (
'
a
L p ^#
d
/ f o ^
[4.5.13,
DYNAMICS AND KINETICS
4.43
or, expressed in terms of PetT J =
_l6^o_(
}
2a\ exp(g)-4 ha X where f°°
4«0aJ
dy
expg-^
a, = 3
4—
[4.5.15]
3K
By comparing [4.5.14] with [4.5.10] it is clear that there is no theoretical foundation for simple additivity of the rates. However, the result of Van de Ven and Mason11 is only valid for small Pe^ where the rate of coagulation is still mainly determined by the Brownian coagulation process. For technical applications the issue of the additivity of the two fluxes is of great importance, especially under conditions where one of the two processes dominates. There are numerous studies on particle deposition. By way of illustration we mention the work of Elimelech and O'Melia2' who studied the deposition of poly(styrene) latex particles from flowing dispersions with the objectives (i) to collect experimental evidence on stability characteristics of homodisperse well-defined colloids over a broad range of particle radii, and (ii) to study the effect of particle size on the corresponding collision efficiency of Brownian particles in deposition. The particle
Figure 4.21. Experimentally observed values for the stability factor W for poly(styrene) latices A and B, with particle radii of 46 and 378 nm, respectively, as a function of salt concentration c. Flow rate 1.36 xlO" 2 m s ~ ] ; temperature is 24°C. ( ) and ( ): theoretically predicted values for A and B respectively. (O) and (•): experimental values as derived from deposition on glass beads. (Redrawn from M. Elimelech and C.R. O'Melia, Langmuir 6 (1990) 1153.)
11 21
T.G.M. van de Ven, S.G. Mason, toe. cit. M. Elimelech, C.R. O'Melia, Langmuir 6 (1990) 1 153.
4.44
DYNAMICS AND KINETICS
breakthrough features of a glass bead collector column were related to the collision efficiency, which is inversely related to the stability factor W. Typical dependencies of W on salt concentration are given in fig. 4.21. The slopes hardly depend on particle radius; neither does the critical deposition concentration. However, agreement with theoretically calculated curves is extremely poor: (i) there is only qualitative agreement in the sense that deposition rates increase with increasing electrolyte concentrations, but (ii) on a quantitative level it is observed that experimental particle deposition rates are up to several orders of magnitude larger than the ones predicted on the basis of incorporation of DLVO type interaction into the convective diffusion equation. This set of results led Elimelech and O'Melia to conclude that for a real colloid, the stability factor (or, for that matter, the collision efficiency) is independent of particle size, and suggests that the electrodynamics of particle-particle interaction, coupled with the hydrodynamics, is the likely cause of the discrepancy. We described this in sec. 4.4b. 4.6 Aggregation and fractals When emulsion droplets coalesce, larger spheres are formed. The diffusivity of the spherical droplets is well defined during the process. However, when solid particles coagulate the shape of the aggregate formed is not well defined and generally depends on the process of attachment. For large aggregates, a detailed description of the coordinate of every primary particle within each aggregate is of no use; no two aggregates will be identical. More convenient is a description in terms of a small number of parameters related to averages. An important experimental observation is that in a coagulating dispersion with a particle volume fraction smaller than 1% a percolating particle network/(gel) may eventually be formed. (See ch. IV.6 for details.) This indicates that the floes formed during the coagulation process have a very open structure. The fractal cluster theories, developed in the 1980s and 1990s 1 ' 23 ' 41 are able to describe such aggregation phenomena. If we plot, for a large number of aggregates, the mass as a function of the aggregate size (larger diameter or radius of gyration for instance) we obtain a straight line, but with non-integer slope d f , called the fractal dimensionality. For coalescing oil droplets df = 3 but for particle aggregates df < 3 . The lower the fractal dimension, the more open the structure. For a linear needleshaped aggregate, we would obtain the lowest possible fractal dimension df = 1. 11
P. Meakin, Fractal aggregates, in Adv. Colloid Interface Set, 28(1988) 249-331. L.G.B. Bremer, Fractal Aggregation in Relation to Formation and Properties of Particle Gels, PhD Thesis, Wageningen Agricultural University, The Netherlands (1992). 31 P. Walstra, Fractal Aggregation in Colloidal Dispersion, Syllabus PhD student course in Han-sur-Lesse (1996). 41 M.T.A. Bos, The Structure of Particle Gels as Studied with Confocal Microscopy and Computer Simulations, PhD Thesis, Wageningen Agricultural University, The Netherlands (1997). 21
DYNAMICS AND KINETICS
4.45
Figure 4.22. Scheme for the algorithm applied in simulating diffusion-limited cluster-cluster aggregation.
The relation between aggregate mass M and size R reads M~Rdf
[4.6.1]
In cases where this equation is valid over a large range of aggregate sizes the aggregate is said to have a self-similar structure. An approach which has turned out to be very fruitful is the numerical modeling of colloidal aggregation. A realistic algorithm was introduced by Meakin1' and Kolb et al.2) in 1983. This algorithm, depicted in fig. 4.22, is called the diffusion-limited cluster aggregation (DLCA). The simulations show that the DLCA algorithm yields clusters that are fractal objects. An important aspect of the DLCA algorithm is the treatment of clusters as particles with a scaled diffusion coefficient Dp ~ JVJ
[4.6.2]
with 0 < v < 1, and JV the number of particles in the cluster. It has been found that the fractal dimension is 1.8 and independent of the diffusion scaling exponent v. The rate of coagulation of course does depend on D . Figure 4.23 gives an example of the projection of 3-dimensional aggregates obtained by the simulation. It is seen that they have a very open, tenuous structure. Figure 4.24 illustrates the trend in the dependence of the number of particles N in the aggregate with radius R . The relation is given by
^P=^o(f) f 11 21
P. Meakin, Phys. Rev. Lett. 51 (1983) 1119. M. Kolb, R. Bolet, and R. Jullien, Phys. Rev. Lett. 51 (1983) 1123.
I4-6-3!
4.46
DYNAMICS AND KINETICS
Figure 4.23. Projection of 3-dimensional aggregates of particles, obtained by simulation of cluster-cluster aggregation (from ref. '', after Sutherland).
Figure 4.24. Example of the relation between the number N of particles of radius a in an aggregate of radius R. The fractal dimensionality dj here is 1.8. (Redrawn from P. Walstra, Physical Chemistry of Foods, Marcel Dekker (2003) p. 494.)
where a is the radius of the primary spherical particles. Due to the randomness of the aggregation process, N is considered to be an average number; /Vo is a dimensionless proportionality constant, generally of the order of unity. The actual value of No depends on the definition of R (and a for non-spherical or soft subunits). R may e.g. be defined as the radius of gyration or as the maximum aggregate dimension. In the following derivations we will set yV0 = 1 for convenience. Equation [4.6.2] is often obeyed over a wide range of aggregate sizes, usually over a few orders of magnitude. This implies that these aggregates are on average self-similar on scales between a and R . An important aspect is that during the aggregation process fractal aggregates are formed from smaller aggregates, with (almost) identical
11
P. Walstra, loc. clt.
DYNAMICS AND KINETICS
4.47
dimensionality. It is difficult to offer a simple explanation for this observation but it appears to be true for many systems, both in the DLCA simulation as well as experimentally. It should be realized that the observed average self-similarity may not be interpreted in the sense that [4.6.2], if applied to a single aggregate, would describe a multiaggregate system that becomes less dense when going from the centre to the periphery. Rather, [4.6.2] applies within an individual aggregate, but over a more limited range and only on average, as outlined above. The number of positions in a spherical aggregate with radius R to be occupied by primary particles, is given by
yv s =f^j
[4.6.4]
and the volume fraction of particles in an aggregate is then *A=TT=
H
[4 6 51
- '
since always df < 3 , the volume fraction will be invariably smaller for larger R ! (I) Rate of growth of fractal aggregates Simple analytical expressions are at hand for the rate of growth of fractal aggregates both for perikinetic and orthokinetic aggregation11. For unhindered perikinetic aggregation, Smoluchowski obtained for the rate at which particles aggregate - ^ =-8nD p ap2
[4.6.6]
where p T is the total number density of the particles. Note the difference with [4.3.5] which expresses the elimination of two primary particles upon one collision; the total number of particles is then reduced by one. For the time necessary to reduce the number of particles to half the original value (compare [4.3.7] and [4.3.9])
tf?=(8"DPPo)~l=2Trap
I4-6-7'
For unhindered orthokinetic aggregation in simple shear, Smoluchowski estimated
In monodisperse fractal aggregates, the number concentration of aggregates pj can be related to the size of the aggregates R ; pT is the initial particle concentration, p0, divided by the number of particles in an aggregate 11
L.G.B. Bremer, P. Walstra, and T. van Vliet, Colloids Surfaces A99 (1995) 121.
4.48
DYNAMICS AND KINETICS
p = — ^adf"3JR~df
[4.6.9]
47T
Differentiation of this equation leads to ^L dR
=
lr%adf-3R-df-i W 4TT °
[ 4 6 1 0 ]
Substitution of these equations into [4.6.6] or [4.6.8] results in relations for d£ that may be integrated from a to some cluster size R . For perikinetic aggregation, this results in
tJiLdR.J^^-v-d.^d,^, J dR 1
i kT ^°
a
a
[4611]
and
^C{(ff-l}
14.6.12]
For orthokinetic aggregation, it leads to df=3:
t = — Ck'lnf— I
d ff < 3 :
t=
7tdff
,( (R\dt-3) J n ' l - -
4y(3-df) ° [ {a) )
[4.6.13]
[4.6.14]
(ii) Gelation The DLCA-algorithm has become widely used as a model for colloidal aggregation11. It has been used in particular as a model for particle gelation. It has been recognized21 that DLCA must eventually result in the formation of very large clusters and gelation. This can be seen as follows. The average volume fraction inside the aggregate
11 L.G.B. Bremer, Fractal Aggregation in Relation to Formation and Properties of Particle Gels, PhD Thesis, Wageningen Agricultural University, The Netherlands (1992). 21 L.G.B. Bremer, B.H. Bijsterbosch, P. Walstra, and T. van Vliet Adv. Colloid Interface Sci. 46 (1993) 1 17.
DYNAMICS AND KINETICS
4.49
(Rg) = a ^ " 3 )
[4.6.151
(*P, g K f / ( d f ~ 3 )
'4.6.16,
where (Rg)^Rf^)V{3-d!)
[4.6.17]
In a gelled system the identity of individual clusters is lost and one may determine the fractal properties of a cluster alternatively by a version of the so-called sandbox method1'21. In this method one considers the number of particles JV(r) in a sphere of radius r around any particle of the cluster. One finds the average values of the fit parameters by linear regression of log(h) versus log r after pre-averaging of JV(r) over all particles. Bos2' gives a modulated curve for JV(r) for a gelled system (see fig. 4.25). For low values of r we see the fractal regime (df ~ 1.8) which ends at the size of the average cluster in the system. This point, or rather the corresponding radius, is identified as the upper cut-off length of the gel. It is an indication of the average size of the aggregates close to gelation. At higher values of r we have the homogeneous regime where simply N(r) = 0{r/a)3 . The aggregation as given by DLCA-algorithms is completely irreversible. The effect of reversibility on the geometry of clusters has been tested in a number of computer simulations derived from modified DLCA algorithms. It has been shown 3 4 5 ' that reversible bonding results in a fractal dimension larger than 1.8, and increasing with time. The reverse is also found in a simulation where DLCA-aggregates are reformed and compacted after they are formed. A different approach is adopting a modified DLCA-type scheme in which, upon collision, bond formation has a certain probability671. In a DLCA algorithm, where reversibility or reorganization is important, universal scaling with constant df no longer holds. In the case of immediate coalescense of aggregated particles (df = 3) the volume fraction <j>A stays constant (see [4.6.20]). For any smaller dimensionality, <j>A will increase during the aggregation process up to unity at the gel point. 11
S. Haw, Physica A208 (1994) 8. M.T.A. Bos, The Structure of Particle Gels as Studied with Confocal Microscopy and Computer Simulations, PhD Thesis, Wageningen Agricultural University, The Netherlands (1997). 31 M. Kolb, J. Phys. A: Math. Gen. 19 (1986) 263. 41 W.Y. Shih, I.A. Aksay, and R. Kikuchi, Phys. Rev. A 36 (1987) 5015. 51 A.H.L. West, J.R. Melrose, and R.C. Ball, Phys. Rev. E 49 (1994) 4237. 61 P. Meakin, Phys. Rev. Lett. 51 (1983) 1119. 71 M. Mellema, Scaling Relations between Structure and Rheology of Ageing Casein Particle Gels, PhD thesis, Wageningen University, The Netherlands (2000). 2
DYNAMICS AND KINETICS
4.50
Figure 4.25. Fractal scaling In DLCA.
The magnitude of (R«) at the gel point is given by [4.6.15]. Substituting this size in [4.6.12] we obtain an approximation for the gelling time for perikinetic aggregation r =
^ , 3 /
d f
- 3
= rpen,odf/df-3
[ 46 l g ]
We note that, in principle, the Smoluchowski approach may be applied only to dilute systems. In the case df = 3 , tg 6n —> <~ and no gel will develop; for lower dimensions a gel will always be formed. For a typical dimensionality df of 2 and an initial volume fraction of 0.05, fPeri = 4007^2'. Here, r\^ is the typical timescale at which we experience the first signs of instability. Typically at that time, monomers, dimers and some trimers are present. When we wait 400 times as long, a percolating network has formed. For orthokinetic aggregation, we obtain
^--J^y-^-^0^-?)
[4.6.19]
which for d f = 2 and
[4.6.20]
and this shows that, under mild stirring conditions a space-filling network can be formed very fast after formation of the first doublets. Moreover, the aggregate formation and break-up equilibrium is of a dynamic nature (see for example Miihle and Domasch11). Many macroscopic features of unstable aggregating systems or gels can be explained fairly well with the above fractal theory. We mention sedimentation, gel rheology, gel permeability for a liquid under a pressure gradient, etc. For laminar flow through a gel, Bremer21 derived for the permeability coefficient B in d'Arcy's law 11 21
K. Miihle, K. Domasch, Chem. Eng. Process. 2 9 (1991) 1. L.G.B. Bremer, loc.cit. (1992).
DYNAMICS AND KINETICS
4.51
Figure 4.26. Dependence of the permeability coefficient ( m 2 ) on the protein concentration (in % w./w.) for acid-induced gels of blocked (open symbols) and unmodified (closed symbols) aggregates of whey protein isolate. B =^ / K
d
f~
3
(m 2 )
[4.6.21]
Figure 4.26 illustrates this via the measured permeability coefficient of a whey protein gel, plotted against the logarithm of the protein concentration11. The gelation was acid induced (cold gelation). 4.7 Applications Generally speaking, the incorporation of kinetics and dynamics Into colloid and Interface science is still at a fairly early stage of development. Smoluchowski's theory for rapid and slow coagulation is the classical paragon of the significance of kinetics in colloidal coagulation rate analysis. On the other hand, the DLVO theory is essentially of a static nature. It basically considers the limiting cases of constant surface potential, where the relevant double layer relaxation processes are fast on the timescale of particle interaction, and that of constant surface charge, where the particle charge density does not relax at all. In the intermediate case between these two limits, colloidal stability is partly governed by kinetic factors. This regime is difficult to access and so far has only been touched upon at a semi-quantitative level of comparing characteristic time constants of potentially operative relaxation mechanisms21. With the aid of powerful computers and suitable numerical procedures it should be possible to further open up this dynamic stability regime and thus to achieve a higher level of understanding of some basic properties of colloids. Kinetics of a variety of interfacial processes play crucial roles in a wide range of applications of colloids. An interesting application, still to be explored, is that of the dynamics of charged colloids. Nowadays this issue is routinely investigated by dynamic " A . C . Alting, Cold Gelation of Globular Proteins, PhD Thesis, Wageningen University, The Netherlands (2003). 21 J. Lyklema, H.P. van Leeuwen, and M. Minor, Adv. Colloid Interface Sci. 83 (1999) 33.
4.52
DYNAMICS AND KINETICS
light scattering, but in the interpretation of the structure factor the dynamics of the double layers is usually ignored. In fact, double layer interaction is often (over-) simplified by only considering a static interaction at fixed 'effective' potential. Electro-rheology may be another application, especially if considered for lubricants under heavy-duty conditions. Likewise, triboelectricity has an undeniable electric component; this also applies to electrospraying. As a special example of a multidisciplinary nature, we may recall here the electrophoretic deposition of colloidal particles onto a macroscopic surface. The topic is highly relevant for such fields as thin-layer coating technology, e.g. in the electronic industry, and painting technology, e.g. in the car industry. After deposition of an arriving colloidal particle at the receiving surface, the reverse osmotic flow of liquid around the attached particle still goes on. Such hydrodynamic counterflow around deposited particles can be utilized to arrange the particles into neatly ordered 2-D lattices", provided the attraction between surface and particle is weak enough to warrant a certain extent of lateral particle mobility. If the receiving surface is an electronic conductor, e.g. a metal, and simultaneously serves the purpose of maintaining the necessary electric field in the dispersion, then interfacial electron transfer reactions come into play as well. For a given particle/surface combination, deposition technologies then derive from subtle tuning of the externally applied potential difference and the redox conditions in the dispersion medium. The result should be that at the effective potential in the double layer at the surface/medium interface: (i) the charge of the particle satisfies the condition of weak attraction by the surface, (ii) the magnitude of the ongoing faradaic current corresponds to the appropriate ensuing field strength, that is, the correct electrophoretic deposition rate, and (iii) the electroosmotic counterflow generates a sufficiently strong driving force for 2D particle rearrangement. Needless to say that a successful design of such particle deposition technology requires the simultaneous and careful consideration of the various kinetic parameters that define the interfacial and electrochemical behaviour of the colloids involved. The topic becomes even more intricate in the setting of lab on a chip' type of devices, or micro- and sub-micro reactors21. In such systems we typically face the flow of electrolyte solution along charged and reactive walls, giving rise to such phenomena as bipolar electrolysis. Then the coupling between transversal interfacial electron transfer kinetics and double layer relaxation on one hand, and lateral flow of reactive medium on the other, subtly directs the effective rates of the various processes. Basic knowledge of such coupling and the pertaining electrokinetic characteristics is essential in optimizing the performance (efficiency) of the device. 11
M. Bohmer, Langmuir 12 (1996) 5747; Y. Solomentsev, M. Bohmer, and J.L. Anderson, Langmuir 13 (1997) 6058. 21 C.J.M. van Rijn, Nano and Micro Engineered Membrane Technology, Aquamarijn Research, The Netherlands, SPI Supplies (2002).
DYNAMICS AND KINETICS
4.53
4.8 General references H.R. Kruyt, Ed., Colloid Science, Elsevier (1952). In particular chapter VII, J.Th.G. Overbeek, Kinetics of Flocculatlon, p. 278. (Outstanding, rigorous treatment of the topic.) J.Th.G. Overbeek, Colloid and Surface Chemistry. A Self Study Course. Part 2, Lyophobic Colloids, MIT, Cambridge Massachusetts (1972). (The famous instructive lectures with the black-board.) A.J. Bard and L.R. Faulkner, Electrochemical Methods. Fundamentals and Applications, 2nd ed., Wiley (2001). (The leading text in fundamental electrochemistry, including electron transfer kinetics, coupling of interfacial reactions to hydrodynamic conditions, coupling of capacitive and faradaic properties of conducting interfaces.) M. Elimelech, J. Gregory, X. Jia, and R. Williams, Particle Deposition and Aggregation: Measurement, Modeling and Simulation, Butterworth-Heinemann (1995). (Technical book which gives a clear explanation of the different aspects related to particle deposition and aggregation.) T.G.M. van de Ven, Colloidal Hydrodynamics, Academic Press (1989). (Emphasis on micro-rheology of colloids with many elaborations.) R.J. Hunter, Foundations of Colloid Science, Vol. II, Clarendon Press (1989). In particular chapter 13, based on material by R.W. O'Brien. (In depth treatment of double layer relaxation in colloids.) J. Lyklema, H.P. van Leeuwen, and M. Minor, DLVO Theory, a Dynamic Reinterpretation, Adv. Colloid Interface Set 83 (1999) 33. (Review, 64 refs., covers most of the contents of this chapter.) P. Meakin, Fractal Aggregates, Adv. Colloid Interface Sci. 28 (1988) 249. (Older, but not dated review, 165 refs., describing various types of clusters and their formation.)
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5
STRUCTURE OF CONCENTRATED COLLOIDAL DISPERSIONS
Agienus Vrij and Remco Tuinier 5.1
The notion of structure and some history
5.2
5.2
Osmotic equilibria
5.4
5.3
5.2a
Link of osmotic equilibria with thermodynamics
5.5
5.2b
The potential of mean force
5.7
5.2c
Pair interactions between colloidal particles
5.8
5.2d
More-component solvents; Donnan equilibrium
5.9
5.2e
Low colloid concentrations; virial expansions
Structure factor and correlation functions
5.13
5.3a
Particles in an external field; inhomogeneous systems
5.13
5.3b
Particle systems in a weak external field
5.14
5.3c
Interactions in different types of external fields
5.15
5.3d
Osmotic compressibility and closure relations for colloidal dispersions
5.4
5.11
The hard sphere (HS) model for colloidal interactions
5.19 5.22
5.4a
Theory of the hard sphere gas
5.23
5.4b
The Percus-Yevick solution for a collection of hard spheres
5.24
5.4c
Experiments with hard, sphere-like interactions in colloids
5.30
5.5
Attractive interactions between colloidal particles; adhesive spheres
5.39
5.5a
Theory of dispersions with adhesive spherical particles
5.39
5.5b
Experiments with adhesive spheres
5.42
5.6
Soft interactions
5.47
5.6a
Theory of repulsive and attractive soft-sphere interactions
5.48
5.6b
Experiments with soft-sphere interactions: Steric interactions 5.55
5.6c
Experiments with soft-sphere interactions: Electrical interactions
5.7
5.8
5.58
Phase stability
5.61
5.7a
Instability and phase separation in atomic fluids
5.63
5.7b
Instability and phase separation in colloidal suspensions
5.68
5.7c
Stability and phase separation in colloidal mixtures
5.74
Phase transition experiments
5.87
5.8a
Colloid particles in a good solvent
5.87
5.8b
Colloid particles in a poor solvent
5.91
5.8c
Phase separation in two-component colloid mixtures
5.92
5.9
Concluding remarks
5.10
General references
5.98 5.100
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5 STRUCTURE OF CONCENTRATED COLLOIDAL DISPERSIONS AGIENUS VRIJ AND REMCO TUINIER
Concentrated colloidal dispersions are defined as dispersions in which the colloidal particles are frequently in contact with each other. Several pair interactions have been discussed in previous chapters, for instance in 1.4 and II.5. Here we raise the question as to how the pair interactions between the particles determine the structure of colloidal dispersions. The focus is thus on the many-body character of the dispersion. Understanding the relation between pair interaction and structure of the dispersion enables one to: a) analyze the particle interactions in concentrated systems from experimental [e.g. scattering) data and b) to predict the stability and phase behaviour of colloidal dispersions. The dispersions we will discuss contain spherical colloidal particles and the volume fractions q> are up to several tens of a percent of the dispersed phase, which may be solid or liquid. In the physical sense, the fact that these particles frequently meet in the dispersion means that the interactions between the particles have to be accounted for explicitly. So, sols in which the pair interaction has a long range but where
Fundamentals of Interface and Colloid Science, Volume IV J. Lyklema (Editor)
© 2005 Elsevier Ltd. All rights reserved
5.2
CONCENTRATED COLLOIDAL DISPERSIONS
5.1 The notion of structure and some history Over a short time span in the 1970s, the science of colloidal dispersions gained a rapid momentum when it was realized that such systems behave as supramolecular Jluids with tunable pair interactions. The analogy between colloidal and atomic fluids was essential, except that for colloids the specific time and length scales are much larger and, hence, experimentally more accessible. A further advantage of colloidal particles is that the interaction between colloids can be modified relatively easily. For instance, by changing the electrolyte concentration, the thickness of a stabilizing polymeric sheath or the molar mass of non-adsorbing polymer chains, the interaction range can be adapted. Before the 1960s not much fundamental knowledge existed about concentrated colloidal fluids, not even for the much simpler atomic fluids. It was recognized in early times, however, that colloidal particles suspended in a low molecular weight solvent behave as simple fluids. From the work of van der Waals it followed that the model structure of a simple liquid is that of a hard sphere system1'. Einstein derived van 't Hoffs law of osmotic pressure for colloids using statistical mechanics2' and McMillan and Mayer showed that the expression for the osmotic pressure of solutions has the same formal structure as that of a simple gas. Onsager4' gave a statistical mechanical formulation for treating colloids as atoms. Still, at that time no theory was available predicting the pressure p or radial distribution function g(r) for, say, liquid argon in terms of pair interactions. This in contrast to the gaseous state (completely random) or the solid state (highly ordered at lattice points). For a number of reasons, this scene changed dramatically over a relatively brief time span in the 1960s. (i) It became possible to simulate fluids on a computer for systems containing sufficient particles (at that time about 102 , nowadays rather 104 ) to obtain statistical averages that were sufficiently reliable to give representative bulk properties. These simulations started from a given pair interaction, but the important progress was that now (approximate) statistical mechanical theories starting with a given (pair) interaction could be tested rigorously. Previously, theories could only be tested directly with experiments, which has the fundamental drawback that it cannot be decided whether a discrepancy between theory and experiment is due to a deficient theory, due to a deficient pair interaction or due to an unknown deficiency in the experiment.51 By
The separate treatment of repulsive and attractive forces was already proposed by J.D. van der Waals in his thesis 'Over de contlnuiteit van den gas en vloeistoftoestand', Leiden University, 1873. 2)
A. Einstein, Investigations on the Theory of the Brownian Movement, Dover Publications, (1956), pi; Ann. Phys., 17 (1905) 549. 3) W.G. Me Millan, J.E. Mayer J. Chem. Phys. 13 (1945) 276. 4) L. Onsager, Chem Rev. 13 (1933) 73. For a brief introduction and use of simulations, see, e.g. 1.3.2, 1.5.2c and II.2.2b.
CONCENTRATED COLLOIDAL DISPERSIONS
5.3
studying systems with model pair interactions, say of the Lennard-Jones type, it became clear that the structure of a concentrated fluid is primarily determined by the hard repulsive part, upon which the softer attractive part is superimposed as a perturbation. Otherwise stated, the model structure of a simple liquid and a stable colloidal fluid is that of a hard sphere liquid11. Before that time, all of this was not obvious. Typically, in 1935 Kirkwood wrote: 'However, the point to be emphasized is that probability distribution functions derived from rigid spheres can scarcely be applied to actual fluids, where the attractive forces play a dominant role in determining the form of the distribution functions.' (ii) The development of synthesized homodisperse model colloids. Model systems were already known to Perrin at the turn of the 19th-20* century, but he had to obtain his latex-like biological particles by arduous fractionation. Nowadays, monodisperse31 sols are made using controlled nucleation and growth (see chapter 2). Alternatively, sterically stabilized particles can be prepared by terminally anchoring long chains. Microemulsions may also serve as models (chapter V.5). (iii) The development of sophisticated experimental techniques, including small angle X-ray scattering (SAXS), scattering of thermal neutrons (SANS), dynamic light scattering. Throughout the development of this field, the co-operation between physicists (who like to call such systems soft matter) and chemists has been most conducive. In conclusion, the structures of colloidal fluids are characterized in terms of statistically averaged densities on the scale of interparticle distances. By their very nature, these averages are static properties, although their establishment is dynamical. As a consequence, they can be treated by (statistical) thermodynamic methods. In this chapter we will first, in sees. 5.2 and 5.3, lay down the theoretical basis for understanding the structure of concentrated colloidal dispersions. The basic description for interactions between colloidal particles is the hard sphere model, which is outlined in sec. 5.4, whereas more complex applications toward adhesive spheres and soft spheres are discussed in sees. 5.5 and 5.6. In sec. 5.7, phase stability as based upon the properties of the structure of colloidal dispersions will be dealt with using a wide variety of origins of the attractions. Throughout the chapter, the theories will be compared with computer simulations and experimental data, except for the phase transition experiments. These will be discussed in sec. 5.8, while concluding remarks end this chapter in sec. 5.9.
21
In sec. II.2.2b, this was also mentioned for a liquid near a hard, solid wall. J.G. Kirkwood, J. Chem. Phys. 3 (1935) 300 (esp. p.312). The terms monodisperse and homodisperse are equivalent.
5.4
CONCENTRATED COLLOIDAL DISPERSIONS
5.2 Osmotic equilibria11 Colloidal dispersions are usually multi-component mixtures of solvent molecules, lowmolecular solutes {e.g. electrolytes) and particles in the colloidal size range. The system is asymmetric in the sense that the colloidal particles are much larger than the low molecular solvent and solute molecules. This property makes it appropriate to consider systems in which the colloidal particles can be held separated from the low molecular solvent and solute components. In this sec, the osmotic equilibrium between a colloidal dispersion and its solvent is discussed. The colloidal dispersion consists of colloidal particles and a solvent that may contain various species. The osmotic equilibrium treatment will be used to uncover a fundamental relation between microscopic and macroscopic thermodynamic quantities. A schematic picture of osmotic equilibrium is given in fig. 5.1. The solution inside the membrane, having a volume V , is called the system. It contains the components: colloid (1), solvent (2) and one solute (3). The outside solution is called the reservoir. It contains only the components 2 and 3. A semi-permeable membrane, which is permeable only for the components 2 and 3, separates the inside and outside solutions. The pressure inside the membrane (the system) is higher than in the reservoir. The difference is osmotic pressure. For an introduction to osmotic equilibria we refer to 1.2.34 and to sec. 2.20 for a derivation of van 't Hoffs law. In passing it is recalled that the nature of the membrane has no influence on the osmotic pressure, provided it is perfectly semi permeable2'3 .
Figure 5.1 Osmotic equilibrium between the system (left part) with volume V and pressure p , containing the components 1,2, and 3 and the reservoir (right part) with pressure p ^ , containing the components 2 and 3. The separating membrane is impermeable to component 1. The volume of the reservoir is much larger than V . The temperature T is always kept constant. l=colloid, 2=solvent, 3=solute.
11 A. Vrij, E.A. Nieuwenhuis, H.M. Fijnaut, and W.G.M. Agterof, Faraday Discuss. Chem. Soc. 65 (1978) 101; A. Vrij, J.W. Jansen, J.K.G. Dhont, C. Pathmamanoharan, M.M. Kops-Werkhoven, and H.M. Fijnaut, Faraday Discuss. Chem. Soc. 76 (1983) 19. It could, therefore, be replaced by a fictitious, repulsive, external field working only on the colloidal particles. This picture was used by Joos and Debye in deriving van *t Hoffs law. 31 A. Prock, G. McConkey, Topics in Chemical Physics (based on the Harvard Lectures of P.J.W. Debye), Elsevicr (1962), p. 155). G. Joos, Lehrbuch der Theoretischen Physik. Akadcmische Verlaggesellschaft, Geest & PortigK.-G., Leipzig, 1956, pp. 528-529.
CONCENTRATED COLLOIDAL DISPERSIONS
5.5
There is no necessity to view a colloidal solution in this asymmetric way. It may in principle be viewed as a multi-component solution in which all the components are treated on the same footing as is done, for instance, in the Flory-Huggins theory (see II.5) for polymer-solvent mixtures, where it is assumed that the solvent molecules have the same sizes as the polymer segments. However, a theory in which particles of very different sizes are present makes a full analysis very involved. The same can be said of computer simulations of such systems. In the next sec, we will introduce tools that are appropriate to analyze the osmotic equilibria in such a way that the properties of colloidal particles and low molecular solvent and solutes can be treated as separate problems. 5.2a Link of osmotic equilibria with thermodynamics Osmotic equilibria have the feature that manipulations can be performed in which one or more chemical potentials of present components can be kept constant. In the example given in fig. 5.1, the concentration of the colloid can be changed while keeping the chemical potentials of solvent and low molecular solute constant, i.e. by taking the reservoir very large with respect to the system inside the membrane. We now invoke relations which were first derived by Klrkwood and Buff11. For an alternative derivation, see Vrij2). They relate chemical potentials to spatial distribution functions. In this way the notion of structure is included and connected with thermodynamics.
H ik (0) = (PiP k ) 1/2 J°°4OT 2 h ik (r)dr
[5.2.2]
h
I5-2-3!
ik(r) = 9ik( r )- 1
The labelling of the components runs over l,2,...,p . In our example p = 3 . Here, //( is the chemical potential per molecule or particle of component i, Nk is the number of molecules or particles of type k in the volume V , and p k = JV k /V 3) . The function <jjk(r) is the radial distribution Junction of the pair i,k . It is defined by the stipulation 11
J.G. Kirkwood, F.P. Buff, J. Chem. Phys. 19 (1951) 774. The r.h.s. of 15.2.1] is connected with number density fluctuations ( ANi=Ni-(Ni) ): r.h.s.= (AJVjAJVk)/V. By using the relations (AJVjAJVk> = kT(dNi/<9/(k) , , first derived by Zernike, the l.h.s. follows. 21 A. Vrij, Proc. Koninkl. Ned. Acad. (Amsterdam) B88 (1985) 221. The r.h.s. of [5.2.1] is connected with particle excesses in membrane equilibrium. The l.h.s. follows by using Maxwell's cross relations in a Gibbs-Duhem relation like [5.2.5]. 3 In order to avoid unwieldy notations, we use in this chapter py. for the number density JVk / V of component k ; elsewhere in this series of books the symbol p ^ is used.
5.6
CONCENTRATED COLLOIDAL DISPERSIONS
that Pjg ik (r) is the local concentration of i at a distance r from a particle k fixed with its center at the origin. Note that gik[r) = g ki (r) • The function hik(r) = gik(r) - 1 is called the total correlation Junction,
see [1.3.9.23], It becomes zero when r becomes
sufficiently large. For an introduction to these functions see sec. I.3.9d and the references in sec. 5.10b. The subscript fj' in the partial derivative of [5.2.1] means that all ji. it jui are kept constant in the differentiation. Further, c>ik is the Kronecker delta (Sik = 1 for i = k and <5ik = 0 for i * k), and kT is the thermal energy with T the absolute temperature. For our system, with p = 3 , we now concentrate on the colloid component and write
^^K^h^o4^'^])1
[5241
'-
A remarkable feature is that 9/^/9/^ is related to h n (r) of the particles 1 only. On the other hand, this derivative is directly related to the osmotic pressure 77. This can be seen as follows. We first rewrite the Gibbs-Duhem relation [1.2.13.4] for the pressure p at constant T in the following way p dp = ^ / 3 j d / / j =Pjd//j + p2dju2 + p3d/i3
+ ...
[5.2.5]
i=l
By partial differentiation with respect to Pj at constant ju2 and /u3 , one finds:
[JP]
JfL]
=Jfil
where 77= p - p R and p R is the pressure in the reservoir. Substitution of this result into [5.2.4] yields
=( 1 + ^ r 2 h " ( r ) d r ) '
^ 1 L
1J
<"2-"3
15-2.7]
V
Thus, the osmotic pressure, see fig. 5.1, is also found to be related to hjj(r) only. Note that the form of this equation is identical to that for a one-component system [1.3.9.32], which is known as the compressibility equation. This implies that [5.2.7] treats the colloid system as a quasi one-component system. The term quasi is used because, as we will also see later, the radial distribution function of the colloid particles not only depends on the colloid concentration, but also on all interactions between particles and the other molecules and on their concentrations. However, in many important cases this problem can be sequestered. Osmotic pressures can, in principle, be obtained from the study of membrane equilibria. Often this is a cumbersome procedure and not applicable when the colloidal particles are too large, and consequently too few, resulting in a too low, experimentally imperceptible osmotic pressure. Therefore, (light) scattering experiments are usually
CONCENTRATED COLLOIDAL DISPERSIONS
5.7
preferred. The light scattering intensity is directly proportional to the osmotic compressibility and thus to (3pj /dfJ)
, see further sec 5.3d.
5.2b The potential of mean force In this sec. we use a function, which was introduced in 1.4.3c and which is related to the radial distribution function, the potential of mean force. Although [5.2.1] is valid for any isotropic system, we will assume here that the colloid particles have isotropic interparticle interactions, which usually implies that the particles are spherical or weakly aspherical. Consider the following 'experiment' in our system given in fig. 5.1. Fix one colloid particle with its centre at the origin and consider a second particle on a position at a distance r, see Hill
or sec. 1.4.3c. The average force the second particle feels (among
the other particles, which are free to move) is then /(r) = -3io n (r)/3r , where wn(r) = -kT In gu{r)
[5.2.8]
Note that when moving a second particle of nature 1 to another position it should be done 'quasi-statically,' which means: so slowly that all the diffusion processes of all the components have sufficient time to equilibrate at each infinitesimal step. Techniques for measuring forces between colloidal particles have been discussed in sec. 3.12. Let us concentrate on the function iOj j (r) . For p^ —> 0 , only pair interactions between the colloid particles play a role and we write -"fi(r) r
9ll( ) = 9fiW =
e
kT
-^iW =e
kT
[5.2.9]
where u>fj(r) = <2^j(r) is the potential of mean force of two particles 1 far removed from other particles 1 2) . It is important to recognize that in some cases the study of gl l (r) or co-y i (r) can be treated as a separate problem. In general, one could remark that our formulation has not helped us much from a fundamental point of view, because there is no simple way to calculate these functions from first principles. These functions will depend in an unknown way on the properties of the solvent and the interactions of the solvent with the particles. Indeed, for the case where the colloid particles would be comparable in size with the solvent molecules, or more realistically comparable in size with a third component, properties of particles and solvent components could not a priori be 11
T.L. Hill, An Introduction to Statistical Thermodynamics, Addison Wesley (1960) p 313, 314. We have chosen for the separate symbol (Uj j(r), instead of u>j*j(r), in order to omit the unwieldy notation of the superscript 0. aij j(r) is the pair interaction of a selected pair of particles 1 in the solution with components 2,3,...p. It is noted that in chapter 3 pair interactions are expressed with the symbol G(h) , were h is the closest distance between the two particle surfaces. Therefore, a>^ j(r) = G(r — 2a) .
5.8
CONCENTRATED COLLOIDAL DISPERSIONS
disentangled. In the next sec. we consider in more detail the pair interactions between colloidal particles. It will become clear, however, that physically convincing and intuitively attractive approximations can be formulated when the (principal) colloid component consists of particles that are much larger than the low molecular solvent components. Many examples of pair interactions between charged colloidal spheres suspended in an electrolyte solution are derived in sees. 3.3-3.9 and collected in app. 2. In such dispersions the role of the solvent is characterized by the concentration of the chargedetermining ions, the concentration and type of indifferent electrolytes present and possibly organic admixtures. 5.2c Pair interactions between colloidal particles The pair interaction of two colloidal particles, as defined in the previous section, is of course very different from the pair interaction of two such particles in a vacuum. In colloid suspensions, a solvent medium, being a condensed liquid, is always present. What can be said of co^ j (r) starting with the case that further components than pure solvent are absent? The range of the particle-solvent interaction is at least comparable with the size of the solvent molecules, but we may say that its range is much smaller than the particle size in many systems. We may thus consider three regions in the colloid pair interaction. a>ll(r) = oo
o < r < la
&)j j (r) = repulsive/or attractive wx j (r) = 0
2a
[5.2.10]
where la is the diameter of the particles and d is the thickness of a surface layer having properties differing from those of the bulk of the solvent. For many cases, d«2a. Further, we do not know very much in detail; yet, we have to describe the properties of these layers. We may assume that the interaction is monotonous because particle 'surface roughness' will eventually average out local extrema. When the suspension is stable, we expect that the pair interaction in the interval 2a < r < 2a + d will be repulsive. Then it will be a good approximation to replace the real pair interaction by that of a 'hard sphere' with diameter, ax. Thus, °h i (r) = °° f° r 0 < r < a, a>l j (r) = 0 for r > a^
[5.2.11].
The value of <7j is approximately known, but often it is expedient to consider it as an adjustable parameter for fine-tuning when interpreting experimental results. When suspensions are not stable, in the sense that colloid particles tend to agglomerate, we may expect an attractive part in the range 2a < r < 2a + d . This will be further consid-
CONCENTRATED COLLOIDAL DISPERSIONS
1
5.9
Figure 5.2. Short-range forces (a) operate in localized patches. The total interaction (2) is equal to the sum of three pair interactions (1). Long-range forces (b) operate over distances comparable with the particle size or longer. Usually the total interaction (2) is slightly different from the sum of the three pair interactions (1). In other words, the pair interaction lb is perturbed by the presence of a third particle.
2
(a)
(b)
ered In sec. 5.5. When the above narrow interaction range is only small with respect to the particle size, i.e. d « l a , the pair Interactions are sufficient to describe the thermodynamic and structural properties. That Is to say, the interaction (free) energy of all the particles in the system is denoted as a pair-wise additive; the total interaction in the system may be described as a sum of pair Interactions. This Is Illustrated In fig. 5.2. 5.2d More-component solvents; Donnan equilibrium In the previous subsections we considered colloids In a pure solvent. However, very often other components are also present as solutes. Well-known examples are low molecular solutes such as electrolytes, which play a role In the case of colloids bearing an electric charge. In addition, high molecular weight components, such as polymers, dendrimers, micelles, microemulsions and proteins are commonly studied nowadays. The last group is In use as stabilizing (protecting) or destabilizing (by bridging or depletion) agents. It will be clear, however, that the last types of systems are on the borderline with colloidal mixtures, of which two examples will be given in sec. 5.711. In the present subsection, we shall consider the situation in which the solvent and all solutes (2,3,...p) can pass the membrane, irrespective of their molecular mass. This type of osmotic equilibrium results In the redistribution of solutes over the two membrane compartments. The redistribution of electrolytes21 was first studied by Donnan at the beginning of the 20th century with protein colloids, and is called after him the Donnan equilibrium^. The Donnan osmotic pressure 77 = p - p R depends on pj, as well as the composition of the solvent (//2.-.-,/O- Below we consider only one extra solvent, component 3. The distribution of this component 3 over the system and reservoir is a function of the In the parlance of physics all solutions, be they low or high molecular weight, arc often called 'complexfluids' We already encountered the redistribution of electrolytes (negative adsorption) by charged colloids, for instance in sec. II.3.5.b. 31 F.G. Donnan, A.B. Harris, J. Chem. Soc. 99 (1911) 1554; F.G. Donnan, Z. Elektrochem. 17 (1911)572; Chem. Rev. 1 (1924)73.
5.10
CONCENTRATED COLLOIDAL DISPERSIONS
same variables, i.e. pl and ^ • / ' s • When a membrane is available that is permeable to the solvent component 2 ('main solvent component') and 3 ('extra component') this distribution can in principle be measured by analytic chemical means. Another possibility is to use the scattering of light (or SAXS and SANS) at different scattering contrasts . These techniques can also be applied when no semi-permeable membrane is available. The Donnan pressure and the composition of the solvent are related to each other by Gibbs-Duhem relations dp = p1dfil+p2dju2+p3dfi3
[5.2.12]
dp R =p£d// 2 +pfdfi3
[5.2.13]
From these equations, the following thermodynamic relation can be derived21. — \dUr>\
= L
p
\ ^ - \
R
MA
[5.2.14] R
Here L3 is defined by
and may be considered as the preferential adsorption of component 3 (with respect to component 2) by the colloid particles31. This can further be clarified by considering the following process in the osmotic equilibrium of figure 1. Add a number dJVj of colloid particles to V . This induces the changes diV2 and d!V3 in the number of particles 2 and 3, taken from or added to the reservoir. In many cases, dJV2 and dN3 will be negative; sometimes d/V3 may be positive if component 3 is strongly adsorbed by component 1. This is the case when component 3 contains charge-determining ions, which are positively adsorbed to the particle surface. The difference in the ratio of the concentrations of component 2 and 3 in V and the reservoir is a measure of the preferential adsorption of 3 over 2 by colloid 1. Equation [5.2.14] shows that there is a fundamental thermodynamic relation between the dependence of the osmotic pressure on the solute activity and the redistribution of this solute as a function of the colloid concentration. This will be elaborated in sec. 5.2e.
11
Sec e.g. A. Vrij, J. Th. G. Ovcrbeek, J. Colloid Set 17 (1962) 570. For the details of this derivation, see A. Vrij, Colloids Surf. 51 (1990) 299. A derivation for the somewhat simpler two-component case will be given in sec. 5.7. We note here that there is a relation between L3 and the notions of surface excess in surface chemistry, see II.2.3. 21
CONCENTRATED COLLOIDAL DISPERSIONS
5.11
As in the case of a two-component case of colloid in a simple solvent, one has to consider how IJ and L3 change with pj at constant composition of the reservoir. For classical membrane experiments, this condition is automatically fulfilled when we take the volume of the reservoir large enough or continuously flush it with solvent of the same composition. In the two-component case of a condensed liquid as the solvent, the redistribution of solvent is simply a volume replacement of solvent by particles being equal to Pjiij , where L>J = (9V73iV|) , the partial specific volume of 1. Since the osmotic pressure is very small, vl does not depend on the pressure difference across the membrane. In applying the theoretical [5.2.7] or interpreting scattering experiments, however, one uses a partial derivative of the osmotic pressure at prepared compositions of 2 and 3 in V which in principle will be different from those in the (fictive) reservoir. A 'natural' process of concentrating and diluting is not so simple here as in the two-component case because one has to consider explicitly the distribution of components 2 and 3, i.e. the value of L3 as a function ofpl, in that L3 may be different from zero. In the next sec. we will consider this further for lowpj. 5.2e Low colloid concentrations; virial expansions For low colloid particle concentrations, one may write down virial expansions for the osmotic pressure and the solute distribution over the membrane (see fig. 5.1). ^
= A + B{ 2 1 )p? + BJ31>p? + ...
[5.2.16]
and L3 =K31)p1 + K 3 2) /f+...
[5.2.17]
Equation [5.2.16] is a variant of [1.2.18.25b]. Here the B s and Ks are called virial coefficients. An expression for the second virial coefficient BJ2' follows from [5.2.7] and [5.2.9] on expanding 3/7/9pj in terms of pl and comparing this with [5.2.16]. Then, one finds11 7 ( -*ii(rn Bpj' = In I r 2 1 - e W dr
o I
[5.2.18].
J
A similar equation exists for L3 . The second virial coefficient BJ2' is a measure of the two-particle interaction of the colloid and is still a function of //3 . A well-known case is that of charged colloidal particles in an electrolyte solution. The electrical charge gives a repulsive contribution to coll, which tends to make BJj (more) positive. But ft>n depends on the electrolyte (component 3) concentration. An increase of p3 gives more screening of the electrical double layer and will make Oj j less repulsive, leading to a Compare this with the similar expression [1.3.9.12].
5.12
CONCENTRATED COLLOIDAL DISPERSIONS
decrease of B (2) n . Such manipulations are often used to influence the stability of the colloid system. The coefficient Kg' is a measure of the preferential adsorption of an indifferent electrolyte for a single particle 1. It is known from electrical double layer theory (see II.3.5b) and experiments that Kg' is negative. This means that when a colloid particle 1 is added to V from its neighbourhood, a certain amount of (electroneutral) electrolyte is removed. In other words, a colloid particle 1 creates around itself an (effective) depletion zone of which the thickness is comparable with the thickness of the ionic atmosphere around the particle 11 . The coefficient Kg2' expresses what happens when interactions between two particles 1 occur. In that case, one has to consider the consequences of the overlap between two depletion zones. The theoretical answer is that two overlapping zones are less effective than two separate zones. This was also found experimentally a long time ago by Moller21 from Donnan membrane experiments with bovine serum albumin, see fig. 5.3.
Figure 5.3. Negative adsorption of chloride ions by negatively charged bovine serum albumin (BSA) (z =20 elementary charges) in six aqueous KC1 solutions (indicated, in M ). Vertical axis: relative expulsion of Cl~ ions: ( p^j — /9Q )//}Q • Horizontal axis: BSA concentration in gdm~ 3 . (Redrawn from: W.J.H.M. Moller, Electrical Transport Properties of Alkali Albuminates, PhD thesis. University of Utrecht, 1959, his fig. V-3.)
This implies that Kg2' will be positive, hence it reduces the negative contribution pjKg" to L 3 . It is interesting to connect these two effects, i.e. the decrease of B j j , with increasing electrolyte concentration and the increase of L 3 with increasing colloid concentration by using [5.2.14]. With equations [5.2.17] and [5.2.16], it follows that 31
For the comparable case of pair interaction thermodynamics, see sec. IV.3.2, in particular [IV.3.2.6]. 21 W.J.H.M. Moller, Ph.D. Thesis, Utrecht University (1959). See sec. V.3. 31 A. Vrij, Colloids and SurJ. 51 (1990) 299. Sec also D. Stigtcr, J. Phys. Chem. 64 (1960) 838.
CONCENTRATED COLLOIDAL DISPERSIONS
5.13
kT—^- = - K , ; 3 d//3
[5.2.19]
which is indeed in accordance with the observations of these two effects. The above thermodynamic equation tells us that the relation we described for charged particles has general validity, in line with the conclusion drawn in sec. IV.3.2. that overlap of depleted volumes leads to an attraction between particles. 5.3 Structure factor and correlation functions In the previous section, we observed that structural information is an important part of our fundamental knowledge of concentrated colloidal dispersions. Therefore, it is necessary to obtain a good conceptual knowledge of the notion of structure (formation) in such systems. In our further treatment, we will focus our attention on a single colloidal particle component with spherically symmetric interactions. Other (solvent) molecules or particles may be present, but we will not consider these explicitly. In the next subsection, we will start with the effect of a conservative external field as a probing tool to provoke (extra) structure on the particle system and investigate what effect this has on the spatial organization of the system. This will finally allow us to define the total pair and direct pair correlation functions. In turn, these can be used to relate pair interactions to the structure factor introduced in sees. I.3.9d and I.7.8f.
5.3a Particles in an external field; inhomogeneous
systems
Consider a large, open volume V in a much larger volume filled with particles. When the system is homogeneous, the singlet number density distribution as a function of position r is a constant nW(r) = /?°
[5.3.1]
Now, we impose on the system an external field v{r), where v has the dimensions of an energy that makes the number density position-dependent. This implies that the system becomes inhomogeneous. A simple example of an external field is the field of gravity, where the number density becomes a function of the height, z . Then n(1) (z) = p° => nSlHz mpgz) where m
[5.3.2]
is the mass of the atmospheric gas molecules or the (buoyancy-corrected)
mass of the colloidal particles. More generally,
1
For small molecules (in the earth's atmosphere), a height of kilometres is required but for colloidal particles a height of millimetres may be sufficient to observe such spatial density variations.
5.14
CONCENTRATED COLLOIDAL DISPERSIONS
n^(r) = p° ^nW(r||u)
[5.3.3]
An example of the last case is the field induced by an (extra) atom or colloidal particle, which is fixed at the origin. Then, v[r) becomes equal to u{r), the pair interaction in a gas of atoms or w(r) the pair interaction of colloidal particles in a dispersion. Then n'1' becomes nM
(r \\v) = p°gW (0 = P°g{r)
[5.3.4]
where g{r) is the radial distribution function (see sees. I.3.9d and 5.2b). In general, u{r) and <w(r) are not weak fields. Nevertheless, the hope is that knowledge about n'1' (r|i>) in weak fields, which can be treated as a perturbation, can teach us something about g(r) . 5.3b Particle systems in a weak external field Consider for simplicity not a continuous space variable r but a discrete one, so the volume V is divided into small volume elements dr, which are numbered in a certain way. The question is now: when the field in a volume element at p is equal to v , what will be the induced (average) change in the number density p° due to that field? A simple and plausible approximation is to assume that the Boltzmann factor is applicable, in which case
n J>= p o e -V^ = p o^j
[535]
for v /kT « 1. We assume that the field is only felt by the particles, but not by other (solvent) components that may be present. From now on, we will write p for n'1' so that one obtains p°Svu
This simple result, however, can only be valid for very small p° . At higher number densities, the number density in elements around the volume element p will change to such a degree that this will indirectly influence the value of p . Hence, we can write Sp
o
P=^
h Pq
(-^)+^Xhpq(-^)^°dr)
- 9 p q - 1 = 0(|'-p-'- q |)- 1
l5 3 7]
--
I5-3-8!
where q runs over all the volume elements in V . The second term in [5.3.7] can be
CONCENTRATED COLLOIDAL DISPERSIONS
5.15
made plausible as follows". The external field in element q- in the neighbourhood of element p - leads to an additional number of particles equal to -Sv p°dT/kT at position q. One extra particle in q gives a density change p°[gpq -1] = Pohpq i n P • s o -Sv p°dr/kT particles give rise to a density change equal to p°h (-Sv /kT)[p°dr). Summation over all elements in V then leads to [5.3.7]. One may write [5.3.7] more compactly as follows,
^P=(P°) 2 I^qf-^l d 7 q
L
I5-3-9'
J
with
Here, S is the 'Kronecker delta,' i.e. S = 1 when p = q and S = 0 when p * q . For p,q = 1,2,...n , the linear functions in 5i> can be written in the form of a correlation matrix,
sPl/(pofdr--htl[-^yhi2[-s^y... ;
•••
[5.3.H]
«^j""-»s.(-£H(-&)Thus, the local number density variations can be calculated when the functions h(r) = g(r) and v(r) are known. In sec. 5.3c we will give some examples. Let us now reverse the question and ask for the values of 5u(r), or Si> , which are needed to induce a given value of 8p . In mathematical terms, -a^-fcTdr^ctp^p
[5.3.12]
P
where the sum over p covers the same elements as before and where the matrix [cpq ] is the reciprocal matrix of [ht ]. Both matrices are symmetrical, i.e. hj = hj ; c qp = cpq • F o r s u c h matrices, the following property is valid: - < V ( P ° d 7 ) 2 =X h PS C sq I 5 - 3 - 13 " s where, again, s covers all elements in V . Analogous to [5.3.10], one can define ^
^
-
^
|5 3 141
- -
"This relation was first derived by J. Yvon, Suppl. Nuouo Cim. 9 (1958) 144; see e.g. J.-P. Hanscn, I.R. McDonald, Theory of Simple Liquids. Academic Press (1976) chapters 5 and 6.
5.16
CONCENTRATED COLLOIDAL DISPERSIONS
Substitution of [5.3.14] into [5.3.13] finally leads to the following result h
pq=CPq+^I>psCsqdT
I5 3 151
- '
s
where we have used the following properties,
I s Vsq = <W X S V V = V I s ^ s q = *pq
I5'3'16]
Let us now return to continuous space variables and write
W = ^ [ - ^ ] + (p°)2JhC-,r')[-^]d3r'
[5.3.17]
S[-v(r)/kT] = Sp(r)/p° - \c{r,r')5p{r')d3r'
[5.3.18]
h(r,r') = c(r,r') + p°jh(r,r")c{r",r')d3r"
[5.3.19]
The last equation is the famous relation of Ornstein and Zernikel\ which can be considered as the defining equation of c{r) in terms of h{r). The function c(r,r'), is called the direct correlation function, whereas h(r,r') = g(r,r')-\ is called the total correlation Junction. In isotropic fluids, these functions depend on (|r-r'|) only. The direct correlation function usually has a (much) shorter range than the total correlation function, see also [5.3.39], 5.3c Interactions in different types of external fields (i) Weak field with a small gradient. Examples of systems in weak fields are colloid particles in the field of gravity, in a centrifugal field or in the electric field of a (charged) macroscopic surface in a medium of low polarity. Such fields vary little over distances comparable with the range of h{r). We assume that the field only changes in the x-direction, i.e. v{r) = v{x). Then, it will be sufficient to represent v(x) as a Taylor series, i.e. v(x') = v(x) + fr'(x) + -£2v"(x) + ...
[5.3.20]
where v'{x) = dv/dx and E, = x'-x. Substituting [5.3.20] into [5.3.17] then leads to [5.3.21], for which the first two order terms in density are l + p°jh{R)4xR2dR
5p{x) = -p°^^-
L
[5.3.21]
o
For symmetry reasons, the term with u'(x) is zero. The expression between the square 11
L.S. Ornstein, F. Zcrnikc, Proc. Koninkl. Ned. Akad. Wetenschap
(Amsterdam) 17 (1914) 793.
CONCENTRATED COLLOIDAL DISPERSIONS
5.17
brackets is related to the compressibility equation, [5.2.7]. It follows, using [5.2.6], that Sp{x) = -p°^dv{x)
= -^Sv{x)
[5.3.22]
Here, // is the chemical potential of the (colloidal) particles. This result is plausible; the number density variation that is induced by a macroscopic field is proportional to the (osmotic) compressibility (l/p°)(3p°/3/7). A well-known example is the atmospheric distribution of molecules in the field of gravity. For low molecular weight systems, the field of gravity is too weak to create a significant gradient on a laboratory scale except near a critical (solution) point, where the compressibility goes to infinity. In a colloid system, however, the influence of gravity may become large enough to reach measurable density gradients. The entire concentration profile of particles can be computed by introducing a theoretical expression for osmotic compressibility and integrating [5.3.22]. This will be done in sec. 5.4b fit) for the case of a suspension of'hard spheres' (see sec. 5.2c.) in the field of gravity. (it) Weak, periodic (sinusoidal) external fields; susceptibility. Let us consider a (somewhat artificial) sinusoidal field with amplitude v* , obeying v(r) = v(x) = v*cos(Kx)
[5.3.23]
Substitution of this expression into [5.3.17] and working out the integral leads (analogous to [5.3.21]) to Sp(x) = -p° — \cos(Kx)-(p°f \h(R)cosK(x+ £)d3R\
[5.3.24]
where we may substitute Jh(R)cosK(x + £)d3fl = h(K)cosKx
[5.3.25]
in which h(K)= \ 4nR2h{R)^^dR [5.3.26] •* KR o We already encountered the last integral in [5.2.4], at least for K = 0 . Hence, [5.3.24] can be written as 5p(x) = -fP ^ ( K ) c o s J C x : = -pO^Mx{K)
[5.3.27]
5.18
CONCENTRATED COLLOIDAL DISPERSIONS
where X{K) = \ + p°h{K)
[5.3.28]
This equation can be interpreted as follows: a weak, sinusoidal field induces a density variation (in counter phase), of which the amplitude is proportional to ;f(K). The quantity %{K] can therefore be interpreted as a K -dependent susceptibility of the system, which reflects how strongly the system yields to an external field. The definition is equivalent to that of the dielectric and magnetic susceptibility in an electric and magnetic field, respectively (sec. 3.10c). It can be regarded as a 'virtual spectroscope,' which detects the collective spatial behaviour of the system. Collective, since the behaviour not only depends on the properties of the individual particles, but also on their interactions. Later, we will observe (see [5.3.44a]) that ;}f(K) is identical to the structure factor S(K) where X(K) = S{K)
[5.3.29]
or ^(q) = S(q), see [5.3.44b]. The susceptibility can thus be experimentally studied by static light scattering, SANS and SAXS, see sec. 5.3d. (in) Ornstein-Zernike equation in K-space. For a sinusoidal external field, the variation Sp(x) is simply proportional to Sv(x) (see [5.3.27]). Let us now do the opposite: introduce a sinusoidal number density variation into [5.3.18] and calculate 8v(x). p(x) = p*cosKx
[5.3.30]
Then, it can be found, analogous to sec. 5.3c.(ii), that poL^W
L
\ = Sp*[\-p°i{K)]cosKx
[5.3.31]
^ J
with c(K) = J4xR2c(R)SmKR o
dR
[5.3.32]
Comparing this result with [5.3.27] and [5.3.28], one finds = l + p°h(K)
= \-p°c(K) F
{
[5.3.33a]
'
or h(K) = c(K) + poc(K)h(K)
[5.3.33b]
Hence, the descriptions in terms of h and of c are symmetrical. Equation [5.3.33b] is the Ornstein-Zernike equation in K-space (the reciprocal space). Combining [5.3.28,29] and [5.3.33a] gives:
CONCENTRATED COLLOIDAL DISPERSIONS
5.19
—!-T = —?-r = l-p°c(K) *(K) S(K) ^ V ;
[5.3.34]
Hence, c(R), through its Fourier transform c(K) using [5.3.32], yields the structure factor S{K). 5.3d Osmotic compressibility and closure relations for colloidal dispersions (i) Behaviour of the correlation functions
at K = 0. At K —> 0 , i.e. at a large
enough wavelength X = Inl K , the above equations must predict macroscopic, thermodynamic behaviour. X(K
= 0) = S(K = 0) = 1 + p°h(O) = \l- p°c(0)T 1 =kT-?L J dn
[5.3.35]
or — — = l-p°a(0) = l-p°j4^ 2 c(R)dR P
[5.3.36a]
0
and kT^P- = l + p0\47TR2h(R)dR 0
[5.3.36b]
The last equation we already knew (see [5.2.7]). Or in terms of the chemical potential H of the colloidal particles, using dIT/dp = pd}i/dp{see [5.2.5]) p + p2h(0) = kT^-
[5.3.37a]
and i_3(0) = — ^ . p
[5.3.37b]
kTdp
There exists a multi-component version of these equations: (l/fcT)O//j/3pk) , = <J jk -C lk (O) and kT{dpk /djut) , =^ i k + H j k (0), see [5.2.1]. These equations are important in discussing multi-component colloidal systems11. (it) Density variation induced by the field of a fixed colloidal particle. We observed above (see [5.3.4]) that the density variation induced by a particle fixed at the origin is given by p°(g(r) -1) = p°h{r). One may now raise the following question: What field is needed to create a density variation Sp(r) = p°h{r) around the origin?21 Let us substitute this expression into [5.3.18] and obtain
11
See e.g. A. Vrij, J. Chem. Phys. 69 (1978) 1742 ibid, 71 (1979) 3267; D. Gazzilo, A. Giacometti, Phys. A304 (2002). 21 See e.g. F. Kohler, The Liquid State, Verlag Chemie (1972) p. 123.
5.20
CONCENTRATED COLLOIDAL DISPERSIONS
[5.3.38]
This integral occurs in the Ornstein-Zernike equation (see [5.3.19]), so one finds ^L-c(r)
,5.3.39,
This result seems queer because we would have expected the pair interaction co{r) instead of c(r). But, we should recognize that co[r) is not a weak field like v(r), and thus we could not expect to find oAr)! It suggests, however, that c[r) must behave like co[r) in some way. Indeed, it may be argued that the range of c{r) is comparable with that of co(r). (itt) Approximate solutions for the total and direct correlation Junctions. In the previous sections, we have observed that the structure of colloidal suspensions may be characterized by the total correlation function h(r) = g{r) - 1 , as well as by the direct correlation function c[r). These functions are related to each other by the OrnsteinZernike relation [5.3.19]. It will be clear that a second relation between h[r) and c{r) is needed (a so-called closure relation) to solve [5.3.19]. Two relations that have been suggested and often used are the Percus-Yevick (PY) closure: c{r) = g(r)[l-eta(r)/kT)
[5.3.40]
and the hypernetted chain (HNC) closure relation: c(r) = g ( r ) - l - l n g ( r ) - ^
[5.3.41]
Here, aAr) is the pair interaction. Thus, it is assumed that the configurational free energy of the system can be described as a sum of pair interactions. It is found by computer simulations, for example, that the PY approximation is better suited for steep pair interactions like the hard-sphere pair interaction; the HNC approximation applies better for soft pair interactions like those found with electrical interactions with an extended ion atmosphere. There are several routes to derive these approximate equations, for which we refer to literature on the theory of liquids11. One of the routes may be formulated as follows. The direct correlation function c{r) was first introduced by Ornstein and Zernike21 by stipulating that the spatial correlations in the number density p in two volume elements separated by a distance r have a direct and an indirect component. In the direct component, only number density fluctuations in the two volume elements are 11
J.K. Percus, G.J. Yevick, Phys. Rev. 110 (1958) 1; D.A. McQuarrie, Statistical Mechanics, Harper and Row (1976) p.276. 21 L.S. Ornstein, F. Zernike, loc. cit.
CONCENTRATED COLLOIDAL DISPERSIONS
5.21
considered, whereas the fluctuations in all the surrounding volume elements are kept fixed. The indirect contribution to the fluctuations in the two volume elements is added when the fixation in the fluctuation in the neighbouring volume elements is released. The two contributions give 9total(r) = c(r) + g indirect (r)
[5.3.42]
Here, gtotal(r) is nothing else than g(r) = 1 + h{r). Let us now write those equations in terms of potentials of mean force (see [5.2.8]) g{r) = exp[-o)(r) / kT] and make the Ansatz ^direct (0 = e ^ ^ ' ^
=e ^ ) - « ] / f c r
|5
3 43]
where
co(r) i-ekT
( co(r)^ =g(r)i-ekT
which is just equal to [5.3.40]. (iv) Scattering of waves as an experimental tool to study the structure of colloidal suspensions. One can imagine that g{r) can be obtained by direct observation of the position of the colloidal particles with a microscope. This has indeed recently become possible for a special type of synthetic compound particles, which is 'transparent' except for the particle core. The positions of such particles can be measured with a confocal microscope when the particles are large enough1' (see fig. 5.4). Another possibility that has been known for much longer is the detection of the scattering of light, X-rays and cold neutrons by the particles. The methods are known as light scattering, small angle X-ray and neutron scattering (abbreviated as SAXS and SANS, respectively), and are frequently used nowadays (see sees. 1.7.8 and 1.7.9). At low scattering angles, certain thermodynamic properties of the system can be obtained as well. The scattering intensity is given by (see [1.7.9.1 ] and [1.7.8.40]): I(q) = pJ2P(q)S(q) Here, /
[5.3.44]
is the particle scattering amplitude, P(q) is the particle scattering factor (or
form factor), which gives the intensity of a single particle, and S(q) is the structure factor. The angular variation is defined by the scattering vector q (|q| = q), which is given by (4W A)sin(0/2), where X is the wavelength of radiation (in the medium) and lf
A. van Blaaderen, P. Wiltzius, Science 270 (1995) 1177; C.P. Royall, M.E. Leunissen, and A. van Blaaderen, J. Phys. Condens. Matter 15 (2003) S3581.
5.22
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.4. Radial distribution function g{r) of a suspension of charged PMMA spheres ( l a = 1.95 urn, fluorescently labelled with rhodamine) in a solvent mixture of cyclohexylbromide ( — 70% w/w ) and cis-decalin ( ~ 30% w/w), determined with confocal scanning laser microscopy (CSLM). Circles: experimental results. Drawn line: Monte Carlo simulation results taking a hard sphere diameter of a= 1.95 urn , using a particle with 760 elementary charges and KG = 1. The volume fraction of the dispersion is 0.00535. (Courtesy of C.P. Royall, ME. Leunissen, and A. van Blaaderen, loc cit, where details can be found.).
0 the scattering angle. The structure factor S(q) is related to the Fourier transform of the total correlation function h{r), H(K) = p j 4 ; r r 2 h ( r ) ( ^ ^ l d r o
[5.3.44a]
S(q) = l + H(K = q)
[5.3.44b]
by
Note that H(K) is the K -dependent version of [5.2.2], H{K) = ph[K) . Equation [5.3.44] is only valid when multiple scattering can be neglected. For light scattering, this requires special values for the refractive index of the particles (nearly matching that of the solvent). For SAXS and SANS, the problem is absent or less severe. In future sections, the structure and thermodynamic properties of (concentrated) suspensions will often be described in terms of quantities measured using these important tools. 5.4 The hard sphere (HS) model for colloidal interactions In section 5.2c, we argued that steeply repulsive pair interactions between colloidal particles in suspensions may be approximated by [5.2.11], which is equal to the pair interaction in a "gas" of hard spheres with diameter a .
1 ' The parameter a in the hard sphere model is the distance of closest approach for hard spheres (see p. 1.7 and p. 1.4.9); also called hard sphere diameter' for obvious reasons.
CONCENTRATED COLLOIDAL DISPERSIONS
Q)HS (r) = °° for 0 < r < a coHS (r) = 0
5.23
[5.4.1]
for r > a
As the formalism, which connects structural to thermodynamic properties (e.g. [5.2.7] and [5.2.8]) is identical to that of a one-component gas or fluid, we may use all the (equilibrium) structural and thermodynamic properties derived for a hard sphere gas to describe those of a concentrated colloidal dispersion defined through [5.2.11]. This implies that, for example, 77, gfjj(r) and Cjjfr) of a dispersion of particles (component 1) may be replaced with p, g(r) and c(r) of a hard sphere gas. It is interesting to observe what happens when the pair interaction deviates from that of a hard sphere i.e. when the repulsion is softer than that of a hard sphere as when (o7r)~12 is supplemented by a longer ranged attractive tail, -(o7r)~6 in the Lennard-Jones interaction. An example is given in fig. 5.5. In this figure, S(q) (in particular for large p) is practically determined by &>HS , except at small K-0, where the Lennard-Jones fluid shows larger isothermal compressibility, see [5.3.35]. It will turn out that for many stable colloidal particles the HS-pair interaction works very well. Figure 5.5. Structure factor of a model fluid with a Lennard-Jones pair interaction 4e[(o7r)12 -(o7r) 6 ] compared with that of a hard sphere (HS) fluid with particle HS diameter a . pa3 = 0.926 ; e = 1 kT . Drawn line: HS in the Percus-Yevick approximation; circles: simulations. Only those simulation data (circles) are shown that deviate from the drawn full curve. (Redrawn from: G.A. Vliegenthart, J.F. Lodge, and H.N.W. Lekkerkerker, Physica A, 263 (1999) 378.)
5.4a Theory of the hard sphere gas There are no exact analytical expressions for the thermodynamic and structural properties of hard sphere fluids, except at low densities. So, one may write a virial expansion for the pressure. We encountered the virial expansion earlier in this chapter, see [5.2.16] -^ = p+B2pz+B3p3+...
[5.4.2]
For hard spheres, these quantities are purely geometric and do not depend on temperature. For the second virial coefficient B 2 , the following expression holds (see [1.3.9.12,13] and [5.2.18])
5.24
CONCENTRATED COLLOIDAL DISPERSIONS
s 2 B" 1-e 2 =2;r r J
o
kT
L
2 dr=^— J 3
dr = 2Jr
Jo
=tHEL n
o
[ 5 .4.3]
3
The third and fourth virial coefficients of a collection of hard spheres, B 3 and B 4 can also be rigorously obtained. Higher order virial terms were calculated numerically, see table 5.1. Likewise, g(r) can be expressed as a virial expansion: g{r)e"WkT
=1 + pgx{r) + p2g2(r) + ...
[5.4.4]
For hard spheres, gi(r) is known for i = 1,2 .^ Table 5.1. Virial coefficients for a hard sphere fluid. Results were derived analytically (B 2 - B 4 ) or from computer simulation (B5-B7 ). CS = [l/3)PY(v) + (2/3)PY(c) 2) Bn/B»~l
anal./simul.
PY(c)
PY(v)
CS
B3IBl
5/8
5/8
5/8
5/8
BJB\
0.28695
0.29688
0.25000
0.28125
B5/B2
0.110252
0.12109
0.0859
0.10938
B%IB\
0.0389
0.0449
0.0273
0.03906
B7IB%
0.0137
0.01562
0.00830
0.01318
However, virial expansions are only valid over a limited density range. For an accurate representation of p, terms including B 6 are needed for a volume fraction It is therefore of eminent value that there exists a solution of the (approximate) Percus-Yevick integral equation for hard spheres, which is reasonably accurate over a large range of densities, say
11
See, e.g., D.A. McQuarrie, Statistical Mechanics, Harper and Row, 1976, section 13-6. K.W. Kratky, Physica A87 (1977) 584. 31 M.S. Wertheim, Phys. Rev. Lett. 10 (1963) 321, E. Thiele, J. Chem. Phys. 39 (1963) 474. 21
CONCENTRATED COLLOIDAL DISPERSIONS
i
5.25
r ~fi
c(r) = -A1-6^-1
.
c(r) = O
;
0
[5 4 5]
r>a
with tp = -7io3p
[5.4.6]
(1-y) 4
N'T The thermodynamic properties can be derived from the pressure if it is known as a function of T and p. According to statistical mechanical theory, there are two relations ('routes') connecting the pressure and the radial distribution function (see sec. I.3.9d). The so-called virial route expression is p = pfcT-^-[r^^g(r)4OT- 2 dr 6 J dr o
(virial)
[5.4.9]
and the compressibility route (c) uses =l + pfh(r)4/rr 2 dr
JcT ^
(compr.)
[5.4.10]
The results for PY are -^=
1
+2
^+ 3 / 2
_P_=l +P + f
(PY virial)
(PY compr.)
[5.4.11]
[5.4.12]
The fact that the expressions are different for the two routes is a result of the fact that the (approximate) PY theory is internally inconsistent on this level. It can be easily verified that B2 and B3 do not depend on the route. The higher Bt 's, however, become different (see table 5.1). There is another accurate equation for p, proposed by Carnahan and Starling (CS)1'. They found that the Bi 's can be approximated accurately by
11
N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51 (1969) 635.
CONCENTRATED COLLOIDAL DISPERSIONS
5.26
Bi=f|CT3T(i2+i-2)
[5.4.13]
A summation of this virial series leads to _p_=
\+
(Carnahan-Starling)
[5.4.14]
It is very peculiar that p is just equal to the sum of (l/3)p (PY, virial) and (2/3)p (PY, compr.). Unfortunately, there is no expression for c(r) on the CS level" as far as we know, so in practice one uses the PY expressions and the compressibility route for the pressure because it is nearest to the accurate CS expression, and it has a clear, obvious connection to structure and scattering formulas. The structure factor S(q) for hard spheres in the PY approximation can be found directly by substituting [5.4.5] for c(r) into [5.3.32], and c{K) subsequently gives S[q) using [5.3.33] and [5.3.28, 29]. For, q = 0 there is a direct connection with dp/dp according to [5.3.36a]:
^ [ ^ j = l-pJc(r)4OT-2dr = l- P a(K = q = 0)
[5.4.15]
Substitution of [5.4.5] into [5.4.15] leads to
kT[dp\
(1 _^)4
Upon integration, this leads to [5.4.12]. Calculated results for S(q) and p are shown in figs 5.6 and 5.7, respectively.
Figure 5.6. Structure factors of hard spheres as a function of the scattering wave vector q normalized with the sphere diameter a for three volume fractions; q> = 0.\ (dashed curve), 0.2 (full curve) and 0.3 (dashed-dotted curve). The results were calculated using the Percus-Yevick result for the c[r) of [5.4.5] using [5.3.32] and [5.3.34].
We note that there is a recent result from density functional theory for the c{r) of a collection of hard spheres: eq. [41) in R. Roth, R. Evans, A. Lang, and G. Kahl, J. Phys. Cond. Matt. 14 (2002) 12063, gives results that are very close to the CS results.
CONCENTRATED COLLOIDAL DISPERSIONS
5.27
Figure 5.7. Pressure of a collection of hard spheres; computations based on the CS [5.4.14], PYC [5.4.121 and PYV [5.4.11] equations (the subs c and v refer to the compressibility and virial routes, respectively). (Redrawn from J.P. Hansen and I.R. McDonald, Theory of Simple Liquids, Academic Press, 1990.)
Figure 5.8. Radial distribution function of hard spheres at q> = 0.49 ; comparison of PY results with computer simulations. (Redrawn from J.P. Hansen and I.R. McDonald, Theory of Simple Liquids, Academic Press (1990).)
In fig. 5.8, the radial distribution function gf(r) of hard spheres in the PY approximation is compared with computer simulation results. Note the accuracy of PY, except at r = 0 where the PY result is too low. We will now give two examples of quantities that can be determined experimentally which are related to the equation of state of a collection of hard spheres. (i) Scattering properties. Equation [5.4.16] can be used immediately to obtain the light scattering intensity at q = 0 or rather for qa « 1 , see [5.3.44] l(q = O) = pJ2P(q = O)S(q = O) where P(q = O) = l and
[5.4.17]
5.28
CONCENTRATED COLLOIDAL DISPERSIONS
S(q = O) = kT^-
[5.4.18]
or
I(q = 0) = PJ^T^ / W J * ),JlZ*>l 3/7
[5.4.19]
Vita6 ) (l + 2
Equation [5.4.19] shows that at low (p the intensity increases linearly with cp , but that at higher
(at
see fig. 5.9.
Figure 5.9. Volume fraction dependence of the scattered intensity at q = 0, following [5.4.19] for a collection of hard spheres. (ii) Sedimentation equilibrium. Another example to which the hard sphere equation of state can be directly applied is the sedimentation equilibrium. In the field of gravity2', the effective weight of colloidal particles in a slab dz (with unit area) must be compensated by the osmotic pressure difference over the lower and upper surfaces of the slab -d/7 = ^ < ^ l - ^ ^ j p H S g d Z
[5.4.20]
where <5HS and <5soivent are the mass densities of the HS particles and the solvent g is the acceleration of gravity and the expressions between brackets are the buoyancy factor31. From [5.4.20], the distribution of particle concentration as a function of height z readily follows when [5.4.16] is used for osmotic compressibility41 d£ = _±(l-q,)4 dz
?g(l
[ 5 4 2 1 ]
+ 2cpf
11
A. Vrij, J. Colloid Interface Set 90 (1982) 110. In a centrifugal field, g should be replaced by ap'x , where x is the distance of the particles from the center of rotation and a> the angular velocity. o)
For a more general formula which is also valid in a multicomponent hard sphere dispersion, see: A. Vrij, J. Chem. Phys. 72 (1980) 3735. 4 Steep gradients can be treated in the density functional description of inhomogeneous fields, see T. Biben, J.-P. Hansen, and J.-L. Barrat, J. Chem. Phys. 98 (1993) 7330.
CONCENTRATED COLLOIDAL DISPERSIONS
5.29
where
JLsSHc-^fllg
b
" 1—
[.
^HS )
[5.4.22]
K L
and C is the so-called gravitational length introduced in [2.3.25] 1 '. Equation [5.4.21] & can be integrated to give 5 1 o \ cp+ (fr -V—\ + ^-Z — [5.4.23]
(
1-PJ
{l-cpf
where the 'constant' is an integration constant, which is fixed by the initial particle distribution in the tube. An example is shown in fig. 5.10. In the same figure the turbidity (see [5.4.38] in sec. 5.4c) is plotted at the corresponding z . A sample in a test tube is shown in fig. 5.11.
Figure 5.10. Calculated volume fraction
Figure 5.11. Sedimentation equilibrium of octadecyl-silica particles in cyclohexane in a test tube in the field of gravity. Note that the turbidity is not highest at the bottom of the tube where the particle concentration is largest, but at an intermediate height. (Courtesy of J. den Boesterd, Van 't Hoff Laboratory, Utrecht, The Netherlands.)
"compare [5.4.21] with [5.3.22] for particles in a weak field. The gravity field has a small gradient because lg is much larger than a .
5.30
CONCENTRATED COLLOIDAL DISPERSIONS
5.4c Experiments with hard sphere-like interactions in colloids (i) Osmotic compressibility obtained from light scattering. First, some early experiments on spherical silica particles in cyclohexane are shown, in which the particles were made lyophilic by covering the surface with a layer of terminally anchored octadecyl chains. This system, which was especially synthesized for this purpose, has the following attractive aspects: - The refractive index of silica (1.45) and cyclohexane (1.42) is nearly equal. Therefore, these dispersions are, also at higher concentrations, quite transparent, which is necessary to avoid multiple scattering effects. - Because of the small refractive index difference, the Van der Waals-London attraction between the particles will be so small that they may be neglected. - Since cyclohexane is a good solvent for octadecyl chains (both are saturated hydrocarbons) the solvated surface layers of two encountering particles will repel each other sterically. This results in a fairly steep, repulsive pair interaction without attractive components and the system is therefore a good prototype for pseudo-hard sphere repulsion. A schematic picture of such particles is given in fig. 5.12. In fig. 5.13, one finds the light scattering results expressed as d/7/d/>. It shows that the non-ideality contribution to the compressibility becomes very large at high concentrations, and is well represented by the HS-model. Thus, the silica particles in cyclohexane are a good example of colloidal hard spheres. The next system we consider is a microemulsion of water droplets in toluene and benzene (containing some hexanol) as a continuous medium . The water droplets were stabilized with a mixture of potassiumoleate and hexanol chains. The light scattering intensity is plotted as a function of concentration in fig. 5.14. The curve shows a typical maximum in the intensity, which occurs at about 0.12 g/ml, see also fig. 5.9. It was found that the experimentally measured intensities could not be fitted with a pure hard sphere model. Therefore, an attractive term -2awc was added in a semi-empirical way to the expression for osmotic compressibility,
Figure 5.12. Schematic picture of a silica particle with octadecyl chains terminally attached to its surface.
11
W.G.M. Agterof, J.A.J. van Zomcrcn and A. Vrij, Chem. Phys. Lett. 43 (1976) 363.
CONCENTRATED COLLOIDAL DISPERSIONS
5.31
Figure 5.13. Derivative of the osmotic pressure obtained from light scattering experiments, AQ = 546 nm (open circles); XQ = 436 nm (closed circles). The line is calculated with the Carnahan-Starling equation for hard spheres, using a specific volume of the particles of 0.73 cm^g"' . The particles are silica spheres covered with a dense layer of octadecyl chains, which are dispersed in cyclohexane. The radius (obtained from the diffusion coefficient) is 22 nm . (Redrawn from A.K. van Helden and A. Vrij, J. Colloid Interface Sci.
78(1980) 312.)
[5.4.24] The experiments could be fitted very well to this equation (see fig. 5.14), which allowed obtaining the values ov,Q and a , see table 5.2. The values of
CONCENTRATED COLLOIDAL DISPERSIONS
5.32
Figure 5.14. Light scattering intensity (in arbitrary units) of a microemulsion of water droplets (stabilized with a surface layer of oleyl+hexyl chains! dispersed in benzene (code Bl in table 5.2) <7HS = 7.50 nm and a w = 5.4 . The specific volume of the particles is equal to (4;z73)(<7HS/2)3(IVAv /M) , where (M/JVAv) is the particle mass. The scattering angle 9 was 90°. Experimental data are indicated as the open circles. The theoretical curve follows [5.4.24] using the Carnahan Starling equation for /7pig • (Redrawn from: W.G.M. Agterof et al., loc. cit.) Table 5.2. Light scattering results of aqueous microemulsion droplets in benzene (code B l , B2) and in toluene (code T l ) .
M/10 2 kgmol"1
nm
Bl
-0.113
1.25
11
6.1
B2
-0.129
5.25
15
Tl
-0.103
3.10
13
9sP
B
2
cm 3 g - 1
cm 3 g~ 1
7.5
1.06
~0
5.4
10.5
11.8
0.99
~0
5.3
8.4
10.2
1.09
3.2
2.4
nm
nm
It is also possible to solve the PY integral equation for mixtures of hard spheres with different diameters11. For the pressure, this gives —±- = ^ 2 _ + — ^ y +
^-3-
(compressibility version)
[5.4.25]
where
£v=f2>rPi
[5-4.26]
i=l
Note that £3 = qp is equal to the overall volume fraction of the spheres. Pressures of mixtures with the same
CONCENTRATED COLLOIDAL DISPERSIONS
5.33
spheres. This implies that the (simple) relation between overall osmotic compressibility -(1/V)dV/dn and light scattering intensity cannot be used any more, and that a more specific solution of the PY equation is needed for the chemical potential derivatives of each component. Let us concentrate here on q = 0 where only thermodynamic properties come into play. Further, we assume that the scattering amplitude of a particle is proportional to of , thus _/j = yaf (other choices are possible). Hence, it can be derived" that /(q = 0) = ^(6/,r) t , - ^ k -
+
^ %
(l-^f
[5.4.27]
Some results are shown In fig. 5.15 for a log-normal distribution of hard sphere diameters, defined by rin((7/(Tn)l v
-
";
1 -d<7
[5.4.28]
[ PJ2 \ 0 Here, y/(a)d
s^Hf^/-, (°7
[5.4.29,
For p«1, p = s o . One observes that the deviation from the monodisperse case (sa = o) is small for s o = 0.1, but large for s o = 0.3 . Note that the deviations are still large compared with a 'monodisperse system with adapted initial slopes' for sa = 0.3, see fig 5.15a. In fig. 5.15b the scattering of a polydisperse system is compared with that of a Jictive system in which only osmotic pressure fluctuations are taken into account by computing S(q = 0) = kT{dp/dFT) = kTN[d{l/ V)/dI7)N from [5.4.25] (/7 = p ; pi = JVj/V ) and substituting this into [5.4.17]. The results show that it is insufficient to take only osmotic pressure fluctuations into account. Fluctuations in composition at constant osmotic pressure also have a great influence . The case q = 0 gives information about some thermodynamic properties of the polydisperse system. To obtain structural information, scattering at finite q , i.e. at finite scattering angles, is needed. Since colloidal dispersions always have some degree
11
A. Vrij, J. Colloid Interface Set 90 (1982) 1 10. Osmotic pressure fluctuations cannot be measured with static light scattering but sometimes with dynamic light scattering, sec P.N. Pusey, H.M. Fijnaut, and A. Vrij, J. Chem. Phys. 77 (1982) 4270. 2
5.34
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.15. (a) Relative intensity of scattered light of a polydisperse system of hard spheres for a log-normal distribution of hard sphere diameters as a function of the overall volume function
of polydispersity, it is important to know how experimentally determined structure factors are 'deformed' by a spread in particle size. Here, we may only touch upon this complicated problem. Fortunately, an explicit analytical solution is available for polydisperse hard spheres in the Percus-Yevick approximation. In general, one may formulate the scattering intensity as p
l(q)=^fJiBi(q)Bi(q)fe^iSii(q)
[5.4.30]
with Sii(q) = Sii+Hii{K = q)
[5.4.31]
and ff
ij(g) = ^ J 4 O T 2 f l i j ( r ) ^ f : d r qr o
[5.4.32]
Note that H()(q) is the finite q version of [5.2.2], and that hj(r) is the correlation of the pair i,j . Further, Jfi^q) is the scattering amplitude of particle i, _/j is the amplitude at q = 0 , and for a solid sphere with radius
CONCENTRATED COLLOIDAL DISPERSIONS
Bi(q) = 3 ( f ) " 3 [ s i n f - ^ c o s ^ ]
5.35
,5.4.33,
The above expression ,5.4.30, for I[q) is only useful when the h (i (r)'s are known, which is usually not the case. It is also possible to formulate I(q) as a function of the direct correlation function c^ir), which is theoretically often better known than the h i( (r), for which we refer to the literature 1 ' 21 . For hard sphere mixtures in the PY approximation, the following formula can be derived 121
J(q) = 2>i|./i B ite) + A(9)f
[5 4 341
--
i=l
Here, A^iq) is a complicated but elementary function of the variables q,/j, B((q), Oj and pi; i = 1,2,..., p. The Aj 's are complex numbers. For other types of interactions, like those between 'adhesive hard spheres,' the solution for the scattering intensity
Figure 5.16. Average structure factor (S(q)) of a polydisperse system of hard spheres, having an overall volume fraction
11
A. Vrij, J. Chem. Phys. 69(1978) 1742; ibid 71 (1979) 3267. A. Vrij, C.G. de Kruif, Micellar Solutions and Microemulsions, S.-H. Chen, R. Rajagopalan, Eds., Springer Verlag, 1990, p. 143.
5.36
CONCENTRATED COLLOIDAL DISPERSIONS
was recently found . Some results 2 ' are shown in fig. 5.16a and b for a log-normal distribution of hard sphere diameters, see [5.4.28]. The (relative) standard deviation s o is defined by [5.4.29]. The overall volume fraction of the hard sphere particles is
[5.4.35]
i=l
is plotted for two (relative) standard deviations, s o =0.1 and 0.3 . Note that the PY solution for hard spheres gives a very reasonable accuracy and that the main peak in S(q) has not changed much for a spread in diameter of 10% compared with the monodisperse case. For a spread of 30%, however, the main S(q) peak is nearly
Figure 5.17. Average structure factors (S(q)) determined with SANS for silica particles (a = 22.5 nm) covered with a layer of octadecyl chains (d = 1.8nm) and dissolved in cyclohexane. The volume fractions are indicated. Symbols: open triangles and open squares, experimental data points at two different detector distances; dashed line, polydispersc PY calculations ( s o = 0 . 1 2 ) . (Redrawn from: C.G. de Kruif, W.J. Bricls, R.P. May, and A. Vrij, Langmuir 4 (1988) 668.) 11
D. Gazzillo, A. Giacometti, Phys. A304 (2002) 202. P. Van Bcurten, A. Vrij, J. Chem. Phys. 74 (1981) 2744; D. Frenkel, R.J. Vos, C.G. de Kruif, and A. Vrij, J. Chem. Phys. 84 (1986) 4625. 21
CONCENTRATED COLLOIDAL DISPERSIONS
5.37
washed out, see van Beurten and Vrij and Frenkel et al, loc. cits. The PY solution is of great help in analyzing experimental results. An example is shown in fig. 5.17 for lyophilic silica spheres dissolved in cyclohexane. Another, more practical example of an application of the hard sphere model can be found in a study of polydisperse caseins, which are present in large quantities in milk. De Kruif et a!.11 measured the structure factor of a concentration series of /?-caseins using small-angle neutron scattering. In fig. 5.18 some of the S(q) data are reproduced, as derived from the scattered intensity data following [5.4.35]. The agreement between the data points and the theoretical predictions is quite reasonable considering the complexity of the protein particles.
Figure 5.18. Average structure factors (S(q)) of /? -casein (radius of gyration: 8 ± 1 nm ) dispersed in D2O at 298 K for three concentrations as indicated: 10 (triangles), 15 (squares) and 20 (circles) g/L corresponding to
(Hi) Turbidity measurements. Results for osmotic compressibility may also be obtained from turbidity experiments. This has the great advantage that it is practically insensitive to multiple scattering and is easy to perform. Already, Zernike and Debye used the method to obtain thermodynamic data from critical opaleseence. More recently, the technique was used again to obtain values of S(q = 0), an important thermodynamic quantity, which for monodisperse systems is equal to kTdp/dfJ2]. The turbidity r is obtained from the attenuation of the light beam measured in a spectrophotometer. 11 C.G. de Kruif, R. Tuinier, C. Holt, P.A. Timmins, and H.S. Rollema, Langmuir 18 (2002) 4885. 2) J.W. Jansen, C.G. dc Kruif. and A.Vrij, J. Colloid Interface Set 114 (1986) 492; M.H.G.M. Penders, A. Vrij, J. Chem. Phys. 93 (1990) 3704; U. Apfcl. R. Grunder, and M. Ballauff, Colloid. Polym. Set 272 (1994) 820.
5.38
CONCENTRATED COLLOIDAL DISPERSIONS
r = ; t " 1 ln(l/T t )
[5.4.36]
Here, Tt is the transmission coefficient equal to the intensity of the transmitted beam divided by the intensity of the incoming beam, and lt is the optical length traversed in the cuvette. Thus, r is the scattered intensity integrated over all angles. n T=2n\l{q)sm6&G
[5.4.37]
0
For small particles, say a < 1/20, there is no angular dependence, except for the polarization factor n + cos 2 #)/2 because of using unpolarized light. Then, with [5.3.44] and [5.4.37] it is possible to derive r = -;zp/ 2 S(q = O)
[5.4.38]
From this, it follows that S(q = 0) can be obtained from S(q = O)=
r/P T
[5.4.39]
( /P)p-»O i.e. from the specific turbidity at the required q> divided by the turbidity in dilute dispersions where S(q = 0) = 1. For (somewhat) larger particles a longer part of the optical spectrum is needed because the turbidity ratio in [5.4.39] becomes wavelength dependent and the function S(q = 0) in [5.4.38] has to be replaced by the general function Z{XQ,(P) . Then, S(q = 0) is obtained from the limiting value of the integrated structure factor Z{AQ,
Figure 5.19. Integrated structure factor Z{A.Q,
CONCENTRATED COLLOIDAL DISPERSIONS
5.39
reference in the caption of fig. 5.19. Experiments on polystyrene latex spheres (a = 35.5nm, s o ~ 0 . 1 1 ) covered with the surfactant Triton X-405 In 0.01M KC1 solutions. cL = c latex = 50gdm~ 3 in the presence of HEC (hydroxyethyl cellulose and M = 172 kgmol" 1 , a = 41.5 nm are shown in fig. 5.19. The intercepts of the curves with the abscissa axis are equal to S(q = 0) = kT{dp/dI7) and increase in the sequence A to D. Apparently the osmotic compressibility of the latex dispersion increases when more HEC Is added. This can be explained by the so-called depletion effect, further discussed in sees. 5.7 and 5.8, and, more extensively, in sees. V. 1.8 and 1.9. 5.5 Attractive interactions between colloidal particles; adhesive spheres In sec. 5.4, we encountered attractive interactions in our discussion of the light scattering of microemulsions. In this section, we will discuss the influence of attractive forces on the structural and thermodynamic properties of colloidal dispersions in more detail. We will restrict ourselves to particles with a spherical shape and attractions with a short range. 5.5a Theory of dispersions with adhesive spherical particles Under 'adhesive attractions,' we define interactions within a range, which is small with respect to the particle diameter of the colloidal particles. In many instances, this will be the case. One may refer to the classical van der Waals attraction in chapters 3 and 4 of this volume. Also in cases where the mechanism of the interactions is less clear, such as, for Instance solvent structure-mediated forces, the range will be short11. (I) Model: hard sphere plus attractive well. The simplest picture of a steep repulsion combined with a short-range attraction is a hard sphere with diameter a supplemented with a narrow square well with a width A2). This pair interaction has the following mathematical form31 co(r) = ~
0
co(r) = -£
a
ft)(r) = 0
r > <7+A
[5.5.1]
When the narrow well is deep, the pair interaction adopts the character of a (negative) delta function 8(r - a), see fig. 5.20.
Attractions of longer range include those between particles due to embedded magnetic permanent dipoles, see sec. 3.10c. One could think of another shape of the interaction curve than a square well, e.g., a triangular well or a well with a Yukawa tail as exp {—yr)/r . The precise shape becomes irrelevant as long as the range is small with respect to a, see, e.g., C. Regnaut, J.C. Ravey, J. Chem. Phys. 91 (1989) 1211. 3
CONCENTRATED COLLOIDAL DISPERSIONS
5.40
Figure 5.20. Pair interaction between two hard spheres with diameter a and a square well with a width A and a depth e. For small colloid concentrations p , the osmotic pressure of the colloidal suspension may be characterized by the second virial coefficient (see sec. 5.2e. [5.2.16] and [5.2.18]). By substitution of [5.5.1] one obtains
a
7 r - ^ i
B2=27i\r2
.
r—
1-e fcT dr = 2;rjr 2 (1-0)dr + 2/r<72 1-eM" A
o L
J
o
[5.5.2]
L
or
2
r^
i
B2 =-TTO3 - efcT - 1 2;r<72A
[5.5.3]
where the second part of the integral in [5.5.2] is replaced by the value of the integrand at a times dr = A . For small p, the structure factor becomes (see [5.3.44], plus [5.2.9] using [5.3.8])
°° r -'"(o i • S(q) = l + 4roj[r 2 e kT - 1
i L
S1
\
"qrdr
[5.5.4]
qr
and with [5.5.1], a
e
1
S(q) = l + 4 ^ [ r 2 ( 0 - l ) ^ ^ d r + 4^ff2 e^T -\ ER3EA J qr J qa
[5 .5.5]
or oi \ T A
3 sinqa-qoxosqa
S(q) = \ - 47tpcr
^2^5 q-^tr3
:2
A
2
7% , sinqa
— + 47rpcTz efcT-i
A
2—A qc
, r _ „, [5.5.6]
From [5.5.3] and [5.5.6], one may observe the effect of attraction on B2 and S{q). Increasing the depth of the well decreases B 2 , which may even become negative. The main effect on S(q) is an upswing at K = 0, due to an increase of osmotic compressibility, see [5.3.35]:
CONCENTRATED COLLOIDAL DISPERSIONS
5.41
Figure 5.21. Structure factor for hard spheres with an attractive well at low colloid concentrations, calculated with [5.5.6]; p = 0.05; A / a = 0.01. S(q = O) = kT^d/7
[5.5.7]
In experiments, this upswing at q = 0 is a characteristic indication that attractive forces come into play, see fig. 5.21. A further increase of the magnitude of e will eventually lead to thermodynamic instability of the suspension. The particles will attract each other to such a degree that phase separation occurs. Phenomena of phase separation will be discussed in sees. 5.7 and 5.8. For higher particle concentrations, the virial series is not adequate and approximate models must be invoked to solve the statistical mechanical problems. A popular model, which is relatively easy to apply, is the 'adhesive hard sphere' model (AHS) of Baxter11. This model will now be described. (ii) Baxter's model of adhesive hard spheres. Baxter replaced the pair interaction [5.5.1] by 0){r) — ^ = °° kT
0< r < a
=ln^^-
<7
[5.5.8]
(7+A
=0
r >
after which, when applied, the limit A -> 0 is taken. Thus, only a single parameter, the so-called stickiness parameter x , characterizes the adhesive strength. A low value of r means strong adhesion. For the second virial coefficient, one finds
11
R.J. Baxter, J. Chem. Phys. 49 (1968) 2770.
5.42
B
CONCENTRATED COLLOIDAL DISPERSIONS
2=VHs{4-\)
I5-5-9!
where VHS = (;r/6)cr3 is the hard core volume. Thus, the range of the attractive well is coupled to its depth in such a way that B 2 stays finite. The pair interaction [5.5.8] is physically unrealistic, but mathematically convenient for calculating thermodynamic and structural properties. Baxter demonstrated that this pair interaction yields a closed analytical solution of the Percus-Yevick (PY) integral equation. This is convenient when analyzing experimental results. The Baxter model can also be extended to polydisperse systems11. Monte Carlo simulations21 show that the PY approximation gives very reasonable results. For instance, for r = 0 . 2 , where B2 =-VrHS, the structure factor is plotted in fig. 5.22 for
Figure 5.22. Structure factor versus qa for an adhesive hard sphere system (also called sticky hard sphere system); the stickiness parameter r = 0.2 and
5.5b Experiments with adhesive spheres Experiments will now be described for adhesive spheres of silica particles covered with a lyophilic' layer of alkane chains suspended in some organic solvents. In these systems, the (macroscopic) Van der Waals forces between the particle cores only play a minor role because of the (nearly) matching refractive indices of colloid and solvent. The interactions are caused by 'steric interactions' of surface layers combined with an exchange of solvent and alkane chain segments. This exchange may lead to attractions
11 C. Robertus, J. Philipse, J.G.H. Joosten, and Y.K. Levine, J. Chem. Phys. 90 (1989) 4482; D. Gazzilo, A. Giacometti, Phys. A, 304 (2002) 202. 21 W.G.T. Kranendonk, D. Frenkel, Mol. Phys. 64 (1988) 403. 31 C. Regnaut, J.C. Ravey, J. Chem. Phys. 91 (1989) 1211. 41 R.J. Baxter, J. Chem. Phys. 49 (1968) 2770.
CONCENTRATED COLLOIDAL DISPERSIONS
5.43
Figure 5.23. Theoretical structure factors for the adhesive hard sphere model of Baxter (PY) at ip = 0.19 and
when the solvent becomes poor, see also chapter V.I and footnote11. The quality of the solvent is temperature dependent. An example from protein science will also be mentioned. (i) Second virial coefficient; temperature dependence of e. Poor solvents can be recognized by a low, and sometimes negative, value and a strong temperature dependence of B 2 . In the model pair interaction used in [5.5.1 ] this will be formally attributed to the temperature dependence of e 2), for which we write £ = -L e=0
[0
IT
1 1 \kT
J
T<0
[5.5.10]
T>0
where L is a constant supposed to be proportional to the overlap volume (7z74)A2cr. This form resembles an expression for the interactions of two polymer clouds in a poor solvent as given by Flory and Krigbaum3)(FK). Two contributions may be attributed to the expression of the strength of attraction e , which has the character of a free energy of mixing in the FK picture. The term Lk0 resembles an energy contribution while LkT resembles an entropy contribution. The situation at T = 0 and e = 0 was denoted as the theta point by the above authors, and 0 the theta temperature. It implies that the repulsive entropy of mixing (of chain segments and solvent molecules) is just compensated by an attractive energy of mixing contribution. For T <0, the solvent quality becomes poor and e becomes smaller than zero, leading to attractive interactions between the colloid particles. For T > 0 , E is positive and the polymer chains behave as in a good solvent. This can be incorporated as a slightly larger value of the hard sphere diameter, in the case A « d . For convenience, one may then define e = 0 and use the HS pair interaction [5.4.1 ] for T > 0 . For (more) discussion of exchange of solvent and chain segments in microemulsions see e.g. J. Bergenholtz, A.A. Romagnoli, and N.J. Wagner, Langmuir 11 (1995) 1559. The temperature dependence of the term kT is much weaker and can be neglected. 31 P.J. Flory, W.R. Krigbaum, J. Chem. Phys. 18 (1950) 1086.
5.44
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.24. Second virial coefficient of lyophilic silica spheres SP23 and SP45 as determined with light scattering. For SP23 (in benzene: squares): la = 68 nm ; L53; and 0 = 357 K . For SP45 (in n-dodecane: triangles): 2a = 83nm; L = 53; and 0 = 315 K. A was chosen to be 0.3 nm . (Curves redrawn from P.W. Rouw, A. Vrij, and C.G. de Kruif, Colloids Surf. 31 (1988) 299.)
The factor L is a constant proportional to the overlap volume (nl 4)^2
[5.5.11a]
where or = 2 4 - e LT J - l
[5.5.11b]
As an example, some results are shown in fig. 5.24 for silica spheres having octadecyl chains terminally attached to the surface and dispersed in benzene and n-dodecane. The second virial coefficient B 2 w a s determined from turbidity measurements and is included in fig. 5.24 as a function of T . It is found that B 2 changes appreciably over a temperature range of about 20°C. The values L and 0 were evaluated from the plots and are given in the caption of fig. 5.24. It is clear from this figure that the decrease of B 2 as a function of temperature is much more gradual for SP23/benzene than for SP45/n-dodecane. This indicates that the attraction between the particles in benzene evolves much more slowly as a function of temperature than the attraction between the particles in n-dodecane. The authors argue that this difference originates from the local mechanisms of the interactions. The temperature dependence of the solvency is probably different in both solvents. (ii) Structure factor. Experiments with adhesive spheres. Some results of structure factors obtained with SANS are shown in fig. 5.25 for three volume fractions and three temperatures. The symbols are measured values and the drawn lines for the 1
P.W. Rouw, C.G. de Kruif, J. Chem. Phys. 88 (1988) 7799; P.W. Rouw, A. Vrij and C.G. de Kruif, Prog. Colloid Polym. Sci. 76 (1988) 1.
CONCENTRATED COLLOIDAL DISPERSIONS
5.45
Figure 5.25. Structure factor of lyophilized silica particles dispersed in benzene at three temperatures measured using SANS. The particle surface is covered with terminally attached noctadecyl chains. The particle diameter 2a = 40nm (s a =0.11), the volume fractions ip are 0.14 (circles), 0.19 (diamonds) and 0.28 (triangles). Plot a): T = 51.6°C; b): T = 40.8°C; c): T =35°C. Drawn lines: Baxter AHS model. (Redrawn from M.H.G. Duits, R.P. May, A. Vrij and C.G. de Kruif, Langmuir 7 (1991) 62.)
Baxter theory of adhesive spheres. This shows reasonable agreement. The curves show the characteristic 'upswing' at small q , which increases with decreasing temperature, i.e. with increasing attractions. It further shows that with increasing q> the S[q) curves have a tendency to shift to the pure hard sphere case. This implies that the hard sphere repulsion of the particle core has a more pronounced effect on S(q) for denser systems, in accordance with the general rule that at high densities the structure is determined by repulsive forces. From an analysis of the curves, values of r, the parameter that measures the adhesive strength could be extracted for different temperatures. To this end, the quantities r and e, A were converted through the second virial coefficient expressions, which gives In 1 + — — \ = L\— - l ]
L 12EdJ
IT
)
[5.5.12]
5.46
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.26. Temperature dependence of the stickiness parameter r. Data points: experiments. Line follows [5.5.12]. (Redrawn from M.H.G. Duits, R.P. May, A. Vrij, and C.G. de Kruif, Langmuir 7 (1991) 62.)
The l.h.s. of this equation is plotted versus 1/T in fig 5.26 (taking A = 0.3 nm ). From slope and intercept, one obtains L = 31.4 and 0 = 350K 1!. Another example taken from the field of proteins is the structure factor (at q = 0 ) of lysozyme as a function of the volume fraction at several temperatures and two salt concentrations (0.1 MNaCl ;0.2MNaCl )2). In fig. 5.27a, the reciprocal structure factor S~l{q = 0) = {l/kT)dI7/dp is plotted as a function of
i \dnAJ\ + 2
(1 _^)4
with
A =6
i+ iz£ T _(fi + i z £ r f _J_f1 + l j l
[5.5.14]
For osmotic compressibility, compare with [5.4.16] for the hard sphere result ( X = 0). With decreasing temperature the initial slope of the curves (at
11
M.H.G. Duits, R.P. May, A. Vrij and C.G. dc Kruif, Langmuir 7 (1991) 62. R. Piazza, V. Peyre and V. Degiorgio, Phys. Rev. E58 (1998) R2733. The data are taken, in fact, in the metastable fluid region, i.e. metastablc with respect to fluid-solid transition. 21
CONCENTRATED COLLOIDAL DISPERSIONS
5.47
Figure 5.27. (a) Derivative of the osmotic pressure dIJ/kTdp = (M I RT)dn/ dc as a function of the volume fraction of hen egg-white lysozyme at pH = 4.7 and c^aQ[ = 0.2 M and at different temperatures: 35.0°C (filled circles), 30.0°C (open circles), 24.5°C (filled squares), 17.2°C (open squares) and 12.2°C (filled diamonds) as measured with light scattering. The specific volume of lysozyme is 0.71 cm3/g . (b) Baxter x parameter as a function of temperature for lysozyme at pH = 4.7 and c NaC1 = 0.1 M (filled circles) and 0.2 M (open circles). (Redrawn from R. Piazza et al, loc. clt.)
Finally, we want to mention a SANS study of very small magnetic (y -Fe 2 O 3 ) nanopartides [a= 4.4 nm) coated with negative citrate groups in salt solutions11. The structure factor shows the characteristic trends displayed in fig. 5.25 when the attractions between the particles are increased, which is accomplished by increasing the ionic strength. The attractions are attributed to magnetic dipole and Van der Waals forces and come into play when the repulsive double layer forces are decreased. Critical phenomena and phase separations are also reported. Near the critical point dlJ/dp vanishes and the typical angular scattering pattern is displayed (see also figs 5.42 and 5.61). 5.6 Soft interactions In the previous section we treated colloidal suspensions, which could be described by steeply repulsive and attractive interactions, i.e., the interactions change appreciably over distances A much smaller than the particle size ( A « 2 a ) . In this section interaction forces of a longer range will be considered where A is on the order of the particle size or larger (A > 2a) . These interactions are described here as 'soft.' The steep interactions, i.e. hard spheres or adhesive hard spheres, could be characterized by a few parameters like hard sphere diameter a, attraction range A , and depth of an attractive well, e. Soft interactions require a functional form for the repulsive or attractive part of the pair interaction instead. These functionalities have been exhaustively described in chapter 1.4 and chapter 3 of this volume. However, in connection 11
F. Cousin, E. Dubois, and V. Cabuil, J. Chem. Phys. 115 (2001) 6051.
5.48
CONCENTRATED COLLOIDAL DISPERSIONS
with the purpose of describing multiparticle interactions, approximated analytical expressions are often used. In this connection, it is recalled that for predominantly repulsive interactions the precise properties of the inner double layer part are not so critical. We will first consider the theory of soft interactions and later discuss experimental illustrations. 5.6a Theory of repulsive and attractive soft-sphere interactions Well-known examples of soft pair interactions are the Lennard-Jones pair interactions used for atomic liquids and the Yukawa pair interaction as a model for repulsions or attractions in colloids. The latter reads % s (r) = °° a)(r)=
;
r < a
;
r > a
B-KT
a>Y(r) = A
[5.6.1]
where a is the distance of closest approach or hard sphere diameter, K is the reciprocal interaction range and A is a measure of the interaction strength ( A > 0 for repulsion and A < 0 for attraction). Repulsive Yukawa pair interactions mimic, for example, interactions of charged colloidal particles in a dilute electrolyte solution. In this case, K~1 is the Debye length. Attractive soft interaction may be found in particles with an imbedded permanent magnetic dipole and in dispersions containing repulsive colloidal particles with long brushes in a (slightly) poor solvent (see also chapter V.I) or dispersions containing star polymers". For a sketch, see fig. 5.28. For small colloid concentrations, the osmotic pressure of the dispersions may be characterized by the second virial coefficient, B 2 . In general, B 2 cannot be formulated in closed form as in sections 5.4 and 5.5 for the hard sphere or the square well pair interactions, except when co{r) has a tail with such a small co(r) value that the exponent can be linearized, i.e. 1 - exp(-oir) / kT) - o)[r)/kT . Substituting [5.6.1] into the second part of [5.5.2] and performing the integration then gives 27rA(l + >ca)e-™ B9-ByHc+— 7y
[5.fa.2|
Take the case, for example, of electrical double layer repulsion of (weakly) charged particles A = 4xe2Ka£0£a2{y/d)2, where e is the relative dielectric permittivity of the solvent, e0 is the dielectric permittivity of vacuum (see 1.5.11), a is the particle radius, which is taken equal to o"/2, y/^ is the (small) diffuse double layer potential, K = {2e2ps Iee0kT)l/2 is the reciprocal double layer thickness and ps the number of (1-1) salt molecules per unit volume. Using the low potential approximation (II.3.5.47] \jA = ze/4rre0ea[\ + !ca) with |g>d| = ze (see [II.3.5.49]), one finds
11
C.N. Likos, H.M. Harreis, J. Phys.: Cond. Matt. 4 (2002) 173.
CONCENTRATED COLLOIDAL DISPERSIONS
5.49
Figure 5.28. Schematic pictures of hard (A, B, (C)) and soft interactions ((C), D, E, F) in polymer colloids. A= hard sphere; B, C, D= cores (sterically stabilized by terminally attached polymer chains), E= star polymer; and F= linear polymer. (Redrawn from A. Vrij, Pure and Appl. Chem. 48 (1976) 471.) (1 + 2KU) Z 2 B 2 -B 2 ,HS +
(1 +
[5.6.3]
ra)24ps
Here, z is the number of charges per particle that compensates for the charge in the diffuse double layer [z = \Qd\/e ; see [II.3.5.51 ]). As in practice, only the diffuse part of the double layer counts, z equals, in fact, the total number of charges minus those in the non-diffuse double layer part. When A IkT is not small, the integral in B 2 must be calculated numerically. An early example11 is shown in fig. 5.29 for the osmotic B2of bovine serum albumin (BSA) in dilute electrolyte as a function of z. One observes that charge greatly increases B 2 above the HS level (corresponding to z = 0 ), in particular, at low electrolyte concentrations. One also observes that the curves have an apparent parabolic shape as predicted by [5.6.3] and that the magnitude is in (semi) quantitative agreement with the experimental points21. The structure factor (on the B2 -level) may be found as previously from [5.5.4]. For small «taiI(r) S(q) = S HS (q)--^
[5.6.4]
where
11
9)
A. Vrij. PhD dissertation, University of Utrecht (1959) fig. V-2. ~
The experiments were reported by J.T. Edsall. H. Edelhoch. H. Lontie, and P.R. Morrison, J. Am. Chem. Soc. 72 (1950) 4641.
5.50
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.29. Osmotic second virial coefficient of bovine serum albumin in (1-1) electrolyte solution as a function of the particle charge z . Electrolyte concentrations are given in mol dm" 3 . Drawn lines: calculated values. The lowest value at z = 0 corresponds to the B 2 value of a hard sphere with a = 6 nm . Symbols: measured by J.T. Edsall et al, loc. cit. (Redrawn from A. Vrij, loc. cit.)
< u ( q ),4;rj^ u (r)^ r 2dr
[5.6.5]
a and using [5.6.1] ft>tail(q)=
K
cos qcr + - s i n qcr\e~ Ka X" + q [ q J 2
2
[5.6.6]
These equations have only a limited value in practice since p must be kept small [pB2 < 1) • For (somewhat) higher concentrations, exceeding the B 2 level, the so-called random phase approximation (RPA)11 may be useful. It is formulated as follows. The direct correlation function is given by c r
( ) = c Hs( r )
c(r) = ^
;
«
0
*
I5 6 71
" "
The hard sphere is here the 'reference state' on which a perturbation -« t a i l (r) is superimposed. Instead of the hard sphere (HS) result, the adhesive hard sphere (AHS) result or another steep repulsion may also be chosen 21 . Upon substitution of [5.6.7] into [5.3.34] and using [5.6.5], one finds
S[q) =
?MsW
[ 5 6 8 ]
I+ S H S ^ ^ ] Note that [5.6.8] is compatible with [5.6.4] on the B 2 level.
2)
J.P. Hansen, I.R. Me Donald, Theory of simple liquids, Academic Press, London, 1976. The second part of [5.6.7] reminds us of [5.3.39],
CONCENTRATED COLLOIDAL DISPERSIONS
5.51
Figure 5.30. (a) Structure factor of a hard sphere (diameter a ) supplemented with a narrow square attractive well (A / o = 0.05) with a small depth , (e = 0.5 kT) plus a long range repulsive Yukawa tail with «7 = 3 and having a contact value of 1 kT. cp = 0.25 . Open circles: Monte Carlo square well plus Yukawa; full line: AHS plus RPA theory; dashed line: only AHS (T = 2.45). (b) As fig. 5.30a, but a contact value of the Yukawa tail of 3 kT and p = 0.12 . (Redrawn from K. Shukla et al., loc. cit.) Results 1 are shown for some cases with A > 0 . In the example, the reference state is not the HS but the AHS with a weak attraction (fBaxter =2.45). In fig. 5.30a,
11
K. Shukla, R. Rajagopalan, Colloids Surfaces A 92 (1994) 197. For an example of a colloid suspension, see, e.g., E.A. Nieuwenhuis, A. Vrij, J. Colloid
Interface Sci. 8 1 (1981)212.
5.52
CONCENTRATED COLLOIDAL DISPERSIONS
the mean spherical approximation, MSA (or RMSA)11. It is the simplest one to apply because of its linear character and even has analytical solutions in some cases {e.g. for the Yukawa pair interaction). It describes a colloid particle with a hard sphere interaction supplemented with an attractive or repulsive soft tail. g(r) = O
;
r
, ^ -»y( r )
[5 6 91
--
None of these closure relations is exact and all lead to certain inconsistencies when thermodynamic properties are calculated along different routes, i.e. via the virial and compressibility route (see [5.4.9] and [5.4.10]). To remove these inconsistencies, the Roger-Young (RY) closure relations can be used, which is a mix of HNC and PY (see section 5.3d). Klein and co-workers21 have investigated such closure relations in detail, in particular in mixtures to study polydispersity effects. Two cases of strongly repelling particles are chosen here as a test for such closure relations. This concerns a dispersion of hard spheres (^9 = 0.185) of a diameter 2a = cr = 125 nm, z = 155 and Yukawa tails with «r = 5.13 (fig. 5.31a; with 0.1 mM 1-1 electrolyte added, corresponding to (Z1 = 16 mV ) and xrr = 2.23 (fig. 5.31b; the salt free case, so the surface potential yfi = 27 mV ). The pair interaction between the spheres is given by [5.6.1], but with K the inverse Debye length, now redefined as sr2 = 4;rtB(/?c + 2ps) with lB = e2 I A7teQekT , pc the number of counterions and ps the number of added salt 1 -1 electrolyte molecules per unit volume. The Bjerrum length (B was set at 0.903 nm (corresponding to a temperature of 294 K and e = 63). Figure 5.31a demonstrates the accuracy of the RY closure relation. The other closure relations are less accurate. In fig. 5.31b, the repulsion is so high (the maximum in the simulation g{r) = 2.5 instead of g(r) = 2.3 for the RY closure) that even the RY closure becomes approximate. For more details, we refer to the extensive review of Nagele3'. Finally, we address (very) soft interactions of particles without a core as encountered before. Examples are polymer chains (depicted as spherical segment clouds) and polymer stars (see fig. 5.28; F and E, respectively). By way of illustration4', we give an early attempt to formulate the direct correlation function of Gaussian segment clouds embedded in a background of segments of the surrounding polymer molecules: -r2
pc(r) = -pDe2a2
11
J.P. Hansen, J.B. Hayter, Mol. Phys. 46 (1982) 651. J.M. Mendez-Alcarez, B.D'Aguanno, and R. Klein, Physica A178 (1991) 421. 31 G. Nagele, Physics Reports 272 (1996) 215-372. 41 A. Vrij, J. Polymer Sci. Symp.Nr. 44 (1974) 169. 21
[5.6.10]
5.53
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.31. (a) Radial distribution function of hard spheres supplemented with a repulsive Yukawa tail. The volume fraction of hard spheres is 0.185. (b) As fig. 5.31a, but for the salt-free case. (Courtesy of A.J. Banchio and G. Nagele.)
pD = m p ( 2 / r ) - 3 / 2 a - 3 ( j ^ - 2 W |
15.6.11]
Here, a is a measure of the cloud size (a ~ ^2/2, a ) , m
is the degree of polymer-
ization, i.e. the volume of a polymer chain divided by the volume of a solvent molecule, T] is the volume fraction of polymer ( r\= pm times the volume of a solvent molecule) and a
is the polymer's radius of gyration, x
is t n e
Flory-Huggins parameter. The
c{K) follows from a Fourier transformation [5.3.32], pc(K) = -mp\-^--2Xr]\e
2
and the osmotic pressure derivative follows as
[5.6.12]
5.54
±.^L
CONCENTRATED COLLOIDAL DISPERSIONS = l_pa(K
= q = 0) = 1+ mpl_l_.2xr^
[5.6.13]
Expression [5.6.13] matches with the familiar Flory-Huggins expression for the osmotic pressure derivative. For p —> 0 : - L | ^ = l + m p (1-2*)»;+...
[5.6.14]
Thus, the second virial coefficient, m (1 / 2 - x) > becomes zero when ^ = 1 / 2 (the theta state); a familiar result. The structure factor may be obtained by substituting [5.6.12] into [5.3.34], which leads to a smooth angular dependence of scattered radiation. In a recent paper on flexible dendrimers, the same formulation was found as in [5.6.10] and [5.6.11], but with a slightly different interpretation11. For a review of soft polymer particles, including polymer stars, we refer further to a recent paper by Likos . Let us make some concluding remarks. From the shape of a single S(q) it is difficult to conclude whether the interactions are soft or steep , let alone their precise nature. This is due to the fact that the structure factor (at least at higher concentrations) is not sensitive to the details of a(r). Attractive interactions, however, can be discriminated by an upswing of the S{q) at small q when the particle concentration is not too large. Soft interactions show, however, a shift in the first maximum of S(q) to lower q -values upon increasing the concentration, which is (much) larger than for steep interactions. Semiquantitatively, the shifted S(q) may often be described with an effective hard sphere with a larger value of a, (a ~ p~ 1 / 3 ). The increase of the osmotic pressure with concentration is (much) less for soft than for steep repulsions. Soft attractions will be considered in section 5.7 in a discussion on phase stability. There it turns out that the width of the attractive tail has a large influence on the type of phase diagram. In that section, we will also pay some attention to colloidal systems with more than one colloidal component. In particular, depletion attractions will be considered. In all our discussions, we have assumed that pair interactions are sufficient to describe the interactions between all particles. Although as we have seen above (see fig. 5.2a and b) this is a reasonable assumption for short-range forces, we should keep in mind that this is not necessarily so for soft interactions4'.
11
C.N. Likos, S. Rosenfeldt, N. Dingenouts, M. Ballauff, P. Lindner, N. Werner, and F. Vogtle, J. Chem. Phys. 117 (2002) 1869. 21 C.N. Likos, Phys. Rep. 248 (2001) 267-439. One could think of obtaining g(r) from S[q) by Fourier transformation, but this requires very accurate data, which arc seldom available. 41 See e.g. C. Russ, H.H. von Gninberg, M. Dijkstra, and R. van Roij, Phys. Rev. E66 (2002) 01 1402 for electrostatic forces or E.J. Meijer, D. Frcnkcl, Phys. Rev. Lett. 67 (1991) 11 10 for depletion forces.
CONCENTRATED COLLOIDAL DISPERSIONS
5.55
5.6b Experiments with soft-sphere interactions: Steric interactions (i) A concentrated latex dispersion in benzene^.
The colloid particles are poly-
methametcrylate (PMMA) latex spheres internally cross-linked and dispersed in benzene. The refractive index difference of PMMA and benzene is very small so that multiple scattering can be ignored, even in concentrated dispersions. The particles are characterized with viscosity, sedimentation, and static and dynamic light scattering in dilute solutions. Some results are given in table 5.3. Here a h is the hydrodynamic radius, a M is the radius of the compact particles, and a R
is the radius of a sphere
having the same radius of gyration as measured. The numbers indicate that the particles are swollen. The swelling increases from A to C. Table 5.3. Dimensions of PMMA latex spheres in nanometer (nm). Sample
ah/nm
aM/nm
A
140
B
120
C
97
a^/nm
a
69
101
2.03
48
68
2.50
31
46
3.13
In fig. 5.32a, the osmotic pressure derivative dfl/kTdp,
h/ a M
as obtained from the
structure factor at zero wave vector S{q = 0), is shown as a function of particle concentration. To describe the softness of the particles, the equivalent hard-sphere-volume fraction 0 HS at each concentration is calculated using [5.4.14] and the scaling factor q HS = ^ H S ^
ls
Pitted in fig. 5.32b. For A, this scaling factor (the specific volume of
the swollen particle) is q HS = 4.6 cm3/g and nearly constant. Thus, A behaves as a hard sphere with a hard sphere volume that is nearly five times the compact volume (in water). Particles B and C are more swollen and 'softer' as inferred from the fact that q HS decreases upon increasing particle concentration. The intensity of scattered light as a function of scattering angle is shown for some particle concentrations in fig. 5.32c, where the normalized scattering intensity R(q) is plotted versus q on a logarithmic scale. In R (q) = In P (q) + In S (q) + const.
[5.6.15]
with const, independent of q . This constant was used to set the theoretical and experimental results equal at q = 0 . The calculated volume fractions range from cpHS = C<
?HS - 0 0 1 5 to 0.25. At the lowest concentrations, where S(q) = l, the plots are
nearly linear over the whole q range, except for sample A where there is some evidence of a downward trend at the highest q values (see dashed lines). Thus, the particle
11
E.A. Nieuwcnhuis, C. Pathmamanoharan, and A. Vrij, J. Colloid Interface 196.
Sci. 8 1 (1981)
5.56
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.32. (a) Reciprocal osmotic compressibility of latexes A, B and C in benzene (see table 5.3. (b) Effective hard sphere volume, q^jS , as a function of the weight concentration of the latices. (c) Light scattering intensities (measured at wave length XQ = 436 nm ) for the samples A, B and C; bottom to top: weight concentrations in 10~2g cm" 3 indicated. The scattering is plotted on a (natural) logarithmic scale in relative units; one vertical unit equals 0.4. Curves at different concentrations are arbitrarily shifted upwards to improve visibility. Dashed lines are the low concentration results. (Redrawn from E.A. Nieuwenhuis et al., loc. cit.) scattering factor has a Gaussian form, P(q) = exp[-(l/3)a 2 q 2 ] 1!. When the particle concentration increases, the curves shift parallel upwards, which suggests that the particle shape does not change appreciably. Only at small q do the curves bow downwards indicating that S{q = 0) becomes less than unity, in accordance with the equation
S(q = 0) = kT[dp/dI7] discussed above, and indicating repulsive forces
between the particles.
This implies that the swollen particles contain a cloud of segments apparently with a decreased density at the periphery. Since the Fourier Transform of a Gaussian is a Gaussian this suggests that the spatial segment distribution in real space is also nearly a Gaussian.
CONCENTRATED COLLOIDAL DISPERSIONS
5.57
Theoretical scattering curves are also included in fig 5.32c, assuming that S(q) may be described by hard sphere interactions. The
Figure 5.33. Structure factors of lyophilized calcium carbonate particles in toluene studied with SANS. Particle concentration: a) 5%, b) 15%, c) 30%, d) 40 vol.%. Open circles: data points; lines: theory. (Redrawn from I. Markovic and R.H. Ottewill, loc. cit.)
This tendency can indeed be explained by taking a soft sphere potential, see E.A. Nieuwenhuis et at, loc. cit.) 21 I. Markovic, R.H. Ottewill, Colloid Polym. Set 264 (1986) 454.
5.58
CONCENTRATED COLLOIDAL DISPERSIONS
data were reported. The smaller core makes it easier to extract S(q) since P(q) is nearly q -independent for small particles (qa « 1) as well as colloid concentrationindependent. Results are given in fig. 5.33 for particle core concentrations of 5 - 4 0 % . The drawn curves are theoretical hard sphere results with an adapted HS-diameter (as a floating parameter), similar to the previous case. The adapted thickness of the surface layer varied from 6 nm for 5% calcium carbonate to 2.8 nm for 40% calcium carbonate, as shown in table 5.4. The trend is comparable with that for the latex in a benzene system, shown in fig. 5.32b. The individual curves follow the hard sphere picture quite well when the hard sphere diameters are assumed to decrease with increasing particle concentration due to compression . Table 5.4 Parameters used for the fitting procedure. R^ = particle radius; t HS = layer thickness. Concentration
Rj/nm
t H S /nm
K H S /nm
10.5 10.3
6.0
15 % 30%
9.8
3.3
40%
9.5
2.8
16.5 15.3 13.1 12.3
0.076 0.194 0.295 0.365
5%
5.0
5.6c Experiments with soft-sphere interactions: Electrical interactions (i) Light scattering of very dilute latex dispersion in very dilute aqueous electrolyte solutions. One of the earliest investigations of liquid structures in colloids with light scattering was on latex spheres, 2a ~ 50 nm, in extremely dilute aqueous electrolyte solutions21. The dispersion shows long-range statistical ordering already at very low volume fractions of the colloid ( q> = 10~5 to 5 x 10~4 ) (see fig. 5.34). The very low particle concentrations are, in fact, required because PS Latex in water shows very strong (multiple) scattering even at, say, (p=0.0l. This is caused by the large difference in refractive index between polystyrene (n = 1.60) and water (n = 1.33) . The particles have a surface charge of about 500 elementary units and have extended, electrical double layers because of the very low electrolyte content. It is concluded that the particles repel each other strongly at all particle concentrations and that, there fore, the structure is that of an expanding 'quasi' solid. This explains the fact that the product <7maxP~1/3 is approximately constant for all samples and equal to = 7.5 . This value is also found in, for example, condensed liquids. (ii) Silica particles in a non-aqueous solvent. Since the refractive index of most solid particles is usually much larger than that of water , other solvents are required For an example from the field of polymeric micelles we refer to: B. Chu, G. Wu, and D.K. Schneider, J. Polymer Sci. B: Polym. Phys. 32 (1994) 2605. 21 J.C. Brown, P.A. Pusey, J.W. Goodwin and R.H. Ottewill, J. Phys. A 8 (1975) 664. Exceptions include fluorcarbon latex particles which have a refractive index of 1.36.
CONCENTRATED COLLOIDAL DISPERSIONS
5.59
Figure 5.34. Structure factors of charged PS latex spheres dispersed in very dilute electrolyte solutions. Particle concentrations in 10 4 gcm~ 3 . 1): 5.08, 2): 3.42, 3): 1.74, 4): 1.00, 5): 0.746. The structure factors have been shifted upwards by one unit for better visibility; for each curve the S = 0 and S = 1 values are indicated. (Redrawn from J.C. Brown et al. loc. cit.)
to obtain a matching of refractive indices. This is necessary for interpreting light scattering studies at higher volume fractions of colloidal particles in order to suppress multiple scattering. Some results are given for silica particles with a radius a = a 12 = 80 nm and an estimated f potential; £ ~ -60 mV in a mixture of toluene and ethanol (70/30 w). No salt was added. The estimated double layer thickness ic~l = 55 nm was obtained from conductivity measurements. The experimental points are compared with the RMSA theory11 where [5.6.2] is used for the tail with A = C<jeKakT. Here, C is a dimensionless prefactor that depends on the diffuse double layer potential. The particles carry only little charge (gi d / e = 500 -.(f1 /e = 0.006 nm"2 ; see II.3.36). The best fit is obtained for KG = 3 (x""1 = 55 nm and c7=160nm) with C as an adjustable parameter giving a value C = 60 ±10% leading to (Z1 = 76 mV , indeed comparable with the estimated C, potential. One observes that a reasonable fit can be obtained. (Hi) Charged latex in water studied by SANS (fig. 5.36). Multiple scattering can also be suppressed by using small particles and SANS21. The polystyrene latex particles have a diameter 2a = 31.2nm and are studied at several particle and electrolyte concentrations. Structure factors are shown for cNaC1 = 10~4, 10" 3 . 5 x 10~3 mol dm" 3 and
A.P. Philipse. A. Vrij, J. Chem. Phys. 8 8 (1988) 6459. J.W. Goodwin. R.H. Ottewill. J. Chem. Soc. Faraday Trans. 8 7 (1991) 357.
5.60
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.35. Structure factors of charged silica particles in a toluene/ethanol mixture. The particle core diameter is 160 nm and the screening length is estimated at 55 nm. The (core) volume fraction of the particles is indicated. (Redrawn from A.P. Philipse and A. Vrij, loc. cit.)
Figure 5.36. Structure factors for PS latex particles (2a = 31.2nm) measured with SANS. The particle concentration (for the three cases) is
CONCENTRATED COLLOIDAL DISPERSIONS
5.61
of | ^ | = 51, 49, and 47 mV from top to bottom, and quite comparable with C, potentials of between -54 to -62 mV, (at lower particle concentrations). The fit with theory is, again, reasonable. Finally, we would like to note that a model system of charged fluorescent PMMA latex spheres, which allows index- and density-matching of the particles with respect to the solvent, has been shown to be very suitable for studying long-range repulsions . In this colloidal suspension, the range of the soft interaction can be tuned by varying the solvent salt concentration. Further, by applying an external electric field, an anisotropic (dipolar) contribution to the pair interaction can be independently imposed. 5.7 Phase stability So far we considered colloidal suspensions that are inherently stable; they do not show signs of instability and phase separation. One sign of instability is the experimental observation that the intensity of scattered radiation (greatly) increases when the system is approaching a point of instability. The intensity of scattered radiation can be attributed to local fluctuations in the colloidal particle concentration. When the intensity increases, it implies that the concentration fluctuations become stronger and extend over larger regions of space when such a point of instability is approached2'. Progressing further, the spatial concentration fluctuations lead to a complete separation of spatial regions with different particle concentrations and (mostly) different mass densities; phase separation is taking place, where the denser phase is accumulating at the bottom of the vessel under the action of gravity. These phenomena are comparable with what happens in simple atomic or molecular fluids. Colloidal dispersions mimic atomic fluids in many aspects in which the colloidal particles act as giant atoms, as noted already in sec. 5.1. So, it will be appropriate to first briefly review the phenomena occurring in atomic liquids, see also sec. 1.2.19. But there are also differences: (i) It is possible to 'tune' the interparticle forces in colloidal dispersions, e.g. by changing the dispersing solvent, the surface coating of the particles, the temperature, the particle charge or by imbedding a smaller particle with a permanent, magnetic, dipole moment inside the particles 3) . (ii) One may prepare colloidal dispersions with particles, which act as 'hard spheres' (HS), i.e. there are no interactions between the particles, except for a steep repulsion at very short distances. We encountered this in sec. 5.4. (iii) Dispersions with colloidal particles may be prepared that act as hard spheres, " A . Yethiraj, A. van Blaaderen, Nature 421 (2003) 513; C.P. Royall, M.E. Leunisscn, and A. van Blaaderen, J. Phus.: Condens. Matter 15 (2003) S3581. 9)
The system has a 'milky' appearance because of the strong scattering of light. This appearance is sometimes denoted as 'streaming clouds.' 31 See e.g. A. Yethiraj, A. van Blaaderen, Nature 421 (2003) 513 for a further discussion.
5.62
CONCENTRATED COLLOIDAL DISPERSIONS
plus an attractive tail, with a (very) short range, i.e. short with respect to the particle diameter, see also sec. 5.5. (iv) One may prepare dispersions of colloidal particles mixed with polymer chains, polymer stars, dendrimers, micelles, micro-emulsions, etc. Particles of the categories (ii) and (iii) are not found in atomic liquids. As a consequence, it is thus not surprising that new phenomena in such systems11 are found that cannot be studied in atomic liquids. So, it is found that purely repulsive forces may give rise to phase separations where crystalline phases appear when the particle density transcends a certain value (q>= 0.494 in the case of hard spheres). On the other hand, changing the range of the attractive forces, say, from the particle diameter 2a to a (small) fraction of it, gives rise to phenomena that were not found in atomic systems. At first this was, in fact, unexpected because the shape of the structure factor was found to be insensitive to the range of attraction. The variation of the range and depth of the attraction is often achieved (indirectly) by mixing the dispersion of colloidal particles and solvent with an extra component, for instance (softly) repelling, non-adsorbing macromolecules, see further sec. 5.7c(iv). For charged aqueous systems, this is achieved by changing c s a l t . A great advantage of colloidal systems is that attraction in a macroscopic system can be achieved by a combination of purely repulsive pair interactions. The range of the ensuing depletion attraction can be modified simply by changing the diameter of the added non-adsorbing macromolecules, its intensity by changing the concentration of them, see sees. VI. 8 and 9. The original dispersion and the solution of macromolecules are stable themselves. This 'attraction through repulsion' mechanism reduces the occurrence of irreversible processes, which might otherwise occur in the classical aggregation of charged colloids by electrolytes. A disadvantage of depletion attraction is that it requires extra attention in the sense that the dispersion now contains two mesoscopic components instead of one, which makes the analysis more complex. The physical experimentation with these concentrated colloid systems has emerged in, say, the last twenty-five years. Because of the relatively large length and time scales involved, one may directly study the positions of an individual particle (see fig. 5.4). Further, external fields like magnetic fields (ferrofluids) may be applied21 and also crystallization of particles in contact with a template31 etc., which can be studied by optical means (CSLM)41.
These phenomena are sometimes denoted as "new states of matter,' see D. Frenkel, Science 296 (2002) 65. 21 Section 3.10 and A.P. Gast, C.F. Zukoski, Adv. Colloid Interface Sci. 30 (1989) 153. 31 E.H.A. de Hoog, L.I. de Jong-van Stccnscl, M.M.E. Sncl, J.P.J.M. van der Eerden, and H.N.W. Lekkerkerker, Langmuir 17 (2001) 5486. 41 E.H.A. de Hoog, W.K. Kegel, A. van Blaaderen, and H.N.W. Lekkerkerker, Phys. Rev. E64 (2001)021407.
CONCENTRATED COLLOIDAL DISPERSIONS
5.63
5.7a Instability and phase separation in atomic fluids^ (i) The picture of van der Waals for fluids. It is instructive to start with the well known, classical (1873) picture of van der Waals2' for the gas-liquid (G/L) transition. For the equation of state, he derived3 N2a
a
P = PHS — ^ 2 " = P H S - ^ 2
,c T ii l5 7 1]
- -
where N is the number of particles in volume V and where v = V/N = l/p, and where p H S is the pressure of a collection of hard spheres. The parameter a characterizes the attraction between the particles. Expression [5.7.1] is nowadays recognized as the simplest case of a perturbation treatment of the pressure of a fluid. For the hard-sphere pressure, van der Waals proposed the following equation ^HS
NkT y_bN
but this expression is only accurate at low densities. Much better expressions are given by the Percus-Yevick [5.4.12] and Carnahan-Starling [5.4.14] equations of state. The Helmholtz energy follows as F = FHS-aN^
[5.7.3]
from which the pressure and chemical potentials follow using standard thermodynamics (see chapter 1.2),
'~(£L "=(sL-"» s - 2a ?
i57 3bi
-
The system is mechanically stable when
(¥-)
<0
[5.7.4]
In other words, in a stable fluid the pressure must decrease when the volume is increased. This is demonstrated in fig. 5.37, where p is schematically plotted versus v . For high temperatures [5.7.4] is always fulfilled, but for T
Here, we partly follow H.B. Callen, Thermodynamics, John Wiley, New York (1959). J.D. van der Waals, On the Continuity of the Gaseous and Liquid States, Vol. 14 of Studies in Statistical Mechanics (North Holland, (1988) with an introduction by J.S. Rowlinson. 31 For an insightful explanation, see B. Widom, Science 157 (1967) 375. For Langmuir monolayers, this was elaborated in fig. III.3.15.
5.64
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.37. Pressure p of a simple Van der Waals fluid as a function of the volume per particle v . Top = highest temperature; bottom = lowest temperature. (Redrawn from H.B. Callen, Thermodynamics, John Wiley (1959).)
Let us now discuss this in more detail. In fig. 5.38, v is plotted versus p . Starting at point a , we slowly increase the external pressure. Until just before b , the function v(p) is single-valued. Transgressing point b , the function becomes multiple-valued because three values of v are possible (e.g. the values at c, 1, n ). What will become of the equilibrium value(s) of v ? At 1, the system is (mechanically) unstable and one is left with c or n . Thermodynamics teaches us that the lowest value of the state function G[T, p) gives stable equilibrium, see 1.2; [2.12.10]. To this end, one plots G/N = g = [i from [5.7.3b] as a function of p, see fig. 5.39. Thus, c will be the stable point. When the pressure is increased to the point d , one finds from fig. 5.39 that d and o correspond to identical values of the chemical potential ji. Thus, both situations are stable. The procedure to locate the pair d, o can be deduced from the Gibbs-Duhem relation, which (at constant T) says that dfi = vdp. Thus, the criterion d/i = 0 can be found
Figure 5.38. Volume per particle v versus the pressure p. Several states are designated by a lower case letter. (Redrawn from H.B. Callen, toe. cit.)
CONCENTRATED COLLOIDAL DISPERSIONS
5.65
Figure 5.39. Free enthalpy per particle as a function of pressure p . The unstable points have been omitted. (Ref. as previous fig.)
from the stipulation Po
[ vdp = 0 Pd
which implies that the (vertical) line d - o should be chosen in such a way that the two shaded regions have the same area (Maxwell construction). The derivatives 3///3P at d and o are equal to v[d) and u(o). The physical process that occurs when the pressure at d is slightly increased is not a sudden drop in v from u(d) to v{o), but the formation of a small amount of the condensed phase with v = v(o), of which the amount increases (all at constant p ) until the whole substance has this value , after which the function becomes single-valued again beyond point r . For every temperature, such a construction can be made, which leads to fig. 5.40. The curve o, o 1 , o11, o m , d ln , d11, d1, d is called the binodal and is represented as the full curve in the (phase) diagram given in fig. 5.41. The binodal is a boundary curve
Figure 5.40. The pressure p versus v for several temperatures T. (Ref. as in fig. 5.38.)
According to the 'lever rule,' v= xv{o) + (1 - x)u(d) upon x increasing from 0 to 1, see sec. 1.2.16.
5.66
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.41. Temperature T versus v phase diagram. Solid curve: binodal; dashed curve: spinodal; cp: critical point; G : gaseous phase; L : liquid phase; shaded area: metastable region.
in the phase diagram. On the left side of the curve in fig. 5.41, one finds the condensed liquid phase ( L) and on the right side, the dilute (gaseous) phase (G ). Under the (full) curve, the system is not homogeneous (in the stable state), but contains both the phases L and G . Above the maximum of the curve, there is only one (stable) 'fluid' phase (F); there is no distinction anymore between L and G. The maximum is situated at what is defined as the critical point (see also sec. 1.2.19). The critical point can be determined from the condition that both (dp/dv)T and (3 2 p/3u 2 ) T are equal to zero. (ii) Metastable states. We return to fig. 5.38 and remark that thermodynamics does not tell us how the condensed phase at o is formed at the transition d -» k -> o. We need to consider the notion from statistical thermodynamics that the system is not static but undergoes thermal fluctuations, which can be probed, for example, by scattering techniques. Thus, one could say that the system probes' these fluctuations if other phase states become more probable than the current phase state. If the fluctuation is large enough, a (stable) nucleus will eventually appear and the L phase will grow. But, if such a nucleus is not present, the system will continue along the curve d , e , h upon increasing the pressure p (see fig. 5.39), where it is locally stable with respect to small fluctuations and the region is metastable. The metastable branch stops at the point h; beyond that point the system becomes unstable and cannot exist as a homogeneous phase. The points h and m are particular in the sense that {dv/dp)T becomes infinite. If we put the points into the diagram of fig. 5.41, one obtains the so-called spinodal line. This spinodal line, which bounds the unstable from the metastable region in fig. 5.41, can in principle be found from scattering experiments. (Hi) Spinodals, increasing fluctuations and the scattering of radiation. From [5.3.44], one has for the scattering intensity I(q) = pf2P(q)S(q) and in the particular case of q —> 0
[5.7.5a]
CONCENTRATED COLLOIDAL DISPERSIONS I(q = 0) = pJ2S(q = 0)
5.67
[5.7.5b]
where S(q = 0) is connected with thermodynamics through (see [5.3.35]) S(q = 0) = kT\ - ^ I {dp)T
[5.7.6]
Using [1.7.7.5] gives
W(Ap)2) _J
(d \
L= kT
££ VdPJT
P
[5.7.6a]
and v((Apf\ = p l + p\47n-2h(r)dr
L
[5.7.6b]
0
where Ap is the number density fluctuation. Expression [5.7.6a] tells us that the fluctuations increase when the (isothermal) compressibility ,^=(9p/p9p)T increases. When a fluid is more compressible it takes less work to achieve deviations from the average density. Spontaneous fluctuations, which require less work, will therefore become more probable. The first term In [5.7.6b] gives the fluctuations in the absence of interactions. The second term, containing h{r), characterizes the overall effect of fluctuations in a central volume element on neighboring volume elements. Near the critical point, in particular, h(r) acquires a long range.
Figure 5.42. (a) Reciprocal SAXS scattering intensity of liquid argon as a function of the squared scattering angle, 6 (Ornstein-Zernikc plot), q = 2/r#/ X . From bottom to top: 0.05, 0.25, 0.45, 1, 2° above the critical temperature (150.85 K); p = 48.3 at. (Redrawn from J.E. Thomas and P.W. Schmidt, J. Chem. Phys. 39 (1963) 2506.) (b) Schematic plot of the reciprocal intensity at q = 0 , obtained from (a), versus temperature for two values of p . The extrapolated points on the T axis arc spinodal temperatures. A is closer to the critical point than B.
5.68
CONCENTRATED COLLOIDAL DISPERSIONS
In fig. 5.42a, the reciprocal SAXS intensity of the homogeneous phase is plotted for the atomic liquid argon as a function of the scattering angle 6. Observe that the intensity and the q -dependence ( q = 2n6l k ) increase greatly when T is lowered. The extrapolated values at q = 0 are plotted in fig. 5.42b. From the phase diagram in fig. 5.41, it will be clear that the spinodal can only be found from an extrapolation over the metastable state. At the critical point the metastable region vanishes, but at p •*• pc the extrapolation to determine the spinodal, see fig. 5.42b, becomes less accurate due to a larger, experimentally inaccessible, metastable region. At the critical density, the spinodal can be approached very closely. Thereby, the scattering near the spinodal diverges and becomes strongly angular- dependent {critical opalescence). The angular dependence is connected with the extending spatial correlations of the density fluctuations. (iv) Phase separation at larger densities. At higher densities a solid phase ( S) will also appear. A schematic diagram is given in fig. 5.43. Traditionally, it has been assumed that for a fluid-solid ( F/S ) phase transition, attractive forces are necessary. It has recently been found that this is not the case, but that particles having no attractive forces may indeed exhibit an F/S transition11. Since atoms (and molecules) with pair interactions without attractive tails are not known in nature, we will discuss this item in the sees. 5.7b and 5.7c where colloids are treated.
Figure 5.43. Schematic phase diagram with T versus p showing G, F, L, and S phases. Subscript c means critical state; subscript t means triple point.
5.7b Instability and phase separation in colloidal suspensions As discussed in the prelude of sec. 5.7, it is possible to 'tune' pair interactions from, say, hard sphere-like to soft or adhesive spheres into colloidal dispersions. This is not possible for atomic fluids. The properties of stable colloidal fluids are not very different from those of atomic fluids in the stable regions of the phase diagram. We will see,
For the unstructured, isotropic phases, the designations G, F, and L are in use. A fluid phase F may have any density, but when it is small one usually speaks of a gas, G . A liquid L is a fluid in a condensed state in coexistence with a gas phase G . When the solid phase S is crystalline, the letter C is often designated.
CONCENTRATED COLLOIDAL DISPERSIONS
5.69
however, that this is not the case anymore in phenomena of instability and phase separation where colloids have richer phase behaviour as compared with atomic liquids. For this reason, colloids are nowadays a common playground for scientists having theoretical interest in the relation between the shape, range, and sign of the pair interaction and the macroscopic structure and phase behaviour. In the following we will first treat theoretical models and simulation results in colloidal systems. Illustrative, experimental results will be described in sec. 5.8 and compared with theory if possible1'. (I) Phase behaviour in purely repulsive colloidal dispersions. In experimental work on colloids with repulsive interactions, at present mainly F/S phase separations have been found. The best and earliest known examples of these equilibria are observed with highly charged latex particles suspended in a dilute aqueous electrolyte solution, giving rise to crystalline structures21. Later, it was also found that the hard-sphere type of colloids can result in crystalline structures31. The formation of colloidal crystals in suspensions of particles having solely hard sphere interactions sounds counter-intuitive, where 'classical wisdom' requires attractive forces to be necessary to spontaneously form a crystal4'. In fact, F/S transitions were already encountered in early computer simulations5'. The results of the F/S transition to hard sphere dispersions are schematically shown in fig. 5.44. It is possible to calculate the phase transition from (approximate) equations by requiring the equality of pressure and chemical potential in the two phases. An accurate equation of a hard sphere fluid is that of Carnahan and Starling [5.4.14] pkT
(\-cpf
and for the pressure in the hard sphere crystal -£- = pkT
-
[5.7.7b]
l-(cp/
with
5.70
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.44. Equation of state and phase diagram for hard spheres; (pF = 0.494 ; (pg = 0.545 and pv01 kT = 6.18 ; pa31 kT = 11.69 . Numbers from W.G. Hoover and F.H. Ree, loc. cit. value for the Glbbs energy In the solid phase for which simulation results11 can be taken, fiex /kT = 19.0455 at ^ = 0.5760. This gives cp = 0.49 and cp= 0.54 and pvo/kT = 6.08; in good agreement with the values found by Hoover and Ree 2 , see also fig. 5.44. From the experimental observation of the F / S transition and the confirmation with computer simulation, it follows that hard spherical particles may gain entropy by creating maximum space in their surroundings. This is attained when the particles are positioned on a (regular) lattice. It turns out that the gain in entropy due to local freedom outweighs the loss of entropy by the ordering of the particles on a lattice. When the hard sphere repulsion is supplemented with a softer repulsive tail, the P p / p s densities have the tendency to shift (say up to 50%) to lower values, which is understandable because this may be regarded as an increase of the effective hard sphere diameter31. (ii) Phase behaviour of dispersions containing particles with repulsions, plus attractions with variable range. Let us start with a common pair interaction in atomic fluids, i.e. a generalized Lennard-Jones pair interaction
flHr) = 4 £ p )
_^j
[5.7.8]
Observe that a>(o) = 0 and rrain I a = 2 1/n . For n = 6 one has the familiar case, often used as a model pair interaction for atomic liquids, see fig. 5.45a. The phase diagram obtained with simulations41 is given in fig. 5.45b. Now we are interested in what happens when the repulsion becomes steeper and the attraction range smaller because interactions between colloidal particles exhibit this feature. The shape of the attractive 11
D. Frenkel, A.J.C. Ladd, J. Chem. Phys. 81 (1984) 3188. W.G. Hoover, F.H. Ree, J. Chem. Phys. 4 9 11968) 3609. 3) For a recent discussion see. e.g., A.-P. Hynninen, M. Dijkstra, Phys. Rev. E, 6 8 (2003) 021407. 4) G.A. Vliegenthart, J.F.M. Lodge, and H.N.W. Lekkerkerker, Phys. A, 263 (1999) 378. 21
CONCENTRATED COLLOIDAL DISPERSIONS
5.71
Figure 5.45. (a) Lennard-Jones pair interaction with variable width. Solid line: n = 6 , long dashes: n = 12 , short dashes: n = 18 . (b) Phase diagram for a Lennard-Jones pair interaction with n = 6. Critical point at fcTcp I e = 1.316 , /a cr3 = 0.316; the curves are smooth lines through the simulation points, (c) Phase diagram for a Lennard-Jones pair interaction with n = 12 . Critical point at fcTcp / e = 0.557 and p^a3 = 0.42 . (d) Phase diagram for a LennardJones pair interaction with n = 18. Critical point at fcTcp / e = 0.43 and p^a3 = 0.43 . The vertical dashed line indicates the critical density line. Redrawn from G.A. Vliegenthart et ah, loc.cit.
well is not so important for the structure factor. So, let us for convenience look at a 'Lennard-Jones' type of interaction with n = 12 and 18 . The phase diagram with n = 12 has undergone a peculiar change as compared with n = 6 , see fig. 5.45c. Several noticeable differences between n = 6 and n = 12 are: - The liquid phase L has just disappeared for n = 12 . There is only a (stable) F/S transition. - The (calculated) G/L transition has been shifted and is now located under the F/S transition for n = 12. This 'hidden' G/L transition has thus become metastable for n = 12 . and has a flatter shape as compared with the n = 6 case. - The critical point (metastable) lies under the F/S binodal and has moved to larger particle concentrations.
5.72
CONCENTRATED COLLOIDAL DISPERSIONS
When the value of n is further increased to 18, the repulsion is again steeper, but more importantly the attraction much narrower. This has the effect on the phase diagram that the hidden (metastable) G/L transition drops even further along the fcT/£-axis as shown in fig. 5.45d. What consequences may one expect for the behaviour of colloidal suspensions? Firstly, whereas in most suspensions attraction (when present) has a range that is much smaller than the particle diameter, one may not expect to find a stable G/Ltransition. And indeed when phase separation occurs, the condensed phase is usually 'solid like,' i.e. there is no 'liquid-liquid' phase separation. Secondly, it is conjectured that the hidden, metastable G/L phase, although not stable, may still influence the following phenomena: - When the critical point is just below the left branch of the F/S diagram, it may nevertheless be feasible to find metastable situations where a metastable G/L transition may persist and is observable11. - It is assumed that metastable states may influence the behaviour of non-equilibrium phenomena when the dispersion is suddenly quenched into the metastable region, e.g. by sudden cooling or adding destabilizing components like (concentrated) salt, etc. (iii) Phase behaviour of colloidal dispersions containing hard sphere, plus attractive Yukawa interactions. In sec. 5.5 we addressed that, besides a (steeply) repulsive part, many colloidal particles also have a (narrow with respect to the particle size) attractive part in their pair interaction. The attractive interactions are a source of destabilization of the dispersion, perceptible by a negative second virial coefficient of the osmotic pressure. The influence of the width of the attractive well on the phase behaviour was discussed in the previous subsection. In particular, it was found that the G/L phase transition 'disappears under the F/S transition' for narrow wells and becomes metastable. In this subsection we will investigate in more detail how attractive forces manifest themselves in phase separations of colloidal dispersions. We will discuss simulation results that were made using a routine that is also convenient for mixtures21. The aim of the calculation is to find values for the Helmholtz energy F as a function of particle concentration and temperature over a large grid of values. Subsequently, phase equilibria can be found by using the common 'tangent construction,' see sec. 1.2.19. The method starts with Monte Carlo simulations in which the interaction energy, U t[N,V;A) , of a collection of JV particles (here 108) in a volume V , is obtained from a canonical average as a function of a charging parameter A; {0 < X < 1 ). The 1 One may think of the crystallization of proteins in solution for which we refer to the literature, e.g.. N. Ashcrie, A. Lomakin, and G.B. Benedek, Phys. Rev. Lett. 77 (1996) 4832; H.N.W. Lckkerkcrker, Phys. A244 (1997) 227, P.R. ten Wolde, D. Frcnkcl, Science 277 (1997) 1975, R. Piazza, Curr. Opinion Coll. Interface Sci. 5 (2000) 38. 21 M.Dijkstra, R. van Roij, and R. Evans, Phys. Rev. E59 (1999) 5744.
CONCENTRATED COLLOIDAL DISPERSIONS
5.73
Helmholtz energy of the system is determined by integration over the parameter A from 0 to 1; in other words, the reversible isothermal work upon the charging process is calculated. For w{r), the attractive Yukawa pair interaction was used11, see also [5.6.1 ].
^J-^expMl-r/a)] [<x>
; r>a
[5 ? g]
; r
Then, from the Helmoltz energy, J(p,T)= F/V can be formulated. Some (schematic) results are shown in fig. 5.46. In panel (a), one has a familiar van der Waals type of phase separation of a substance in a gas phase (a ) and a liquid phase ((3 ). This leads to a phase diagram as in fig 5.41. In (b), the a phase is a gas and the (3 phase a solid. There is no continuous curve between the two minima as in the van der Waals case (a). This leads to a phase diagram as in fig. 5.45b, top-right part. In (c), the a phase is a gas, the (3 phase a fluid, the y phase the same fluid with a larger density and the 5 phase a solid. This leads to a phase diagram as in fig. 5.45b. In (d), the a phase is a gas, the f} phase a solid, but there is a metastable G/L transition. This leads to a phase diagram as in fig. 5.45d2).
Figure 5.46. Helmholtz energy per unit volume as a function of volume fraction (
5.74
CONCENTRATED COLLOIDAL DISPERSIONS
5.7c Stability and phase separation in colloidal mixtures Under this title we will consider the theory of a) mixtures of hard spheres and b) mixtures of hard spheres and macromolecules. For macromolecules one may think of linear polymer chains, star polymers, dendrimers, micelles, microemulsions, protein molecules, etc. Adding non-adsorbing macromolecules to dispersions of colloidal particles leads to depletion attraction". But, (smaller) hard spheres used as a depletion agent for larger hard spheres will also induce a similar effect. The pair interaction between the large spheres due to the smaller ones was already derived a long time ago21. This may lead to a decrease of the second virial coefficient of the Donnan osmotic pressure of the colloidal dispersion . But, usually, thermodynamics on the B 2 level is insufficient for phase separation studies. Therefore, we will turn our attention to a more general approach. (I) Osmotic approach. The osmotic approach was already introduced in sec. 5.2. Osmotic equilibrium is schematically depicted in fig. 5.4741. The thermodynamic relation for this situation in compartment can be formulated as follows ( T = constant)
dF = -pdV + //1diV1+//2diV2
[5.7.10]
from which one can find the grand potential, which has the required variables di2 = d(F-// 2 iV 2 ) = -pdV + /^dJVj -N2d/u2
[5.7.11 ]
with
fl^-1
=-N2
[5.7.11c]
The independent variables in fig. 5.47 are Va, IV™ , and //" ; the dependent ones are 1 The interaction between two parallel plates in a solution of non-adsorbing ideal chains was calculated by S. Asakura, F. Oosawa, J. Chem. Phys. 22 (1954) 1255. Also see II.chapter 5 and V. chapter 1. 21 S. Asakura, F. Oosawa, J. Polymer Set 33 (1958) 183. 31 A. Vrij, PureAppl. Chem. 48 (1976) 471. The background solvent component 0 docs not play an explicit role in this colloidal model. The criterion is, in fact, whether the changes of p in Va have any influence on the pair interactions ft)] j , ^12' fi)22 • m CC|U°idal and macromolecular solutions, these changes Ap = A/7 are so small that their influence may be neglected. The effects would only become perceptible when pressures of thousands of atmospheres are involved (see e.g. C. G. dc Kruif, J.A. Schouten, J. Chem. Phys. 92 (1990) 6098, who induced phase separations in this way). The presence of component 0 will further be omitted in the thermodynamic equations.
CONCENTRATED COLLOIDAL DISPERSIONS
5.75
Figure 5.47. Osmotic system consisting of solvent (0), colloidal particles (1) and "depletion agent' (2). The top membrane is impermeable to particles 1 and the bottom membrane is impermeable to particles 1 and 2. The pressure difference is pa - pP = 77 (Donnan colloid osmotic pressure). The chemical potentials of the component that can permeate (component 2), / ^ , is identical to compartments a and /} (and ji0 is constant in a, /? and y). The volumes V$ and V"t are taken to be very large with respect to Va .
77 = pa - pP and N% . The quantities p\ , pP and ^ | can be kept constant in the differentiation by choosing very large V$ (reservoir R). Here, p" = JV™ / V™ and pP=Jvf/vP. We will further use the nomenclature: pi = p° (1 = 1,2); V = V a ; p | = p^ ; p = pa ; p R = pP and 77 = p- pR 1!. The coexisting phases (binodals) in a phase-separated dispersion obey fil' = Ml"
[5.7.12a]
77'= 77"
[5.7.12b]
Here the superscripts ' and " are symbols for the separated phases designated as ' and " 2) . The Gibbs-Duhem relation (see 1.2 and [5.2.5]) reads dp = Pjd/Zj + p2d//2
[5.7.13]
and by changing to pj as the primary variable d(p-// 1 p 1 ) = -//jdp 1 +p 2 d// 2
[5.7.14]
Now, two relations readily follow. From [5.7.13]
M
-Pl\p]
[5.7.15,
and from [5.7.14]
IT1-} - 4 1 ^ 1 ^2
15 7 16]
--
and from cross-differentiating [5.7.14] Because only pressure differences with respect to the constant pressure in compartment y are relevant here, we take pY = 0 . Then, pa and pP become numerically equal to the osmotic pressures with respect to compartment y . 2 The third condition, i.e. //2 ' = //2 " is automatically fulfilled because of the equilibrium with the reservoir B .
5.76
CONCENTRATED COLLOIDAL DISPERSIONS
[ djU, 1
\rp-\ duo
[3 On
= -U^ dp.
[5.7.17]
Substitution into [5.7.16] gives
JT-1 = *-*\ir]
15 7 181
--
These relations will be used in the following approximate model. (ii) Thermodynamic equations for an approximate model. For the two-component colloid/colloid and colloid/polymer mixtures in osmotic equilibrium, as shown in fig. 5.47, we adopt an approximate model to describe the depletion effect11. For given pj in V and p2 in the reservoir, the particles 2 will (re)distribute over the system and reservoir. The following factor a - the partition coefficient for the concentration of particles 2 in the system with respect to the reservoir - is defined by p2 = a(p1)p2l
[5.7.19]
where it is assumed that a is not a function of p^. This assumption may be tested with computer simulations. Substitution of this equation into [5.7.18] then gives, using d//2=dpR/p£ p = p°(l)+ a-p^lp*
[5.7.20]
where p°(l) = p(p2 = 0) is the pressure of the pure component 1. This expression has the typical form as would result from a van der Waals-like perturbation treatment where the second term in [5.7.20] is the perturbation on the first term. In the same manner, one derives from [5.7.17] ^=^°m-^pR
[5.7.21]
dpj
where ju^{1) = p.l{p2 = 0) . For small px , a Taylor series expansion leads to the expression a(p1) = l + alp1 + a2p12+...
[5.7.22]
where n! « n is the 'nth' derivative of a after px at pj = 0. Then, one finds that the osmotic pressure becomes n = p-pR = p°(l)-a2pRpf+...
[5.7.23]
Note that the linear term in [5.7.22] cancelled. The second virial coefficient for the osmotic pressure becomes 11
H.N.W. Lekkerkcrkcr, Colloids Surf. 51 (1990) 419.
CONCENTRATED COLLOIDAL DISPERSIONS
B 2 =B 2 °(1)
5.77
^—
[5.7.24]
where B2°(l) is the value for p2 = 0 . Further, one has ju1=/u°a)-[al+2a2p1
+ ...]pR
[5.7.25]
For higher px, these virial expressions are, however, insufficient and more specific particle models are needed. Therefore, we will first consider a mixture of hard spheres. (i) Mixtures of large and small hard spheres. It was believed for a long time that mixtures of hard spheres of any size ratio are always thermodynamically stable. One reason for this was that the Percus-Yevick (PY) theory predicts that such mixtures are stable. However, the PY theory is not accurate for mixtures of many-component hard spheres with large size ratios. Therefore, the assertion was questionable and further taken up by Biben and Hansen
who used the improved, but still approximate,
integral equation theory that gives reasonable evidence to the existence of a spinodal instability for j = a2 I <J\ ^ 0.2 . They recognized the depletion effect due to the small spheres as the driving mechanism. Later, analytical calculations of the pair interaction between two large spheres in a sea of small spheres in the Deryagin approximation , as well as density functional theory (DFT) results31, supported this view. Recent studies based on computer simulations of such mixtures reveal that there may be Lj/L2 phase separations for low enough j < 0.1, but these are all metastable4'. However, simulations usually do not give the insight we would like to have and therefore we return to the approximate model of the previous subsection (approximate in the sense that a is assumed not to be a function of the chemical potential of the depletion agent (2)). To this end, we will introduce the so-called free volume theory, which is flexible in the sense that it can be applied to mixtures of particles 1 and particles 2 (in a common solvent), where particle 1 is a colloidal particle and particle 2 can be a colloidal or other particle, such as a macromolecule. In the free volume theory, a is equal to the fraction of the volume V , which is freely available51 in a suspension of hard spheres 1 for a particle 2, Vfree = Va = Vp21 pR • for a hard sphere with diameter a2 = j <7j the following approximate expression has been formulated, obtained with the scaled particle theory (SPT). a(pl)=
11
free Pl
= (l - i)exp(-Ay - By2 - Cy3)
[5.7.26]
T. Biben, J.P. Hansen, Phys. Rev. Lett. 66 (1991) 2215. Y. Mao, M.E. Cates, and H.N.W. Lekkerkerker, Phys. A222 (1995) 10. 31 R. Roth, R. Evans, and S. Dietrich, Phys. Rev. E62 (2000) 5360. 41 M.Dijkstra, R. van Roij, and R. Evans, Phys. Rev. Lett. 81 (1998) 2268; Phys. Rev. E59 (1999) 5744. 51 H.N.W. Lekkerkerker, Colloids Surf. 51 (1990) 419; H.N.W. Lekkerkerker, W.C.K. Poon, P.N. Pusey, A. Stroobants, and P.B. Warren, Europhys. Lett. 20 (1992) 559; E.J. Meijer, D. Frenkel, J. Chem. Phys. 100 (1994) 6873. 21
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CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.48. Simulation results of Vfree(Pi)/V for j =0.496. Data points: simulation results for hard spheres plus ideal chains; the curve follows [5.7.26], (Redrawn from E.J. Meijer andD. Frenkel, loc. clt.)
in which ys—^—
[5.7.26a]
and A = 3j + 3 j 2 + j 3 ;
B = 9 j 2 / 2 + 3J 3 ;
C = 3J3
[5.7.26b]
The Taylor expansion coefficients in 15.7.22] follow from differentiation of [5.7.26] giving «j =-(l + j)3{xof 16) and a2 = (6J 3 + 1 5 j 4 / 2 + 3 j 5 + j 6 /2)(/rof / 6 ) . These can be used in [5.7.23], [5.7.24] and [5.7.25]. The results calculated with expression [5.7.26] may be checked with computer simulations. An example for a free volume simulation result 11 for j = 0.496, see fig. 5.48, demonstrates the accuracy of the SPT for this case. For much smaller j , more relevant for phase separation studies, a large number of simulations were performed by Dijkstra et al., loc. cit. for mixtures of large (1) and small (2) hard spheres. The distribution of small hard spheres 2 over the system and reservoir are shown in fig. 5.49 for j = 0.05 and j = 0.10. The fit is good for small qj^, but deviations appear at higher
11 21
E.J. Meijer, D. Frenkel, J. Chem. Phys. 100 (1994) 6873. H.N.W. Lekkerkerker, A. Stroobants, Phys. A195 (1993) 387.
CONCENTRATED COLLOIDAL DISPERSIONS
5.79
Figure 5.49. Concentration of particles 2,
Figure 5.50. Spinodal curve following free volume theory for a mixture of hard spheres with j = 0.10. Circles are integral theory results of T. Biben and J-P. Hansen, Phys. Rev. Lett. 66 (1991) 2215. (Redrawn from H.N.W. Lekkerkerker and A. Stroobants, loc. cit.)
Figure 5.51. Phase diagram for the pressure in the reservoir pR = pR(;K7]3 / 6kT) versus the colloid concentration for an asymmetric mixture of hard spheres with j = 0.10. Squares are computer simulations of M. Dijkstra, J.M. Brader, and R. Evans, J. Phys. Condens. Matter 11 (1999) 10079. (Redrawn from H.N.W. Lekkerkerker, S.M. Oversteegen, J. Phys. Condens. Matter 14 (2002) 9317.)
(iv) Mixture of colloidal particles and non-adsorbing macromolecules. It was already found a long time ago that colloidal particles (rubber latex spheres) dispersed in an aqueous phase could be concentrated into a cream layer when small amounts of natural lyophilic colloids (weakly charged polyelectrolytes like carrageenan) were
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CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.52. Phase diagrams in the '
Figure 5.53. Phase diagrams in the q>2 = oup^ representation for a mixture of hard spheres calculated with the help of computer simulations for j =0.10 and j =0.20. (Redrawn from fig. 15 in M. Dijkstra, R. van Roij, and R. Evans, loc. cit.)
added to the suspension. The concentrated cream and the dilute subphase were separated by a sharp interface11. The process was found to be reversible; dilution with water redispersed the colloid again. A very early explanation for the underlying mechanism of this phase separation was given by Asakura and Oosawa2 , who argued that the (centres of) the macromolecules are repelled by the colloid particle surface. In other words, there is a (depletion) zone around the colloid particle with a thickness comparable with the radius of the macromolecule. When two colloidal particles approach each other, the two depletion zones start to overlap, which results in the 1
J. Traube, Gummi Zeitung 39 (1925) 434; C.F. Vester, KolloidZ. 84 (1938) 63. S. Asakura, F. Oosawa, J. Chem. Phys. 22 (1954) 1255. 31 S. Asakura, F. Oosawa, J. Polymer Set 33 (1958) 183.
21
CONCENTRATED COLLOIDAL DISPERSIONS
5.81
depletion zone of each particle pair, which is smaller than that of two individual particles, see fig. 5.54a. Thus, the suspension can lower its Gibbs energy by clustering the particles, which implies that phenomenologically the particles feel an attractive (average) interaction. Asakura and Oosawa made a calculation of this effective pair interaction (1958, loc. cit.) by replacing the macromolecule with a hard sphere 11 (or rod-like particle in the case of stiff macromolecules). This is usually called the AO-pair interaction. In the previous section, we observed that the depletion mechanism is not sufficiently strong to lead to a stable G/L transition in a mixture of hard spheres. The effect of the small spheres may be enhanced (very) much by taking, instead of small, hard spheres, macromolecules as the depletion agent. Macromolecules, like polymer coils, may still create a depletion zone of a width = a , the polymer's radius of gyration, around a colloid sphere. At the same time, the interaction between two macromolecules may be (far) less than that of two hard spheres with the radius a . The reason is that two macromolecules may partially overlap because of their looser structure . In the past we have proposed the following model pair interaction scheme for the separate components: U n (r) = oo
r
U n (r) = O
rgcr
U12(r) = oc
r<(l/2)(cr 1 +0 2 )
U12(r) = 0
rg(l/2)(cr 1 +(T 2 )
U22(r) = 0
all r
[5.7.27]
Note that this U22 (r) is zero for all r and mimics two polymer coils in the so-called 0 state 4 '. In other words, particles 2 are freely overlapping spheres (FOS), which differ from the model of Asakura and Oosawa for two hard spheres. The formula for the potential of mean force of only two particles 1 in a sea of particles 2 may also be derived from a simple force picture 5 6) (see fig. 5.54b) and is given by ft)n(r) = «>
r<(T ]
r
t
«nM = -p|(a 1 + c x 2 , 3 | l - | - ^ - + ^ - | - j ffiij j (r) = 0
11
\
3
j
a l S r S ( a 1 + a2)
[5.7.28]
r > (Tj +
and leading to [5.7.28] for low p2 . Formally one may say that the situation is comparable with a mixture of spheres but with socalled non-additive diameters, i.e., Cj j = (l/2)(tjj + Cj); O"12 = (l/2)(o"[ + CT2) ; av,2 + (l/2)(a 2 +
5.82
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.54. (a) Colloid particles 1 and macromolecules 2. Depletion zone (dashed) around particles 1 in which the centre of particle 2 may not enter. Left: two well-separated particles 1. Right: the particles 1 are situated in each other's neighbourhood and the depletion zones overlap (lens-shaped), (b) Two hard spheres 1 of which the left one is fixed. The average force on the right sphere 1 is the uncompensated osmotic force caused by particles 2. The right sphere 1 feels on its right side the osmotic force over the whole surface area, whereas on its left side part of the surface area is screened off by the left sphere 1. Thus, the right particle 1 is pushed towards the left particle 1. Arrows point to the region where the osmotic pressure is unbalanced. with p = p2kT . This formula is exact for the just-mentioned model but not exact for a mixture of (additive) hard spheres. Let us first apply this to a discussion of the second virial coefficient. The second virial coefficient for the (Donnan) osmotic pressure follows from [5.2.18]. Substitution of the AO/V pair interaction [5.7.28] into this equation gives11
B2(p2) = 2n
\
r2 l - e x p | - ^ -
dr
[5.7.29]
The integral must be evaluated numerically21 and some illustrative values of B2 are plotted in fig. 5.55 for j = 1/7 and 1 . These values of B2 may be used to obtain an approximate spinodal curve defined by 11
A. Vrij, Pure Appl. Chem. 48 (1976) 471; H. de Hek, A. Vrij, J. Colloid Interface Set 84 (1981) 409. For oij/fcT << 1 , however, the exponent [5.7.29] may be linearized. This leads to an expression, which is identical to [5.7.24] and the same a2 as derived from the free volume model [5.7.26].
CONCENTRATED COLLOIDAL DISPERSIONS
5.83
Figure 5.55. Reduced second virial coefficient of colloidal spheres 1 in a dispersion of macromolecules (freely overlapping spheres). Solid curve: j = o~2 I O\ = 1 : broken curve: j = 1/7. Note that (p^ can exceed unity. (Redrawn from H. de Hek and A. Vrij, toe. cit.)
0=
1+2
^lf= ^
[5730al
--
or
Expression [5.7.30b] is only valid for small px . It will be used in sec. 5.8, fig. 5.64b. The value of the second osmotic virial coefficients of the Donnan osmotic pressure also seems to play a role in the prediction of the enhancement of globular protein crystallization upon salt, organic solvent or polymer11. For higher colloid concentrations, this approach is insufficient and another route is required, which is not restricted to low pl. Gast, Hall and Russel2' incorporated the VAO/V -pair interaction as a perturbation on the hard-sphere pair interaction of the colloidal particles, using a so-called thermodynamic perturbation theory (TPT). This made it possible to formulate the thermodynamic properties of a quasi one-component system allowing calculation of the phase diagram. For 2a /ai>0.3 they found a G/L, an L/S and a G/S transition of the type given in fig. 5.45b. For la I ax < 0.3 , only an F/S transition appeared of the type given in fig. 5.45d. Here, e, plotted as kT/ e along the vertical axis, plays the same role as the concentration of the depletion agent 2 in the VAO/V-pair interaction [5.7.28] (see also fig. 5.56a found from the free volume theory given below). This picture would stay unaltered through the following 11 see e.g. A. George, W. Wilson, Acta Crystall. D50 (1994) 361; D. Rosenbaum, P.C. Zamora, and C.F. Zukoski, Phys. Rev. Lett. 76 (1996) 150; G.A. Vliegenthart, H.N.W. Lekkerkerker, J. Chem. Phys. 112 (2000) 5364. 21 A.P. Gast, C.K. Hall, and W.B. Russel, J. Colloid Interface Sci. 96 (1983) 251: Faraday Discuss. 76 (1983) 189.
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CONCENTRATED COLLOIDAL DISPERSIONS
twenty years. Another perturbation approach was proposed by Vincent and coworkers1'. An intrinsic drawback of these perturbation schemes is, however, that the system is considered to be an (effective) one-component colloid, i.e. the depletion agent and its partitioning over the separated phases cannot be discussed explicitly. Therefore, we will return to the free volume model, which was already used in the previous section . The thermodynamic equations to be used are the same as discussed previously, i.e. for a, equations [5.7.19] and [5.7.26], for /7, [5.7.20] and [5.7.23], and [5.7.21] for //,. Note however that there is no upper limit here for cp2 = {nItyo^p^, as in the previous case where component 2 consisted of (small) hard spheres. In the present case, the spheres of component 2 may freely overlap each other31. The equilibrium conditions [5.7.12a and b] are used to compute the binodals to obtain the phase diagrams.
Figure 5.56. Phase diagrams for colloid/ macromolecular mixtures (HS + FOS).
CONCENTRATED COLLOIDAL DISPERSIONS
5.85
The phase diagram topology for short-range depletion forces, e.g. j = 0.1, shown in fig. 5.56 (left panels), is similar to that of short-range Lennard-Jones type attractive forces (see fig. 5.45) when one recognizes that q>2 plays the role of a reciprocal temperature11. This emphasizes that the range of attraction is more important as compared with the type of the attractive forces. The type of phase diagram for longerrange depletion forces, e.g. j = 0.4 , is shown in 5.56 (right panels). It is similar to that of the longer range Lennard-Jones type attractive forces, see fig. 5.45b, and reflects the type of phase diagram found in common liquids. The G/L transition is stable. A critical point (cp) and a triple line (tl) are also shown. The range of the attractive forces, i.e. the value of j , must apparently also be large enough (j > 0.32) in order to give a stable G/L transition. For j ' = 0.1, the G/L transition is metastable and hidden under the stable F/S coexistence, as sketched in fig. 5.56 (a) and with a critical point located at
5.86
CONCENTRATED COLLOIDAL DISPERSIONS
5.3d. The free-volume theory, that was addressed earlier, has been extended by adjusting the size of the freely overlapping spheres (FOSs) to the depletion thickness and uses the osmotic pressure of interacting polymer chains instead of the ideal van't Hoffs law . Further, a Gaussian core model2' was proposed that follows the Ansatz of Flory and Krigbaum who suggested representing the polymer coil by a single spherical particle with a radius on the order of the polymer's radius of gyration. The colloidpolymer mixture is then described as a mixture of Gaussian cores plus hard spheres. Density functional theories are suitable for inhomogeneous situations, but require step functions for the interactions31. Taking into account the polymer-polymer interactions leads to a less effective depletion effect and, thus, to a larger polymer concentration needed to obtain a G/L phase separation. The theories considered for interacting polymer chains are valid for the limiting case of good solvent qualities (full excluded volume limit). Analytical expressions for the depletion thickness and polymer density profiles near a wall and around a sphere are available for a range of solvencies between a good and a theta solvent in a mean-field approximation4'. In sees. V. 1.8 and 1.9, a mean-field approach for arbitrary solvent quality in combination with the free volume theory will be elaborated. In the next section, experiments on phase transitions in colloids will be discussed. Before doing so, it is appropriate to mention another important feature not discussed so far, but which presents itself immediately when carrying out experimental studies on adhesive colloids. (vij Gelation and percolation. In experimenting with adhesive particles having a (very) short, attractive interaction range, one encounters the phenomenon of gelation. Before the phase separation phenomena treated in the previous subsections can occur, the system may get 'stuck' in a gel-like state when, for example, the colloidal suspension is cooled down or when depletion agents are added. In fact, the same phenomena occur in simulation studies of phase equilibria. Clusters of particles appear in the simulation, which may eventually grow so large that they span the whole (simulation) volume. This phenomenon is called percolation. Various workers5 were able to construct percolation lines in the phase diagram in this way. Figure 5.57 shows that the critical point and the right side of the coexistence curve (B) are in a region that is deeply buried in the percolation area, i.e. at the r.h.s. of the percolation line. " D . G . A . L . Aarts, R. Tuinier, and H.N.W. Lekkerkerker, J. Phys. Condens. Matt. 14 (2002) 7551. 21 P.G. Bolhuis, A.A. Louis, J-P. Hansen, and E.J. Meijer, J. Chem. Phys. 114 (2001) 4296; P.G. Bolhuis, A.A. Louis, and J-P. Hansen, Phys. Rev. Lett. 89 (2002) 128302. 31 M. Schmidt, M. Fuchs, J. Chem. Phys. 117 (2002) 6308; M. Schmidt, A.R. Denton, and J.M. Brader, J. Chem. Phys. 118 (2003) 1541. 4) G.J. Fleer, A.M. Skvortsov, and R. Tuinier, Macromolecules 36 (2003) 7857. 51 N.A. Seaton, E.D. Glandt, J. Chem. Phys. 86 (1987) 4668; W.G.T. Kranendonk, D. Frenkel, Mol. Phys. 64 (1988) 403; M.A. Miller, D. Frenkel, Phys. Rev. Lett. 90 (2003) 135702.
CONCENTRATED COLLOIDAL DISPERSIONS
5.87
Figure 5.57. Coexistence curves calculated for the adhesive hard-sphere model (AHS). (A): PYcompressibility; (B): simulated points, with hand-sketched line and critical point at rc = 0.113; pc = 0.508; (C) PY energy. Circles: simulated percolation points; (D): percolation line obtained from PY theory; r is the 'reduced' temperature in the AHS model; p is the number density; a is the sphere diameter. (Redrawn from M.A. Miller and D. Frenkel, foe. cit.) At this moment, there are significant theoretical developments in the field of colloidal glasses and gelation. For the moment, we refer to reviews" and short papers 2 1 that may help interested readers.
5.8 Phase transition experiments In describing experimental phase separation phenomena in colloidal dispersions, we will consider three types of systems: a) Colloidal particles in a good solvent giving rise to a particular type of crystallization phenomena induced by purely repulsive interparticle interactions; b) Colloidal particles in a poor solvent giving rise to a more common type of flocculation, coagulation or phase separation induced by attractive, interparticle forces and giving rise to clustering of particles; c) Colloidal particles mixed with other colloidal particles or with macromolecules in a good solvent and giving rise to attractive interactions due to depletion forces. 5.8a Colloid particles in a good solvent (i) Crystallization due to soft repulsive forces. Some thirty years ago Hachisu and co-workers31 were the first to study crystallization in charged colloids in great detail. 1 W.C.K. Poon, J. Phys.: Condens. Matt. 14 (2002) R859; J. Bergenholtz, W.C.K. Poon, and M. Fuchs, Langmuir 19 (2003) 4493. 21 F. Sciortino, Nature Materials 1 (2002) 145; D. Frenkel, Science 296 (2002) 65. 31 S. Hachisu, Y. Kobayashi, and A. Kose, J. Colloid Interface Sci. 42 (1973) 342.
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CONCENTRATED COLLOIDAL DISPERSIONS
Their simple, elegant experiments give good insight into the systematic procedure they used, which would be followed by many investigators. They also suggested that crystallization was due to what is now called the 'Kirkwood-Alder' transition, first found by Alder and Wainwright" using computer simulations, see sec. 5.7b. A series of polystyrene latex (a = 85 nm) dispersions were first deionized and then mixed with the appropriate amount of KC1 and closed off in stoppered tubes. Systems prepared at the lower latex concentrations are mainly unordered but appear to become ordered at some higher particle concentration, which can be observed from the twinkling appearance of the iridescent scattering due to the multi-crystalline facets of the polycrystalline material. Some of the ordered phases at the lower electrolyte concentration and at very small colloid volume fractions appear instantaneously and fill the whole dispersion. After about half a day, the phase separation process was clearly seen in cases where an interface appeared between coexisting phases. The phase diagram is shown in fig. 5.58. Preparations with the higher electrolyte concentrations are amorphous except for the highest latex concentrations. The transition from an amorphous to an ordered state occurs at volume fractions close to the two drawn curves in fig. 5.58. This trend is easy to understand because the range of the repulsive electrostatic forces is a function of the electrolyte concentration.
Figure 5.58. Phase diagram of polystyrene latex (a = 85nm) dispersed in aqueous KC1 solution. The initial (=ovcrall| volume fraction of latex is plotted versus the C K Q ( mol dm" 3 ; logarithmic scale). Some details are omitted. Two drawn lines: calculated boundaries of the coexistence regions for a = lOOnm and particle charge = 5000 elementary units. At very low C KC1 anc^ ' o w f' * ne structure becomes BCC instead of FCC (not shown). Filled circles = ordered state; open circles = disordered state; half-filled circles = state of coexistence. (Redrawn from W.B. Russel, D.A. Savillc, and W.R. Schowaltcr, 'Colloidal Dispersions', Cambridge University Press., USA (1989) p. 347, fig. 10.8 including the experimental data of H a c h i s u el ah, loc.
11
cit.)
B.J. Alder, T.E. Wainwright. J. Chem. Phys. 27 (1957) 1208.
CONCENTRATED COLLOIDAL DISPERSIONS
5.89
So, at a constant latex volume fraction
W.B. Russel, D.A. Saville and W.R. Schowalter, 'Colloidal Dispersions', Cambridge Univ. Press. (1989); Y. Monovoukas and A.P. Gast, J. Colloid Interface Sci. 128 (1989) 533. 21 E.B. Sirota, H.D. Ou-Yang, S.K. Sinha, P.M. Chaikin, J.D. Axe, and Y. Fuji, Phys. Rev. Lett. 62 (1989) 1524. 31 A.P. Philipse, A. Vrij, J. Colloid Interface Sci. 128 (1989) 121; C. Smits, W.J. Briels, J.K.G. Dhont, and H.N.W. Lekkerkerkcr, Prog. Colloid Polym. Sci. 79 (1989) 287.
5.90
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.59. Theoretical phase diagram for (colloidal) hard spheres. The volume fraction of freezing is 0.494, melting is at 0.545, and the (non-equilibrium) glass transition at —0.57. The percentage of crystal in the dispersion versus the initial volume fraction is plotted, (redrawn from Verhaegh and Lekkerkerker, loc. cit.j
In the latter case, the volume fraction of the intervening solvent is (much) smaller than in the former. This hampers the motions of solvent and particles with the result that the speed of nucleatlon and crystal growth is (much) smaller as compared with more dilute dispersions with soft repulsive particles. Thus, it is often found that the first stage of the phase separation is the formation of an amorphous phase designated as a glass. The subsequent processes may take hours to weeks before the typical appearance of faceting iridescence takes place, often not starting in the bulk but induced at a surface, e.g. at the top of the sedlmented amorphous phase. At a volume fraction of 50%, the crystallization may still be reasonably fast, but at volume fractions of about 57%, the particles come to a motional arrest. This point is called the glass transition point with (pqlass ~ 0.57, see fig. 5.59. This may make the study of such phase transitions rather time consuming and cumbersome11. Phase separation studies were reported by Pusey and van Megen2', including phase diagrams similar to those in fig. 5.58.
Figure 5.60. Profile of volume fraction q> in the field of gravity for polystyrene latex spheres (la = 720 nm, A ~ 0.7% in aqueous electrolyte solution (3mMHCl) ). A discontinuity in
11 For more details the reader is referred to some overviews, e.g.. by A.P. Gast. W.B. Russel, Physics Today, December 1998, p. 24; and V.J. Anderson, H.N.W. Lekkerkerker, Nature 416 (2001) 811. 21 P.N. Pusey, W. van Megen, Nature 320 (1986) 340.
CONCENTRATED COLLOIDAL DISPERSIONS
5.91
The nearly hard sphere F/C phase transition is also found in fig. 5.60 where a polystyrene latex in aqueous electrolyte solution is in equilibrium in the field of gravity11. The particle density profile is measured with X-ray densitometry. The transition is clearly seen as a discontinuity in cp. With an effective HS diameter crHS = 730±30nm the curve closely follows the theoretical one based on the Carnahan-Starling equation [5.4.23] (see sec. 5.4) for osmotic pressure in the fluid branch [
Figure 5.61. Reciprocal scattering intensities (analogous to fig. 5.42a) as a function of q 2 for ij» = 0.19 at nine temperatures from 17.98°C (bottom) to 18.66 °C (top). The scattering intensity increases greatly when T becomes lower. The experimental points, which are closely described by the drawn lines, are omitted here for sake of clarity. (Redrawn from H. Verduin and J.K.G. Dhont, loc. cit.)
In fig. 5.61, the reciprocal light scattering intensity of Clg-silica spheres (a = 39 nm) dispersed in benzene is plotted versus q 2 for cp= 0.19 at nine temperatures from 17.98°C (bottom) to 18.66°C (top). The scattering intensity and its angular dependence increase greatly when T goes into the direction of the spinodal, where it would become infinite. At still lower temperatures, a phase separation takes place21. An example is given in fig. 5.62 for the same silica/benzene system. Upon slowly cooling the dispersion, its phase behaviour is observed simply by eye. At lower
5.92
CONCENTRATED COLLOIDAL DISPERSIONS
Under the binodal at lower
Figure 5.62. Phase diagram of octadecylcoated silica particles (a = 39nm±12%) in benzene. Open triangles = binodal from direct obsrvation; closed circles = spinodal from turbidity measurements; diamonds = spinodal points from dynamic light scattering (DLS), closed triangles = percolation from DLS and the open squares refer to the percolation line from direct observation, q = 0.69 cm^/g. I = critical volume fraction. (Redrawn from H. Verduin and J.K.G. Dhont, J. Colloid Interface Sci. 172 (1995) 425.)
The spinodal can be obtained, e.g., by the technique of pulse-induced critical opalescence (PICS), see M.H.G.M. Penders, A.Vrij, and R. van der Haegen, J. Colloid Interface Sci., 144 (1991) 86. This leads to a similar plot as fig. 5.42b. 21 P. Varadon, M.J. Solomon, Langmuir, 19 (2003) 509. 31 See e.g. M. Carpineti, M. Giglio. Phys. Rev. Lett. 68 (1992) 3327; N.A.M. Verhaegh, D. Asnaghi, H.N.W. Lekkerkerker, M. Giglio, and L. Cipelletti, Phys. A242 (1997) 104 or the review by W.C.K. Poon and M.D. Haw, Adv. Colloid Interface Sci., 73 (1997) 71.
CONCENTRATED COLLOIDAL DISPERSIONS
5.93
of the particles is redistributed over the separated phases. In two-component colloidal dispersions or colloid-polymer mixtures, the composition of the mixture may also be redistributed after phase separation. Let us first consider mixtures in which both colloids are of the hard sphere type. (i) Colloidal mixtures of hard spheres. We shall address several mixtures of (nearly) hard particles of quite different nature, such as lyophilic latex and lyophilic silica particles in organic solvent and mixtures of screened charged latex or silica particles in aqueous or highly polar organic solvents. Further mixtures of emulsion droplets and soap micelles will also be considered. Mixtures of PMMA spheres with different diameters (
11
P. Bartlett, R.H. Ottewill, and P.N. Pusey, J. Chem. Phys. 93 (1990) 1299. A.D. Dinsmore, A.G. Yodh, and D.J. Pine, Phys. Rev. E52 (1995) 4045. 3) J. Bibette, D. Roux, and F. Nallet, Phys. Rev. Lett. 65 (1990) 2470. The scattering was measured at different scattering angles in very thin cuvettes (thickness 20 urn ) to avoid multiple scattering. 21
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CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.63. (a) Light scattering intensity (in arbitrary units) of oil globules (2aj = 460nm , P! = 0.17) in an aqueous solution of SDS micelles (2a 2 = 4.0 nm ) as a function of the scattering wave vector q. Case B: globules without micelles (
at small q , due to the increasing osmotic compressibility of the oil globules dispersions, indicates that SDS induces attractions between the oil globules. The attraction can be explained by depletion forces between the oil globules caused by the presence of the small micellar spheres. (Note that the globules, as well as micelles, bear a negative charge and, hence, electrostatically repel each other.) The dotted curves are the structure factors calculated using the mean spherical approximation assuming a square well interaction with well depth e , which is considered a free parameter and the micelle diameter a2 as the well thickness. The obtained well depth E/kT is plotted versus the micelle volume fraction cp2 in fig. 5.63b. The straight line is in accordance with the depletion-induced pair interaction [5.7.28], which has a depth proportional to (p2. At higher
CONCENTRATED COLLOIDAL DISPERSIONS
5.95
the largest amount of depletion agent is required as expected, see [5.7.28]. The solid phase gives light diffraction patterns that are consistent with an FCC structure. The solid curves were compatible with a simple G/S equilibrium model. The phase separation studies were extended to a procedure of fractionating according to particle sizes of the globules". Fluid-solid ( F / S ) phase separation occurred in a mixture consisting of two types of silica particles
11
J. Bibette, J. Colloid Interface Set 147 (1991) 474. A. Imhof, J.K.G. Dhont, Colloids Surf A122 (1997) 53. Also the dynamics of the colloids were studied by marking them with a fluorescent dye. 41 M.H.G. Duits, R.P. May, A. Vrij, and C.G. de Kruif, J. Chem. Phys. 94 (1991) 4521. 51 J.S. van Duijneveldt, A.W. Heinen, and H.N.W. Lekkerkerker, Europhys. Lett. 21 (1993) 369. 61 H.G. Bungenberg de Jong in Colloid Science, H.R. Kruyt (Ed.) Elsevier, 1949, Vol. II, p.244. 71 H.R. Kruyt (1882-1959) obtained part of his education in the Amsterdam School of H.W. Bakhuis Roozeboom (1854-1907) on phase equilibria. 21
5.96
CONCENTRATED COLLOIDAL DISPERSIONS
Figure 5.64. (a) Photograph of a test tube containing a phase-separated mixture of polystyrene fiVf = 2.4 10 3 kg/mol) and C18-silica (a = 21nm) in cyclohexane. Initial colloid concentration = l%(w/v) and PS concentration ~25g/dm . The limiting concentration, below which no phase separation is found, is 17g/dm 3 . Note the sharp interface between the two demixed phases. Picture is rescaled from H. de Hek and A. Vrij, J. Colloid Interface Sci. 70 (1979) 592. (b) Cj8-silica (a = 46 nm) in cyclohexane, concentration l%(w/v). The limiting polystyrene concentrations (circles with dot), below which no phase separation occurred, is plotted versus the molar mass of the polystyrene. Hatched region: theoretical limits between which the spinodal curve is situated. (Redrawn from H. de Hek and A. Vrij, J. Colloid Interface Sci. 84 (1981)409.)
A more recent example, depicted in fig. 5.64a refers to a well-characterized colloidal mixture, i.e. octadecyl silica (2a = 42 nm) and polystyrene ( M = 8 - 2400 kg/mol; diameter = 2.25a = 6 - 9 3 nm in cyclohexane . Both separated phases are fluid. In fig. 5.64b, results of a quantitative study are shown. The limiting c , below which no phase separation occurs in a solution containing l%(w/v) C18-silica, is plotted versus the molar mass of the added polymer (M = 10-10 3 kg/mol). Note the low concentrations of both colloid and polymer involved. It is found that less polymer is required when the molar mass is larger. For the smallest molar mass, the separated phase was gel-like instead of a fluid. This experimental trend can be predicted by using the spinodal condition [5.7.30b], but this is only a semi-quantitative test because in fact the binodal condition is required. It is then found that the trend mentioned above is also followed when the binodal is calculated. Still, the FOS model (see sec. 5.7c) overestimates the depletion effect. There are now better models for the depletion agent available, which incorporate effective interaction between the macromolecules, see sees. V. 1.8 and 9 for a mean field treatment of the polymer chains on the level of pair interaction. When colloidal particle concentration is increased the situation becomes (much) more complex, depending on M . er and c . er . Two experimental studies with rather different systems will be mentioned. The first one21, refer to (stabilized) poly11 21
H. de Hek, A. Vrij, J. Colloid Interface Sci. 70 (1979) 592; 84 (1981) 409; 88 (1982) 258. F. Leal Calderon, J. Bibette, and J. Biais, Europhys. Lett. 23 (1993) 653.
CONCENTRATED COLLOIDAL DISPERSIONS
5.97
styrene latices (a = 60-80-95 nm), plus hydrophllic polymer (hydroxyethyl cellulose (M = 1.6 x 102 kg/mol)) in an aqueous NaCl -solution (\IK < 1.5 nm). The size ratios of the polymer radius of gyration over the radii of the colloid are j = 0.33 -0.25-0.21. The second study11 uses a PMMA latex stabilized with PHSA chains (a = 228 nm), plus dissolved polystyrene molecules M = 0.39-2.85 -14.4-103 kg/mol as the depletion agent with j = 0.08-0.24 and 0.57, respectively, dispersed in decalln. Thus, in both cases the ratio j = r^/a varied, although In a different way. The trends In both studies are as follows. For low c t er and high colloid concentrations [
Figure 5.65. Schematic phase diagram for a system described in the text. The capitals have the following meaning: F = fluid phase; S = solid phase; G / L = gas-liquid coexistence; G / C = gas-crystal coexistence; G/L/C = gas-liquid-crystal coexistence (3 phases). (Reconstructed from F. Leal Calderon et al., loc. cit, P.N. Pusey et al, loc. cit.).
11 S.M. Ilett, A. Orrock, W.C.K. Poon, and P.N. Pusey, Phys. Rev. E51 (1995) 1344; P.N. Pusey, W.C.K. Poon, S.M. Ilett, and P. Bartlett, J. Phys. Condens. Matt. 6 (1994) A29. Fj / F2 is also called G/L because it mimics the gas-liquid phase separation in low molecular systems. For phase nomenclature, see also a footnote in sec. 5.7a. For phase diagrams and details, see ref. .
5.98
CONCENTRATED COLLOIDAL DISPERSIONS
starts to gel because it is in an arrested, non-equilibrium state. This may be concluded from studies on the diffusive motion of the colloidal particles. Another non-equilibrium state is found when at low c . vmer and high (p (q>> 0.58) the colloidal particles come to a motional arrest because they are closely surrounded by neighbours that hinder the particle motions. Such a system is usually called a glass11, also found in the onecomponent system of pure hard spheres. It has been found recently that in some cases adding more polymer leads to more freedom of the particles and, thus, to a collapse of the arrested structures21. 5.9 Concluding remarks In this chapter, we have addressed the equilibrium structure and stability of dispersions that contain spherical colloidal particles. The aim was to discuss the items at a basic level. The interest in concentrated colloidal dispersions had been initiated by physico-chemists, who closed the gap between classical colloid science that was explored by chemists and (theoretical) physicists who became interested in colloid science at the end of the 20th century. An important step was to link the complexity of dispersed chemical particles to simple models from physics on liquid state theory. When new colloidal particles are being synthesized, the role of physico-chemists remains similar: they will have to connect (new) particle syntheses to physical experimentation. The above explains why the chapter has been written from a physicochemist's perspective and aims at gaining interest from (physico-)chemists and chem.ical engineers. This also explains the level of mathematics involved. The amount of formal, statistical thermodynamics (or statistical mechanics) used has been limited31 as much as possible for that reason. In order to meet these criteria, we have chosen to limit ourselves to spherical colloidal particles. Since the end of the 20th century, there is an increasing amount of attention for non-spherical particles, such as rod-like41 and plate-like51 colloidal particles. The description of suspensions of such anisotropic particles is, however, much more involved. Still61, a good understanding of colloidal dispersions of spheres is at the base of understanding dispersions with particles having more complex shapes. 11
K. Dawson, Curr. Opinion Colloid Interface Sci. 7 (2002) 218. K. Dawson, G. Foffi, M. Fuchs, W. Gotze, F. Scortino, M. Sperl, P. Tartaglia, Th. Voigtmann, and E. Zaccarelli, Phys. Rev. E63 (2000) 011401. For instance the focus has been on two-particle correlations g(r) rather than triplet correlations g ^(r,,^,^) etc., cluster expansions as well as ensemble theory (essential however for computer simulation results used in our chapter) were omitted. 41 G.J. Vroege, H.N.W. Lekkerkerker, Rep. Prog. Phys. 55 (1992) 1241. 51 F.M. van der Kooij, H.N.W. Lekkerkerker, J. Phys. Chem. B102 (1998) 7829; F.M. van der Kooij, K. Kassapidou, and H.N.W. Lekkerkerker, Nature, 406 (2000) 868; D. van der Beek, H.N.W. Lekkerkerker, Europhys. Lett. 61 (2003) 702. While noting that 'shape matters'. 21
CONCENTRATED COLLOIDAL DISPERSIONS
5.99
As an introduction, we have given a short historical overview of the development of the current understanding of the structure of concentrated colloidal dispersions. Subsequently, the basic physics required in this chapter were explained, where first use was made of the concept of osmotic equilibria. This allows considering the properties of colloidal particles and the low-molecular solvent as separate problems. Using the osmotic equilibrium concept and the radial distribution function, a microscopic quantity can be related to (macroscopic) osmotic compressibility. This connects experiments (scattering, osmotic pressure measurements) to the structure of a colloidal dispersion. The relation between correlation functions and the structure factor that can be measured in a scattering experiment was addressed using the concept of particles in external fields. It led to a definition of the radial distribution function in terms of direct and indirect correlations. A few well-known closure relations, allowing a computation of the structure factor given a certain pair interaction and colloid volume fraction, are discussed. The generalized framework of the thermodynamics of colloidal pair interactions and their relation to the structure and thermodynamic properties, such as osmotic pressure, compressibility and the second virial coefficient are applied to several types of colloidal model particles, i.e. hard, adhesive and soft particles that serve as representative examples of 'practical' colloidal particles. The hard sphere pair interaction serves as the starting point from which the pressure can be calculated up to large colloid volume fraction. Using the Percus-Yevick closure relation, analytical expressions for the direct correlation function can be derived, which are accurate up to a volume fraction of 40%. The scattering structure factor and sedimentation equilibrium profile can be calculated very accurately for hard spheres. The Percus-Yevick theory can easily be extended to include size polydispersity. For adhesive hard spheres, the case of a square well is treated in which the structure factor can be calculated for small volume fractions. One can, for mathematical reasons, make progress by assuming an infinitely short-ranged attraction (the Baxter model), yielding simple expressions for the second virial coefficient in terms of the effective 'temperature.' Using the Baxter model, one can calculate the structure factor using the Percus-Yevick closure, yielding accurate predictions for the structure factor, also for the polydisperse 'Baxter' case. Soft spheres are particles with a relatively long-range repulsion or attraction, and are more complex to describe theoretically. For weak Yukawa interactions, some results are given for the second virial coefficient and the structure at small volume fraction. The Percus-Yevick closure relation is not accurate for these longer-ranged attractions and repulsions and various closure relations appropriate for soft spheres are discussed instead. More complex interactions, such as the pair interactions between two colloids mediated by other colloidal species or polymer chains11, dendrimers, polymer stars etc., were considered to be too specialized and are omitted. The pair interaction as mediated by polymers will be discussed in detail in V.chapter 1.
5.100
CONCENTRATED COLLOIDAL DISPERSIONS
In the end, the pair interactions determine the thermodynamic state of a concentrated colloidal dispersion, resulting in phase transitions for specific colloid concentrations. The thermodynamics of these phase transitions are treated rigorously and applied to atomic systems, followed by colloidal dispersions. Basically, the fluid-solid transition of the pure hard sphere and soft repulsive systems are discussed, followed by the fluid-fluid and fluid-solid in colloid-colloid and colloid-polymer mixtures are addressed. Throughout the chapter, experimental illustrations on colloidal systems are given. In line with the structure of FICS, we have not discussed here the experimental techniques that are commonly used to characterize the structure and phase behaviour of colloidal dispersions; several relevant techniques are addressed in other FICS chapters. We have aimed at describing the equilibrium properties of colloidal dispersions. The dynamics, i.e. transport properties such as diffusion, viscosity and phase separation kinetics, were described in chapters 4 and 6. We further refer to the work of Dhont11. 5.10 General references 5.20a General reviews Aut. Div., Concentrated Colloidal Dispersions, in Faraday Discuss. 76 (1983). (A variety of papers all related to the theme of this chapter.) K. Akhilesh, K. Arora, and B. V. R. Tata, Interactions, Structural Ordering and Phase Transitions in Colloidal Dispersions, in Advances in Colloid and Interface Science, 78 (1998) 49-97. Ordering and Phase Transitions in Charged Colloids, A.K. Arora, B.V.R. Tata, Eds., VCH Publishers (1996). (Collection of papers, related to the present chapter but biased towards so-called Sogami potentials.) J. Bibette, F. Leal Calderon, and P. Poulin, Emulsions: Basic Principles, Rep. Prog. Phys. 62 (1999) 969-1033. (Review on the physical properties of emulsions with emphasis on the phase behaviour, stability criteria and structure of emulsions including the influence of non-adsorbing polymers.) D. Frenkel, Soft Condensed Matter, Phys. A313 (2002) 1-31. (Physics point of view explanation of the basics of colloidal dispersions.) A.P. Gast, C.F. Zukoski, Electrorheological Fluids as Colloidal Suspensions, in Advances in Colloid and Interface Set, 30 (1989) 153-202. (Review on the electrical field-induced arrangements of colloidal dispersions.) 11
J.K.G. Dhont, An Introduction to Dynamics of Colloids, Elsevier (1996).
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5.101
P. Salgi, R. Rajagopalan, Polydispersity in Colloids: Implications to Static Structure and Scattering, Adv. Colloid Interface Set., 43 (1993) 169-288. (Extensive review on the influence of polydispersity on the scattering properties of colloidal dispersions.) N.A.M. Verhaegh, H.N.W. Lekkerkerker, Phase Transitions in Colloidal Suspensions, in: F. Mallamace and H.E. Stanley, Eds., The Physics of Complex Systems, International School of Physics 'Enrico Fermi', Course CXXXIV, IOS Press, 1997, p. 347. (Overview of the structure and kinetics of phase-separating colloidal dispersions.) G.J. Vroege, H.N.W. Lekkerkerker, Phase Transitions in Lyotropic Colloidal and Polymer Liquid-crystals, Rep. Prog. Phys., 55 (1992) 1241-1309. (Classic review on the phase behaviour of colloidal dispersions of rod-like colloids and polymers.) 5.20b Liquid state theory J.P. Hansen, I.R. Me Donald, Theory of Simple Liquids, Academic Press (1976). (A classic.) V.I. Kalikmanov, Statistical Physics of Fluids. Basic Concepts and Applications, Springer (2001). (Clear elaboration of the use of statistical thermodynamics applied to fluids.) C.N. Likos, Effective Interactions in Soft Condensed Matter Physics, Physics Reports 348 (2001) 267-439. (State-of-the-art description of the physics description of colloid and polymer solution as well their mixtures in terms of effective interactions; with emphasis on colloid/star polymer mixtures and depletion.) Y. Marcus, Introduction to Liquid State Chemistry, John Wiley (1977). G. Nagele, On the Dynamics and Structure of Charge-stabilized Suspensions, Physics Reports 272 (1996) 215-372. (Extensive review on the dynamic properties of charged colloids in electrolyte solutions.) 5.10c Depletion interaction in colloid-polymer mixtures P. Jenkins, M. Snowden, Depletion Flocculation in Colloidal Dispersions, Adv. in Colloid and Interface Sci., 68 (1996) 57-96. (Review that focuses on pair interactions and mean-field methods; somewhat out of date by now.) W.C.K. Poon, The Physics of a Model Colloid-polymer Mixture, J. Phys. Condens. Matter 14 (2002) R859-R880. (Review on colloid-polymer mixtures with emphasis on the work done in the Edinburgh group.)
5.102
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R. Tuinier, J. Rieger, and C.G. de Kruif, Depletion-induced Phase Separation in Colloid-polymer Mixtures, Adv. Colloid Interface Sci., 103 (2003) 1-31. (Review on colloid-polymer mixtures aiming at a qualitative description on classical and ongoing work.) 5.10d Dynamics of colloids R. Buscall, The Sedimentation of Concentrated Colloidal Suspensions, Colloids Surf. 43(1990)33-53. J.K.G. Dhont, An Introduction to the Dynamics of Colloids, Elsevier (1996). (Classic book that introduces all basic concepts required to understand the dynamics of colloids in dispersion.) W.C.K. Poon, M.D. Haw, Mesoscopic Structure Formation in Colloidal Aggregation and Gelation, Adv. Colloid Interface Sci., 73 (1997) 71-126. (Comprehensive review on aggregation and gelation in colloidal dispersions.) 5.10e Relevant experimental techniques Y. Liu, S-H. Chen, and J.S. Huang, Small-angle Neutron Scattering Analysis of the Structure and Interaction of Triblock Copolymer Micelles in Aqueous Solution, Macromolecules 31 (1998) 2236-2244. (Extensive set of SANS data and analysis of copolymer micelles.) J.S. Pedersen, Determination of size distribution from small-angle scattering data for systems with effective hard-sphere interactions, J. Appl. Cryst. 27 (1994) 595608; Analysis of small-angle scattering data from colloids and polymer solutions: modelling and least-squares fitting, Adv. in Colloid and Interface Sci. 70 (1997) 171210. (Reviews that explain how to analyze scattering data of concentrated colloidal dispersions.) K. Schatzel, Light scattering - Diagnostic Methods for Colloidal Dispersions, Adv. Colloid Interface Sci., 46 (1993) 309-332. (Introduces the basic concepts of using light scattering to characterize colloidal dispersions.) 5.10f Computer simulations of concentrated colloidal dispersions M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press (1989). (Comprehensive review on simulation methods to study the chemistry and physics of liquids.) M. Dijkstra, Computer Simulations of Charge and Steric Stabilised Colloidal Suspensions, Curr. Opin. Colloid Interface Sci., 6 (2001) 372-382. (update review, 150 refs.)
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5.103
D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press (2001). (State-of-the-art reference for both students and experts use to learn or improve performing computer simulations.) D.P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press (2000). (Book on all aspects of Monte Carlo simulations; from condensed matter to polymer solutions.) I.K. Snook, W. Van Megen, K.J. Gaylor, and R.O. Watts, Computer Simulation of colloidal dispersions, Adv. Colloid Interface Set, 17 (1982) 33-49. (Early reference explaining the powerfulness of using simulation methods to understand the dynamics of colloidal dispersions.) For an update, see I.K. Snook, The Generalised Langevin Approach to Dynamics, in Curr. Topics Colloid Interface Set. 2 (1997) 111.
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6
RHEOLOGY
Ton van Vliet and Hans Lyklema 6.1
Basic notions
6.2
6.2
Rheological quantities, types of deformations
6.6
6.3
Descriptive rheology, phenomenology
6.10
6.3a
Equilibrium behaviour
6.11
6.3b
Non-equilibrium behaviour
6.14
6.4
Dynamics: the role of time scale
6.5
Yield and fracture
6.17
6.6
Measuring methods
6.19
6.6a
Tests at constant strain. Stress relaxation
6.19
6.6b
Tests at constant stress. Creep
6.22
6.6c
Tests at constant strain rate
6.25
6.6d
Oscillatory measurements
6.25
6.7
6.15
Measuring apparatus
6.30
6.7a
Capillary viscometers
6.30
6.7b
Rotational rheometers
6.36
6.8
Relationship between structure and rheological properties
6.45
6.9
The viscosity of dilute sols
6.47
6.9a
Einstein's law and its extensions
6.48
6.9b
Electroviscous effects
6.52
6.10
Viscosity of concentrated dispersions of particles
6.57
6.11
Dilute and semi-dilute macromolecular solutions
6.60
6.12
Concentrated macromolecular solutions
6.65
6.13
Effects of colloidal interaction forces
6.71
6.14
Gels 6.14a
6.15
6.78 Polymer networks
6.79
6.14b Particle networks
6.81
6.14c
6.84
Large deformation behaviour
General references
6.86
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6 RHEOLOGY TON VAN VLIET AND HANS LYKLEMA
Rheology is, briefly, the science of deformation and flow. Rheological properties of a material are noted when a force is exerted on It, and as a result of which it deforms or flows. The extent to which a material deforms under a certain force depends strongly on its properties. Air flows very fast under a small force, water much less, while for obtaining fast flow of honey or sugar syrup a large force is needed. A good mattress deforms considerably under your body weight, while a concrete floor does only to a negligible extent and therefore It is very uncomfortable to sleep on. In this chapter, the deformation and flow of three-dimensional systems will be discussed. Some basics of bulk rheology have already been treated in volume I, sees. 6.1 and 6.3 and volume III, sec. 3.6, preceding the discussion on interfacial rheology. In volume I the discussion focused on transport phenomena and limited to a phenomenological treatment of flow of simple Newton liquids. In rheology, the flow and deformation behaviour of more complicated systems is studied. In this chapter, some parts of the discussion in volumes I and III will be repeated to allow it to be read on its own. In daily life everybody is aware of the fact that some materials only deform if a force is applied {elastic behaviour), whereas others start to flow {viscous behaviour). For still other materials the reaction is less unequivocal. For example, most cheeses and bread- and cake dough will only deform temporarily under a force of short duration, but they exhibit flow over a longer course of time, implying permanent deformation. These materials react viscoelastically to an applied force; so do most paints. Their reaction to an applied force is partly elastic and partly viscous. It is typical that time is an important parameter determining their reaction to a force. To stress the importance of time in rheology one could include this notion in a descriptive definition of rheology; 'Rheology concerns the study of the relations between forces exerted on a material and the ensuing deformation as a function of time'. For various reasons rheology plays an important role in colloid science and in the industrial and household application of colloid science; 1. Knowledge of rheological properties is essential for designing and constructing machines for manipulating colloid systems such as milk, paint, and slurries of minerals. Fundamentals of Interface and Colloid Science, Volume IV J. Lyklema (Editor)
© 2005 Elsevier Ltd. All rights reserved
6.2
RHEOLOGY
2. Rheology controls the application of colloid systems, e.g., the spreadability of paint and butter 3. Rheology helps to obtain information on the structural properties of colloidal systems. Understanding the relation between structure and rheology helps in designing products with specific rheological properties. For the first two reasons it is sufficient to determine the phenomenological rheological properties, but for the last one the relationship between structural properties of colloidal systems and rheological properties must be known. To a certain extent, the measurement of rheological properties is pre-eminently suited because, in contrast to techniques such as microscopy or turbidimetry, it provides information on the interaction forces between the structural elements in the colloidal system. However, to that end, theoretical relationships based on models are also required, and the combination of rheological results with those of other measurements is often essential for a complete understanding. The difference between the two approaches in using rheology is related to the two distinct ways in which one can interpret rheological properties, viz. phenomenologically or in relation to the molecular and/or colloidal structure of the material. In the former case, no structure-based interpretation is offered, while in the latter such an interpretation is the very goal of the study. In this chapter we will first discuss phenomenological aspects (sees. 6.1-6.7) and in later sections the relationship between molecular and colloidal structure and rheological properties. Colloidal aspects will be emphasized. 6.1 Basic notions In rheology one deals with the relationship between forces and deformations. To correct for variations in size and shape of the materials, forces have to be normalized to forces per unit area [stress, SI units N m~2 ) and the accompanying deformation is taken relative to the size of the specimen [strain, dimensionless). Depending on whether a material deforms under an applied stress, or flows, or shows a combination of both, they belong to different categories of rheological behaviour. To a first approximation, two extreme, ideal types of rheological behaviour can be distinguished, viz, ideally fluid or viscous behaviour, and ideally solid or elastic behaviour. Viscous materials flow at a certain rate when a stress is applied and after removal of the stress they retain the shape they had at the moment the stress was taken away. Elastic materials deform instantaneously to a certain extent when a stress is applied, and regain their original shape after the stress has been removed. These two groups of materials differ with respect to the fate of the energy supplied to the material by the external force. This energy is the product of the stress (N m~2 ) and the strain (relative displacement (-)) and measured in J m" 3 . For an ideally elastic material this energy is stored, and upon terminating the stress it is fully and
6.3
RHEOLOGY
immediately released. For an ideal fluid, all the energy supplied is immediately and completely dissipated as heat and is not available anymore for allowing the material to regain its original shape after terminating the stress. Forces are characterized by both a specified magnitude and a direction in space; they are vectors. The direction has to be fixed relative to a co-ordinate system and characterized by decomposing the force into three components, one normal to, and two in the plane of the surface at the point on which the force acts. The first gives rise to the so-called normal stresses, and the other two to the so-called shear stresses. The plane on which the force acts also has a certain magnitude and spatial direction, and can be characterized by a vector perpendicular to the plane. The relationship between the two vectors is given by a second-order tensor, the stress tensor.
Figure 6.1. Notation for components of stress. For clarity, the stress components acting on the back face of the block are not shown.
For describing all stresses acting on a material it is useful to consider a cubeshaped infinitesimal volume element of the material in a linear orthogonal (Cartesian) co-ordinate system (fig. 6.1). On each face a force may act that can be decomposed into x, y and z components. Each face of the volume element is characterized by the co-ordinate axis perpendicular to that plane. Then, nine stress components can be distinguished, each characterized by two indices, i and j , the first denoting the direction of the vector normal to the plane on which the force acts, and the second the direction of the force component. If the volume element is small enough to neglect gravity and inertial forces, nine components are needed to describe completely the stress(-es) acting on the volume element. These nine components together constitute the stress tensor 1. In Cartesian co-ordinates, xx T
xy
xz
T
*=TyX yy yz
zx
r
I6-
1 1
!
zy zz
They are shown in fig. 6.1. In [6.1.1] the three diagonal components (TXX, T ,
6.4
RHEOLOGY
r
zz ' a r e *-ne n o r m a l stresses acting perpendicular to the six faces of the cube. A pressure or isotropic stress applied on a material is characterized by three equal, normal stress components. Application of such a stress to an isotropic material will only lead to volume compression or expansion. The other six components of the stress tensor, known as shear or tangential stresses, act parallel to- and in the plane of- the six faces of the cube. Based on the law of conservation of momentum, the components at the two sides of the diagonal terms are equal to each other. Moreover, the isotropic pressure can always be chosen in such a way that one of the normal stress components becomes zero, leaving two normal stress differences, r - r , = JV, and r , - r,, = No , known as the first and JCJC
yy
'
yy
^^
-^
second normal stress difference, respectively. This leads to five independent components in the stress tensor, two normal components and three shear components. As stated above, if a material has a stress applied it will deform. The extent of deformation will depend on the properties of the material and the magnitude of the stress, but also on the size and shape of the material. To obtain a quantitative measure of the deformation, this quantity is always taken relative to a characteristic length of the material. Both the deformation and this characteristic length are characterized by a size and a direction, so they are vectors. The relative deformation is given by the quotient of their absolute values. For the case where the material is inhomogeneous, the relative deformation will vary with the position in the material, and one can only define the relative deformation unambiguously for an infinitesimal length. The relationship between the spatial vector, that defines the position of a point and the deformation vector, is given by the deformation gradient or strain tensor. There are various ways to define strain. Here, we will follow the classical definition of infinitesimal strain, which is strictly valid only for small-amplitude deformations. For more general definitions, we refer to textbooks11. Consider two points, A and B, in a volume element of a material at a distance h apart, and situated in a Cartesian coordinate system (see fig. 6.2). As a result of an applied force, material will be displaced and deformed, resulting in a change of distance between the points A and B
Figure 6.2. planes.
11
Shear between two parallel
R. Darby, Viscoelastlc Fluids, An Introduction to their Properties and Behaviour, in Chem. Processing Eng., L.F. Albright, R.N. Maddox and J.J. McKetto (Eds) Vol. 9, Marcel Dekker (1976). A.S. Lodge, Body Tensor Fields in Continuum Mechanics, with Applications to Polymer Rheology, Academic Press (1979).
RHEOLOGY
6.5
(B moves to Bj) and in a rotation of the points with respect to each other. Rotation implies movement of the material in the co-ordinate system, without change in mutual positions, and does not depend on the rheological properties of the material. However, the relative change in position does so depend. Considering only the elongational component, the relative deformation of the material between the points A and B is given by (AB1- ABJ/AB = Ax/x . For the present case, only involving a change in length, the strain er is defined as the infinitesimal relative elongation of x or, in general, as the relative deformation gradient, so, e,=^ dx
[6.1.2]
In real three-dimensional systems the situation regarding deformations owing to a shear stress is more complicated, as is illustrated for flow in sec. 1.6.4.a. Consider in fig. 6.2 the line AB connecting the parallel plates. If a shear stress r
is applied to the
upper plate, the line AB is not only elongated, but also rotated. The rotational part is given by {dAx I dy) - {dAy I dx) and the elongational part by (3Ax/3y) + (3Ay/3x). For a shear displacement, (dAx/dy) = (dAyldx) so that
3Ax = l(^ + ^ W — - ^ 1 dy
2{dy
dx )
2{ dy
|6 . 13]
dx )
The second term on the r.h.s. represents the net rotation of B with respect to A, and the first term the relative separation of the points, and hence this is the proper measure of the infinitesimal relative deformation or strain. By analogy to [6.1.1], for the force exerted on a material, its deformation is fully described by a strain tensor, e , consisting of nine components, which can be written as: 23Ax/3x e = dAy/dx + dAx/dy 3Az/3x + 3Ax/3z
dAx/dy+ dAy/dx
3Ax / 3z + 3Az / dx
2dAy/dy
dAy/dz + dAz/dy
3Az/ dy + dAy/3z
23Az/3z
[6.1.4]
The same applies for the flow of a viscous material, which is described by a rate of strain tensor, e , consisting of nine components, see [ 1.6.1.13]. 23u x /3x 1 = dvy/dx + dvx/dy dvz I dx + dvx I dz
dvx /dy + dv /dx
3u x /3z + 3i>z Idx
2dvy/dy
duyldz + dvjdy
dvz/dy + dvy/dz
2dvz/dz
[6.1.5]
Equation [6.1.4] implies the separation of the displacement gradient into a
6.6
RHEOLOGY
symmetric and an anti-symmetric part. The strain tensor is symmetric, whereas the rotation tensor is anti-symmetric. For simple shear, the rotation and the strain (elongational flow) component contribute equally to the flow. In formal rheology, relationships between the stress, the strain, and the rate of strain tensor are formulated. This requires tensor formalisms that we consider beyond FICS. Therefore, in the following we shall restrict ourselves to the main types of stress-strain relations, whereby the tensors can be read, 'term by term'. 6.2 Rheological quantities, types of deformations When defining rheological quantities it is again useful to make a distinction between elastic and viscous behaviour. In addition, a distinction has to be made between the various ways in which a material can be deformed. Below, we shall distinguish three main types of deformation. (a) All-sided, or isotropic, compression; for isotropic materials this will only lead to a change in volume of the material, whereby the normal components of the strain tensor ev are equal to two thirds of the relative change in volume, [AV/V = (3 / 2)23Ax / 3x = 33Ax / dx {= 3d Ay /3y = 33Az/ 3z}].
[Y , dAx, V , dx +
dx dt/H
3Ayj V , te, \ , , , 1 — dy
dz +
dx ){ y dyy){
AV _ [{ V
dz -dxdydz
3z )
y
\
dxdydz ( 3Ax =
H
\ dx 3Ax dx
3Ay 3Az ^ ( 3Ax 3Ay 3Ax 3Az 3Ay 3Az 3 Ax dAu 3Az ^ —+ + —H + —— H — dy dz ) \ dx dy dx dz dy dz dx dy dz J
3Ay 3Az dy dz
[6.2.1]
The ratio between the stress and the relative change in volume is called the compression modulus K (units N m~2 ). T XX
= Tyy = Tzz=P
= -K£V
I6-2-2'
For small ev, K is a constant which characterizes the material. For most solids and liquids it has a value of about 109 Pa (N m~2 ). For gases K is about 105 Pa. For a gas, the resistance to a volume decrease stems from a lowering of its entropy, while for a condensed material it is a result of an increase in enthalpy owing to short range repulsion between the molecules. This difference allows the gas content of a given heterogeneous material to be determined by measuring its compressibility. (b) Shear, upon which parts of the material are shifted parallel to other parts. It occurs when a shear stress is applied. If inertial forces can be neglected, for a material that is homogeneous down to a size << volume element considered, each volume
6.7
RHEOLOGY
Figure 6.3. Laminar linear flow between two parallel plates. The lower one is fixed, the top one moves at constant rate vx = dx/ dt in the x-direction, under the influence of a shear stress il,x . The arrows give the velocity as a function of the distance y from the lower plate.
element will be subjected to the same relative deformation. A common measure of the shear y is: y = tana = Ax/Ay
[6.2.3]
where a is the angle of shear (fig. 6.2). If a shear stress is applied to an elastic material it will deform, leading to a shear strain y while a viscous material after a brief inertial period will flow at a certain shear rate, y= dy/dt.
Following the usage in rheology, the dot above a parameter
denotes the time derivative. Consider a fluid confined between two parallel plates of which the lower one is fixed (fig. 6.3). On application of a shear stress, r
, on the
upper plate it will move with a velocity, vx . Under steady-state conditions {dy/dt = 0), T
is uniform throughout the fluid, and for a homogeneous material, vx will change
linearly with y. The partial derivative dvx/dy
is constant, and is known as the shear
rate or rate of shear. Hence, y = dy/dt = dvx /dy = d(Ax/Ay)/dt
[6.2.4]
To obtain some feeling for the practical range of y, see table 6.1. Next, Theological quantities can be defined which characterize the material's behaviour. For an ideally elastic material this is the elasticity modulus, G^ [N m~2] T
U =G 1J {i(3A J /at + aA1/a/) + i(3A J /3t-3A 1 /3/)} = G1J71J = ^
[6.2.5]
where i and j can be either x, y or z. For an ideally viscous fluid, h = n{^{dvj/di + dvi/dj)+l{dvi/di-dvi/dj)\=T]r=Tji
[6.2.6]
where rj is the (dynamic) shear viscosity [N m~2 s]. (c) Uniaxial compression or extension, upon which only one side of the material is subjected to a normal compressive or tensile stress. No external stress is applied to the other two sides (except atmospheric pressure), so towards these sides the material is allowed to adjust itself to the internal stresses. If the volume of the material stays constant during uniaxial deformation in the x direction the strain e( is given by:
RHEOLOGY
6.8
Table 6.1. Shear rates typical of some familiar materials and processes. Situation
Typical range of shear rates
Application
Sedimentation of fine partic-
10"6 -10~ 4
Medicines, paints, several drinks
10-2 _ 1 Q -1
Paints, printing inks
Draining under gravity
10"1 -10 1
Paints and coatings Toilet bleaches
Extruders
10°-10 2
Polymers
les in a liquid Levelling of a liquid owing to surface forces
2
Chewing and swallowing
IO'-IO
Dip coating
IO^IO 2 1
Foods Paints, confectionery 3
Mixing and stirring
10 - 1 0
Pipe flow
10 - 1 0 3
Pumping, blood flow
Spraying and brushing
103-104
Spray-drying, painting, fuel atomization
Rubbing
104 - 1 0 5
Application of creams and lotions to
Milling pigments in fluid bases
103-105
Paints, printing inks
High-speed coating
105-106
Paper
Manufacturing liquids
the skin
3
Lubrication
10 -10
7
Gasoline engines
(Adapted from H.A. Barnes, J.F. Hutton and K. Walters, An Introduction to Rheology, Elsevier (1989)).
ec=— 2
2dAx/dx 0 0
0 9Ay/y 0
0 ,. 0 = dx dAz/z
1 0 0 -0.5 0 0
0 0 -0.5
[6.2.7]
The rheological quantity characterizing ideally elastic material behaviour in uniaxial elongation or compression is Young's modulus E, which is defined as the ratio of the exerted normal stress over e{ in that direction. E = Tu/aA(/Ai = r i l /£ i i
[6.2.8]
For a fluid, the relationship between the normal stress and the rate of strain tensor reads as
RHEOLOGY
6.9
where t]e is the elongational
viscosity^ . The shear and elongational viscosities are
related. (d) Relationship
between
rheological quantities.
For a homogeneous material,
relationships can be deduced between the rheological quantities, K, G and E, where the numerical values depend on the volume change of the material upon deformation in elongation. A measure of the change in volume upon uniaxial elongation or compression is the Poisson ratio ji which for elongation in the x direction reads
dAx/dx
dAx/dx
2\
Vde)
Under conditions where the strains produced by the stresses are additive 21 an isotropic tensile stress, r u , will give strains {e-2/ue) in each co-ordinate direction. The fractional increase in volume, dV/V, will be [l + e(l -2//)] 3 - 1 = 3e(l -2/i). The bulk modulus K can then be written as K = ^- = ^ ev 3e{l-2ju)
=
E. 3(1-2/;)
16.2.11]
In order to derive the relationship between E and G, consider in a thought experiment the deformation of a cube of isotropic Hooke material by simultaneous application of a tensile stress, xxx, in the x direction and an equal compressive stress r
in the y
direction (fig. 6.4). The cube deforms into a rectangular block in such a way that there is no change in dimension in the z direction, and the longitudinal strains in the x and y direction are of equal magnitude, but of opposite sign. The square, ABCD, will be deformed into a rhombus, A'B'C'D1, without any rotation of AB and CD. The elongational strain in the x direction will be E=
Ixx-+nIyy_ E E
[6.2.12]
The prism APD will be in equilibrium if a shear stress, r, acts on AD towards A. The x component of the force will be proportional to TPD to the right, and rAD/V2 = rPD to the left. Such shear stresses act on all faces of the square, ABCD. The linear strains, ex = DD'/OD = - AA'/OA, correspond to a shear strain equal to 2ex 3 ) so we get 2ex = r / G . In combination with [6.2.12] this gives E = 2G(1 + //)
[6.2.13]
Note that in surface rheology the corresponding quantity is called the dilational viscosity. Additivity here means that if several stresses are applied consecutively, the resulting total deformation at time t equals the sums of the individual deformations, had they been applied independently [Boltzmann superposition principle). 31 See R.W. Whorlow, Rheological Techniques, Ellis Horwood (1992), henceforth in this chapter referred to as Whorlow, loc. cit., sec. 1.4.3. 21
6.10
RHEOLOGY
Figure 6.4. Cross section of a cube before and after the application of equal tensile and compressive stresses, t, in perpendicular directions. (Redrawn after Whorlow, loc. cit). For an incompressible material, dVIV = 0, which requires K = °° . Then one obtains ju = 0.5 and E = 3G . The relationship E =3G may be extended to a Newton fluid by interpreting E and G as being the ratio of a stress to the corresponding strain at time t. In shear G =4 =7 yt t and in tension E =
it=^t~
16.2.14]
[6.2.15]
It follows that ?7e=3/7
[6.2.16]
The ratio of the elongational viscosity over the shear viscosity is known as the Trouton ratio. For a Newton fluid, [6.2.16] is not limited to small strains. On deformation, such a fluid does not remember its previous shape and [6.2.16] will apply for each successive small strain. For non-Newton fluids, such as polymer solutions, the Trouton ratio may become very large (up to 104 ) and will depend on the strain and the strain rate applied. For other flow types, e.g. for biaxial extension, the Trouton ratio may be different. 6.3 Descriptive rheology, phenomenology Phenomenologically it is possible to classify materials according their rheological behaviour. As a starting point, we will first consider ideally viscous and ideally elastic
RHEOLOGY
6.11
behaviour. A first complication is that material properties often depend on strain or strain rate (non-linear behaviour). Moreover, the properties of a material may depend on the deformation time, resulting in non-equilibrium behaviour. Finally, for many materials, the reaction on a stress or strain generally consists partly of a viscous contribution and partly of an elastic one; they behave viscoelastically. The ratio between these two contributions mostly depends on the speed of deformation. Below, first equilibrium behaviour will be discussed, followed by non-equilibrium behaviour and rate effects. Except where mentioned specifically, the discussion will be limited to shear deformations.
6.3a Equilibrium behaviour In fig. 6.5, a graphical overview is given of the simple basic relationships between stress and strain rate for fluids and fluid-like materials subjected to a shearing deformation. In practice, a combination of these simple relationships can often be observed, particularly when the mechanical behaviour is studied over a large range of shear rates or stresses.
Figure 6.5. Shear stress-rate of strain diagrams for a number of liquid-like materials. (1) Newton behaviour; (2) shear thinning; (3) shear thickening; (4) Bingham fluid; (5) plastic flow behaviour. Curve 1 in fig. 6.5 illustrates the relationship for a linear viscous fluid [6.2.6 and 6.2.9]. Fluids obeying such behaviour are called Newton fluids. Only one material parameter, the viscosity rj, suffices to define fully their rheological behaviour under shear. The viscosity is given by the slope of the line, and is independent of shear rate and shearing time. For non-Newton fluids the relationship between r and y is not linear. A shear rate-dependent viscosity is obtained, called apparent viscosity, TJ
, with 77
=
J(y) • If 77app decreases with increasing y, one speaks of shear thinning (or pseudoplastic) behaviour (curve 2 in fig. 6.5), and when an increase is observed, of shear thickening behaviour (curve 3). If the latter is accompanied by a volume increase of the material, the term dilatant behaviour is often used. In daily practice, the above terms are often used in a rather imprecise way. According to the given
6.12
RHEOLOGY
definitions, it would be more precise to speak of shear rate-thinning/thickening, respectively or, if the phenomenon is observed for other flow types, of strain ratethinning/thickening. Using these terms avoids confusion for a number of products including bread dough and some polymer melts, for which the resistance against deformation increases with increasing strain, so-called strain hardening, but decreases with increasing strain rate [strain rate-thinning). Many products exhibit shear thinning and/or shear thickening over a limited range of y. Often the relationship between r and y can be modelled by the empirical relation T=kyn
[6.3.1]
where k (the so-called consistency index, with the strange units of Nm~ 2 s n ) and n (the power law index) are constants. The advantage of this power law equation is that it is relatively easy to use in either analytical or numerical calculations for complex flow situations. Its main drawbacks are that for constant values of k and n it predicts an unlimited increase or decrease of n infinite 77
with increasing y, and either zero or
values for y —> 0 for n greater or less than 1, respectively. However, in
practice k and n are only constant over a limited range of y, mostly over one order of magnitude or even less. The consistency index, k, is in fact a measure of 77
at
1
y = 1 s" ; n = 1 for a Newton fluid, n< 1 for shear rate-thinning, and n > 1 for shear thickening behaviour. To avoid the dimensional problem that the units of k depend on n one can rewrite [6.3.1 ] as n-l V
where /
y 'P
is a reference shear rate, usually taken to be 1 s"1 . Thus, k
always has the
dimensions of viscosity. Many common materials such as margarine, tomato ketchup, buttermilk, many paints, and clay suspensions behave like solids under small stresses, and are liquidlike under a large stress. Clear flow is only noticed above a certain threshold value, the so-called yield stress, r . For such systems one speaks of plastic flow behaviour. Above the yield stress, they exhibit a shear thinning character. Often the flow behaviour of these materials is modelled as T T
=
+ 7 y
lBr
[6.3.3]
where rjB is a constant, the so-called Bingham
viscosity
(curve 4 in fig. 6.5). In
practice, Bingham flow behaviour is never observed; but often for stresses clearly above r , [6.3.3] describes flow behaviour well enough to be used for engineering calculations. In fact, r
and r;B are fitting constants, and r]B is not even a viscosity
because, for an ideal Bingham fluid, 77
decreases with increasing y. For r > r ,
RHEOLOGY
6.13
napP =nB+(Ty/y), and for r < ry , //app = ~. A commonly used alternative to [6.3.3] for modelling plastic flow behaviour is the Hershel-Bulkley model1', which reads as r = ry + kyn
[6.3.4]
Above the yield stress, the flow curve is not straight as for the Bingham model. Establishing r may be problematic because its value often depends on time (see below). This implies that, depending on the purpose for which one wants to know r , one may have to accept different values for different times. However, this does not imply that r cannot be a useful material characteristic.
Figure 6.6. The shear stress, T , as a function of the shear strain, y, for various materials with solid-like character. (1) linear elastic, (2) non-linear elastic, (3) plastic, and (4) plastic-like fluid.
A corresponding overview of the various relationships between stress and strain that can be observed for predominantly solid materials is presented in fig. 6.6. Just as for fluids (fig. 6.5), in practice a combination of these simple relationships is observed when mechanical behaviour is studied over a large range of strains and or time. For an ideally elastic material (Hooke solid), the relationship between r and y is linear (curve 1), as in [6.2.5]. Such behaviour is seen for all solid materials at low y, although the strain range over which the relationship is linear varies greatly, e.g., 2-3 for rubbers, 0.2-1 for most polymer gels, ~1 for gelatin gels, —0.003-0.03 for many particle gels (yoghurt), and —0.0002 for bread dough, margarine, and cast iron. Only brittle materials such as cast iron, ceramic products, potato crisps, and several hard biscuits, are linearly elastic up to the point where they fracture. For most other materials the behaviour becomes non-linear at larger deformations (curve 2), and the product may even start to flow at stresses above the yield stress (curve 4, fig. 6.6b). For non-linear materials, G and E are functions of strain, but the material behaves reversibly with respect to deformation. This reversibility is lost for so-called plastic 11 W.H. Hcrschel and R. Bulklcy. Proc. Amer. Soc. Testing Mater. 26 (1926) 621, W.H. Herschel and R. Bulklcy. Kolloid-Z. 39 (1926) 291.
6.14
RHEOLOGY
materials that behave according to curve 3. During deformation, their behaviour is apparently non-linear elastic, but the material does not return to its original shape after the stress is removed. Part of the deformation is permanent, indicating a viscous contribution; the material behaves viscoelastically. This also applies for systems obeying the trend of fig. 6.6b. Plastic and many plastic fluid materials under low stresses behave viscoelastically. One speaks of linear viscoelastic behaviour if the elastic component can be described by [6.2.5] and [6.2.8] and the viscous component by [6.2.6] and [6.2.9]. 6.3b Non-equilibrium, behaviour For many non-Newton fluids and for products with a plastic flow character the relationship between r and / depends on the flow time and flow history. Two main types can be distinguished, thixotropy and antithixotropy (sometimes referred to as rheopexy). In the first case, 11 decreases with increasing time of flow, and in the second case it increases. Both thixotropy and antithixotropy are reversible phenomena. The apparent viscosity has to return to its original value during rest or during shearing at a lower y. In general, thixotropy and antithixotropy will be recognized as such when the time scale of the structural changes leading to this behaviour is longer than l/y and the response time of the apparatus and shorter than about 103-104 s, depending on the patience of the person carrying out the experiment. These two phenomena may not be confused with situations in which 77 changes irreversibly owing to irreversible changes in structure during shearing. A direct consequence for these types of behaviour is that 77 is affected by all handling involving flow of the material, such as filling the measuring body of a rheometer. Thixotropy is a common phenomenon for many food materials such as tomato ketchup and mayonnaises, cosmetics, and pharmaceutical products, but also for many concentrated suspensions (e.g., of bentonite) and polymer solutions. For paints, thixotropy is often required because they should behave like fluids upon application with a brush but must not sag or slide after that.
Figure 6.7. Hysteresis in a thixotropic material.
RHEOLOGY
6.15
Thixotropic behaviour is often studied by subjecting the material to increasing y or r followed by a decrease at a known rate. If the time during which the material is sheared at each y or r is shorter than that required to reach the equilibrium value of ?7
, the relationship obtained between r and y shows a so-called hysteresis loop
(fig. 6.7). Its size will depend on the rate at which y or r was increased and reduced and, hence, depends on measuring conditions. Modelling of thixotropic behaviour is usually done by introducing a structure breakdown and re-formation function, with its equilibrium depending on the shear rate. 6.4 Dynamics: the role of time scale As mentioned above, dynamic features play an important role in the mechanical behaviour of many materials. The consequences of the duration of a material deforming under constant conditions of shear rate or stress have been introduced above, in sec. 6.3b. Below, we will discuss the effect of time scale, i.e., the time that a stress of a certain magnitude and direction acts on a material. For a steady state experiment, in which a certain compressive stress is applied to an ideally elastic material, the applied time scale and the duration of the experiment coincide. However, this is no longer the case if the material is subjected to a varying stress (e.g., to a sinusoidally oscillating stress). The time scale of the experiments is then roughly equal to the reciprocal frequency of the oscillation in rad s" 1 . Upon shear flow, the time scale and the duration of shear stress
or shear rate application are also (usually) completely
different. Owing to the rotational component in a shear flow, dispersed (aggregates of) particles or macromolecules will start to rotate. This implies that, owing to the elongational component of the flow, a certain point on the surface of the particle will be subjected successively to a tensile- and a compressive stress equal to the shear stress. The time scales of these stresses depend on the rate of rotation and, hence, on y.
The time scale is roughly a quarter of the rotation time of the particle
= 7i/4y= y'1 . An example from daily life, showing the importance of time scale for rheological behaviour is that the inner part of a mature soft cheese such as Camenbert flows if one stores it for some time after slicing the cheese. However the inner part of the same cheese can be cut with a knife, and one has to chew on it during mastication, implying elastic behaviour over the time scale of cutting and chewing. In general, one sees for viscoelastic systems that they behave relatively more elastically over short time scales and more viscous over longer ones. The origin of time scale-dependent behaviour is in the structure of a material. A characteristic of ideally elastic materials is that they recover their shape after release of the applied stress. This implies that all energy supplied during the deformation was stored in the material, for example in deformed bonds, and that this stored energy is
6.16
RHEOLOGY
used to let the material return to its original shape. Hence, the bonds between the structural elements of the material must be permanent over the time scale of deformation. To allow flow, bonds have to break and re-form, stress-free, during the time scale of the experiment; all supplied energy is dissipated as heat. In viscoelastic materials the process of spontaneous disruption and stress-free re-formation of the bonds proceeds over time scales between seconds and days. It can easily be followed by first deforming the material to a certain extent and then measuring the stress which is required to keep this deformation constant. Except for ideally elastic materials, stress will decay at a certain rate; it relaxes. The time required for the stress to decrease to 1/e (i.e., 36.8%) of its value at the moment the deformation was stopped is called the relaxation time (see fig 6.10 below). Relaxation processes can be visualized by considering a Maxwell element consisting of a spring and dashpot in series, representing elastic and viscous deformation, respectively (sec. III.3.6i and fig. 6.11 below). If such an element is stressed over a short time scale, the resulting deformation will be almost exclusively due to deformation of the spring, whereas for long lasting stresses, deformation will be caused by displacement of the piston in the dashpot. For intermediate time scales, both contribute to the reaction of the element. The ratio of both contributions depends on the 'Maxwell' modulus, GM of the spring, and the effective resistance/friction (which can be modeled as a viscosity TJM ) experienced by the piston in the dashpot. If a Maxwell element is subjected to a stress deformation, the rate of deformation of the spring and dashpot are additive, in shearing terms
dZ= _ L * I + J _ r dt
,6.4.!,
G M dt /7M
For constant deformation dy/d£ = 0 and one obtains T=Toe~lGM/7lMK
=
T o e-
t/r
rel,M
[6.4.2]
where r rel M is called the relaxation time of the system. The stress, r 0 , directly after deformation is GM x yQ , the deformation at t = 0 . From [6.4.2] it follows immediately that, for t« r rel M , the reaction of the element is primarily elastic whereas it is primarily viscous for t » r rel M . For liquids, observed relaxation times are very short (e.g., about 10~13 s for water) while for solids, r rel is very long. For viscoelastic materials, r rel is in-between and corresponds roughly with the human time scale. It indicates that the ratio between the relaxation time and the characteristic time scale of observation, t obs , is important for the observed rheological behaviour of a material. The ratio is expressed in the Deborah number (De), defined through11 (see sec. 1.2.3)
11 M. Reiner, Physics Today 17 (1964) 62. After the Old Testament, Judges chapter 5, 5 where in the original version, Deborah sings, 'The mountains flowed before the Lord'.
RHEOLOGY D e s
W'obs
6.17 [6.4.3]
The rheological behaviour of materials with one single relaxation time can be classified according to their Deborah numbers as follows: elastic or solid behaviour when De » 1, viscous or liquid behaviour when De « 1, and viscoelastic behaviour when De is of the order of 1. The important conclusion is that the distinction between solid and fluid behaviour not only depends on an intrinsic property of the material but also on the duration of observation. In practice, most materials contain structural elements connected by various bonds each with different r r e l . Such materials cannot be characterized by a single r rel , but a spectrum of relaxation times is needed to describe their mechanical behaviour. Similarly, many viscoelastic materials exhibit predominantly viscous behaviour over longer time scales, but in practice the relationship between viscous and elastic behaviour as a function of time scale is often more complicated. A spectrum of relaxation times can be modelled as a series of Maxwell elements in parallel. 6.5 Yield and fracture As mentioned in sec. 6.3a, many materials exhibit solid-like behaviour under a low stress whereas they may flow under larger applied stresses: they yield above the yieldstress. This phenomenon shows, on one hand, a clear resemblance with fracture of real (brittle) solids but, on the other hand, these materials behave as liquid dispersions after yielding. The main phenomenological difference is that the material does not fall apart in different pieces, but upon yielding remains a coherent mass. For gels containing much liquid this distinction may be arbitrary. Under large stresses the gel network will fracture; however, if this fracture process is accompanied by expulsion of
Figure 6.8. Stress concentration at the tip of a notch. L is the notch length. In practice the top of the notch is not indefinitely sharp but has a radius r (not shown). Schematic picture for the two-dimensions case.
6.18
RHEOLOGY
liquid, the system as a whole yields. The end result is a liquid dispersion with remnants of the gel network dispersed in it. Such a process occurs during the stirring of milk gels as part of yoghurt manufacturing and, less clearly so, during spreading of butter and margarine. In general, for fracture to occur, all bonds between the structural elements in a certain macroscopic plane have to break, leading to structure breakdown of the material over distances much larger than the relevant structural elements, and finally to falling apart of the material. For yielding and flow, only the first of these prerequisites has to be fulfilled. The initial processes leading to fracture and yielding are the same. The most important inherent properties of a material determining its fracture behaviour are the mechanical properties of the structure elements, the strength and number of the bonds inbetween, and the inhomogeneity of the material. These inhomogeneities can be considered as tiny cracks. Stresses exerted on the material will be concentrated at the tip of these cracks by a factor ~ -J~L / r , where L is half of the crack length, and r is the tip radius. These cracks will start to grow if the local stress exceeds the bond strength between the structure elements. The growth mechanism of cracks can be understood by starting with the following energy balance1'2 W = W' + W" + (Wk)
[6.5.11
where W is the amount of supplied energy, W the amount of energy elastically stored in the material, W" the amount of energy dissipated, not owing to the fracturing process itself, and Wfr the specific fracture energy. During deformation, energy is supplied and, for an ideal elastic material, completely stored in the product as elastic energy W . From the moment a crack starts to grow, the energy stored directly around the crack becomes available owing to stress relaxation of the material. If the differential amount of energy that becomes available upon crack growth exceeds the differential energy required for the fracturing process, the crack will propagate spontaneously. This is the case if the crack has grown beyond a critical length, (c (fig. 6.9). The result is that the material falls apart. Energy-dissipative processes other than for fracturing, W" , directly affect the amount of energy available for crack propagation. If energy dissipation is too high, no extensive crack propagation will occur and, in combination with extensive growth of the small cracks, this will generally lead to yielding rather than to fracture. As the fraction of the supplied energy that is dissipated depends on the rate of deformation, whether a material will yield or fracture also depends on this rate. For
11
See, for example A.G. Atkins, Y.M. May, Elastic and Plastic Fracture, Ellis Horwood (1985). A.A. Griffith, Phil. Trans. Roy. Soc. (London) A221 (1920), 169; T. van Vliet, P. Walstra, Faraday Discuss. 101 (1995) 359. 21
RHEOLOGY
6.19
Figure 6.9. (a) Area of stress relaxation with the formation of a fracture interface, (b) The amount of fracture energy necessary, the amount of energy released by stress relaxation, and the sum of both as a function of the length of the fracture for a breach passing through a material.
example, in various types of Swiss and Dutch cheese, holes can be formed, caused by bacterial gas production. This process is very slow, resulting in yielding of the cheese mass. If gas production is too fast, cracks can be formed, and this is an indication that the cheese manufacturing process has not proceeded well. 6.6 Measuring methods One has to make a distinction between measuring methods and measuring techniques, the latter dealing with apparatus. In this section we discuss how stress or strain is applied as a function of time, and how the system can respond. A first classification of measuring methods can be made, depending on whether the test material is subjected to a stress r, a strain y or s, or a strain rate y or e. Another distinction can be made between methods involving the application of a steady t, y (orf) or y (or e), and methods in which the applied variable varies in time (often sinusoidally), the so-called dynamic tests. The latter division is somewhat arbitrary; experiments in which the reaction of a material upon a sudden change in r, y (or e) or y (or e) is determined (so-called transient tests), can be considered as either dynamic or steady state tests. An excellent overview of rheological techniques and methods is found in the book by Whorlow". 6.6a Tests at constant strain. Stress relaxation Suppose a certain strain is applied instantly to the material, and the t required to maintain the deformation is measured as a function of time. Ideally the material is 1!
R.W. Whorlow, loc. dt.
6.20
RHEOLOGY
deformed in a step function but, in practice, the deformation always takes some time. For a fluid, r will be finite during deformation and zero for all positive t (fig. 6.10). For an ideally elastic material, r stays constant in time, whereas for other materials r will initially decrease relatively rapidly and then more slowly. Ultimately, r may become zero, or approach some limiting value.
Figure 6.10. Stress relaxation at fixed strain y . At t = 0 a strain y , leading to a stress TO is applied instantly; its relaxation is shown for different rheological materials. From the stress response the stress relaxation modulus G{t) can be calculated, G{t)=y/r(t)
[6.6.1]
In practice, systems can rarely be represented just by G{t). In fact, we are Interested in the relaxation spectrum
H(r re |), representing the consecutive relaxations. To that
end it is useful to consider a Maxwell element11 consisting of a single spring and dashpot (fig. 6.11a). During deformation at any time the stress in the spring and the dashpot will be the same; however, the strain redistributes itself over the spring and dashpot. Initially, it will reside completely in the spring (elastic deformation, yel) but eventually it may be entirely in the dashpot (viscous deformation,
7visc )• During
deformation, Y = Ye\ + 7viSC . which gives, y=T/GM+{T/j]M)t
[6.6.2]
On integration one obtains for a single Maxwell element, where rjM and GM stand for the viscosity and stress relaxation modulus, respectively, of a system having the properties of a Maxwell element T(t) = GMye-t/T^M
Equivalent circuits have been introduced in sec. III.3.6i.
[6.6.3]
RHEOLOGY
6.21
Figure 6 . 1 1 .
Maxwell
model of vlscoelastic behaviour, (a) Single clement, (b) Parallel set of Maxwell elements plus a spring.
and r r e | M = ?7M/GM is the relaxation time of the Maxwell element, as in [6.4.2], Note that r denotes stress whereas rrel stands for a relaxation time. For a set of parallel Maxwell elements and one element consisting of a single spring (fig 6.1 lb) one obtains for the total stress experienced by the material after a sudden deformation y is applied r(t) = y(Ge + ^G n e" f / T r d ' M i)
[6.6.4]
i=l
This gives for the relaxation modulus G(t), G(t) = G e +JTG n e~ (/rrel - M i
[6.6.5]
i=i
Next, it is convenient to introduce a modulus Junction H(rrej) in such a way that r 0 H(r re |)dlnr re | represents the contribution to the initial stress r 0 of elements having logarithms of relaxation times between lnr rel and (lnr rel + dlnr rel ) . If the number of Maxwell elements becomes infinite, one obtains Git) = Ge+ j H(r rel )e- t/r reid In rrel
[6.6.6]
For liquids, Ge will be zero as r approaches zero for large t. Usually H(rrel) is called the relaxation spectrum (spectrum of relaxation times), although strictly speaking it is a spectrum of modulus densities. A complicating factor in relaxation tests is that stress relaxation already starts during the finite time required to attain the applied deformation. This relaxation process will be more extensive in the case of fast relaxing bonds and/or slow deformation rates. The result is a slower relaxation process directly after the intended deformation has been reached, and a longer apparent relaxation-time. Moreover, this leads to the fact that the relaxation time can be calculated accurately only for t > > deformation time, roughly about ten times as large.
6.22
RHEOLOGY
6.6b Tests at constant stress. Creep In these tests a constant stress is applied instantaneously to the material and the ensuing strain measured as a function of time. For an ideally viscous material the strain will increase linearly with time and a viscosity can be calculated using [6.2.6] and [6.2.9]. For an ideally elastic material the observed strain will be instantaneous and will remain constant in time. A shear or Young's modulus can be calculated by using [6.2.5] or [6.2.8], respectively. The material will regain its original shape after removal of the stress.
Figure 6.12. Principle of a creep measurement for a viscoelastic material. At t = 0 a constant shear stress is (instantaneously) applied and maintained until t = te . The shear strain y is followed as a function of time. At t = te this stress is instantaneously removed. Idealized behaviour.
A particular type of test carried out by applying a constant stress is the so-called creep test. These tests are particularly useful for viscoelastic materials and allow the study of effects over long time scales. The applied stresses are such that no fracture occurs. For a viscoelastic material there is, as illustrated in fig. 6.12, first an instantaneous increase, AB, in the strain, the elastic response, followed by a 'delayed' elastic response, BC, and a viscous response, CD, which in the ideal situation is linear In time. The term, 'delayed elastic response' is used to indicate that it does not happen Instantaneously after sudden application of the stress. When the stress is suddenly removed at t = te, the purely elastic part of the stored energy is immediately released, after which the delayed part follows more slowly. In the strain vs time diagram these recoveries are reflected In the drops DF (= AB) and FG (~ BC), respectively. The energy dissipated due as a result of flow cannot be recovered; it gives rise to the remaining deformation GH (=CD'). From the slope the (apparent shear) viscosity can be calculated. Figure 6.12 Is idealized; in practice the sections AB and DF and BC and FG are not always identical. For some systems, DF decreases with increasing £e because of the slow breakdown of network structures.
6.23
RHEOLOGY
Usually, creep test results are expressed in terms of the compliance J[t) = y(t)/ r • For an ideally elastic material, J = l/G; however, for viscoelastic materials, J{t) * 1/G(t). General relationships exist by which J[t] can be converted into G{t), and vice versa , but to make this conversion accurate J{t) has to be known over quite a long time.
Figure 6.13. Kelvin or Voigt model of viscoelastic behaviour, (a) Single element, (b) set of elements in series with an additional Maxwell element. The springs are labeled with the appropriate stress/strain ratio and the dashpots with the stress/strain rate ratios.
A retardation spectrum can be calculated from J{t) in a manner analogous to the calculation of a relaxation spectrum from stress relaxation data. Consider for this purpose a Kelvin (or Voigt) element consisting of a parallel spring and dashpot (fig. 6.13a). If a stress is suddenly applied, the spring cannot respond immediately because of the resistance caused by the viscous flow in the dashpot (delayed elasticity). So the increase in strain is retarded. After cessation of the stress, the energy stored in the spring relaxes, again with a rate determined by the viscosity of the fluid in the dashpot. Behaviour like this is semi-solid. In the limit where the effective viscosity of the dashpot, ?/k —> 0 ideal elastic behaviour is attained. The total strain experienced by the model after sudden application of a stress, z , is y(t)=TJK(l-e
t/T
ret.K)
[6.6.7]
where r r e t K = J^TJK is the retardation time of the Kelvin element21. This equation can be derived by realizing that at any time the strains of the spring and the dashpot are the same; during the experiment the stress redistributes over the 11 J.D. Ferry, Viscoelastic Properties of Polymers, 3 rd ed. Wiley, (1980) chapter 3. Henceforth this book will be referred to as Ferry, loc. cit. In rheology it is customary to reserve the terms retardation and relaxation for creep and stress relaxation, respectively.
6.24
RHEOLOGY
two branches representing elastic and viscous behaviour, respectively. During the experiment r = rel + rvisc . This gives r= y/JK + riKdy/dt
[6.6.8]
Integration using r r e t K = Jk??K leads to [6.6.7]. For a set of Kelvin elements and one Maxwell element, characterized by a compliance J and a viscosity rj in series, the total strain after a sudden application of a stress r is given by n
X(t) = T J g + - t + r ^ J r l ( l - e " f / r r e t K i )
[6.6.9]
The compliance, J , represents the instantaneous elastic deformation of the material. For an infinite number of Kelvin elements (and considering that for most materials the range of retardation times to be considered is very wide), it is more convenient to choose a logarithmic scale and write ^ret)d7ret=^ret)dln7ret
I 6 - 6 - 10 !
where L[rTet) = f ret 7(r ret ) is usually called the retardation spectrum (spectrum of retardation times) although, strictly speaking, it is a spectrum of compliance densities. The creep compliance may then be written as J{t) = Jg+-+
JL(r r e t )(l-e- f / T ret)dlnr r e t
[6.6.11]
If the retardation spectrum is very low at long retardation times, say r r e t x , and stays so for t > 7 r e t x , the contributions of the term L(rret)e"t/Tret will be small. All significant contributions to the delayed strain have then reached equilibrium and the creep curve becomes linear. For a solid-like material the strain will be constant in time, but for materials with a liquid-like material strain it will increase linearly with time. The creep compliance is then given by J(t) = J g + - + j L ( r r e t ) dinr r e t
(exp(-t/r ret ) « l)
[6.6.12]
After unloading, the curve for the strain versus time has to be the inverse of the deformation curve after loading, except for the deformation as a result of flow. If either the relaxation spectrum or the retardation spectrum is known over the entire range of time scale, together with certain limiting values such asG e , J g and 7], the other of these two spectra can be calculated". Treatment of these interrelations is outside the scope of this chapter but we note the formal analogy with dielectric spectroscopy (see 1.4.4.31 and 32], the so-called Kramers-Kronig relations. 11
J.D. Ferry, loc. cit., ch. 3e.
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6.25
Figure 6.14. The stress r as a function of time t at three different shear rates y for a material exhibiting stress overshoot.
The relaxation modulus and the creep compliance are related by the relations1 t
jG(r)J(t-r)dr=£
[6.6.13a]
and ( jj{T)G(t-r)dr=t, o
[6.6.13b]
from which it follows that J{t) G(t) < 1 , and hence J{t) * 1/G(t). 6.6c Tests at constant strain rate A set strain rate is applied, usually as a shearing deformation, and the resulting stress is measured as a function of time. For systems with a yield stress T , T may display an overshoot (fig. 6.14). Such materials are first deformed 'elastically' until r exceeds r and the breakdown progresses. The material yields. Upon continued deformation the structure of the material is broken down further resulting in a decrease in TJ
with measuring-time (thixotropic behaviour). After a longer shearing time a
steady-state situation may be reached. If r (steady state) is plotted as a function of the shear rate, a graph characteristic for shear thinning behaviour is obtained. Often fracture experiments on solids or semi-solids are performed by deforming them in uni-axial compression or tension at a set strain rate. For a further discussion of these experimental methods see, e.g.2'. 'Overshoot' measurements are useful for the determination of the structure breakdown of plastic fluids upon large deformations.
6.6d Oscillatory measurements A small periodic, mostly sinusoidally oscillating stress, strain, or strain rate, is applied to the material at an angular frequency a> (rad s" 1 ), and the resulting strain 11
J.D. Ferry, foe. cit. ch. 3e. T. van Vliet, Rheological Classification of Foods and Instrumental Techniques for their Study in Food Texture, Measurement and Perception, A.J. Roscnthal, Ed., Aspen Publ. (1999) 65.
6.26
RHEOLOGY
or stress measured. In these experiments, the time is replaced as a variable by the frequency, a) . For viscoelastic systems, upon increasing a> the response tends to become increasingly more 'elastic'. Periodical measurements have, compared to continuous ones, the advantage that relaxation phenomena can be studied on one welldefined timescale which can be varied easily by changing the frequency. The mathematics of the elaboration of oscillatory Theological measurements is identical to that for dielectric spectroscopy, as discussed in sec II.4.8a. We will discuss below the derivation of the equations for an applied harmonic oscillation of a shear strain y y(t) = yQ cos mt
16.6.14]
where y0 is the maximum shear strain. For a linear viscoelastic material, the ensuing stress amplitude will, at small maximum strains, be proportional to the strain amplitude, and oscillate at the same frequency but out of phase with the strain (fig. 6.15). It is convenient to write the stress and strain as complex variables, which gives for the strain y=yoeio*
[6.6.15]
where y is the complex shear strain. The complex stress r is then given by f=
TQei(cot-8)
[6.6.16]
where r 0 is the maximum shear stress and 9 the phase lag between the strain and stress oscillation. Using Euler's law, e~ie = cos#-isin#, the complex shear modulus G = il y can be written as G = ^-e- ie =^_(cos<9-isin(9)
r0
[6.6.17]
To
The first, or real, part of the right hand side of [6.6.17] is the part which is in phase {0=0,
cos 9=1 and sin 9=0) with the strain. It is called the shear storage
modulus,
G', and is a measure of the amount of energy stored during a periodic application of a strain or stress. The out-of-phase {9={n + ^)n where n is an integer, c o s 0 = 0 and s i n # = l ) , or imaginary, component is the shear
loss modulus,
G", which is a
measure of the energy dissipated during the periodic application of strain or stress. The in-phase and out-of-phase contributions are sketched in figs. 6.15b and c, respectively. Mathematically, these two moduli are given by G' = — cos0 To
[6.6.18]
In fact, time and frequency can be interpreted as each other's Fourier-transforms, as is explained in Vol. I, appendix 10b.
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6.27
Figure 6.15. Response of a viscoclastic material to a sinusoidally varying strain, (a). Panels (b), (c) and (d) represent the elastic, viscous, and total, response of the stress. Schematic.
G" = ^-sin8
[6.6.19]
ro and G = G'-iG"
[6.6.20]
The ratio of r 0 / / 0 gives the absolute shear modulus G G =-^_ = V(G')2+(G")2)
[6.6.21]
The quantity G is also called the modulus of G , i.e., the modulus of a modulus. Also, G equals 6\ew, G' = |GCOS0, and G" = |Gsin0. Furthermore, the ratio G"/G' is called the loss tangent, or
6.28
RHEOLOGY
tan0 = ^H = 9:
G1
cos (9
|6.6.22]
where 9 is the loss angle. The quantity tan 6 characterizes the extent to which the behaviour of the material studied is elastic or viscous. This can be illustrated further by starting with [6.6.14] and also writing the ensuing stress in goniometric, language r{t) = r0cos(a>t + 6) = ro(cos0cos<wt -sin#sin<wt) = yQ[G\co)coscot -G"{co)sin cot], giving r(t)(visc) = r 0 sinOsincot. In fact, if the strain varies as cos cot it follows from differentiation that the rate of strain, y, varies as -co sin cot. As r(£)(visc) = rfy ,where r[ is the (dynamic) viscosity", one obtains the identity, G"=7fco
[6.6.23]
which gives tan6> = - ^ G'
[6.6.24]
for the loss tangent. As with the stress relaxation modulus, dynamic moduli can also be written as functions of the relaxation spectrum. As described in sec. 6.6a, for a single Maxwell element the relationship y=ye\ + Yvisc holds. Differentiation with respect to time gives drAlt G
+
r
| 6 6 2 5 ]
n
M
(compare [6.4.1]) or T
=rrel^+W
[6.6.26]
Substituting for y and r , [6.6.15] and [6.6.16], respectively, gives r o e- i 9 (l + i«r rel ) = icoGrTelyo
[6.6.27]
From this equation it follows directly that for the complex modulus
G = -2 Y0
=
E£L l + i«rre]
[6.6.28]
whereas for t h e storage a n d loss m o d u l u s
G'(ffl) =
Gco2r2 ^f-
[6.6.29a]
and G"(co) =
Gcorro, f^~
For a Newton liquid if =
[6.6.29b]
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6.29
respectively. For the loss tangent tan6>=—"— = -^L = - L WT
rel
T
rel
[6.6.30]
De
For more complicated materials having a spectrum of relaxation times, the relationship between De and tang1 is much more complicated; rather, a spectrum of De numbers is required. Equations [6.6.28a and b) can be generalized by considering an infinite array of Maxwell elements. The results are1'21
G'[co) = Ge+la>2T?elfl^)d\nTrel
G"(ffl)= 1
m H(
^ Y]
dlnr rd
[6.6.31a]
[6.6.31b]
In the above derivations the moment when the shear strain was at maximum was taken arbitrarily as the origin of the time measurements. Alternatively, the same derivation can be given with the maximum of the stress as the origin, resulting in the definition of a storage and a loss shear compliance3^". Another pair of quantities is obtained if the strain rate is taken as the starting point. This choice may be convenient if essentially liquids are considered41. Such a derivation results in the definition of a dynamic viscosity, rj' = G'la>, (the in-phase component of the complex viscosity) and an out-of-phase component of the complex viscosity, rj' = G"l a>. For an ideally viscous material, rf = r\ and TJ" = 0 . The main advantage of dynamic tests is that the contribution of both the elastic and the viscous component in the reaction of a material to an applied stress or strain can be determined over a large range of time scales (frequencies). For a relatively simple and ambiguous interpretation of the data, measurements should be available in the linear region. It is generally assumed that, over this range, measurements neither affect the structure of the material studied nor the structure, formation and rearrangement processes. Over recent years, techniques have been developed allowing the interpretation of measurements carried out somewhat beyond the linear region.
11
21
R.W. Whorlow, loc. cit. sec. 8.5
J.D. Ferry, loc. cit. chapter 3. 31 R.W. Whorlow, loc. cit. sec. 6.1. The choice between these alternatives is formally equivalent to that in electrodynamics in terms of complex conductivities or complex dielectric permittivities: sec sec. II.4.8a. The two are equivalent through the Kramers-Kronig relationships, which one to choose being a matter of practical convenience.
6.30
RHEOLOGY
6.7 Measuring apparatus
A large variety of instrumental techniques exists which allow the determination of Theological properties. In colloid science, the two main classes are tube- or capillary viscometers and rotational instruments. The latter can be equipped with a range of different measuring geometries such as coaxial cylinders, cone and plate, parallel plates, or vane geometry. Capillary instruments are well suited for determining accurately the viscosity of Newton liquids, and rotational instruments for studying rheological properties of non-Newton liquids and materials that can solidify in the rheometer (e.g., gels). For equipment having a parallel plate geometry they can also be used for studying solid materials. 6.7a Capillary viscometers
In a capillary viscometer, the flow rate of a material caused by a pressure gradient (Ap/I) is determined, or conversely. For a low-viscous fluid, flow is mostly due to gravity. Capillary viscometers are popular because they are cheap and easy to use and to thermostat. Moreover, they give reproducible data. The derivations of the equations to give the shear rate, shear stress, and viscosity, from a measured relationship between flow rate and pressure gradient parallel those given in sec. I.6.4d. The starting point is the Navier-Stokes equation for incompressible Newton fluids, which reads as (1.6. lb) p—
= i]W2v - grad p + pg
[6.7.1]
where bold characters denote vectors; p is density; v, velocity; p , pressure; g, acceleration owing to gravity. Dv/ Dt is the substantial derivative of v, which is defined as, Dv/Dt = dv/dt + {v-grad)v. Next, one can elaborate the required terms of [6.7.1] for flow through a cylindrical shell of length I and thickness dr, at a distance r from the axis, which is narrow enough for vz to be constant (fig. 6.16). For low dr, the momentum terms at the inner and outer sides of the shell are then equal; the remaining contributions are the viscous and pressure terms (2nrlTrz)r+dr-(2nrlTrz)r+2nrdr(p0-p1)
+ 2nrdrlpg = 0
[6.7.2a]
or
r— = I
C?- = \rrj —M dr dr[ \ dr ) \
6.7.2b
where Ap is an abbreviation for p0- p^+ pgl: p 0 -Pjis the pressure drop over the cylinder from z = 0 to z = I. If the cylinder is inclined at an angle, a, to the vertical, the gravity contribution must be multiplied by cos a. Integration of [6.7.2] with respect to r leads to
RHEOLOGY
6.31
Figure 6.16. Flow through a cylinder. Its length l» a .
and the following parabolic velocity distribution (fig. 6.17)
v=
' -¥r\l-lL)2}
|6 7 41
--
According to [6.7.3], the shear stress varies linearly from zero at the axis (where dvz Idr = 0 ) to its maximum value, Apa/2l, at the cylinder wall. The velocity has its maximum at the axis; vz{r = 0) = Apa2 I Aril, which is greater by a factor of 2 than the average velocity
( O = - L / j a ^ ( a 2 - r 2 ) 27trdr = T r x
'
Tta2 J 4rjr
'
[6 7 51
--
8T]1
r=0
As for the shear stress, the shear rate r
=^
=^ P
[6.7.6]
is also linear In r. The volume-flow rate Q Is obtained by integrating vz over the cross section of the tube. Q = ]2nrvzdr = -^P^87]l o
[6.7.7]
6.32
RHEOLOGY
Figure 6.17. Position of fluid particles a short time after being in the same cross-sectional plane. (a) Newton fluid; (b) power law fluid, n > 1 (full line), n < 1 (dotted line); (c) Bingham model.
For non-Newton liquids, the relationship between r and y has to be known for one to derive the extensions of [6.7.4] to [6.7.7]. To obtain such relationships it is convenient to express vz and Q in terms of r rather than r. Then ^-^-r^-^Jirrz] dr dr dr Ap Apr z
[6.7.8,
Integrating [6.7.8] from radius r to the wall, assuming no slip at the wall, one obtains
V =
* % J/(T-)dT
[67 91
'-
where raz is the shear stress at the wall (= Ap a/21) The volume-flow rate, Q, follows from integration over the cross section of the tube a
r
T
T 21 f 21 V
Q= \2nrvzdr= •I
23MrJ-f-
J
Ap
0
Ap J ^ °
1 21
/(rrz)dr f-dr Ap J
[6.7.10]
Since Ap/l = 2 r a z / a , we obtain
^
= ^rK2z/(Wdr
[6.7.11]
RHEOLOGY
6.33
As the right hand side of [6.7.11] is a function of raz and/(r r z ) only, a graph of Q/na3
as a function of rQZ is a unique curve, characteristic for a given material and
independent of the pressure gradient and tube radius. Equation [6.7.11] can be integrated when /(f r z ) is known and has a tractable form. The simplest case is, of course, that for a Newton fluid where J{rrz) = y = T/rj, with r\ constant, resulting in [6.7.7]. For a power law fluid obeying [6.3.1] or y = (l/fcjrd/n) = / ( r r z ) , where k is a constant, substitution in [6.7.11] and integration gives , Tf- T(2n+l)/n TUn az -^- =- i - f - ^ dr= [6.7.12] jta 3 r3 J k fc(3n + l) az o For a power law fluid, the flow profile will deviate from the parabolic form observed Q
for Newton liquids, the extent and direction depending on n (fig 6.18 panel b). For a Bingham fluid one has [6.3.3], from which y = (T-Ty)/r]B
for r > r , and
(x - r ) / ?/B = 0 for r < T . The contribution to the integral in [ 6.7.11 ] is zero for r < r so,
JL i Y ^ i i ^ ™
3
J r
^
d T = ^k_^LiKf
^B
^B
4
3
4
11
[6.7.13,
UzJ
Equation [6.7.13] was originally derived by Buckingham11. It does not lead to a simple linear plot of experimental data if Q/na3
is plotted as a function of Ap/l
or in
another form. The flow profile (fig. 6.17 panel c) has a truncated parabolic form, the more, so when raz is closer to r . The material near the axis moves as a plug, whose radius decreases as the pressure gradient increases. For raz approaching r , the flow profile will approach that for plug flow21. A well-known example is toothpaste squeezed out a tube. For liquids whose relationship between the shear stress and shear rate is not known, an expression for the shear rate at the wall of the tube can be derived starting from [6.7.11]. Rearranging and differentiating gives /(,az) = _
L ^ _ ( ^ l =^ ( dT
raz az{™
3
3
3+^
) ™{
l ^ ]
[6.7.14,
3
™ dI"Taz)
Equation [6.7.12] is often given in a simplified, formal form by introducing an effective shear rate at the wall, / w N = 4Q/na3 , obtained by combining [6.7.6 and 7.7], which stricltly only applies for Newton fluids, and by defining dlnfo = din ywN / d l n r Q Z ; [6.7.14] can then be written as
11
E. Buckingham, Am. Soc. Test. Mater., Proc. 21 (1921) 1154. Plug flow is like the movement of a piston through a cylinder.
6.34
AV = > - W N ( ^ P )
RHEOLOGY
I6-7-15!
The parameter b may depend on the range of shear rates considered. For a power law liquid, b = 1/n . A profound discussion on different types of capillary viscometers and the manipulation of these instruments has been given by Whorlow (1992) . Below, we will only discuss those that are most relevant for colloid science, and their major sources of inaccuracies. Well-known examples of capillary viscometers are the Ostwald and the Ubbelohde viscometers (fig. 6.18), which essentially consist of two bulbs connected by a capillary. In these viscometers, an accurately known volume of liquid flows, driven by gravity through the capillary, and the required time is measured. This time is a direct measure of the kinematic viscosity /J = r\l p where p is the density of the liquid (Ap is proportional to p and vz proportional to lit). Hence H = Ct
[6.7.18]
where C is a constant depending on the geometry of the capillary, and especially on its radius a, because of Poisseuille's law, C ~ a4. The bulbs should have the same diameter to minimize errors owing to surface tension. For opaque liquids, reverse-flow viscometers are available (Whorlow 1992)21. The main sources of errors are: Entrance effect. The velocity profiles in the bulb and the capillary are different; a certain length of the capillary is required to obtain the equilibrium velocity profile. Moreover, at the entrance of the capillary the flow is strongly convergent; it contains a large elongational component. This is incorporated into the calibration of the viscometers for Newton liquids, but for non-Newton liquids with a Trouton ratio larger than 3 (e.g., polymer solutions), this may lead to a larger energy dissipation at the entrance than is accounted for in the calibration. Kinetic energy correction. This concerns an exit effect. Depending on the construction of the viscometer and the properties of the liquid, part of the driving pressure is converted into kinetic energy of the liquid as it leaves the end of the capillary. The kinetic energy generated per second is of the order aQp{vz}2 , where or is a factor of unity, which depends on the velocity profile and on the exit conditions. To correct for this, Ap in [6.7.7] should be corrected by a term proportional to ap{vz)2 , which modifies it into S =^-(P-«P<^>2) and [6.7.16] to
11 21
R.W. Whorlow, toe. cit., ch. 2. R.W. Whorlow, loc. cit.. sec. 2.7.1.
[6.7.17]
RHEOLOGY
6.35
Figure 6.18. (a) Ostwald, and (b) Ubbelohde viscometcrs (not to scale).
c9 v = Clt
-
[6.7.18]
where Cj and C2 are calibration constants for a particular viscometer. The kinetic energy correction is, in practice, only important for low viscosity fluids at high flow rates. For long flow-times the correction is small. Turbulence. The flow velocity must be small enough to avoid turbulence, which starts at a Reynolds number ( 2pa(vz)l r\ (1.6.4b, table 6.2) above about 2100. Particle migration. Particles in a dispersion flowing through a tube exhibit a tendency to move to the centre of the tube, creating a particle-depleted layer with reduced viscosity along the wall. Partly this may be due to an excluded volume effect, because the particle centres cannot become closer to the wall than their radius, a . Another more significant radial motion of particles (the 'tubular pinch' effect) has been described by Segre and Silberberg11. According to this principle, the particles move radially to concentrate eventually at 70% of the inner cylinder radius. However, even an extreme redistribution of particles only produces a change in flow rate equivalent to the presence of a particle-free layer of thickness 0.7 a . Wall slip. This is due to the formation of a lubricated layer along the wall. Most common is apparent wall slip owing to, e.g., particle migration, or exudation of a low viscosity liquid, and alignment of polymer molecules. 11
G. Segre, A. Silberberg, J. Fluid Mech. 14 (1962) 115; J. Colloid Sci. 18 (1963) 312.
RHEOLOGY
6.36
Viscous heating. This will only play a role for highly viscous liquids at high shear rates.
6.7b Rotational
rheometers
Rheological measurements are often performed in rotational rheometers in which the test material is deformed between two coaxial cylinders, cones, plates, or cone and a plate. This group of instruments has two fundamental advantages over capillary viscometers. First, for an appropriate geometry, the shear strain and shear rate are almost uniform over the test material. Second, the sample can be sheared for as long as desired, allowing the study of time-dependent behaviour. On the other hand, these types of instruments are less accurate for determining the viscosity of low, or moderate, viscosity liquids, and are more liable to viscous heating at higher shear rates for moderately- and highly viscous materials. We shall use the term, 'rotational rheometers' instead of rotational viscometers, because these instruments can be, and are, used for more than just the determination of the viscosity. They are well suited for studying such properties as gel formation by oscillatory measurements, and other gel properties, in addition to creep tests. The most frequently used geometries are those in which the sample is sheared between two coaxial cylinders, or a cone and a plate, or two parallel plates (fig. 6.19). Other variants are discussed by, e.g., Whorlow1', and Ferguson and Kemblowski21. Most instruments are based on the rotation relative to the common central axis.
Figure 6.19. (a) Concentric cylinder; (b) cone and plate, and (c) plate-plate geometry.
Concentric cylinders In these types of measuring devices the sample is confined between two cylinder surfaces, of which one can rotate. In the Couette system the outer cylinder rotates, 11 21
R.W. Whorlow, loc. cit. ch. 3. J. Ferguson, Z. Kemblowski, Applied Fluid Rheology, Elsevier (1991) ch. 4.
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while in a Searle system
6.37
the inner one rotates. The equations relating rotation to
torsion are the same for the two systems. In the following derivation it is assumed that end and inertial effects may be neglected and that there is no slip at the cylinder surfaces. Moreover, we assume that the material elements move in circles around the common axis with an angular velocity co (rad s"1 ), which is a function only of the radial position r, and the flow is associated with a shear stress r , at radius r. The flow is conveniently described in terms of cylindrical co-ordinates, centered on the axis of the cylinders. The flow field can then be written in terms of the velocity components, vf = vz = 0 and u0 = r
= ^i^ro)
16.7.19]
Integration gives directly rr0=^
[6.7.20]
where Cj is an integration constant, which can be evaluated by considering the balance of moments acting on any cylindrical surface with radius r in the material. The net applied torque T being the product of the force times the lever arm at which it is measured, must be balanced by the moment owing to the shear force developed within the material T = Tre2nrLr,
[6.7.21]
where L is the length (height) of the cylinder. Equation [6.7.21] applies for all values of r for which R} < r < R2 . Combination of [6.7.20] and [6.7.21] gives for the constant, Cj = T / 2nL . Hence the shear stress at any point r in the test material becomes rre = ^ - ,
[6.7.22,
Just as for tube flow, the shear stress distribution in a Couette geometry is determined only by the equation of motion, and is hence independent of the test material properties. It is proportional to the reciprocal square of the radius. Only if there is a very small gap between the cylinders, compared with the cylinder radii, say a few percent, may rr9 be considered constant over the gap. Then, the two cylinder walls may be considered as parallel, and in the absence of wall slip the shear strain in the gap can be approximated by
11
R. Darby, loc. cit., sec. 4.3.2.
6.38
y =———a a2-ax
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[6.7.23]
and for laminar flow, the shear rate, / , by y = ———Q,
[6.7.24]
where a is the angular displacement of the rotating cylinder (rad), Q the angular velocity (rad s^1), and a a an average of ax and a 2 , the radius of the inner and outer cylinders, respectively. However, this requires very precisely positioned cylinders because the gap width has to be constant and large compared with the dimensions of the suspended particles. For many suspensions this requires very large cylinders.
Figure 6.20. Concentric cylinder viscometer. (a) Horizontal section, (b) deformation of a fluid element.
For obtaining an expression for the strain (rate) we consider two material points A and B, a distance dr apart; see fig. 6. 20 panel b. During shearing over a time dt, the radial line AB moves to A' B', whereas if the material had rotated as a rigid body around the central axis it would have remained radial at A'C . Because BB' = {r + dr) {co + dco) dt and BC = {r + dr) codt
the shear strain becomes y = B'C/CA' = {r + dr)da)dt/dr
[6.7.25]
where co is the angular velocity of the fluid element. In the limit where dr —> 0 , the strain rate is given by f =r ^ [6.7.26] dr By combining [6.7.22], [6.7.26] with an expression relating vTd to y, expressions can be obtained for the shear rate as a function of r, for a^
RHEOLOGY rd« =
6.39
^ _ 2%r\Lrl
dr
[ 6 ? 2 7 ]
Since co - 0 for r = ax and
tf
2
[r" 3 dr
[da> =
[6.7.28]
a,
0
Hence, n = ^-r\\-X\
[6.7.29]
4n77L^af a;|J or
n = -r^— \ \ - \ \
[6.7.30]
From [6.7.22] and [6.7.30] we obtain for the shear rate at the inner and outer cylinder walls rrO
2a%Q 2
7i = — =
9
9
[6.7.31a]
and 7
2
2a?Q = ^ ^
[6.7.31b]
respectively. In a similar, way expressions can be obtained for the shear strain in a Hooke solid. This leads to the following pair of expressions for the shear strain at the inner and outer cylinder walls T
,
re
2a a
2
[6732al
and
2a? a y =—-L—
[6.7.32b]
a£-a? respectively. From these equations it follows that over the gap, both y and y decrease as r 2 : so does rr(). For a power law fluid, r = kyn [6.3.1] or / = (l/fc)r(1M),
r^-H^J'n dr
k{2nLr2)
Integration as before gives
[6.7.33,
6.40
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' ^
(
\l/n (
,
,
—1 \~4r~-4jn
2nL) [af'
[6.7.34]
n
a^ )
Combining [6.3.1 ] and [6.7.34] gives for the shear rate 212 r=
n
ln
^ W ~«22'n)
'6'7'351
Substitution of a^ or a 2 for ai gives the shear rate at the inner- and outer wall, respectively. From [6.7.35] it follows that for shear thinning fluids, /decreases more strongly with r than for Newton fluids. For instance, for a fluid characterized by n — 0.4, y~r~5 . For a material behaving according to the Bingham model we have, r = r + r\By [6.3.3] or y =(T - T )/ r)B for T> r , and y = 0 for r < T . Three distinct situations can be distinguished. One extreme is that
r
>T
everywhere in the gap between the
cylinders, and the other is that t < T , implying that no flow occurs at all. The third case refers to situations where flow takes place only in that part of the gap where T>T
For the first case, substitution of [6.7.26] and [6.7.22] into y = (r-rM r\ gives r ^ =— ^ - ^ [6.7.36] dr
2nLr]Br2
T?B
[6.7.36] can be integrated to give T £2 = —
(l -g
1 ^1 T V «9 ln — r "—
[6.7.37]
So, a plot of Q against T will be linear with a slope (a^2 -a^\l AKLT]B , provided that T > 2nLa^Ty . As soon as T falls below this value, the yield stress at the outer side has not been exceeded, i.e., material near the outer cylinder remains solid. In the third situation, integration of [6.7.36] now only makes sense between the limits o = 0 at r = ax and co = Q at r = {T/2nLr ) . The result is n=—
-V
~\
^
l n
o\
[6.7.38]
Figure 6.21 shows the angular velocity as a function of the applied torque for a material that behaves according to the Bingham model. As this model describes idealized behaviour, such a relationship is seldom, if ever, observed. However, the formation of a stationary layer of material near the outer layer has been observed. Several torque versus angular velocity relationships have been derived for other types of flow equations. Moreover, approximate solutions have been given for the case
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6.41
Figure 6.21. Angular velocity-torque graph for a Bingham fluid. that the flow curve is not known. For a further discussion of these, see the textbooks . The main sources of errors in using a concentric cylinder geometry are as follows (beside obtaining the proper setting of the measuring bodies). End-effects. The resistance against deformation of the material below (and above) the inner cylinder is often neglected compared to the contribution of deformation between the two cylinders. A correction can be obtained by determining the torsion as a function of the immersed height of the inner cylinder. The end effect will be quantitatively different between Newton and non-Newton liquids, i.e., larger for shear thinning liquids and smaller for thickening ones. The end effect can be reduced by using specially designed measuring bodies, e.g., double concentric cylinders. Another solution is to use an inner cylinder with a cone at the bottom end, where the angle of the cone has been chosen in such a way that y below the cone is the same as in the gap between the concentric cylinders. Departures of the streamlines from circular geometry. In the section above we have assumed that the flow is laminar and occurs in a circular path around the axis of rotation of the instrument. Departure from such a flow regime may arise from inertia resulting from the centrifugal displacement. If the inner cylinder rotates, centrifugal forces cause the (relatively) fast-moving liquid near the inner cylinder to move outwards. Since this is not possible for the liquid, en masse, localized secondary flow patterns develop, the so-called Taylor vortices2). Such vortices are drawn in fig. 1.6.8a. (Note, however, that they should be as high as they are wide and that their shapes rather resemble squares.) For Newton fluids they occur when
11 21
R.W. Whorlow, loc. cit. ch. 3, p. 110-113, also see R. Darby, toe. cit. ch. 4. G.I. Taylor, Phil. Trans. Roy. Soc. A223 (1923) 289.
6.42
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\a9 -a. Re— -
>41.3
[6.7.39a]
a
2
where Re is the Reynolds number and v = Qr is the fluid velocity. Using pi rj=l/ ji, where // is the kinematic viscosity, gives v(a0 ^
312 -a,) l >41.3
jua2
[6.7.39b]
The flow is still ordered. Turbulence will occur at still higher Reynolds numbers, depending strongly on al/a2 (sec. 1.6.4b, table 6.2). Taylor vortices cause extra energy dissipation, resulting in an increase of the torque and of the apparent value of rj. They will not occur if the outer cylinder rotates, because centrifugal action then stabilizes the flow: v is zero near the inner cylinder and at its maximum near the outer one. Concentric cylinders cannot be used easily for dynamic measurements at high frequencies because the inertia of the oscillating body inhibits that. Generally, for simple shear, the expression for r(t) below [6.6.22] has to be replaced by 7(t)
+G
=T $ 2
=
A dt md^ + A dt2
> + -^7
[6 7 401
'-
co dt ^ fe^dx co dt v '
where m is the mass of the moving measuring body and A the adjoining area of the test piece. The inertia (first term on the r.h.s.) increases with ft)2 . Cone and plate geometry This measuring device consists of a circular plate and a cone, with a radius R, having its axis perpendicular to the plate and its vertex in the plane of the surface of the plate (fig. 6.22). Mostly, the point of the cone is flattened to avoid direct contact with the plate. If the angle 0c between the cone and plate is small (< 5°), T , y, and y are about uniform over the material inberween. This is the main advantage of this measuring geometry compared to the concentric cylinders or plate-plate geometries. It makes the cone-and-plate geometry well suited for the study of strongly shear thinning liquids and of materials with a yield stress. With respect to expressions for the strain and strain rate we will restrict ourselves to values of 0c that are so small that the difference between sin 9C and 0C may be neglected. Moreover, we assume that the material is moving in concentric circles around the axis of rotation of the cone. For 0C smaller than 5° this will normally be the case11. The strain y at radial distance r will be
11
R.W. Whorlow, loc. cit, p. 114.
6.43
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Figure 6.22. (a) Vertical section through an ideal cone and plate viscometer; (b) section when the cone is truncated. y = —^?—= — rsin0c 9c
[6.7.42]
and the strain rate y=
rQ
~ rsin6»c #c
[6.7.43]
Because y and / are uniform the shear stress
exerted on the cone (r e , in
spherical co-ordinates), will also be constant everywhere. The total torque on the cone is obtained from a
.
T
2
=j 2 " r r S d r = 3 M \
I6-7-44'
o When the flattened tip has a radius Rx, [6.7.44] becomes T=\2nr\lfdr = -n(a*-a3)%
[6.7.45]
R
i
As long as aL is less than 0.2 a the torque is reduced by less than 1% compared to the unflattened cone device. The cone and plate geometry also permits the determination of the first normal stress difference1'21. Plate-plate geometry For some materials it can be advantageous to use a plate-plate geometry (fig. 6.23) because the test materials are subjected to smaller stresses whilst they are introduced 11 21
R.W. Whorlow, loc. clt., sec. 3.8. K. Walters, Rheometry, Chapman & Hall (1975) ch. 4.
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6.44
between parallel flat plates than between a cone and a plate or between concentric cylinders. Moreover, one may set the distance between the plates freely, within certain limits, whereas for a cone- and plate- geometry this has to be done very precisely. Disc-plate geometry also has certain advantages for oscillatory studies because the formulae are more precise than those for the concentric cylinder and cone- and plategeometries".
Figure 6.23. Disc and plate viscomcter.
During a measurement, one of the plates is rotated around the axis through its centre. If the distance between the plates is H, the strain at radius r will be nv y=— H
[6.7.46]
and the strain rate y= — H The total torque on each of the plates is
[6.7.47]
R
T = j2nr r rzadr o
[6.7.48]
However, because /and /depend on the radius, rZOL is now a variable depending on r. For a Hooke solid and a Newton fluid, Tza changes linearly with y and y, respectively. The expressions are T =^
^ 2L
[6.7.49]
and T =
B ^ 1 2H
[6.7.50]
for a Hooke- and a Newton- type material, respectively. For non-Newton liquids one can obtain an expression for the torque by taking the
11
K. Walters, R.A. Kemp, in Polymer Systems, R.E. Wetton, R.W. Whorlow, Eds., Macmillan (1969) 237.
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6.45
shear rate as the variable in [6.7.46], Using [6.7.45], T= J2n^y3r](y)dy o
[6.7.51]
or Y3T
H
r3niy)dy
-^-=
[6.7.52]
0
This gives for the viscosity at ya 1 /
7 5'aa) =
d 3
( dT
2Ita lvdra
3T
+
)
T
=
(•,
1+ ^— 3
y a j 2na ya{
]
dlnT)
3dlnyJ
[6.7.53]
However, for non-Newton fluids it is usually better to use a concentric cylinder geometry, since shear rate variations throughout the sample and edge effects are larger for the plate-plate geometry. On the other hand, the latter geometry has advantages for the determination of normal stress differences and, as mentioned above, for the study of strain-sensitive materials, for example in oscillatory measurements. 6.8 Relationship between structure and Theological properties In the following sections the relationship between structure and rheological properties will be discussed. The emphasis will be on relatively simple colloidal systems. In this context, structure is defined as the spatial arrangement of the structural elements, i.e., the physical building blocks of the material. A structural element may be heterogeneous in itself, containing structural elements by itself. Rheological properties do not depend only on the distribution in space of the structural elements, but also on the interaction forces between them and, in various cases, also on the rheological properties of the structural element. Therefore, to establish the relationship between structure and rheological properties, theoretical relationships based on models are required, and often a combination of rheological results with those from other measurements is required. In the forthcoming sections the starting point will be a Newton liquid containing a low volume fraction of dispersed spherical particles, which are hard, smooth and impermeable for the liquid. Unless explicitly mentioned otherwise we will assume shear flow. The description of the relation between the structure of the material and the rheological properties will become more complicated when: - The shape and nature of the dispersed particles is no longer smooth, spherical, hard, or impermeable.
11
R.W. Whorlow, loc. clt., sec. 3.8.
6.46
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- The volume fraction of particles is so high that the flow pattern around a given particle disturbs the flow pattern around the neighbouring particles. Colloidal interaction forces between the particles cannot be neglected any more compared with the hydrodynamic interactions. When attractive, they may lead to aggregates and even to gel formation. Friction between anisometric particles may not be neglected with respect to hydrodynamic interactions. - The continuous phase is no longer a simple Newton liquid. Often it is not directly clear what the continuous phase is. For a dispersion of emulsion droplets in an aqueous macromolecular solution the water is the continuous phase for the macromolecules, but the macromolecular solution is that for the emulsion droplets. In other cases, the continuous phase can be a macromolecular gel with large particles dispersed (entrapped) in it. Table 6.2. Main factors determining the rheological properties of a liquid dispersion of particles. Type of interactions Between
Particles and continuous phase
Factors of importance
Nature
Of the particles
Of the continuous phase
Hydrodynamic
Volume fraction Size distribution Shape Surface smoothness Flow of solvent through particles Deformability
Rheological properties such as; Apparent viscosity Elasticity Yield stress
Flow situations such as; laminar flow: flow rate elongational component Turbulent flow: power density Hydrodynamic Particles
Colloidal
Hydrodynamic interaction between particle and continuous phase, insofar as the particles affect the flow pattern around each other. Size Hamaker constant Zeta potential Charge and charge distribution Adsorbed amount of low and high molecularweight substances
Via effect on particleparticle interaction Ionic strength Type of ions pH
Dielectric permittivity Solvent quality; in the presence of polymers: depletion flocculation
6.47
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A short overview of the main factors determining rheological properties of a liquid dispersion is given in table 6.2 The viscosity of macromolecular solutions can partly be considered along the same lines as that of solid dispersions. However, a few essential differences deserve separate consideration. First, many macromolecules are easily deformed owing to the flow and, secondly, permeation of the continuous solvent through the macromolecule can occur and, finally, macromolecules may mutually interpenetrate. In the following sections the above features will be considered. 6.9 The viscosity of dilute sols For many practical purposes it is important to know the extent to which the viscosity of a given solvent r]s increases if colloidal, or other particles are added. As long as the rheological behaviour is of the Newton type (laminar flow, high Peclet number, no links between the particles, etc.) the viscosity is the sole quantity required to describe the system Theologically. This viscosity can be measured by one of the techniques described in sec. 6.7. Table 6.3 Glossary of definitions and symbols for the viscosity of Newton liquids1'. Symbol
Name
n
(dynamic) Viscosity
Si-unit 2
Ns m
Viscosity ratio, or relative viscosity
-
Relative viscosity increment
-
Notes
= Pa , a)
"s
Reduced viscosity, or viscosity number ^inc/P
Ibid, on volume fraction basis
[77 ] = lim|^j
Intrinsic viscosity Ibid, on volume fraction basis
m 3 kg"1
m 3 kg"1
b)
-
b)
a) The obsolete c.g.s. unit, the poise, corresponds to 0.1 Pa s. b) Clearly, \n] = (q>/c){f]} = p r\, where p is the particle density. Analysis involves the relationship between the viscosity r\ of the sol and t]s, which can be described in some of the ways given in table 6.3, either on the basis of the mass concentration c or the volume fraction
We mostly adhere to the IUPAC recommendation in sec. 6.15a.
6.48
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subsections we shall discuss what that means. We may recall that, for a homogeneous fluid, sometimes the term kinematic viscosity, fi=rj/p
(in m 2 s"1, where p is the
4
density of the solution) is used. The SI unit is 10 times larger than the c.g.s. unit, the Stokes. Here, we shall avoid this quantity because we emphasize heterogeneous systems. In the literature two types of intrinsic viscosities occur, depending on whether the concentration is expressed as mass concentration c (in kg m~3) or as volume fraction
6.9a Einstein's law and its extensions The dispersion of particles in a liquid leads to an increase of the viscosity. Part of the liquid is replaced by solid, non-flowing material (we ignore here fluid drops for which the rheological boundary conditions are a topic of their own). The sheared fluid must make a detour around the particles, which leads to an extra energy dissipation which, in turn, is observed physically as an increase of viscosity. The solution to this problem of fluid dynamics is not simple. One of the difficulties is that the dissipation has partly a shear origin, and partly stems from dilation and compression. Other complications are that this dissipation depends on particle shape and concentration. The basic problem has been solved by Einstein. For a collection of hard spherical particles with radii which are large compared to those of the solvent, and assuming no slip at the SL border, creep flow, and such a dilution that adjoining particles do not interact hydrodynamically, we have the Einstein equation^ r]=ris(l + 2.S
W p = 2.5
lrj] = 2.5/pp
[6.9.1]
where q> is the volume fraction of the colloid and p the particle density. It has been a comfort for many that in his original paper Einstein made an arithmetical error, leading to a factor of unity instead of 2.5. One of the present authors learned about that when colleagues at the Department of Physics of Wageningen University carried out viscosity measurements in dilute suspensions and found that the experimental coefficient of cp strongly exceeded Einstein's original factor of one. However, a student of Perrin's had already found the same at the beginning of the 20th century, after which Einstein himself corrected his familiar equation. In [6.9.1] the size of the particles does not occur; only the volume that is blocked from flow counts. Nevertheless, the average viscometric particle volume VPMSC-^~
[6-9.2]
" A . Einstein, Ann. Phys. 19 (1906) 289; 34 (1911) 591; Kolloid-Z. 27 (1920) 137. R. Simha, Kolloid-Z. 76 (1936) 16, demonstrated [6.9.1] to also be correct for flow in a capillary.
RHEOLOGY
6.49
can be obtained, where a visc is the average uiscometric particle radius. This radius is slightly larger than the physical radius because, upon tangential displacement of a fluid with respect to a hard wall, a thin liquid layer behaves as if it were stagnant. This is the stagnant layer in electrokinetics. So, a yisc = aeikin ~ (a + d)
as
in sec. 3.9, with
the slip plane acting as the physical border between flowing and not-flowing solvent. Einstein's equation has also been invoked for the measurement of the thickness of adsorbed polymer layers, and in sec. 6.11 we shall discuss the dependence of the viscosity of macromolecules on concentration. Equation [6.9.1] has been extensively subjected to experimental tests and proved correct within a few percent for q> < 0.01 . The problem is to obtain perfectly impenetrable particles without hairs or asperities. Usually, measurements are carried out with decreasing (p until the linear range is attained. The situation becomes progressively more difficult when the particles are not spherical, because rotation then has to be taken into account. The point is that energy dissipation depends on the orientation, which in turn depends on the rate of shear. The extent to which this happens is determined by the rotational Peclet number Pe =— 1
[6.9.3]
D
r
where Dr is the rotational diffusion coefficient (s" 1 ); see sec. I.6.5g. Because of this, an additional method is required for obtaining the particle size, but information on the asphericity is then also obtainable. Examples of such additional methods include dynamic light scattering experiments11, theoretically accounting for higher terms in the series development as a function of cp and simulations2 . In practice it is common to adapt the Einstein equation empirically for nonspherical particles by replacing the factor 2.5 by v, where v depends on the shape of the particles, and also on the flow rate. Generally speaking, rods/prolate ellipsoids increase the viscosity more than do discs/oblate ellipsoids at equal concentration (fig. 6.24). Alignment of anisometric particles in the direction of the main flow-component lowers the disturbance of the flow pattern, and reduces the energy dissipation, and hence the viscosity. Because the extent of flow alignment increases with shear rate it leads to shear rate-thinning behaviour. If the changes in the extent of alignment with changing shear rate take longer than the reaction time of the apparatus the dispersion will also behave thixotropically. For Pe r —> °° particle alignment will be at a maximum, irrespective of flow rate, and the viscosity will be independent of shear rate (fig. 6.24).
11 21
M. Rasmusson, S. Allison and S. Wall, J. Colloid Interface Set 260 (2003) 423. F.M. van der Kooij, E.S. Bock, and A.P. Philipsc, J. Colloid Interface Set 235 (2001) 344.
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6.50
Figure 6.24. Intrinsic viscosity as a function of the rotational Peclet number. In elongational flow, the effect of strain rate is different. For this flow type an increase in elongational viscosity is observed with increasing strain rate starting at Pe r = 1, in which the effect is larger for prolate than for oblate ellipsoids. Finally, it should be repeated that [6.9.1] and its generalized form for non-spherical particles only holds for very dilute suspensions, i.e., for (rj/ rjs < 1.03). For higher volume fractions of spherical particles, a number of extensions to [6.6.1] have been proposed. There is something to be said for developing r]{(p) into a power series, such as; 77= Tjs{l + 2.5(p + k2(p2 +k3
[6.9.4]
similar to the virial expansion for the osmotic pressure. The (p1 term then accounts for binary particle interaction, the
11
L.A. Spielman, J. Colloid Interface Set 33 (1970) 562. See, T.G.M. van de Ven, Colloidal, hydrodynamics. Academic Press (1989); Adv. Colloid Interface Set 17(1982) 105. 31 R. Ball, P. Richmond, Phys. Chem. Liquids 9 (1980) 99. 21
J.W. Goodwin, R.W. Hughes, Rheology Jor Chemists; an Introduction, Roy. Soc. Chem. (2000).
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6.51
Table 6.4 Some semi-empirical extensions of the Einstein equation. Ref.
Eq.
Vand (1948)
Equation n=nB d + 2.5^7.35^)
1)
[6.9.5]
Simha (1940)
77= % (1 + 2.5^+3.73(p2)
2)
[6.9.6]
Guthand Gold (1938)
77= 77S (1 + 2 . 5 ^ 1 4 . 1 ^ ,
3)
[6.9.7]
4)
[6.9.8]
5)
[6.9.9]
6)
[6.9.10]
7)
[6.9.11]
Name and year
Mooney (1951)
( 2.5
(1.35 < Jc<1.9) Batchelor (1977)
77= n& (\ + 2.S
Krieger-Dougherty (1959)
I
DeKrulf etal. (1986)
77= 77 [l + 2.5
1) V. Vand, J. Phys. Chem. 52 (1948) 277; 2) R. Simha, J. Phys. Chem. 44 (1940) 25; Science 92 (1940) 132; 3) E. Guth, O. Gold, Phys. Rev. 53 (1938) 322; 4) M. Mooney, J. Colloid Set 6 (1951) 162; 5) G.K. Batchelor, J. Fluid Mech. 83 (1977) 97; 6) I.M. Krieger, T.J. Dougherty, Trans. Soc. Rheol. 3 (1959) 137; 7) C.G. de Kruif, E.M.F. van Iersel, A. Vrij and W.B. Russel, J. Chem. Phys. 83 (1986) 4717. [6.9.10]) and the 'self-crowding factor' [k in [6.9.8]) both depend on particle shape, dispersity and shear rate. So, it is not surprising that the applicability of these equations changes from system to system. In the literature there is a plethora of experiments and of theoretical analyses. In the absence of, or for weak colloidal interaction forces, a main factor determining k2 is the relative importance of the effect of hydrodynamic forces and of Brownian movement, as expressed in the translational Peclet number Pet=
^—[6.9.12] kT This gives the ratio of the time taken for a particle to diffuse over a distance equal to its radius divided by a time characterizing the flow-field (1 / / ) . For Pe t « 1 the distribution of particles is not, or is only slightly, altered by the flow. For that case, Batchelor derived [6.9.13] for dilute hard-sphere homodisperse suspensions 1
((p<0.l)l) r\= i]s(l + 2.5
[6.9.13]
11 G.K. Batchelor, J. Fluid Mechanics 41 (1970) 545, Ann. Rev. Fluid Mech. 6 (1974) 227, J. Fluid Mechanics 83 (1977) 97.
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For uniaxial extensional flow, k2 in [6.9.4] was found to be equal to 7.6. The factor 6.2 partly stems from hydrodynamic interactions and partly from Brownian motion (thermodynamic forces). Flow distorts the structure from its equilibrium form, and Brownian motion acts as an entropic restoring force driving the structure back towards equilibrium. For high Pe t (high shear rates), the equilibrium microstructure will be significantly affected by the flow, the Brownian contribution becoming less important. This will give rise to shear thinning behaviour, more strongly as the volume fraction becomes higher11. Then, k2 and higher terms are smaller than for low Pe t . 6.9b Electroviscous effects The term, electroviscous effect denotes the influence of charge on particles or macromolecules upon the Newton viscosity2'. Three classes can be distinguished, viz. the; (i) primary electroviscous effect, that Is the influence of an electric double layer on the viscosity in the Einstein region; (ii) secondary electroviscous effect, the influence of pair interaction, electric repulsion particularly on the viscosity; (iii) tertiary electroviscous effect, i.e., the influence on the (intrinsic) viscosity of solutions of polyelectrolytes, and polyelectrolyte-stabilized particles as a consequence of conformational changes. A typical example is the swelling/compression of coils as a function of pH and c s a j t . However, for these phenomena the term, 'tertiary electroviscous effect' is not generally used. They will be discussed in sec. V.2.4. The primary and secondary electroviscous effects both lead to a viscosity increase. The distinguishing feature is that the former is a typical singlet phenomenon (~ q>), whereas the latter is a pair interaction phenomenon (~ q>2 ). So, generally [6.9.4] can be extended for charged particles to become TJ= J]ll + 2.5(l + fcel x)(p + {k2 + Jce]2)(P2 + ...]
[6.9.14]
where ke] j and fcel 2 are respectively the (dimensionless) primary and secondary electroviscous coefficients. As before, 77S is the viscosity of the solvent, but as double layers play their roles predominantly in aqueous solutions T]s may invariably be identified with //w . The computation of neither kei j nor kel 2 is simple. Here, the discussion will be limited to the main features and some results. (i) The primary electroviscous effect essentially stems from the energy dissipation
11
J.W. Goodwin, in Colloidal Dispersions, J.W. Goodwin, Ed., Royal Soc. Chem. (London) (1982) p. 165. No confusion is expected with the viscoelectric effect, the influence of an external field on the viscosity of a liquid. This phenomenon has been invoked in the interpretation of electrokinetic potentials, see sec. II.4.4.
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caused by distortion of the double layer by shear. Computation of this dissipation requires insight into the composition of the ionic cloud around the central particle and the friction these ions experience upon displacement. The basic equations include the Navler-Stokes equation, Poisson's equation, and expressions for electroneutrality, and do not differ in principle from those employed in the theory of electrokinetics (see sec. II.4.6). One of the boundary conditions (for r —» °° ) is that the shear field is unperturbed by the particle; the other depends on the chosen double layer model; mostly Stern layer dynamics are ignored, so the particle with its Stern layer (r = a + d ) behaves as an impenetrable sphere with surface charge
^-j(eoe£)2
(Smoluchowski)
[6.9.15]
where K is the bulk conductivity. The fact that K occurs in the denominator can be understood intuitively: when the conductivity is high, it is difficult to build up nonequilibrium double layers. Smoluchowski's arguments were mainly intuitive; at that time little was known about double layers. Gouy theory stems from 1916; and what we now call C, was, in Smoluchowski's language, the potential of the shearable part of the double layer. As is now known, [6.9.15] is dimensionally and qualitatively correct, but quantitatively it overestimates the effect by several orders of magnitude. This became particularly evident from a careful study by Booth21, following the general lines described above. For low f and symmetrical electrolytes, Booth's result can be written as
(e eCi2 kel , = Jbca) ° „ K/?wa2
(Booth)
[6.9.16]
where f(Ka) is a decreasing function of tea , given by Booth. Such an expression can only be computed with some knowledge of electric double layers. It transpired that fbca) is << 1. Overbeek3 has given numerical illustrations. For very large Ka most of the hydrodynamics takes place beyond the double layer, so that the electroviscous effect vanishes. For low Ka , kel
1
can acquire values exceeding 2.5, but for very small
radii the Einstein model is no longer valid. We may consider as the limit for such small particles (ions) the fact that the system behaves as an electrolyte solution, for the viscosity of which we have the empirical Jones-Dole equation [1.5.3.5]
11
M. von Smoluchowski, Kolloid-Z. 18 (1916) 190. F. Booth, PTOC. Roy. Soc. (London) A203 (1950) 533. 31 J.Th.G. Overbeek in Colloid Science, H.R. Kruyt, Ed., Vol. I, Elscvicr (1952), sec. IX, 2c. 21
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ri=riw[\ + Acxl2+Bc+..)
[6.9.17]
In this (electrolyte) case the interaction term, Ac1/2 comes before the singlet contribution Be . This term is much less than unity (see table 1.5.9); even in 1 M electrolyte, Be never exceeds 0.4. Booth's theory has been generalized by Watterson and White11 to give a wider range of applicability for f. The Watterson-White results for (1 -1) electrolytes are in agreement with those obtained by Sherwood, following an alternative route21. The results are represented in graphical form. For low £ the results are identical to those derived by Booth; for higher f they tend to be lower. A remarkable feature is that under certain conditions kel
l
passes through a maximum, see fig.
6.25. Experimentally the maximum is not easily observable because it requires either very high £ 's (beyond what is usually attainable) or high Ka 's (where kel j is very low). We recall from chapter II.4 that similar maxima were also predicted for the electrophoretic mobility, but that these became less pronounced, or even vanished completely, if conduction behind the shear plane was taken into account (fig. II.4.3.1)). Rubio-Hernandez et al.3) suggested that agreement with experiments (latices) became better if the Watterson-White theory is extended by incorporating this additional conduction. In this connection it may be mentioned that Garcia-Salinas and de las Nieves
found, for sulphonate and carboxylate latices at low electrolyte concentration,
Figure 6.25. Primary electroviscous coefficient according to Watterson and White, loc. cit. y e k = F£/RT .
11
I.G. Watterson, L.R. White, J. Chem. Soc. Faraday Trans. II 77 (1981) 1115. J.D. Sherwood, J. Fluid Mech. 101 (1980) 609; E.J. Hinch, J.D. Sherwood, ibid 132 (1983) 337. 31 F.J. Rubio-Hernandez, E. Ruiz-Reina, and A.I. Gomez-Merino, J. Colloid Interface Sci. 226 (2000) 180; Colloids Surf. A192 (2001) 349. 41 M.J. Garcia-Salinas, F.J. de las Nieves, Langmuir 16 (2000) 7150. 21
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There is again an analogy to electrokinetics, where such maxima disappear if the proper conductivity is substituted; that is, by including the Stern layer conductivity, see figs, II.4.29 and -30. Generally speaking, there is room for more experiments. In those carried out (e.g., listed in ref. ) it is mostly found that in the singlet range the coefficient exceeds 2.5. For some systems, agreement with (Booth) theory was observed, for example with silica sols" and latices2', but in others the experimental kel [ exceeds the theoretical prediction too much. The caveat is that a perfect control of the surface chemistry is mandatory; swelling (of some oxides such as ludox3 , and short charged hairs on the surface (for latices) may also lead to substantial pH- and c salt -dependent coefficients of cp . For a review, see van de Ven4).
Figure 6.26. Orbits for colliding pairs of particles in a shear field. uncharged particles, separating orbits; ibid, closed orbits; charged particles. (Redrawn from Goodwin, loc. cit. p. 184, with permission.)
(ii) The secondary electroviscous effect, Is less studied than the primary one. This is not surprising because the window of q> where pair interactions become relevant, but multi-particle interactions have not yet, is narrow. The secondary electroviscous effect causes the excluded volume of the particles during a collision to be larger than that for uncharged particles. The particles stay at a greater distance from each other. As a result, there are no closed trajectories of the particles anymore during the collision (see fig. 6.26). The paths of collision and recession become different from each other. This departure of the streamlines gives rise to extra energy dissipation which is observed macroscopically as a higher viscosity. The most powerful expression, derived by Russel5' reads as:
[ 11
B l + 2.5
1
^
[6.9.18]
E.P. Honig, W.F.J. Punt, and P.H.G. Offermans, J. Colloid Interface Sci. 134 (1990) 169. J. Stone-Masui, A. Watillon, J. Colloid Interface Sci. 28 (1968) 187. 3) J. Laven, H.N. Stein, J. Colloid Interface Sci. 238 (2001) 8. 41 T.G.M. van de Ven, Colloidal Hydrodynamics, Academic Press (1989). 51 W.B. Russel, J. Colloid Interface Sci. 55 (1976) 590. 21
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where B =(ln—I V \na)
[6.9.19]
and a is a measure of the ratio between the electrostatic and Brownian forces (a » 1). The precise shape of a depends on the expression chosen for the interaction Gibbs energy, i.e. one of the equations of app. A.2a. Suppose that Gel{r) can be written as Ae~*T / KT , a equals Ge(r)//cT . The parameter a contains (y d ) 2 or a more advanced function of y d ; this parameter enters because interaction at fixed diffuse potential is assumed. These equations may be condensed into1 , U(y)= J1\l
+ 2.5(p+2.5(p2
+— l ^ a s J
^2
+
[6.9.20]
where rmin Is the minimum centre-to-centre separation between a colliding pair of particles. The reason why this parameter enters the equation is that upon shear the particles are swept towards each other until the electric repulsion prevents further approach. A transient, rotating doublet is formed. So, rmin is determined by the balance between shear and electrostatic forces. (When the particles are small, the random force of Brownian motion also has to be accounted for.) We note that /c el2 is very sensitive to r min , i.e., the computed effect depends sensitively on the correctness of the double layer interaction model. As rmin is a function of / , so is r\. In other words, this hydrodynamic particle interaction imparts a non-Newton contribution; the suspension becoming shear thinning. A further improvement of the theory leading to [6.9.20] is to describe particle interaction under shear by applying an appropriate regulation model (sec. 3.5). At high particle concentration and/or sufficiently low electrolyte concentration, rmln may become > 2a(pi^3 . Multiparticle interactions may ensue, which can be so strong that Brownian movements can no longer disturb the spatial arrangement of the particles. With increasing concentration and homodisperse spherical particles the degree of order changes from one which is analogous to a gaseous state, to an ordered liquidlike state (see chapter 5). When the pseudo-lattice spacing is comparable to the visual wavelengths, diffraction may lead to iridescence as is exhibited by concentrated homodisperse latices31. The systems exhibit elastic behaviour upon (fast) deformations, and/or a very high viscosity at (slow) deformations under low forces, so they behave viscoelastically. At larger forces, the particle arrangement will be destroyed and shear thinning behaviour is observed. Another remarkable aspect of these systems is that upon an increase in attraction between the particles the viscosity will drop. 11
W.B. Russel, J. Rheol. 24 (1980) 287. J.W. Goodwin, R.W. Hughes, Rheologyjor Chemists, Roy. Soc. Chem. (2000), eq. [3.60]. 31 J.W. Goodwin, in Colloidal Dispersions, J.W. Goodwin Ed., Royal Soc. Chem. (London) (1982) p. 165. 21
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6.10 Viscosity of concentrated dispersions of particles For hard sphere suspensions Batchelor derived [6.9.9] for the low shear rate limit". However, the applicability of [6.9.9] is limited up to cp = 0.15. Many semi-empirical equations have been developed for describing the viscosity as a function of
[6.10.1]
In this equation 77 becomes infinite for 1/k =
11 G.K. Batchelor, J. Fluid Mechanics 41 (1970) 545, Ann. Rev. Fluid Mech. 6 (1974) 227, J. Fluid Mechanics 83 (1977) 97. 21 R. Ball, P. Richmond, J. Phys. Chem. Liquids 9 (1980) 99.
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Figure 6.27. Illustration of the polydispersity effect for a binary spherical particle system with diameter ratio 5:1. Transition P -> Q illustrates the fifty-fold reduction in relative viscosity when a 60% (V/V) suspension is changed from mono to bimodal; P —> S shows the 15% increase in attainable phase volume for the same change in dispersity, retaining the same viscosity. (Redrawn from Goodwin, loc. cit.)
However, at stresses above the yield stress they start to flow. Then the viscosity is for a large part determined by the stiffness of the particles, as has been illustrated, for example, for starch particles by Steeneken11. Even more strongly than for semi-dilute suspensions at high Pe t (high shear rates), the equilibrium microstructure is significantly affected by the flow. At low shear rates and in the absence of gravity the distribution of non-aggregated particles in the suspension is random, owing to Brownian motion. This involves a large resistance to flow. With increasing shear rate, the flow induces an orientation of the particles with respect to each other, which allows them to move alongside each other more freely. At still higher shear rates, particles form layers separated by layers of the continuous phase. These layers act as a kind of slip film between the solid-like particle layers. This arrangement of the particles in a layer-like structure proceeds faster and more extensively when cp is higher, and when the particles are smoother and better homodisperse. The transition of the particles from a random distribution to an ordered layer structure gives rise to shear thinning behaviour, which is more pronounced when the volume fraction of particles is higher. This effect has been observed experimentally (fig. 6.28)21. As the formation of an ordered structure takes time, this phenomenon also leads to thixotropy. When shearing is stopped, the flow-induced 11 21
P.A.M. Steeneken, Carbohydrate Polym.ll (1989) 23. I.M. Krieger, Adv. Colloid Interface Sci. 3 (1972) 111
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Figure 6.28. Relative viscosity for 50% poly(styrene sulphate) latices in various solvents as a function of the Peclet number (derived from Krieger).
structure will gradually disappear, owing to Brownian motion. Flow-induced structures may also be formed in dilute dispersions by anisometric particles, as discussed in sec. 6.9.1. The formation of ordered structures might also lead to enhanced wall slip. Wall slip is a general phenomenon for suspensions because particles are depleted from the wall. For a suspension with (p < 0.25 wall slip can generally be neglected If the diameter/gap width ratio is more than ten times the particle diameter11. For higher
with y is then relatively smooth. Recent analyses and experi-
ments provide strong evidence that, In shear thickening systems, clusters are formed owing to hydrodynamie lubrication forces between particles in close contact. Orderdisorder transitions and shear thickening are considered to be independent phenomena. Shear layering, as may occur for suspensions stabilized by repulsive forces, will suppress such cluster formation, leading to the occurrence of a, 'sudden cluster formation21 above the order-disorder transition, resulting in a fast increase of 77app with y. The extent of shear thickening depends on
11 21
H.A. Barnes, J. Non-Newtonian Fluid Mech. 94 (2000) 213. J.W. van Egmond, Current Opinion Colloids Interface Sci. 3 (1998) 385.
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again at extremely high shear rates . Both the degree of shear thickening, and the slope with which the apparent viscosity increases against shear rate increase with
Figure 6.29. Flow curves for latices, dispersed in a variety of solvents with viscosities 77S varying from 0.018 to 14 Pa s. (Redrawn from R. Hoffman3'.) 6.11 Dilute and semi-dilute macro molecular solutions In solution, macromolecules can assume a variety of shapes, depending on the nature of the chain (flexible, stiff, branched, ...) and their interaction with the solvent (good or bad?). These differences are reflected in the viscosity and other rheological properties. Some shapes are sketched in fig. 6.30. In the present section solutions of flexible, uncharged
polymers will be discussed
for
the dilute or semi-dilute
range.
Concentrated macromolecular solutions follow in sec. 6.12, polyelectrolytes in chapter
' H.A. Barnes, J.F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier (1989) ch. 7. L.M. Walker, Current Opinion Colloid Interface Set 6 (2001) 451. 31 R.L. Hoffman, Trans. Soc. Rheol. 16 (1972) 155. 21
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linear macromolecular chains
coils; size and shape stochastically determined
flexible rod
helix
stiff rod
non-linear macromolecular chains
Figure 6.30. Sketches of some structures that dissolved macromolecules can assume. V.2 (with sec. V.2.4 dedicated to viscosity) and in sec. V.3.7 surface rheology is addressed. The Theological behaviour of dissolved globular proteins has some analogies to that of charged particles (sees. 6.9 and 10), but that of random coils is vastly different. The main difference stems from the fact that macromolecular coils are not hard, impermeable entities, but allow for fluid flow through them to an extent that is determined by the molar mass M and the quality of the solvent (via the FloryHuggins parameter %). In connection with this, distinction has to be made between the intrinsic volume (fraction) of the chains with (hydrodynamically) bound excluded volume (fraction) of chains with (hydrodynamically bound solvent. To that end it is appropriate to briefly review some basic elements of the statistics of polymer solutions. Part of this can also be found in sec. II.5.2 and more information is available in the standard texts 1 ' 2 . Expansion of coils is generally determined by statistics (i.e. by the probability of finding chain elements at a given position and in a given orientation), by the excluded volume and the solvent quality %. The simplest, hypothetical, case is that there is no excluded volume and that there is no net interaction between the segments (athermal solvent, % = 0 )• m that case Kuhn statistics apply, according to which the root-meansquare (r.m.s.) end to end distance of a chain, consisting of JV freely-jointed segments
21
H. Yamakawa, Modern Theory of Polymer Chemistry, Harper and Row (1971). P.J. Flory, Principles of Polymer Chemistry, Cornell University Press (1953).
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of length I obeys (r2)1/2=lN"2
[6.11.1]
For the radius of gyration, under the same limiting conditions,
To describe real systems a linear expansion coefficient a may be introduced. It modifies [6.11.2] to a =af — g 6
[6.11.3]
I )
Especially in a good solvent (% = 0 ) a » 1 because of the excluded volume effect. As the solvency decreases, the mutual attraction between the segments partly compensates the excluded volume and a is closer to unity. As a itself depends on N, the relation between a and N or, for that matter, between a and M, obtains a power differing from 0.5. Flory derived a5 -a3 «vNl/2 [6.11.4] where v is the dimensionless excluded volume parameter, introduced in [II.5.2.5], which equals u = l-2;r
[6.11.5]
For % = 0 (athermal solvent) v = 1, a is large, a 5 » or3 , so a ~ JV°-1. Although this is a small exponent, it does lead to large expansions when JV is large. It leads to the famous OTg ~ N 3 / 5 proportionality, see [II.5.2.8]. For % = 0.5 , i.e. under 0 conditions a=l and ag—JV1^2; here the expansion due to the excluded volume is just compensated by the segment-segment attraction and random-flight statistics apply. According to [6.9.1 ] we have for the intrinsic viscosity of hard particles [77] = 2.5/ p where p is the particle density (i.e., the density of the pure (undissolved) polymer. This equation would apply when the polymer chains were collapsed. In reality they are statistical coils, with a hydrodynamic volume Vh ~ a3 which is much higher than the collapsed volume Vc = Nl3. As a consequence, the volume fraction
Vc
[6.11.6]
N
The intrinsic viscosity is also higher by this factor. One could also write [t]\ = 2.5/pc , where the effective density pQ of a coil is lower than p by the same factor
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/r2\3/2
63/2a3
[6 n 71
w^Hr-^-tr
--
3
1
Here, <£>o is the Flory-Fox constant. For [77] expressed in m kg" , r i n m and M in kg mole" 1 0O = 2 . 5 x l O 2 3 . For a theta solvent ag~Mxl2 solvents a ~ aMxl2
where a~Mx,
and [?7]~M 1 / 2 . For better
with x = 0.1 under very good solvency condi-
tions, as discussed above. Now a0 ~ M 1 / 2 + x and [77] ~ M 1 / 2 + 3 X . [rj\ = KMa
[6.11.8]
which is kown as the Mark-Houwink equation, originally introduced as an empirical equation. In most cases a = 0.5 + 3x is in the range 0.5^a^0.8 . For heterodisperse polymers, taking [77] as additively composed of the various fractions, M has to be replaced by the viscosity-averaged molecular mass
leading to [T]] = KM«
[6.11.10]
Intrinsic viscosities apply to (extrapolated) zero concentrations. In practice there is room for expressions addressing finite concentrations. A common way is by writing a series expansion similar to [6.9.4] r}= T]s(l + lTfic + kHlT]]2c2 + ...)
[6.11.11]
where kH is the so-called Huggins constant. We can also extend [6.9.4], written in terms of volume fractions. If the series expansion for swollen coils is needed, obviously the volume fraction cpc occupied by the coils enters V=%(l
+ l%
+ ...)
[6.11.12]
However, as the extent of expansion is not usually a priori known, there is no other option than writing the expansions in terms of c. the corresponding intrinsic viscosity [77] is given in table 6.3. The Huggins constant is dimensionless, both in [6.11.11 and 12]. The trend is that JcH is larger for poorer solvents. For polyelectrolytes fcH will be discussed in sec. V.2.4a. Equation [6.11.11] can be used to obtain [77] by extrapolation to c -> 0 , usually by plotting {>]-1SV%C a s
a
function of c. An alternative empirical
way is via the so-called Martin equation In \ri~T]s
I c??s J
=ln[77] +
fcH[77]c
[6.11.13]
which can be derived from [6.11.11] by taking logarithms and approximating ln(l + x) ~ x . Note that [6.11.13] contains the logarithm of a dimension-having quantity
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which requires an additional constant term ~ lnc~', where cs is some standard concentration). Here, k'H is the Huggins constant as obtained from a Martin plot. Equation [6.11.14] was found to describe the r/[c) behaviour satisfactorily for a variety of polymers over a wide range of concentrations1 . The intrinsic viscosity can be considered as a measure of the hydrodynamic size of a macromolecule; hence ^['/L is a measure of the total volume fraction occupied by the macromolecules. The implication is that a system is fully packed at an overlap fraction
A way to estimate c* is the following. A homodisperse sol of fully-packed spheres has a volume fraction of 0.74. For a macromolecular solution the same is valid provided we choose cpc for the volume fraction ^=T
J V A V C
*^
[6.11.14]
which can be combined with [6.11.7] to eliminate a
which for qfc = 0.74 leads to c, =
L08
(kgnr 3 )
[6.11.16]
Dilute and semi-dilute polymer solutions exhibit shear rate-thinning behaviour. The main reason for this can be understood qualitatively as follows. In the absence of flow, a dissolved polymer will, on average, have a random coil structure. Upon shearing, liquid may flow through the coil, whereby each segment may experience a frictional drag f (in N S m"1) causing a small disturbance of the conformation of the polymer chain. The chain will try to rearrange itself by diffusion to regain its equilibrium conformation. This elastic response can be interpreted as being caused by an elastic spring. In the oldest models, the polymers were considered to be free draining31; later, the hydrodynamic interaction between the beads was also taken into account41. In both models the polymer is described as a chain of beads each representing a part of the molecule, which are connected by entropic springs. The movement of (parts of) such chains can, just as for a vibrating string, be described by a number of " F. Sakai, J. Polym. Set A2 6 (1968) 1659. In chapter II.5 the symbol (pov was used for this quantity. 31 P.E. Rouse, J. Chem. Phys. 21 (1953) 1272. 41 B.H.Zimm, J. Chem. Phys. 24 (1956) 269.
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movements, each characterized by a characteristic time, r . Depending on the lengths and spring- stiffnesses of the parts considered, the springs will relax with different relaxation times. Mass diffusion of the whole chain is the p = 0 mode. Higher modes involve bending and rotational motions of parts of the chains, with a characteristic relaxation time r . According to the Rouse model, the longest relaxation time, fj, is given by fJVa2 r , - - ^
[s,
[6.11.17,
where JV is the number of beads per molecule. Shorter relaxation times are given by Tl I p 2 for 1 < p < N . Equation [6.11.17] shows that r; is proportional to M 2 . Upon shear, a polymer molecule is continuously rotated and elongated in one direction and compressed in the other one (see sec. 6.2). This has the result that the solvent is locally expelled from it and locally imbibed. At low shear rates the molecule is fully deformed twice during every completed rotation. This causes additional energy dissipation and hence the viscosity is relatively large. With increasing shear rate the time scale of the deformation (roughly \/y) becomes shorter than that of the longest relaxation time, implying that the molecule no longer deforms fully during a rotation, leading to lower energy dissipation and consequently to a lower viscosity. By using the Rouse-Zimm theory the intrinsic viscosity of a polymer solution at low shear rate can be expressed by using characteristics of the polymer molecule. For the elaboration, see the original literature and Goodwin-Hughes (toe. cit.) sec. 5.6.2.
6.12 Concentrated macromolecular solutions Macromolecular solutions with concentrations far above the overlap concentration c* are considered to be concentrated. The boundary is often taken at c[rj] ~ 5-10 with [77] as a reciprocal concentration. The marked difference between dispersions of hard particles and random coil macromolecular solutions is that, in the latter, coils can interpenetrate. In the concentration region were coil overlap starts the expanding effect of a good solvent becomes more and more screened by the segments from neighbouring molecules and the chains begin to collapse back to the dimension they have in a 0 solvent11. As discussed above, the value of c [77] can be considered as a measure of the extent of overlap. This variable can be used to describe the dependence of the relative viscosity rj/rjs on the concentration and molecular weight. Using equation [6.11.10] for [7;], one can write -^- = / ( c M a )
11
J.W. Goodwin, R.W. Hughes, loc. cit., chapter 5, p. 183.
[6.12.1]
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Figure 6.31. Double logarithmic plot according to [6.12.1] for various poly(vinylchloride) solutions in cyclohexanone. (Redrawn from C. Blom et al., loc. cit, fig. 3.14.)
Figure 6.32. Viscosity of some polymer melts as a function of molecular weight, a) poly(dimethylsiloxane), b) poly(isobutylene), c) poly(butadiene), d) polyfmcthylmetherylate), e) poly(vinyl acetate), f) poly(stryrene). The constants for the horizontal and vertical axes are different for each polymer. (Redrawn from C. Blom et al., loc.)
Experimental results show that the viscosity of of given polymer solutions at various M and c at low shear rate reduces to a single master curve if plotted as function of cMa
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(fig. 6.31)". For larger values of c Ma , the relationship between log// and logcMa becomes linear, so that one may write 77 ~ (cpMa)" = c«M^
[6.12.2]
where f} has been found to be a rather universal constant, with the value 3.4 for most polymers. For a = 0.6 , a would be 5.7; however this value varies much more with the nature of the macromolecule than /? does (fig. 6.32). At concentrations far exceeding c* , so-called entanglements are formed between the molecules (fig. 6.33). It is generally assumed that, at concentrations and/or molar masses below the kink in curves of log// versus logc or logM , entanglements do not play a role, whereas above that, they do. The critical concentrations c* and molecular weight M* at which the kink is found are related in the sense that c* M* is a constant. At concentrations above the kink, the concentration of polymer in the solution is fairly even, down to distances between inter-chain crossings, at the so-called correlation length [II.5.11]. In concentrated solutions, this concerns length scales clearly far below the size of the polymer coil; rather this length is of the order of the segment length; it acts as a characteristic length scale. In this concentration regime, simple mean field arguments can be invoked to describe the rheological properties of the system. We will limit ourselves to a qualitative discussion of the effect of entanglements on the rheology of polymer solutions, considering a more profound discussion to be beyond the scope of this chapter. For a more quantitative discussion we refer to refs.2'3'41. Entanglements will restrict the motion of the polymer, and therefore increase the relaxation time. For flow to occur, entanglements have to disentangle. This costs a relatively large amount of energy dissipation which is measured macroscopically as a high viscosity. As the number of entanglements increases strongly with polymer con-
Figure 6.33. An entanglement.
Taken from C. Blom, R.J.J. Jongschaap, and J. Mellema, Inleiding tot de Reologie, Kluwer (1986). J.W. Goodwin, R.W. Hughes, Rheology Jor Chemists, an Introduction. Roy. Soc. Chem. (2000) chapters 5 and 6. 31 P.G. de Gennes, Scaling Concepts in Polymer Physics. Cornell University Pressf 1979). 41 M. Doi, S.F. Edwards, The Theory of Polymer Dynamics. Oxford University Press (1986).
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centration, and at constant weight concentration with polymer dimensions, this explains qualitatively the strong dependence of viscosity on concentration and molecular weight. At low shear rates and steady state conditions, as many entanglements will be formed by Brownian motion as are disentangled, per unit time, and Newton viscosity ensues, i.e., r\ is independent of the shear rate (first Newton plateau) (fig. 6.34). With increasing shear rates the extent of disentangling increases, and above a certain shear rate this can no longer be compensated by entanglement formation. The time available for formation of new entanglements by Brownian motion during shearing is roughly the reciprocal of the shear rate y, whereas the time needed for disentanglement is less dependent on y. The latter time is determined by the relaxation time of the polymer molecules, which is roughly proportional to the viscosity at the shear rate involved. A new steady state develops in which the number of entanglements between the polymer molecules is lower. This implies that, for further shearing to occur, fewer entanglements have to be loosened, resulting in lower energy dissipation and so in a lower apparent viscosity. Moreover, the energy dissipated during disentangling will also become lower, it is roughly proportional to the prevailing apparent viscosity. The end result is that both the actual number of entanglements present per unit volume during shearing, and the amount of energy dissipated in the disentangling process, decrease with increasing shear rate. The polymer solution exhibits shear rate-thinning behaviour. At high shear rates the time available for formation of entanglements becomes too short, and the polymer molecules remain fully disentangled. Then, the viscosity will be much smaller than at low shear rates, and again independent of shear rate implying, Newton behaviour (second Newton plateau, see fig. 6.34). At constant weight-concentration the extent of shear rate-thinning increases with concentration and with molecular weight (fig 6.35). As the relaxation time for the formation of new entanglements decreases with viscosity, and also does so with polymer concentration, the shear rate at which the shear thinning region sets in will increase with decreasing polymer concentration.
Figure 3.34. Apparent viscosity of a 1 % guar solution in water as a function of the rate of shear, exhibiting two Newton plateaus.
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Figure 6.35. Apparent viscosity of a 2% Na-carboxymethyl cellulose in water solutions as a function of the rate of shear. Me is the weight-averaged molar mass (courtesy H. Beltman).
Another aspect is that disentangling and reformation of the entanglements will take some time. This may cause concentrated polymer solutions to be thixotropic. The picture given above described a concentrated polymer solution as being a temporary network that is destroyed during flow. The concept implies that, upon fast deformations which are too small to cause disentangling, the system will have solidlike (gel-like) properties whereby the entanglements behave as the temporary crosslinks. Various theories have been developed to estimate the relaxation time of these crosslinks. Rather successful is the reptation approach. The central idea is that a polymer molecule in a concentrated polymer solution cannot move freely in all directions, because of the topological constraints imposed by its neighbours, but that it can move along the contour of its own chain in the 'tube' formed by its neighbours11. The chain is folded within the tube (fig. 6.36) and the motion of these 'folds' allows the molecule to diffuse through the tube in a twisting, snake-like motion. De Gennes dubbed this process 'reptation'2'. Related to this process, two dominant relaxation processes can be distinguished, a short-time process due to fluctuations of the polymer within the tube and a longer-time process related to the disengagement of the polymer from the tube. It is generally assumed that the longest relaxation time of the former process can be described by the longer relaxation time according the Rouse model [6.11.17], whereby for N£ the friction coefficient of the whole chain in the tube must be taken. Adaptation of the tube to an applied strain or stress takes much 11 21
M. Doi, S.F. Edwards, The Theory of Polymer Dynamics. Oxford University Press (1986). P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press (1979).
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Figure 6.36. Visualization of reptation (modified after Goodwin and Hughes, loc. cit., p. 198).
longer. The tube disengagement time is given by
[612 31
^=i§
-
where L is the length of the tube. One should note that the tube disengagement time is proportional to the square of the tube length, and not to the root mean square end to end distance, or to the radius of gyration, as would hold for the longest Rouse relaxation time. Since both L and JVf~M, equation [6.12.3] predicts rd to be proportional to M3 , while rx [6.11.17] is proportional to M2 . Both relaxation processes differ considerably with respect to their characteristic relaxation times. At times between the two relaxation processes the polymer behaviour is independent of time. In this region, gel-like behaviour prevails, with a modulus GN roughly independent of frequency. Assuming that the viscosity is dominated by the reptation mode, one can deduce for the viscosity at / -» 0 2
n(r -> o) = Y^G N r d
[6.12.4]
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Figure 6.37. The storage modulus as a function of frequency. Trends predicted by the reptation model. Arbitrary units, logarithmic scale.
As GN is independent of M (see sec. 6.14), [6.12.4] predicts j]{y = 0)~M3. The deviation from the experimentally observed M3 4 is probably caused by tube length fluctuations11. Figure 6.37 illustrates the GN{a>) trend predicted by the reptation model. 6.13 Effects of colloidal interaction forces The discussions in sees. 6.9 and 6.10 were mostly limited to systems in which colloidal interaction forces could be neglected in comparison with the effects of Brownian motion and mechanical forces due to imposed flow. The effects of dominant repulsion forces between the particles have already been discussed for the case of electrostatic repulsion in sec. 6.9b, in the part on the second electroviscous effect. If the repulsion is of a steric nature the main effect is a small increase in the effective volume fraction of the particles, tpe{{ , equal to {{a + h)/a}3
S.T. Milncr, T.C.B. McLcish, Phys. Rev. Letters 81 (1998) 725.
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ductory remarks, we will discuss the effect of flow on doublet formation and break-up, and afterwards that for large floes. Irrespective of whether doublet or floe formation is transient or permanent, its direct result is an increase of the effective volume fraction. Even in a doublet, part of the continuous phase is hydrodynamically trapped in the aggregate, and at the macroscopic scale this will lead to an increase in the viscosity. This effect will, in general, be stronger for larger floes. A second effect is that breaking of bonds between colloidal particles also costs energy and therefore leads to a higher viscosity. However, the amount of energy involved in breaking bonds between the particles is usually much lower than the extra energy dissipation as a result of the flow process. For a suspension of particles with a diameter of 1 |im and
[6.13.1]
The translational Pe number defined in [6.9.12] is obtained by multiplying [6.13.1 ] by a, and dividing by the Brownian energy kT. Equation [6.13.1] shows that the hydrodynamic force increases linearly with shear rate and viscosity, and thus with shear stress. The result is that, for particles aggregated by a relatively small attractive force, the aggregate will break up above a certain shear rate, whereby the exact shear rate will also depend on aggregate structure (see below). The prediction of aggregation and break-up is more complicated for suspensions in which the interparticle interaction is characterized by repulsion at intermediate distances and attraction at small distances between the particles. Besides the size of the different forces one has to consider the time during which these forces act in a certain direction. As discussed in sec. 6.1 above and in 1.6.4, a shear flow consists of a
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rotational and an elongational part. If a particle catches up another one in a shear flow field, they are not only pressed together, but also start to rotate around each other with a characteristic time roughly equal to 1 / y. During rotation, the effect of the elongational flow component changes from pressing the particles together to pulling them apart. If, during the time that the particles are pressed together, they do not approach each other so closely that they have passed the repulsion barrier they will separate again. Besides, by the colloidal repulsion forces at intermediate distances the rate of mutual approach of the particles is also limited by the rate at which the continuous phase can flow out of the small gap between the particles. At small distances, the shear rate of the liquid from the gap between the particles will already be high for low approach rates implying a high energy dissipation. The phenomenon leads to the hydrodynamic correction to the rate of coagulation which has been treated in sec. 4.5. For two spheres the effect becomes substantial, i.e., a limitation of the effective diffusion coefficient by more than a factor of two if the separation is less than 0.25 of the radius, and by a factor of ten if the separation is about 3% of the radius. For details see chapter 4. We have seen that increasing the shear rate has two main effects on the rate of aggregation; namely, increasing the force with which particles are pressed together, and reducing the time they are pressed together. For most systems the former effect, which is proportional to a2rjy, is the most important. The latter causes the orthokinetic aggregation to be proportional to y08 rather than to y as predicted by simple theory. For a review of this matter see Elimelech et al.11. Next, we consider suspensions in which particle-particle interaction is characterized by a secondary minimum at larger particle separations, a primary minimum at small separation, and a repulsion maximum in-between, i.e., a typical DLVO- or
Figure 6.38. Apparent viscosity of a sol with DLVO or DLVOE-type interactions as a function of the rate of shear.
M. Elimelech, J. Gregory, X. Jia and R. Williams, Particle Deposition and Aggregation, Butterworth-Heinemann (1995).
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DLVOE- type of interaction (see sec. 3.9). Moreover, we assume that, at rest, all particles have formed doublets by aggregation in the secondary minimum. For such systems a plot of the apparent viscosity as a function of shear rate could exhibit a form shown in fig. 6.38. At low shear rates the doublets will move in layers as much as possible. Both the doublets and the spheres forming the doublets will rotate. This phenomenon is possible because smooth spherical particles can rotate independently of each other if bound in the secondary minimum. Compared to a rigid doublet, the apparent viscosity will be higher, as a result of the extra contribution to the energy dissipation by the individual spheres. At higher shear rates, doublets will break up, the faster the less deep is the secondary minimum. The apparent viscosity reduces to that of a suspension containing non-aggregated repelling particles. With increasing shear rate, the apparent viscosity will continue to reduce somewhat until hydrodynamic forces become large enough to cause aggregation in the primary minimum. Then, independent rotation of the constituent spheres in the doublet is inhibited, causing the apparent viscosity to remain lower than at low shear rates. At very high shear rates such doublets may be broken up again. Then there will be continuous formation and break-up of doublets. Mostly, shear rates required for disaggregation of particles aggregated in the primary minimum have to be much higher than those needed for causing aggregation, owing to the asymmetry in the force-distance curve. It is recalled from sec. 3.9 that, according to DLVOE, the primary minimum is less deep than predicted by DLVO. Rheological studies can therefore be helpful to discriminate. By way of illustration, table 6.5 gives calculated shear rates needed to give aggregationdisaggregation in a suspension of latex particles with DLVO-type interactions. Experimental results showed that boundary shear rates increased with electrolyte concentration and particle diameter1'. Table 6.5. Calculated shear rates giving aggregation-disaggregation of latex particles of 1 [im diameter in 10 mol dm" 3 sodium chloride (source, J.W. Goodwin, Some Uses of Rheology in Colloid Science in 'Colloidal Dispersions', J.W. Goodwin Ed., (1982) p. 190). C (mV)
Secondary doublets Singlets to primary Primary doublets to singlets to singlets doublets yfs- 1 )
rts"1)
fis-1)
h
min = 4 nm
h
min =
0 4
n m
5
75
0.1
1,300
5,000
5xl0
50
0.2
330
5,000
5xlO5
25
0.3
20
5,000
5xlO5
11 T.G.M. van de Ven, S.G. Mason, J. Colloid Interface Set, 57 (1976) 505; G.R. Zcichncr, W.R. Schowalter, Am. last. Chem. Eng. J., 23 (1977) 243.
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For most systems, not only doublets will be formed, but also larger aggregates; at rest, even a continuous particle network may be formed, which provides the undisturbed system with gel-like properties. These systems behave like solids at low shear stresses, but exhibit shear thinning and normally thixotropic behaviour at larger stresses. The yield stress is determined by the spatial structure of the network and the interaction between the particles (see sec. 6.14). A complicating factor is that the aggregate structure often depends on the aggregation process, and may be affected by the previous flow regime. Modeling of the apparent viscosity as a function of shear rate often requires three constants; the low shear rate viscosity plateau r]0, the high shearrate viscosity plateau n^ , and a critical shear stress or shear rate. The last mentioned characteristic determines the region in which the transition from low hear rate to high shear rate behaviour occurs.11 The resulting equations are of the shape 7 h ~ '? V(Y) = Va,+— — n
1 + (cPe)
[6.13.2]
where n is a constant and c depends on the details of the chosen model. To include thixotropy, a memory function should also be included. In the 1980's some theory for shear rate-thinning was developed in which the aggregate structure Is more explicitly accounted for. Computer simulation and experimental work have shown that aggregation of spherical particles at rest leads to floes with a fractal structure2^. It is a characteristic of fractal structures that they are selfsimilar irrespective of the length at which they are considered, and that the number of particles nf in the floes scale with the floe radius, a f , as nf~ -M
[6.13.3]
where df , the so-called fractal dimensionality, is smaller than 3. It is a measure of change of packing density as a function of the size of the floe. Equation [6.13.3] can only be used if df is independent of floe size, i.e. if the structure is self-similar. Otherwise stated, the packing must be scale invariant. The average radius of the primary particles a is taken as the lower cut-off length for the fractal geometry to apply. Equation [6.13.3] implies that the floe density decreases with increasing size. Real floes are, in general, not exactly self-similar, i.e. they are not exactly scale invariant like deterministic fractals, i.e., parts of the floe are not an exact copy of the floe itself. However, on average for practical reasons [6.13.3] can often be applied because it
11 21
J.W. Goodwin, R.W. Hughes, loc. cit. sec. 6.2.1. P. Mcakin, Phys. Rev. Lett. 51 (1983) 1119; M. Kolb, R. Boltet, and R. Jullien, Phys. Rev.
Lett. 51 (1983) 1123; D.A. Weitz, J.S. Huang, Self-similar Structures and the Kinetics of Aggregation of Gel Colloids, in Kinetics of Aggregation and Gelation, F. Family, D.P. Landau, Eds., North Holland (1984) p. 19-28.
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Figure 6.39. Bremer).
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Distinction between (a) deterministic and (b) stochastic fractal (courtesy L.
offers the possibility of describing random looking geometries and because computersimulation and experimental work have shown that the value of df is related to the mechanism of aggregation. For rapid aggregation, called DLCA for diffusion limited cluster aggregation, one finds df = 1.75, and for slow aggregation due to a lower sticking probability (called RLCA, for reaction-limited cluster aggregation), df = 2.01'. In both models it is assumed that particles aggregate to doublets and these to larger clusters, etc. For particle-cluster aggregation, dfwill be higher. For cluster-cluster aggregation df was found to increase systematically with the Fuchs stability factor (see sec. 4.5.7) in the range from 1-50 to about 2.4 . The latter value is in agreement with experimental values for df if a substantial activation energy for aggregation was present. The volume of a floe scales as (af / a) 3 so that the volume fraction of particles in a floe
11
P. Meakin, Adv. Colloid Interface Sci. 28 (1988) 249.
21
M. Mellema, J . H . J . v a n O p h e u s d c n , a n d T. v a n Vliet, J. Chem.
Phys.
I l l (1999) 6 1 2 9 .
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6.77
[6.13.4]
U) implying that
'
[6.13.5]
For high interaction forces and sufficiently high cp, a gel will be formed, while in the case of weakly interacting particles the system will exhibit a low shear rate Newton viscosity. Here, we consider the latter situation and, for the strongly interacting particles, the situation that the shear stress is larger than the yield stress. Breakdown of the floes at higher shear rates, as a result of hydrodynamic forces, will lead to a lower floe radius and so to a lower
f r
~ A ^
[6.13.6!
where Fc is the bonding force between the primary particles, and x an exponent which depends on the structure of the fractal cluster. For brittle clusters, undisturbed after formation, x will be 4/(3-d f )), whereby the numerical factor 4 depends on the precise way in which the fractal cluster deforms and fractures31. The viscosity of the suspension is given by
where the left term of the right hand side corresponds with the Krieger-Dougherty equation [6.9.10] and the right term accounts for the yielding of a gel network, which
11
L.L. Hockstra, R. Vreeker, and W.G.M. Agterof, J. Colloid Interface Sci. 151 (1992) 17. A.A. Potanin, J. Colloid Interface Sci. 145 (1993) 399; R. de Rooij, A.A. Potanin, D. van den Ende, and J. Mellema, J. Chem. Phys. 99 (1993) 9213; A.A. Potanin, R de Rooij, D. van den Ende, and J. Mellema, J. Chem. Phys. 102 (1995) 5845. 31 M.Mellcma, J.H.J. van Ophcusden, and T. van Vlict, J. Rheol. 46 (2002) 11. 21
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may be present at rest. The quantity (pe{i is calculated from the break-up criterion in combination with equations [6.13.6 and 7]. The formation and/or breakdown of floes may be rather time consuming, either at rest or during flow. This implies pronounced time effects in the apparent viscosity, leading to thixotropy. 6.14 Gels A clear-cut definition of a gel is hard to provide. In general, gels are characterized by a preponderance of solvent and the presence of a three-dimensional network of connected molecules or particles, at least over the time scale considered. Rheologically they are characterized by a predominantly elastic behaviour over the time scale considered and a modulus that is relatively small (generally < 107 Pa) compared with real solids. Various types of gels can be distinguished, where the precise division may depend on the criteria used. A division based on gel structure is, partly after Flory : 1. Polymer networks; a further subdivision can be made between covalent, e.g., cured rubbers, and physically crosslinked networks, e.g., entangled networks and gelatin gels, and between networks with long flexible polymer chains between the crosslinks, e.g., gelatin gels and those with stiff polymer chains between the crosslinks, e.g., gels of many polysaccharides. 2. Particle networks; a further subdivision can be made between networks of hard particles, e.g., gels of latex or silica particles, and of deformable particles, e.g., milk gels such as set yoghurt. 3. Well-ordered lamellar structures, including gel mesophases. The rheology of this last type of gels will not be discussed. With respect to their rheological behaviour, the extremes are gels with long flexible polymers between the crosslinks, and gels of hard particles. One should be aware that macromolecules such as globular proteins, (e.g., soy proteins) behave in this respect more as deformable particles than as long, flexible polymers. Therefore, we restrict the term polymer gel, and not macromolecular gels, to gels characterized by long flexible molecules between the crosslinks. Below, we will first discuss small deformation rheology of gels, in which the main focus will be on the rheology of these extremes. Gel formation theories will not be discussed, but polyelectrolyte gels will be discussed in sec. V.2.3d. At small deformations, gels are characterized by a modulus. First, a general expression will be derived for the modulus, based on a simplified picture of a gel. In this picture the gel is assumed to be built of strands which are mutually crosslinked. A strand can consist of a polymer chain or a linear chain of aggregated particles. A force 11
P.J. Flory, Faraday Discuss. 57 (1974) 8.
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Js applied to such a chain results in a reaction force in the chain which is proportional to the deformation, Ax times the change in the interaction force with a change in distance between the crosslinks/particles, dfs/dx . By multiplying both terms by the number N of elastically effective strands per cross section of the gel one obtains the following equation a=-N^-Ax dx
[6.14.1]
The local change in distance can be recalculated to a macroscopic strain by dividing it by a characteristic length C determined by the geometry of the network. In formal rheology, C is a tensor. Since / s can generally be expressed as - d F / d x , where dF is the change in Gibbs energy11, one obtains d2F a=CN—Ty
[6.14.2]
Since G = <j/y, and dF = dff - TdS , where dH is the change in enthalpy and dS the change in entropy G = CN^
= CNd(dH-TdS)
[6.14.3]
For gels with long flexible polymer chains in-between the crosslinks, the enthalpy term may be neglected compared with the entropy term, whereas for particles the reverse holds. For gels with, 'stiff polymers in-between the crosslinks, it will depend on the bending stiffness of the chains and their lengths which term prevails, and whether one of the two may be neglected. Equation [6.14.3] is valid for identical strands; in most real systems this condition is not satisfied. For example, the bonds between particles in a particle gel may be size-dependent. 6.14a Polymer networks Based on statistical thermodynamic arguments, a relationship has been derived between the modulus of gels consisting of ideal crosslinked long flexible polymers, and the number of chains between the crosslinks v2]. It is based on the idea that, in the undisturbed state of the gel, the chain between crosslinks have such a conformation that their entropy is at a maximum. Each deformation then leads to an average conformation that is less probable. Moreover, it is assumed that deformation is affine, i.e., the strain is everywhere the same down to the change in distances between the crosslinks. Next, the number of chains with a specified end-to-end distance can be calculated for the undisturbed- and the deformed state v° and v, respectively. From such statistical considerations, for the change of entropy
21
Temporarily we write here F for the Gibbs energy to avoid confusion with the G for modulus. P.J. Flory, loc.cit. chapter XI.3.
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AS
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= k
Zj \vi,ln—\
[6.14.4]
Elaboration leads to G = vkT
[6.14.5]
where v is the number of chains between two crosslinks per unit volume. This equation can also be written as: G = —RT
[6.14.6]
where Mc is the average molecular weight of the chain between two crosslinks. Both equations predict that the modulus is proportional to the concentration but, in practice, stronger dependencies are found. Moreover, for example, te Nijenhuis11 found for 1.95 w/w% gelatin gels a modulus of 500 Nm"2, implying Mc to be 94 kg mol" 1 , although the weight-average M of the whole molecule was only 70 kg mol" 1 . The conclusion must be that part of the molecules do not contribute to the elasticity of the network, and/or that part of the crosslinks are elastically ineffective, for example because they are parts of dangling ends or closed loops. A correction for dangling ends has been given by Flory, resulting in21
G = ^iL 2 M Mc ^
[6.14.71
MN)
where MN is the number-average molecular weight prior to crosslinking. Other complications are the presence of a sol fraction consisting of molecules not connected to the main network (which occur especially at low concentrations), and the formation of entanglements (which occur especially at higher concentrations). The latter will also cause the modulus to depend on the time scale of the deformation and, hence, on the frequency in oscillatory measurements. Various theories have been developed to estimate the shear modulus and its frequency dependence, based on entangled concentrated polymer solutions. Here, we will discuss only some results for the Doi-Edwards model. As indicated in sec. 6.12, a plateau in the shear modulus can be observed over the frequency range between the tube disengagement-time and the time re at which tube constrains start to affect the (longer) Rouse-type relaxation of the chains. This will occur at shorter times (higher frequencies) than the longest Rouse relaxation time. This plateau modulus GN is given by
11 21
K. te Nijenhuis, Colloid Polymer Set 259 (1981) 522. P.J. Flory, loc. cit., ch XI.2
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GN=pcfcT^j
6.81
[6.14.81
where pc is the number concentration of polymer molecules, and a is a parameter related to the tube dimension; = M 2 / L = (r2)/L , where L is the contour length of the tube and N the number of links with length (. From [6.14.8] it follows that GN ~ M°. The length of the plateau in terms of time (frequency in an oscillating experiment, see fig. 6.37) increases with molecular weight as l/r d is ~ M~3, and the transition from the Rouse-like relaxation is ~ M"2 . At times shorter than the longest Rouse relaxation time, the storage modulus increases as ft)1/2 , as observed for the Rouse model, while for times longer than r d . G'~
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Figure 6.40. Effects of length scale (distance r from a 'central' particle of radius a ) on the fractal properties of a particle gel. (a) Log of the particle number itf in a sphere of radius a^/a. (b) Fractal dimensionality, d, as a function of log(Oj / a) . Schematic. (Redrawn from P. Walstra, Physical Chemistry of Foods, Marcel Dekker (2000), 722.)
afg = a(»1/df~3
[6.14.9]
where a f is the (3 - df)- average radius of the fractal clusters forming the gel. The contact area between two floes will scale as a%. If we assume the clusters building the gel to be scale-invariant, the number of links between two adjacent clusters will be independent of their size and, hence, independent of the original particle concentration. This implies that the number of links JV between adjacent clusters per cross section of the gel will be ~ a^2 . This number may be equated to JV in [6.14.3], which relates G to some structure parameters of a gel. It results in the next relationship between JV and system properties; iV~af~2~a-V/(3~df)
[6.14.10]
Depending on the structure of the strands, an expression relating C to afg can also be deduced. Here, we will only give the result for two situations, viz, for the case where the strands connecting the cluster are fractal, and for the case where they are straight as a result of some reorganization process. For the latter case, C will be independent of a f . For a straight chain, the relationship between the local extension and the overall strain (C in [6.14.3]) will be independent of its length. Since d 2 F/dx 2 is independent of a f one obtains for the modulus G of a given system1'
11
L.G.B. Bremer, T. van Vliet, and P. Walstra, J. Chem. Soc. Faraday Trans. 1 85 (1989) 3359.
RHEOLOGY G~K(p2l{3-Ai]
6.83 [6.14.11]
A fractal chain connecting the cluster has a length ~ cij-j, where x is the chemical dimensionality with 1 < x < 1.3 . Moreover, deformation of such curves will be by bending and not stretching. It causes Cd 2 F/dx 2 to be proportional to a^'1+x). Combination of these proportionalities with equation [6.14.3] gives G = Ky3+x)/(3-df)
[6.14.12]
For the case where the clusters may be considered to be rigid compared with the link between the clusters, and the links are deformed by stretching, the power becomes l/(3-d f ) . Intermediate powers can also be found for structures in-between those which form the basis of [6.14.11 and 12]21. The exponent in [6.14.11] is the same as that obtained in percolation models for the case of isotropic interaction forces between the interacting units. In that model, a network of connected units is considered in which, at random, connections are removed and then the remaining percolation (or 'conductivity' in electricity terms) is determined, or vice versa. Below a certain number of connections, the conductivity will be zero. The concentration of bonds at which conductivity increases from zero to a finite number is called the percolation threshold. By replacing the number concentration of bonds by the weight concentration of molecules, or the volume fraction of particles, a percolation concentration or volume fraction can be defined. It results in an equation of the form G~{(p-(ptrf
[6.14.13a]
G~{c-c^.f
[6.14.13b]
or
The occurrence of the same power as in [6.14.11] can be understood if one realizes that the conductivity problem is only related to the number of elastically effective strands, just as for [6.14.11], and not to details regarding how it is deformed. Kantor and Webman considered the percolation problem in relation to bending deformation of the strands 3 '. They arrived at a power of about 4 in an equation equivalent to [6.14.13], similar to the situation considered in the derivation of [6.14.12]. It was shown that, starting from their approach, and assuming fractal aggregation, [6.14.11 and 12] and all intermediate relationships between G and
W.H. Shih, W.Y. Shih, S.I. Kim, J. Lui, and I.A. Aksay, Phys. Rev. A42 (1990) 4772. L.G.B. Bremer, B.H. Bijsterbosch, R. Schrijvcrs, T. van Vliet, and P. Walstra, Colloid Surf. 51 (1990) 159; M. Mellema, J.H.J. van Opheusden and T. van Vliet, J. Rheol. 46 (2002) 11. 31 Y. Kantor, I. Webman, Phys. Rev. Lett. 52 (1984) 1891. 41 M. Mellema, J.H.J. van Opheusden, and T. van Vliet, J. Rheol. 46 (2002) 11. 21
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fraction will be a coarser gel (see [6.14.9]). In practice, however, because of disturbances by flow or sedimentation, etc., a critical volume fraction for gelation will be observed11. Another requirement is that the vessel size must be larger than the floe size predicted from [6.14.9]. In practice, this is usually the case. An exception is the formation of a fat crystal network in emulsion droplets. Another important feature for the occurrence of a critical (p is the occurrence of rearrangements in the configuration of the aggregating particles during the aggregation process. The primary particles may rearrange to compact 'aggregates' during the time scale of the flocculation process. Because the time required for this process increases strongly with the size of the aggregating particles/'aggregates', above a certain size of these 'aggregates' network formation proceeds faster than the rearrangement process, a 'fractal' cluster will be formed, and eventually a gel. The size of the primary particles forming the fractal cluster then has to be replaced by the size of the small compact 'aggregates' formed as a result of the rearrangement process. For acid-induced milk gels, as present in yoghurt, it was shown that by increasing the acidification temperature from 20 to 40 °C the size of the effective primary building blocks increases from that of the protein particles present originally to building blocks containing about 40-50 primary protein particles21. If rearrangements proceed very extensively, precipitation may occur instead of gel formation, depending on the primary particle concentration. It leads to a measurable critical volume fraction. Rearrangement after gel formation may lead to loss of the fractal character of the clusters forming the gel31.
6.14c Large deformation behaviour Although the basis, in terms of a change in Gibbs energy, for the resistance against a small deformation is completely different between polymer networks and particle networks, a simple plot of the dynamic moduli against angular frequency often looks very similar (compare figs. 6.41a and b). In contrast to this similarity, the size of the Table 6.6. Linear region in shear for various gels. Material
Type of gel
Linear region
Vulcanized rubber
Gel with flexible chains between crosslinks.
1-3
Gelatin
ibid
0.5-1
Alginate
Gel with stiff chains between crosslinks
-0.2
Casein (milk) gels
Gel from deformable particles.
-0.03
Polystyrene latex gels
Hard spherical particle; gels a~50 nm
-0.01
Margarine
Hard particle gel of anisometric fat crystals
-0.0003
11
L.G.B. Bremer, P. Walstra, and T. van Vliet, Cod. Surf. A 99 (1995) 121. J.A. Lucey, T. van Vliet, K. Grolle, T. Gcurts, and P. Walstra, Int. Dairy J. 7 (1997) 389. 31 M. Mcllema, P. Walstra, J.H.J. van Opheusdcn, and T. van Vliet, Adv. Colloid Interface Sci. 98 (2000) 25. 2)
RHEOLOGY
6.85
Figure 6.41. Comparison between the frequency dependence of a particle gel, (a) , and a polymer network, (b). The particles are latex sols at different volume fractions (indicated) and the polymer is gelatin, 1.95% (w/w) at various ageing times.
Figure 6.42. Comparison between the storage modulus as a function of the shear strain for (a) a particle gel and (b) a polymer network. At the ends of the curves the gels were fractured. In (a) the particles are latices at different volume fraction (indicated), in (b) the polymer is gelatin, 4% w/w, after ageing for four hours at 15°C. linear region (table 6.6) and the large deformation behaviour, above all, the fracture strain is very different between these two types of gels (fig. 6.42). Flexible polymer chains can be deformed strongly before they will rupture, the more so the longer they are. Fracture strain is large, and related directly to the ratio of the (contour) chain length between crosslinks and the distance between them. At large deformations, the polymer chains can no longer be considered as random coils 11 and the alignment may even result in 'strain-induced crystallization' 21 . Both mechanisms
11 21
L.G.R. Treloar, The Physics of Rubber Elasticity, Clarendon Press, Oxford 3 rd ed. (1975). P.J. Flory, loc. cit. chapter XI.
6.86
RHEOLOGY
lead to an increase in the resistance against deformation (fig. 6.42b). For gels in which polymer chains are crosslinked covalently, fracture involves the breaking of covalent bonds in the crosslinks or in a polymer chain. If the polymer chains are physically crosslinked (e.g., through microcrystalline regions in polysaccharide gels), large deformation may lead to, 'unzipping' of these junctions. Unzipping of the bonds takes a certain time and this may cause the fracture parameters to become deformation rate-dependent11. For all polymer gels, fracture stress and strain will depend on the stochastic nature of the structure and on the fracture force of the bonds. For physically crosslinked gels the latter parameter will also vary in a stochastic manner, resulting in a relatively large scatter of the results. For particle gels the fracture strain depends strongly on the structure of the gel and on the deformability of the particles. It is easy to imagine that for straight strands of hard particles that are deformed in tension, the fracture strain will be proportional to Ax /(x + 2a), where x is the equilibrium distance between the particles and Ax the increase in distance up to bond rupture. If Ax does not differ too much between different gels, the fracture strain will be larger for a gel consisting of smaller particles. A second important factor is the curvature of the strand. The stronger this is, the larger is the fracture strain. For gels composed of fractal clusters, the strand curvature also determines, among other things, the dependence of the fracture strain, yfT , on the particle volume fraction. The following general equation has been derived2 yb ~ a " V / ( 3 " d f )
[6.14.14]
where j3 depends on strand curvature and the ratio between the stiffness of the fractal cluster over that of the bonds in-between; fi varies from -2 for gels with curved flexible strands between the fractal clusters to 1 for gels where deformation is located in the bonds between the fractal clusters. As particles are normally much less deformable than polymer molecules, and the strands of aggregated particles are much less curved and flexible, the fracture strain for particle gels will be much smaller than for flexible polymer gels (fig. 6.42). The fracture strain of gels consisting of stiff macromolecules and of deformable particles will assume an intermediate position. 6.15 General references 6.15a IUPAC recommendation Manual of Symbols and Terminology for Physico chemical Quantities and Units. Appendix II, Definitions, Terminology and Symbols in Colloid and Surface Chemistry. Part 1.13, Selected Definitions, Terminology and Symbols for Rheological
11 21
T. van Vliet, P. Walstra, Faraday Discuss. 101 (1995) 359. M. Mellcma, J.H.J. van Opheusden, and T. van Vliet, J. Rheol. 46 (2002) 11.
RHEOLOGY
6.87
Properties. Prepared for publication by J. Lyklema and H. van Olphen, Pure Appl. Chem. 51 (1979) 1213.
6.15b Other general references Aut. Div., Gels, Faraday Discuss. Roy. Soc. 101 (1995). (Gives an excellent overview of many aspects of gels including gel formation, structural aspects and rheology for both polymers and particles gels.) Aut. Div. Non-equilibrium Behaviour of Colloidal Dispersions, Faraday Discuss. Roy. Soc. 123 (2002). (Contains several contributions on the rheology of colloids and macromolecular systems.) H.A. Barnes, J.H. Hutton and K. Walters, An Introduction to Rheology, Elsevier (1989). (Elementary, especially for the non-expert.) J.K.G. Dhont, An Introduction to the Dynamics of Colloids, Elsevier (1996). (Advanced, mostly theoretical.) M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press (1986). (Provides a comprehensive overview of the most important theories regarding dynamic properties of polymer solutions; covers topics such as linear and non-linear viscoelasticity, diffusion, dynamic light scattering, flow and electric birefringence for flexible and rigid rodlike polymers.) Kinetics of Aggregation and Gelation, F. Family and D.P. Landau (Eds.), North Holland (1984). (This multi-author book contains a series of papers on fractal concepts in aggregation and gelation. Together these contributions provide a good introduction into this field.) J.D. Ferry, Viscoelastic Properties of Polymers, 3 rd ed., Wiley (1980). (Classical work on the rheology of viscoelastic polymer systems.) P.J. Flory, Principles of Polymer chemistry,
Cornell University Press (1953).
(Classical work on polymer solutions and gels, also contains rheology.) P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press (1979). (Classical) J.W. Goodwin and R.W. Hughes, Rheology for Chemists, an Introduction, Royal Soc. of Chem. (2000). (Covers many topics dealt with in this chapter in a clear way.) W.-M. Kulicke and C. Clasen, Viscometry of Polymers and Poly electrolytes, Springer (2004). (A good introduction of the topic starting at an elementary level. Both
6.88
RHEOLOGY
measuring techniques and relations between (intrinsic) viscosity and polymer characteristics are discussed.) R.G. Larson, The Structure and Rheology of Complex Fluids. Oxford University Press (1999). ('Complex' includes colloids and (particularly) polymers. Clear presentation, various practical illustrations; some emphasis on the relation between rheology and structure.) M. Mours and H.H. Winter, Mechanical spectroscopy of Polymers in Experimental Methods in Polymer Science, T. Tanake, Ed., Academic Press (2000). (The authors clearly discuss the ins and outs of mechanical spectroscopy, including data analysis for polymer systems, e.g. the calculation of relaxation spectra from dynamic moduli as a function of oscillation frequency.) N. Phan-Thien, Understanding Viscoelasticity, Basics of Rheology, Springer (2002). (Provides a short review (145p) of non-Newton flow behaviour, starting with a short introduction in tensor calculus. Then the kinematic and constitutive equations and constitutive modeling of polymer solutions and suspensions follow. Some familiarity with tensor calculus is advantageous.) R.I. Tanner, Engineering Rheology 2nd ed., Oxford Engineering Series, Oxford University Press (2000). (Profound mathematically-based discussion of the behaviour of non-Newton fluids with engineering applications. Many aspects of both viscometric (shear) and elongational flow are discussed, including the measurement of flow properties. The book is well written. Some prior knowledge of tensor calculus is useful.) R.W. Whorlow, Rheological Techniques, Ellis Horwood (1992). (Standard work on the measurement of rheological properties. Well written and provides a good overview of the advantages and disadvantages of the various techniques.)
Appendices 1 The moment expansion and some applications 2 Approximate analytical expressions for the electrical components of the interaction Gibbs energy and the disjoining pressure between two diffuse double layers 3 Non-retarded Hamaker constants; update
This Page is Intentionally Left Blank
APPENDIX 1 The moment expansion and some applications Assuming that (electron) microscopy can be employed to the colloids under study, typically 1,000-2,000 counts are needed to obtain a representative size distribution, which no longer changes shape when more particle sizes are sampled. A variety of theoretical distributions for data fitting are available which, together with extensive nomenclature, are treated in detail in the literature 123 ', see also 2.3f. Here we focus on some general features of distributions, which do not depend on the applicability of any theoretical fit. We do this on the basis of two data sets in fig. A 1.1, namely a size distribution of magnetite particles (from now on called the 'M-distribution,' l.h.s. of this figure) and one of silica spheres (the 'S-distribution,' r.h.s.). The M-distribution is a typical example of an asymmetric distribution with a significant tail of relatively large particles, whereas the symmetric S-distribution agrees very well with a Gaussian. A variety of averages may be defined for these distributions, the type obtained depending on the technique by which the sol is investigated. These averages can be estimated fairly accurately from the relative dispersity sa. Below we will explain this estimate and test it for the distributions in fig. A 1.1. We start with the n"1 moment of a discrete distribution, which is generally defined as 1
N
(an) = - X a . n
[A.I.I]
For a continuous distribution this equation may be replaced by [2.3.39]. In [A. 1.1] the brackets denote a number average over a total of JV particles with radii ai. The first moment is obviously the number-averaged radius (a) and the second moment determines the relative dispersity sa defined as
(a)2
(a)z
where o~2 = ( a 2 ) - ( a ) 2 is the standard deviation, or absolute dispersity, see sec. 1.3.7a. Here, ai = ai -(a)are the fluctuations around the average sphere radius, which by definition cancel each other such that (a) = 0 . From [A. 1.1 ] we obtain
(a)" \ l (a)) I To make an expansion in the fluctuations, we make use of the binominal theorem, 11
A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw-Hill, (1965). M. Alderliesten, Mean Particle Diameters Part I, Part. Part. Syst. Charact. 7 (1990) 233241; Part II, Part. Part. Syst. Charact. 8 (1991) 237. 31 P. Walstra. Physical Chemistry of Foods, Marcel Dckkcr (2002). 2)
A1.2
which states that for any positive integer n n (l + y)n = y
yk
[A. 1.4]
From [A. 1.3] and [A. 1.4] we find the following expansion, which contains a leading term of order s^ , but no linear sQ -term because aa = 0
(a) n
2
a
^
(n-k)!k!
The shape of the size distribution is, at this stage, still arbitrary. If the distribution is symmetrical around the average ( a ) , positive and negative fluctuations in [A. 1.2] occur with equal probability, so (<7k} = 0 for odd values of k. Consequently, the third moment for any symmetric distribution exactly equals
whatever the width of the distribution. For distributions, which are sufficiently narrow such that s^ « 1, we can truncate the expansion [A. 1.5] to obtain a result already reported in [2.3.44] for the special case of a log-normal distribution:
^).
1 + £i^lls2
2
a
S2 Kl a
[ A .l. 7 i .
This useful approximation (already reported in a light scattering study1') relates higher moments of a distribution to the average particle size and dispersiry. We have already seen one application in the expression for the apparent Guinier radius in [2.3.8], which is obtained by the application of [A. 1.7] to [2.3.7]. In a similar fashion, we find the apparent hydrodynamic radius from dynamic light scattering in [2.3.14] a
h = ( 1 + 5s a)< a >
S
a<
[A-1-81
a«!
[A-l-9]
whereas the specific surface area yields a s = ( l + 2s2)(a)
s
In the context of sedimentation (sec. 2.3d), we encountered the z-averaged particle mass
which for spheres with identical mass density corresponds to an apparent radius
11
P.N. Puscy, H.M. Fijnaut, and A. Vrij. J. Chem. Phys. 77 (1982) 4270.
A1.3
a =
= 1+7s (a>
^ [w>\
( ')
S <<1
[AM11
*
-
again after expanding up to terms of order s^ . Table A. 1. Test of [A. 1. /] for the data in fig. Al. 1. M/1 M Distribution calc.31 data21 data21 calc.31 (a) /nm sQ/%
4.3
-
3.6
-
S
data
21
117.7 9.23 1.074 122.5 124.0 124.4 116.6
calc.31 -
26.04 36.87 1.077 1.610 1.466 2.320 2.213 P 4.4 122.8 6.9 7.3 4.8 ah/nm 4.5 124.2 7.4 8.1 5.2 aG/nm 124.7 5.3 4.6 7.8 8.4 a z /nm 3.7 a/nm Mf is the M distribution in fig. A 1.1 after removal of particles with radius a > 5 nm. 21 Directly from particle counts. 3) Calculated using [A. 1.7] and the relative dispersity s a from this table.
Figure Al.l. Size distribution of 1,081 magnetite (M) particles obtained from conventional TEM images (courtesy M. Rasa) and 2,892 silica (S) spheres imaged by cryogenic-TEM (courtesy P. Homan). The M-distribution is a fairly broad log-normal one, the S-distribution is a Gaussian. Indicated are the number-average radius (a) the z-avcragc radius az , the harmonic mean a and apparent radii, which would be found from static a G and dynamic light scattering a h . See also table 2.1.
A1.4
We now put [A. 1.7] to the test for the distributions in fig. Al.l with results summarized in table A.I. It turns out that [A. 1.7] quantitatively applies for the S distribution even though the silica spheres have a dispersity of 9 percent. For the magnetite particles, with a significant dispersity of 37 percent, approximation [A. 1.7] still provides a reasonable estimate. Table A.I also shows the results of a hypothetical (but practically not unfeasible) fractionation by which large magnetite particles with a radius a > 5 nm (286 out of a total of 1,081 particles) are removed from the M distribution, to illustrate the effect of the tail of large particles. Removal of this tail substantially reduces the discrepancy between the static (aQ) and dynamic (ah) light scattering radius, and the number averaged radius (a). Note in table Al that [A. 1.2] correctly predicts the trend in radii going from the M to the Mf distribution. Some further remarks on the basis of this table are the following. A frequently used, dimensionless dispersity measure (also for polymers) is the ratio p between the weightaveraged molecular mass and the number-averaged molecular mass. For spheres with identical mass density, this ratio equals [A L12]
p=-r^2 (asy
-
where the a 6 term stems from the square of the particle mass. Substitution of the approximation [A. 1.7] shows that to leading order in s^, p equals p = l + 9s2
s
a<<1
[A.I.13]
Clearly, the ratio p is not a very sensitive measure in comparison with the relative dispersity sa itself, as also becomes clear from the numbers in table A. 1.1. Another observation from this table is the marked increase relative to (a) due to polydispersity of the Guinier radius a G and the hydrodynamic radius a h obtained from static and dynamic light scattering, respectively. An increase by a factor of two is to be expected for a polydispersity near s a = 40% . In table A. 1, the harmonic mean a=(a~1)-1
[A.I.14]
is also included as an example of an average, which emphasizes the small-particle tail of a distribution. This tail will be relevant for Ostwald ripening and the amount of dissolved material due to the Gibbs-Kelvin effect, which depends on the reciprocal particle radius, see [2.2.51].
APPENDIX 2 Approximate analytical expressions for the electrical components of t h e interaction Gibbs energy and t h e disjoining pressure between two diffuse double layers 11
In many cases exact analytical expressions are not available and are often not needed. Therefore, for easy reference, this appendix summarizes some of the 'easier' equations; we restrict ourselves to simple geometries. Unless specified otherwise, in these equations h is the distance between the two outer Helmholtz planes, i.e. h only covers the two diffuse double layer parts. Twice the thickness of the Stern layer ( 2 d ) h a s to be added in order to obtain the distance between the solid surfaces. Although the formulae are described for solid plates or particles, they can also be used for fluid phases (such as emulsion droplets, or wetting films). Unless specified otherwise, all equations
apply to symmetrical
( z - z ) electrolytes. Equations
containing
Bessel
functions a n d elliptical integrals are avoided. Note: many of the approximations are not good at short distances. Moreover, regulation across the Stern layer then becomes more critical. Hence, one should be careful when offering a physical interpretation for deviations at short h .
a. Symbols used Ggf1
Total electrical Gibbs energy at constant (diffuse) potential
[J]
G^
ibid, at constant (diffuse) charge density
[ J]
^ael'fr'
Electrical Gibbs energy per unit area, at constant (diffuse) potential
[Jm~ 2 ]
G^]el{h)
ibid, at constant (diffuse) charge
[J irr 2 ]
h
Distance between plates, or shortest distance between
[m]
spheres (8eoecRT)-1/2
p y yd y' K
Dimensionless potential = Fy//RT ibid, for diffuse double layer = Fy/d I RT ibid, for inner Helmholtz plane = Fy/1 /RT Reciprocal Debye length
[m2^1] [-] [-] [-] [m"1]
As regulation requires a multi-parameter-multi-equation set we refrain from presenting closed analytical expressions.
A2.2
APPENDIX 2
/7gj (h)
Electrical contribution to the disjoining pressure at
[N m~2 or N]
fixed (diffuse) potential nl^(h)
ibid, at constant (diffuse) charge
[N m" 2 or N]
DH means Debye-Hiickel approximation
b. Flat diffuse double layers at constant potential, yd For rigorous solutions for G ^ t h ) , see ref.11, eq. [38], p. 81, and our figs. 3.5-3.9 G
!Tei 'h> = 6 4 c RT t a n h 2 {zyd)e~yh
iA2b1'
Source: Verwey and Overbeek (V-O)11. See Main text [3.3.18]. Validity range, jrh>2 depending on y d , see figs. 3.7 and 3.8. The accompanying disjoining pressure is
^r ; (fr>= 6 4 z C 2 R T t*"h 2 ( Z ^W h
IA2b.2]
Main text [3.3.27]. Validity as for [A2b.l].
Gi>,^tann2(^){l-tanh(f)} Main text [3.3.28b]. Generally valid for not-too-strong overlap. The corresponding disjoining pressure is obtained by differentiation with respect to h . nlj}{h) = lScRTtanh 2 ( ^ - j sech2 (^)
[A2b.4]
The low potential variants are obtained by replacing tanh(zyd / 4) by zy d / 4. Corresponding equations have been derived by Usui . For the DH approximation, see [3.3.41] 7ffJ(h)= el
C R 7 V ) 2
[A2b.5,
cosh(x-h/2)
and
GS,,,.^^[f-arctan{sinh(f)}] [A2b.6] is obtained from [A2b.5] by integration with respect to h , using j(coshx)"' d x = arctan(sinhx).
APPENDIX 2
A2.3
For flat plates at constant potential, according to Ohshima19'
GW^frglJAj
1+ e yh
;
l+ 2cosh^-Vc^n^-ll
[A2b.7]
In the derivation, the PB equation was linearized with respect to the deviation of y from y d . The approximation holds close to the surface, so [A2b.7] is valid for low Kh . Hetero-interaction, DH-approximation (HHF equation5', see main text [3.6.11], G ^ t h ) = £°£K[Rp
r| (y d)2 +(y d)2 j (i _Coth(x-h) + 2yfyd cosec(rh))]
[A2b.8]
For equal potentials, this reduces to (see [3.6.12]),
GS^^^^(y
d 2
) f1-tanh(f)j
which is the low-potential limit of [A2b.3]. From [A2b.8], by differentiation with respect to h, [{(yf)2 + (y^)2}cosec2(rh) -2yf yd cosec(x-h)coth(rh)]
n^'ih) = -^
A2b.9]
The corresponding set for constant cr*1 is a poor approximation at low Kh because the potentials become too high for allowing the DH approximation. (Main text, [3.6.13] and discussion). For the same reason we do not include the equation of Kar et al. ' for the mixed situations ( of and y d constant). For a simple general algorithm to be used for asymmetrical electrolytes, see Chan1". c. Flat diffuse double layers at constant charge
ad = -^^RT
[A2c.i] j
s i n h [ ^ | =- 2 ^
O T
sinhf^j
[A2c,i]
For equations in the DH approximation simply, see [3.3.45, 46], yd =
[A2c.iil] eoSKRT
, crd =
£^£KRTud ^_
[A2c.iv]
A2.4
APPENDIX 2
Rigorous solution, fig. 3.10. For low potentials and weak overlap (DH approximation):
Gi>,--^(y d f [coth(f)-l] See main text [3.4.14] and ref.4). The corresponding disjoining pressure is [3.4.13] ni°)(h) = cRT(ydf
cosec2f—)
[A2c.2]
At low Kh , these are poor approximations because the required increase of potentials is incompatible with the DH approximation. For an improvement of this pair by Gregory31, see [3.4.24 and 23]
oir>)=—[W B+ ^ (irft/2) lK
L
V
1+
y
)
[A2C.3]
In (y d ) +cosh(x-h) + Bsinh(/rh)\ + ich\ jl + (y d ) cosec2 —
-1
where B 2 = l + ( y d f cosec2 (—1
[A2c.4] [A2c.5]
The validity is excellent for y d < 2 down to Kh ~ 0.2 . Another attempt for improving [A2c. 1 and 2] involves replacing the y d 's by the corresponding hyperbolic tangents. This comes down to a post Jactum correction to the y{x) behaviour after the computation has been carried out for the low potential limit. The results are, (see [3.4.13a and 14a])
°i>> = ^
tanh2
( ^ ) [ c o t h (f) "l]
[A2C61
-
^(^^? t anh 2 (^)[cosec 2 (f)] An alternative, derived by Ohsima from [3.4.17] for fairly low potentials2', (see main text, [3.4.19 and 20]) ,
G^(W = 1 ^ Z : { ( P ^ ) 2 - [ l - ( P ^ ) 2 ] 1 / 2 J
IA2C.81
APPENDIX 2
A2.5
f )
„ d
n£ (h) = 128cRTUpo )
2l1/2l
r 1
-^-(pcf )
I
[A2c.9]
In the DH approximation we derived [3.3.44] n^>(h) = ^ ( p ^ f i el ; z2 v sinh 2 (rh/2)
[A2C.10]
from which, by integration, using Jcosech 2 xdx = -cothx ,
Oh) =
^-[i"
coth
[A2c J1 ]
(Tjj
-
The constant charge equivalent of [A2b.7], which is relatively good at low Kh i
20)
equals
G w (h)= g£gz:|.g_J
1+e yh
;
A wealth of equations for G ^
l-Vc^v+i+^slnh^l and G ^
[A2c.12]
for different situations of hetero-
interaction has been derived by McCormack et al.18); several of these contain elliptic integrals. d. Two spheres (ss) and sphere-flat surface (sf) Deryagin approximation, applied to [3.3.28a], see [3.7.19]
G ,¥) {h)J_^^2^Ltanh2(^_)lnll el K-2(ai+a2)
I 4 J
l
+ e -,M ;
(ss)
[A2dA]
For two equal spheres,
G
y (h) = 64™cRT tanh2 i^f\
ln(l + e-*)
(ss)
[A2d.la]
sphere and plate, G^[h]=\26nacRT
tanh2
f ^ j
l n ( l+ e -.h)
(sf)
(A2d.lb]
An illustration is given in fig. 3.26. The corresponding forces follow from differentiation with respect to h , using (dln(l + e~'rfl)/dh = Kt~Kh I (1 - e^' 1 )
A2.6
APPENDIX 2 ,„,,
/ r (
A 0( zu \
128^a,a9cRT
^64^KT
£,
t a n h 2
e"**
^_^
128^cKT t a n h 2 r £ M i ] _ ^ L r
[A2d . 2b]
(sf]
^ 4 J 1 - e"™
The corresponding low-potential variants of these six equations, [tanh2(zyd /4)]—> [z 2 (y d ) 2 /16], do not add anything new, but as they are often used in the literature we give them below G
y d In l + e-*M
^ Kz(ay+a2)
K
aC RT
'
GM(h)=^
v
(ss)
[A2d.3]
'
f {y«f
ln(l + e-")
(ss)
[A2d.3a]
GW(h]=8™cfRT(ydf
m(l+ e-^)
(sf)
[A2d.3b]
M
^d.4bl
/
Sometimes, these last three equations are also written with only the numerator for the Kh -dependency. Clearly, this is a further simplification, valid only at large values of Kh . Regarding the quantitative difference, see fig. 3.31 of the main text. Following numerical solutions from the same group12', Sader et al.13) have given a number of accurate analytical formulas, valid up to moderately high potentials. For any Kh , with large ra and yd < 4 for the Gibbs energy and force between two identical spheres, we have Gtf(h)=EJ—}
Y2(h) ^
ln(l + e-^)
[A2d.5a]
and
J^W^-y)
Y2{h]
^rf\ln(]
+ h +Ki2a +h)
^ )
T^^\
[A2d5bl
APPENDIX 2
A2.7
respectively, where y = 4e-*-h/2 arctan e"*"'172 tanh —
[A2d.6]
The authors give a detailed comparison with the exact numerical results. We recall that in sec. 3.7d we have given exact numerical results from this group for heterointeractions . For dissimilar spheres (radii a and a 2 and oHp potentials y d and y^ ) in the above scheme, but with the diffuse layer in the DH approximation, G
ir ) ^^(f) 2 (^^)[( y f + y 2 d )^n(l + e--) + (yf-y2dfln(l-e--)] [A2d.7]
which may be considered an improvement over the Hogg-Healy-Fuerstenau expression, [3.6.11 ] in the main text. A similar expression, at constant o^ 's, has been derived by Ohshima151, who accounted for the resulting deviation from the lateral equipotential distribution by taking the polarization of the two solids into account. His result is
(
PT\ 2
(RTf +
p-Kh
I d\2 2
e-2**
a a
f
™°i-y) («f) ? *{ai+a2+h){ai+h)[-—^(h
£p2 l^h + al+a2\
+ aOf
Here, E, E . and E 2 are the relative dielectric constants of the electrolyte solution, of particle 1 and of particle 2, respectively. In reality the tangential variation of y d is of course not determined by polarization of the solid but by regulation. Ohshima shows that for h « al and h « a 2 , validating the Deryagin approximation, this equation reduces to that by Wiese and Healy161, with the next term being O{e~3Kh). For the constant potential case17' polarization of the solids does not occur: £ J = E 2 = 0, and [A2d.8] reduces to
(
PT \2
r,
p -2«h
y, af a0 -. 2 —. r F ) \ > 1 (al+a2+h)(al+h) yi
(RT)2i d]2 e-2«h -n£-£\ yS a,ai-22 o { F ) \*2) l (ai+a2+h)(a2+h) [A2d.9]
A2.8
APPENDIX 2
which, in the Deryagin approximation, reduces to the HHF equation [3.7.23]. e. Interactions involving cylinders Crossed cylinders in the Deryagin approximation, starting from [3.3.18] according to Sparnaay71 Gi)(h)=
128cKTatanh2r^e_yh
(J)
[A2el]
For parallel cylinders of length ( , ignoring end-cap effects , G,y)(h)=64cKT^tanh2^je^
(J)
|A2e2]
For parallel cylinders of radii a{ and a2 , Ohshima8' gives analytical expressions in terms of Bessel functions, which for /raj » 1 and Ka » 1 reduce to RT r[Vel](h) fir {— \ Cr (n) -- -Jn
e yh 2 a Fosl\ ff\ Z^I2y 9 /o",,d,,d A- „ —^ ~ -\y/,.d\ l \Ka2r £ 1 y2yjica1a2 x j — L
e-2/ch+ , [A2e.3]
and to
+ (y?l a,
2— O
[lU
Wr"a2 1
/ d\ 2
\KalT
i,
£ 2f
^
^ —nrr-
\7tKa2(r-a2)\
l I
r
r1/2
[A2e.4]
1 e-2^h
for constant potential and constant charge, respectively. For the conversion of y d into (jd and conversely, see sec. II.3.5f. The two dielectric permittivities ex and E2 enter through Gauss' law at the border of the diffuse layer. As Ohshima ignores Stern layers, these refer to cylinders 1 and 2. In the presence of Stern layers, E and e2 would stand for the two outer Helmholtz layers and an additional Gauss' law term at the iHp would be required. The distance dependence is complicated because it occurs through h and through r = h + ax+a2 (plus Id if Stern layers are accounted for). This is felt when 17{h) is derived by differentiation. From [A2e,3 and 4] one can simply derive the equations for a, = a 2 and for y = y2 . We only give the results when both pairs are equal. They read d
APPENDIX 2
A2.9
G{f{h) = 2 ^ —
2
eof— ( y d f V2^F e ^ - J
F
r
v
'
Ka
e~2«h
[A2e.5]
\ 1 - (a / r)
and
G
2
£o£— ( y d f V^F e"^ + r
I +
ica
[A2e.6] 2e, I
4
1
r
4
Vl-(a/r)
9/rh e
e \nxa(r-a)
for constant potential and constant charge, respectively. The pairs [A2e.l and 2] and [A2e.5 and 6] are not identical because they have been derived for different premises: the former for arbitrary potentials and the Deryagin approximation, the latter for the DH approximation but without the Deryagin approximation. However, the low potential variant of [A2e.2] coincides with the low Kh variant of [A2e.6]8'. Ohshima notes that the next-order curvature correction to the Deryagin approximation goes with (rar 1 / 2 , as compared to (m)" 1 for spheres. In passing, it is mentioned that Ray and Manning computed the interaction of two parallel cylinders, assuming the interaction mediated by strongly bound ('condensed') counter-ions; not surprisingly this leads to an additional attraction. We shall discuss condensation in chapter V.2. The interaction between a cylinder (2) and a flat surface (1) can, in principle, be obtained from [A2e.3 or 4] by letting ax -> °° . Ohshima gave an ab initio derivation, using the DH (but not the Deryagin-) approximation9'. For sra2 » 1 his results are 2
G$\h) = 2V2^ —
eoet Jm^ y?y$ e~Kh + [A2e.7]
+
i^(«?)a+(^)a^he"Ml
+
°(6"3")
and Gd (h) - 2V2^ —
eoe^a2
Uly2c
+^{yl)
*-——* [A2e.8]
+_ l ( j ^ f 2V2^^2^
l
- ^ - l-g^ a2+h s ^K{a2+h) p
The other condition is that — .
2
e- -
=
c
I
h
+O(e" l
-M '
1
= « 1 and —
£ ^]7nc(a.2 +h)
3
£
« 1. ^7tKa2
When £j and £2 —> <» (plate and surface both metallic)
A2.10
APPENDIX 2
r
1
-,
[A2e.9]
References 1.
E.J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids. The Interaction of Sol Particles, having an Electric Double Layer. Elsevier (1948). Also available as a Dover reprint (2000).
2.
H. Ohshima, Colloid Polym. Sci. 252 (1974) 158.
3.
J. Gregory, J. Chem. Soc. Faraday Trans. II 69 (1973) 1723.
4.
S. Usui, J. Colloid Interface Sci. 44 (1973) 107.
5.
R. Hogg, T.W. Healy and D.W. Fuerstenau, Trans. Faraday Soc. 62 (1966) 1638.
6. 7.
G. Kar, S. Chander and T.S. Mika, J. Colloid Interface Sci. 44 (1973) 347. M.J. Sparnaay, Rec. Trav. Chim. Pays Bas 78 (1959) 680.
8.
H. Ohshima, Colloid Polym. Sci. 274 (1996) 1176.
9.
H. Ohshima, Colloid Polym. Sci. 277 (1999) 563.
10. J. Ray, G.S. Manning, Langmuir 10 (1994) 2450. 11. D.Y.C. Chan, J. Colloid Interface Sci. 245 (2002) 307. 12. S.L. Carnie, D.Y.C. Chan and J.S. Gunning, Langmuir 10 (1994) 2993. 13. J.E. Sader, S.L. Carnie and D.Y.C. Chan, J. Colloid Interface Sci. 171 (1995) 46. 14. J. Stankovich, S.L. Carnie, Langmuir 12 (1996) 1453. 15. H. Ohshima, J. Colloid Interface Sci. 170 (1995) 432. 16. G.R. Wiese, T.W. Healy, Trans. Faraday Soc. 66 (1970) 490. 17. H. Ohshima, J. Colloid Interface Sci. 162 (1994) 487. 18. D. McCormack, S.L. Carnie and D.Y.C. Chan, J. Colloid Interface Sci. 169 (1995) 177. 19. H. Ohsima, Colloids Surf. A146 (1999) 213. 20. H. Ohsima, J. Colloid Interface Sci. 212 (1999) 130.
APPENDIX 3 Non-retarded Hamaker constants; update This compilation is an update of the previous tabulation, Vol. I, appendix 9. The previous results are not repeated here, unless for the sake of comparison. The data apply to the non-retarded range, unless specified otherwise. References8' '"' refer to general remarks about origin, validity, etc., numbered references to the open literature. Theoretical data are in roman font, experimental ones in italics. Measurements of Van der Waals forces rarely come on their own; usually an interpretation step, accounting for other forces, is needed. In view of the growing interest in hetero-interactions, to a large extent stimulated by the advent of AFM techniques, we have adopted the following nomenclature, which differs from that in Lapp.9. A12(3) interaction across 3 between 1 and 2 (general) A]2(G) = A12 interaction between 1 and 2 across a vapour A12(W) interaction across water (or dilute electrolyte) between 1 and 2 A10(W) interaction for water film on 1 At ](3) homo-interaction between 1 across 3 A,1(G) = A,, homo-interaction between 1 across a vapour. All constants are given in units of kT at room temperature (4.07xl0"21 J = 4.07 zJ 1!). Differences between theoretical data obtained by different authors mostly reflect differences in the quality of (and extrapolations accepted in) the 4-(i£) spectra.
Table A3.1 Water. Given is Avm across vapour. 7.1-14.2 12.28 13.6
11
Range of previous data Lifshits Lifshits
I zJ = 1 zeplojoule = 10 ~21 J .
a) 2)
3, b)
A3.2
Table A3.2 Inorganic materials. Homo-interaction across vacuum and water. Substance
A KG)
a-A^CXj (hexagonal) a-Al9O^ (hexagonal)
37.4
a-Al 2 O 3 (hexagonal)
•"ll(W)
Ref.
9.02
1
8.11
4
6.76
6
<X-A12O3 (hexagonal)
40.5
14.0, 14.7
14
a-Al,O, (sc)
39.3
13.1
7
a-Al,O., (sc) a-Al9O^ (not specified)
36.9
12.8
7
38.1
10.2
a
93-112
50-90
a
76
17
19.7
1 1
Au Au
115
BaTiO., (tetragonal
44.2
BeO (hexagonal)
35.6
C (diamond)
72.7 69.8
33.9 34.4
a
24.8
32.0 3.54
4 1
3.22
4
2.56
a
C (diamond) C (diamond) CaCO 3 (trigonal) CaCO., (trigonal) CaCO., (calcite)
17.7
CaF9 (cubic)
17.1
CaF? (cubic)
8.23
1
1.20
1
0.96
4
CdS (hexagonal)
28.0
8.35
1
CdS (hexagonal) CdS (hexagonal)
37.6
11.9 8.80
Csl (cubic)
19.7
2.95
KBr (cubic)
13.8
1.35
a 4 1 1
KBr (cubic) KBr (cubic) KC1 (cubic)
1.74
4
16.5 13.5
1.33
a
1.01
1
KC1 (cubic)
15.2
a
LiF (cubic)
15.6
0.76 0.88 0.66 0.91 5.43 4.32 4.96
4
LiF (cubic) MgF2
14.4
MgO (cubic)
29.7
MgO (cubic)
26.0
MgO (cubic) Mica (monoclinic) Mica Mica
24.2
1 4 1 1 a
3.29
1. 7
2.90 4.68±0.07
4 19
A3.3 Table A3.2 (continued) Substance Mica Mica (unspecified) Mica (unspecified) NaCl (cubic) NaCl (cubic) NaF (cubic) PbS (cubic) Si (amorphous) Si (amorphous) Si
6H-SiC (hexagonal) 6H-SiC (hexagonal) p-SiC (cubic) p-Si^N4 (hexagonal) P-SiaN4 (hexagonal) Si^N4 (amorphous) Si^N4 (amorphous) Si,,N4 (amorphous) SiO9 (quartz, trigonal) SiO9 (quartz, trigonal) SiO2 (quartz, trigonal) SiO9 (quartz, trigonal) SiO2 (silica, amorphous) SiO9 (silica, amorphous) SiO9 (silica, amorphous) SiO2 (silica) SiO9 (fused) SrTiO., (cubic) Ti
TiO, TiO, TiO, TiO, TiO, TiO, Y,O.,
(rutile) (rutile) (rutile) (rutile) (rutile) (rutile) (hexagonal)
Y2O._j (hexagonal)
IMG)
21-27
11(W)
4.68 4.7-6.3
b
15.9 9.95 20.1
21.8 17-22
16.8 16.0 16.0 16.0 16 36.4 61.4 37.6
36.7 37.9 32.7 31.9
20b a 9
1.28 1.45 0.76 12.2 21.7 24
45.8 60.9 61.4 60.4 44.2 44.2 41.0
Re/
23.9 26.8 31.9 26.3 13.4 17.2 11.9 11.2 14.9 2.51 2.6-3.4 1.06 0.39 1.63 1.13 2.04 1.89 2.09 11.72 31.9 14.7±5; 17.2±2.4 13.14 13.3 14.7 13.7 15.6 7.44 9.83
1
4 1 1 4 5 7 1 7 1 1 7 1 4
5 1 a
4 5 14 1 7
7 a 1 17 22 1 4 6 14
15, b 1 7
A3.4 Table A3.2 (continued) Substance
!
ZnO (hexagonal) ZnS (cubic) ZnS (hexagonal) ZrO9 (tetrahedral)
;
• '
A,1|G)
A
1 1(W)
22.6 37.3 42.2 49.1
4.64 11.79 14.10 21.6
Ref. 1 1 1 7
Table A3.3. Inorganic materials. Homo-interaction across non-aqueous media. Ref. Substance Medium (3) 11(3) a-Al,O., (sc) Au
MgO 6H-SiC SiO9 TiO9 (rutile) Y 9 O, ZrO 9 (tetrahedral)
n-dodecane n-dodecane 'hydrocarbon' n-dodecane n-dodecane n-dodecane n-dodecane n-dodecane n-dodecane
12.8 13.1 68.8 7.86 32.0 2.03 14.05 9.82 21.6
Table A3.4 Organic materials. Homo-interaction Substance A KG) n-Pentane
9.21
n-Pentane
6.37
Cyclopentane
7.35
n-Hexane
10.00
n-Hexane
6.79 7.37
n-Heptane Polyester
15.0
Polycarbonate
12.5
Poly(methylmethacrylate)
14.3
Poly(styrene)
15-20
Poly(styrene) 'Hydrocarbon'
17.2
Poly(styrene)
13.7
Poly(tetrafluoroethylene)
14.72
Poly(tetrafluoroethylene) Polv(tetrafluoroelhvlene)
(L.D.)
12.4
Poly(letrafluoroethylene)
(H.D.)
14.8
7 7 17 7 7 7
15, b 7 7
A ^ll(W)
Ref.
0.83 1.07 0.96 0.95 1.00 0.95 1.00 0.86 0.36 0.9-3.4 3.34 1.54 0.78
a 13 13
a 13 13
14 14 14 a
11,b 17 14 a
9.07 1.29 1.82
12 13 13
A3.5
Table A3.5. Inorganic materials: hetero-interaction, across vacuum or water. For each pair: top, Ay2{mc),
bottom, A
2(w) •
Substance
Silica
Si-nitride
A12O3
Mica
Ref.
BaTiO 3
24.8 1.52 23.8 2.33 33.7 4.20 19.8 1.70 16.5 1.11 19.7 1.77 14.6 0.91 15.1 0.88 21.7 1.99 19.7 1.69 15.8 1.08 13.2 -0.20 30.6 3.73 26.5 2.87 18.6 1.55
40.5
37.3
1
11.89 37.8
26.0 2.87 32.2 7.67 23.4 1.79 23.9 1.62 34.9 8.00
8.72 36.4 8.60 51.8 17.32 30.2 5.33 25.3 2.70 29.5 5.28 22.1 1.25 23.1 1.69 33.2 6.85
31.4 6.02 25.3 2.87 21.8 1.57 49.9 17.7 42.5 12.6 29.7 5.09
30.0 5.28 24.0 2.16 19.4 -0.49 47.2 14.9 40.5 10.9 28.5 4.50
30.5 4.86 29.2 4.86 41.8 9.90 24.4 3.32 20.3 1.79 24.2 3.51 17.9 1.13 18.6 1.22 26.8 4.15 = homo = homo 19.5 1.62 16.1 -0.074 38.1 8.70 32.4 6.41 23.0 2.85
37.8 9.95 37.8 10.47 36.6 9.33
4.42 34.9 7.32 34.9 7.64 34.4 7.64
BeO C (diamond) CaCO, CaF, CdS KC1 MgF9
MgO Mica NaCl PbS 6H-SiC P-Si,N, SiO 9 (quartz)
9.51 54.5 19.51 31.7 6.22
SiO, (Stober) SrTiO, TiO, Y,O,
23.2 1.40 23.2 1.69 22.7 2.18
28.5 4.15 28.5 4.50 28.0 4.64
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 21 1 1 1
A3.6
Table A3.5 cont. Substance ZnO
ZnS (cubic)
Silica
Si-nitride
A12O3
Mica
Ref.
18.1 1.43 23.8 2.51
29.5 5.65 38.6 11.2
27.3 3.88
22.3 2.65 29.2 5.38
1
Substance 1
Substance 2
Ag Ag Ag
Cu
Cu Cu
SI,N 4 Si,N. Si,NA Si Si
SiO, (silica)
SiO, TiN
SiO9 TiN Si
SiO9 (silica) Mica SiO, (silica) Mica Mica
35.9 8.72
Across vacuum 83.4 32.9 42.5 34.9 34.9 41.4 25.5 31.4 25.2 31.0 19.7
Across water
16.6 4.36 8.35 4.72 8.55 2.92
1
Ref. 18 18 18 18 18 7 7 7 7 7 7
Table A3.6. Inorganic materials. Hetero-interaction across non-aqueous media. A Medium (m) Ref. Materials 12(m) Si^N4-pSi^N4 Si,N a -pSi,N d Si3N4 - SiO, Si^N4 - SiO,
di-iodomethane 1 -bromonaphthalene di-iodomethane 1 -bromonaphthalene
2.46 6.88 -1.97 -0.49
8, c 8, c 8, c 8, c
A3.7 Table A3.7. Inorganic and organic materials. Hetero-interaction across various media. Substance 1 Substance 2
Medium
a-AlpOc,
Poly( tetrafluoroethene)
cyclohexane
a-Al9Os
Poly (tetrafluoroethene)
cyclohexane
a-Al ? O s
Teflon AF
cyclohexane
Ag
Poly( tetrafluoroethene)
vacuum
Au
Poly(tetrafluoroethene) (amorphous)
cyclohexane
Au
Polyltetrafluoroethene) (crystalline)
cyclohexane
Au
Poly(tetrafluoroethene) (crystalline)
cyclohexane
Au
Polyl tetrafluoroethene) (amorphous)
p-xylene
Au
Poly(tetrafluoroethene) (crystalline)
p-xylene
Au
Polyl tetrafluoroethene) (crystalline)
p-xylene
Au
Polyl tetrafluoroethene) (amorphous)
bromobenzene
Au
Polyl tetrafluoroethene) (crystalline)
bromobenzene
Au
Poly(tetrafluoroethene) (crystalline)
bromobenzene
Au
Polyl tetrafluoroethene) (amorphous)
perfluorohexane
Au
Polyl tetrafluoroethene) (crystalline)
perfluorohexane
Au
Polyl tetrafluoroethene) (crystalline)
perfluorohexane
Au
Polyl tetrafluoroethene) (amorphous)
water
Au
Poly(tetrafluoroethene) (crystalline)
water
Au
Polyl tetrafluoroethene) (crystalline)
water
Cu
Polyl tetrafluoroethene)
vacuum
Glass
Polyl styrene)
water (+ salt)
SiO9
Polyl tetrafluoroethene)
cyclohexane
SiO9
Polyl tetrafluoroethene)
cyclohexane
SiO9
Teflon AF
cyclohexane
TiO9(rutile)
Polyl methylmethacrylate)
water
TiO9(rutile)
Polyl styrene)
water
TiO?(rutile)
Polyl styrene)
dodecane
^12(m)
-3.37 -2.36 -5.81 30.9 - 12.9 3.73 - 13.5±3.7 - 11.89 5.03 12.3+3.7 - 18.7 -4.03 13.5±3.7 - 13.3 3.25 J. 72 ±0.74 -5.48 13.0 attract. 27.5 1.45 - 1.33 - 1.00 -2.13 4.58 4.68 3.13
Ref. 10, b 10, b 16 18 12
12 12 12 12 12 12 12 12 12 12 12 12 12 12 18
11,b 10, b 10, b 16
15, b 15, b 15, b
A3.8 Lettered references a. Data according to previous tabulation (exp. and theory), for comparison. b. The authors (also) give the Hamaker function. c. AFM-experiments in which assumptions had to be made about other types of interaction forces. Numbered references 1. L. Bergstrom, Adv. Colloid Interface Set. 70 (1997) 125. Full Lifshits. This author was not aware of our previous tabulation but his data are mostly complementary; he also gives constants for two familiar approximations. 2. A.V. Nguyen, J. Colloid Interface Set 229 (2000) 648. 3. R.R. Dagastine, D.C. Prieve and L.R. White, J. Colloid Interface Sci. 231 (2000) 351. 4. J.M. Fernandez-Varea, R. Garcia-Molina, J. Colloid Interface Sci. 231 (2000) 394. 5. T.J. Senden, C.J. Drummond, Coll. Surf. A94 (1995) 24. 6. H.D. Ackler, R.H. French and Y-M. Chang, J. Colloid Interface Sci. 179 (1996) 460. 7. L. Bergstrom, A. Meurk, H. Arwin and D.J. Rowcliffe, J. Am. Ceram. Soc. 79 (1996)337. 8. A. Meurk, P.F. Luckham and L. Bergstrom, Langmuir 13 (1997) 3896. This work also contains experiments which can at least confirm the sign. 9. A.M. Stewart, V.V. Yaminsky and S. Ohnishi, Langmuir 18 (2002) 1453. 10. S.-W. Lee, W.M. Sigmund, J. Colloid Interface Sci. 243 (2001) 365. 11. M.A. Bevan, D.C. Prieve, Langmuir 15 (1999) 7925. (Particle levitation method; analysis of surface structure). 12. A. Milling, P. Mulvaney and I. Larson, J. Colloid Interface Sci. 180 (1996) 460. (The data for amorphous (block) or crystalline (colloidal) poly(tetrafluoroethene) are probably extremes). 13. C.J. Drummond, G. Georgaklis and D.Y.C. Chan, Langmuir 12 (1996) 2617. 14. R.H. French, J. Am. Ceramic Soc. 83 (2000) 2117. 15. R. Buscall, Colloids Surf. A75 (1993) 269. 16. S.-W. Lee, W.M. Sigmund, Colloids Surf. A204 (2002) 43, Tefion-AF is a noncrystallizing Teflon ex Dupont with lower /; 's. The paper also gives AFM-experiments. 17. T. Ederth, Langmuir 17 (2001) 3329. 18. S. Eichenlaub. C. Chan and S.P. Beaudoin. J. Colloid Interface Sci. 248 (2002) 389. 19. J.N. Israelachvili. G.E. Adams. J. Chem. Soc. Faraday Trans. (I) 74 (1978) 975. 20. V.E. Shubin. F>. Kekicheff. J. Colloid Interface Sci. 155 (1993) 108.
A3.9
21. I. Larson, C.J. Drummond, D.Y.C. Chan and F. Grieser, Langmuir 13 (1997) 2109. 22. I. Larson, C.J. Drummond, D.Y.C. Chan and F. Grieser, J. Am. Chem. Soc. 115 (1993)11885.
This Page is Intentionally Left Blank
CUMULATIVE SUBJECT INDEX OF VOLUMES I (FUNDAMENTALS), II and III (INTERFACES) AND IV and V (COLLOIDS) In this index bold face print refers to chapters or sections; app., and fig. mean appendix, and figure, respectively. The roman numerals I, II, III, IV and V refer to Volumes I, II, III, IV and V, respectively. When a subject is referred to a chapter or section, specific pages of that chapter or section are usually not repeated. Sometimes a reference is made even though the entry is not explicitly mentioned on the page indicated. Entries in square brackets [..] refer to equations. The following abbreviations are used: (intr.) = introduced; (def.) = definition of the entry, ff = and following page(s). Combinations are mostly listed under the main term (example: for negative adsorption, see adsorption, negative), except where only the combination as such makes sense or is commonly used (example: capillary rise). Entries with 'surface1 are often also found under 'interface' except where one of the two is uncommon. Entries to incidentally mentioned subjects are avoided. For the spelling of non-English names, see the preface to each volume. To avoid undue expansion of this index, chemical substances are mostly grouped together; for instance, for butanol, palmitic acid, sodium dodecylsulphate, hexane and dimyristoylphosphatidylethanol amine (DMPE) look under alcohols, fatty acids, surfactant, (anionic), alkanes and (phospho-)lipids. absorption bands; 1.7.14 absorption coefficient; 1.7.13 absorption index; 1.7.13 absorption (of radiation); see electromagnetic radiation acceptor (in semiconductor); II.3.172ff 'acid rain'; II.3.166, II.3.221 acoustic waves; 1.7.44, II.4.5d acoustophoresis; II.4.5d activity coefficient; 1.2.18a, 1.2.18b, 1.3.50 Debye-Hiickel theory; 1.5.2a, I.5.2b of (single) ions; I.5.1a, 1.5.1b, I.fig. 5.2, [1.5.2.28] (Davies) activator (in flotation); HI.5.97 additivity (in coagulation); IV.3.9k adhesion; II.2.5, II.5.97, III.5.4, III.5.2 (also see: wetting, adhesional, work of adhesion) adhesive joints; III.5.17 adiabatic (process); 1.2.3 (def.) admittance spectrum; II.3.93, II.fig. 3.30, II.3.97 adsorbate; I.1.17(def.), 1.3.17 ideal; 1.1.17 thickness; II.2.63, II.2.76, II.2.79 ellipsometric; 1.7.1 Ob
2
SUBJECT INDEX
(see further: adsorption of polymers) adsorbent; I.1.18(def.) adsorption; I.1.4ff(intr.), II.chapters 1-3 and 5, III.chapter 4, V.chapter 1 and diffusion; I.6.5d, I.6.5e, II. 1.6, see adsorption, kinetics in emulsions; see V.chapter 8, energy; I.1.19(intr.), I.3.23ff, [1.4.6.1], II.1.22, II.1.3c, II.1.3f, II.1.44ff, II.1.50ff, II.3.6d-e, Il.chapter 5, V.chapters 1, 3 for heterogeneous surface; II.1.104ff enthalpy; II.1.3c, II.1.3d, II.1.3f, II.2.5b, II.figs. 2.26-28, Il.figs. 3.60-61 isosteric, II.1.3c, II.1.3d, II.1.28, Il.figs. 1.8-1.10, II.2.26, II.2.49 entropy; 1.3.30, II.1.21, II.1.22, II.1.3c, II. 1.29, II. 1.3f, Il.fig. 1.11, II. 1.43, II. 1.52, Il.fig. 1.16, II. 1.65, V.chapters 1, 3 from solution; 1.2.73, 1.2.85, Il.chapters 2 and 5, IH.chapter 4, V.chapters 1, 3 basic features; II.2.2, Il.fig. 2.1 composite nature; II.2.2, II.2.3-2.6 dilute solutions; II.2.4, II.2.7, II.5.7, II.5.8 electrosorption; II.3.12 exchange nature; II.2.1 experiments; II.2.5, II.5.6, III.chapter 4, V.chapters 1, 3 in emulsions; V.8.2d functional; 11.1.18,11.1.33 Gibbs energy; 1.2.74, II.1.3e, II.1.3f, II.1.45, IH.chapter 4, IV.chapter 3 V.chapters 1, 3 heat; II.1.3c, II. 1.28, Il.fig. 1.7, Ill.fig. 4.16 (also see: enthalpy) Helmholtz energy; II. 1.21, II. 1.23 heterogeneous surfaces; II. 1.7 hysteresis; Il.fig. 1.13, II.1.42, II.1.82, Il.figs. 1.31-33, Il.fig. 1.35, Il.fig. 1.39, II.1.6e, II.5.26, II.5.7d, V.figs. 3.16-17, V.3.38ff, V.fig. 3.20 kinetics; II.1.45ff, II.2.8, II.5.3c, III.4.5 localized; I.3.5b, I.3.6d, II.1.7, II.1.5a, II.1.5b, II.1.5d, Il.l.Se, II.1.5f, Il.fig. 1.21 mobile; I.3.5d, II. 1.7, II. 1.5c partially mobile; Il.l.Sd, II.1.5e, Il.fig. 1.18, Il.fig. 1.21 negative; I.1.3(intr.), 1.1.4, 1.1.5, 1.1.21, 1.2.85, 1.5.93, II.3.5b, II.3.7e, Il.fig. 3.40, II.5.20ff, II.5.3e, Il.fig. 5.6, Il.fig. 5.9, IV.fig. 5.3 physical; II. 1.18, Il.fig. 1.13 presentation of data; II. 1.4, Il.fig. 1.12
SUBJFXT INDEX adsorption (continued), residence time (adsorbed molecules); II.1.46ff specific; see specific adsorption standard deviations; II. 1.45 superequivalent; II.3.62(def.), IV.3.9J, see further: specific adsorption t-plot; Il.figs. 1.27-28, II. 1.88, Il.fig. 1.34 (statistical) thermodynamics; I.2.20e, 1.2.22, II.1.3, II.2.3, II.3.6d-e, 11.3.12,11.5.5 a-plot; II. 1.90 (also see: adsorption isotherm (equation), calorimetry; for adsorption at liquidfluid interfaces, see (Gibbs) monolayers. For specific examples see under the chemical name of the adsorbate) adsorption of: atoms; 1.3.5b, 1.3.5c, II.chapter 1 biopolymers; 1.1.2, Il.fig. 5.26b, Il.fig. 5.29, V.chapter 3 gases and vapours; II.chapter 1 functional; II. 1.18, (for relation to wetting, see III.5.3b, Ill.fig. 5.16 ions; 1.1.20, II.3.6d-e also see: double layers, electric polyelectrolytes; II.5.8, V.2.3c, V.fig. 2.19 charge compensation; II.5.85 chemical and electric contributions; II.5.84ff, Il.fig. 5.32;, Il.figs. 5.38-39 grafting; V.2.3c isotherms; Il.figs. 5.32-33, Il.figs. 5.35-40 multilayer; V.2.6e, V.fig. 2.37 multi-Stern layer; II.5.87 profiles; Il.fig. 5.34 theory; II.5.5g polymers; 1.1.2, 1.1.19, 1.1.23, I.fig. 1.18, 1.1.27, 1.2.72, Il.fig. 4.42, II.chapter 5, Il.fig. 5.1, Il.fig. 5.6, V.chapter 1 applications; II.5.9 bound fraction; II.5.18, II.5.71, Il.fig. 5.22, V.chapter 1 dispersity effects and fractionation; II.5.3d, Il.fig. 5.8, II.5.7c, Il.figs. 5.29-31, V.1.91 energy parameter; II.5.28, V.chapter 1 equilibrium aspects; V.fig. 1.51 experimental techniques; II.5.6 film rupture (effect on); V.8.88 hysteresis; II.5.26, II.5.7d, V.1.12d isotherms; Il.figs. 5.7-8, Il.fig. 5.22. Il.figs. 5.25-28
4
SUBJECT INDEX
adsorption of polymers (continued), kinetics; II.5.3c layer thickness; II.5.6b, Il.fig. 5.19, Il.figs. 5.23-25. II.5.72ff, V.fig. 1.51 electrokinetic; II.4.128, Il.fig. 4.42, II.5.63ff ellipsometric; II.5.64ff hydrodynamic; II.5.61ff, Il.figs. 5.24-25 steric; II.5.65ff (also see, profiles) loops; II.5.18, II.5.32, figs. II 5.19-23, II.5.7Off, V.fig. 1.6, V.1.6c negative (= depletion); II.5.20, II.5.3e, Il.fig. 5.9, III.2.58 (self adsorption), V.1.8, V.I.9 profiles; II.5.18, Il.fig. 5.6, Il.fig. 5.10, II.5.40, Il.figs. 5.15-16, II.5.6c, Il.figs. 5.19-20, III.3.8e, V.fig. 1.1, V.I.6, V.1.5, V.1.8, V.1.9, V.I.11, V.fig. 1.30. scaling; V.I.68, V.2.3 tails; II.5.18, II.5.32 , Il.figs. 5.19-23, II.5.70ff, V.fig. 1.6 theory; II.5.4, V.chapter 1 diffusion equation; II.5.32, [V.I.4.1] (Edwards) excluded volume effect; II.5.5b GSA (ground state approximation); V. 1.2 lattice theories; II.5.30ff Monte Carlo; II.5.30 scaling; II.5.4c Scheutjens-Fleer theory; II.5.31, II.5.5, see self-consistent field theory square gradient method; II.5.33 statistics; II.5.29 trains; Il.fig. 2.27, II.5.18, II.5.32, Il.fig. 5.19, Il.figs. 5.21-23, II.5.70ff, V.chapter 1 proteins; V.chapter 3 competition; V.3.8 dispersion forces; V.3.20 driving forces; V.3.3c electrostatics; V.3.19-20 equations of state (2D); V.fig. 3.24 fluid interfaces; V.3.7 hydration/dehydration; V.3.21-22 hydrophobic interactions; V.3.4b hysteresis; V.3.3 and 3.5 kinetics; V.3.3a, V.fig. 3.6 principles; V.3.1-3 reconformation; see structural alterations
SUBJECT INDEX adsorption of proteins (continued), relaxation; V.3.3b, V.3.32ff, V.fig. 3.18 reversibility; see hysteresis and relaxation structural alterations; V.3.4, V.3.22ff, V.flgs. 3.15-19, V.3.6c, V.fig. 3.28 surfactants; 1.1.25, III. 1.14b ionic; II.3.12, III.4.6d non-ionic; II.2.7d, III.4.6c adsorption isosters; II.1.3d, Il.flg. 1.9, II.1.37, Il.flg. 1.12f, II.2.26 adsorption isotherm (equation); I.1.17ff(intr.), I.fig. 1.12, II.1.3ff, Il.flg. 1.12, II.1.5, Il.app. 1 Brunauer-Emmett-Teller (BET); I.3.5f, H.l.Sf, [II.1.5.47], [II.1.5.50], Il.flg. 1.24a classification, adsorption from dilute solution; II.2.7b, Il.flg. 2.24 gas adsorption; II. 1.4b surface excess; II.2.3c, Il.flg. 2.8, II.2.4 composite; 1.2.85 (see surface excess) Dubinin-Radushkevlch; [II. 1.5.56] electrosorption; II.3.12b. II.5.5g Frenkel-Halsey-Hill; [II. 1.5.55] Freundlich; 1.1.19, II. 1.2, [II. 1.7.7] (generalized), Il.flg. 2.24c Frumkin-Fowler-Guggenhelm (FFG); I.3.8d, II.1.5e, Il.flg. 1.19, II.1.64, Il.flg. 1.43, [II.A1.5a], II.2.65, H.3.195 for surface excess isotherm; 11.2.4d for specific adsorption of ions; II.3.6d Harkins-Jura; [II. 1.5.57] Henry; 1.1.19, 1.2.73, 1.6.65, II. 1.2, [II.Al.la], Il.flg. 2.24a heterogeneous surface; Il.flg. 1.43 high affinity; 1.1.19, Il.flg. 2.24d, Il.figs. 5.7-8, Il.flg. 5.26, Il.flg. 5.29, Il.flg. 5.31, V.chapter 1, 3 Hill-De Boer = Van der Waals; see there individual; see partial Langmuir; 1.1.20, 1.2.74, I.3.6d, I.fig. 3.2, 1.3.46, II.1.2, II.1.28, II.1.4a, II.1.5a, II.1.5b, Il.figs. 1.14-17, Il.l.Sd, Il.l.Se, [II.1.7.7] (generalized), [II.Al.2a], Il.flg. 2.24b, II.2.86, II.3.196 binary mixture; II.2.4b, II.2.4c, Il.flg. 2.11 local; II. 1.104, Il.flg. 1.43, II. 1.108 one-dimensional; 1.3.8a Ostwald-Kipling; II.2.3b, [II.2.3.6], [II.2.6.1] partial (= individual); Il.flg. 2.9, Il.figs. 2.11-14
5
6
SUBJECT INDEX
adsorption isotherm (continued), partially mobile; II.1.5d, II.fig. 1.18 potential theories; II. 1.73 quasi-chemical; I.3.8e, II.1.57, [II.A1.6a], II.3.196, V.2.21 standard deviation; 1.3.36 statistical thermodynamics; II.1.3ff, II.1.5, II.1.6, II.1.105, II.2.4, II.3.12, II.5.5, V.chapter 1 surface excess; II.2.3, Il.figs. 2.11-14, II.figs. 2.18-23 molecules of different sizes; II.2.4e, Il.fig. 2.14 relation to interfacial tension; II.2.4f multilayer; II.2.44ff Szyzskowski; [III.4.3.14] Temkin; [II. 1.7.6] thermodynamics; I.2.20e, II.1.3, II.2.3, II.3.6d-e, II.3.12, II.5.5 Van der Waals (= Hill-De Boer); II. 1.59, [II. 1.5.27], Il.fig. 1.20a, Il.fig. 1.23, [II.A1.7a] virial; I.3.8f, II. 1.58, [II.A1.4a] Volmer; II.1.5c, Il.figs. 1.15-17, [II.A1.3a] (also see: Gibbs' adsorption law) adsorptive; I.1.17-18(def.) Aerosil; see silica aerosol; I.1.5(def.), 1.1.6, V.8.33 AES = Auger electron spectroscopy AFM = atomic force microscopy = SFM, scanning force microscopy ageing; 1.2.99 aggregation; 1.1.2, 1.1.6, IV.fig. 4.23 colloids; IV.2.2c, IV.4.5 emulsions; V.8.3c surfactants; see V.chapter 4 (see the pertaining systems) agitation (foam formation); V.fig. 7.9 air bubbles, electrophoresis; II.4.130ff, IV.3.115 floating; I.fig. 1.4 foam stability; V.7.10, V.7.19, V.fig. 7 submersion of; 1.1.1, 1.1.2, 1.1.11, Ill.fig. 5.19 albumin, adsorption; V.fig. 3.20, V.fig. 3.22, V.figs. 3.24-27, V.fig. 3.28, V.fig. 3.30 negative adsorption; IV.fig. 5.3 second virial coefficient; IV.fig. 5.29
SUBJECT INDEX alcohols, surface dynamics and rheology: III.figs. 3.46, 47, III.4.3c. III.figs. 4.12-16. III.table 4.1 surface entropy; III.fig. 2.15 surface tension; III.4.3c Volta potential; Ill.fig. 4.14, III.fig. 4.24 Alexander-de Gennes model (brushes); V. 1.56, V. 1.60 alkanes, surface entropy; Ill.fig. 2.15 surface dynamics and rheology; III.figs. 46, 47 surface tension, data; Ill.fig. 2.17, Ill.table 2.3, Ill.fig. 4.10 simulations; III.2.41-43, III.figs. 2.12-13 lattice theory; III.2.60 ff, Ill.fig. 2.17-19 alkylpolyglucoside microemulstions; V.5.6b aluminum oxide; IV.fig. 2.2d adsorption of HC1; Il.fig. 1.10 adsorption of water vapour; Il.fig. 1.28 boehmite preparation; IV.2.4c double layer; Il.table 3.8 gibbsite preparation; IV.2.4c point of zero charge; Il.app. 3b alveoles; V.6.8 amphipathic; I.1.23(def.) amphiphilic; I.1.23(def.) amphipolar; I.1.23(def.) analytical ultracentrifugation; IV.2.54ff analyzer; 1.7.27, I.fig. 7.7, 1.7.98 anionic surfactants, see surfactants anisotropic media; 1.7.14 birefringence; 1.7.97, 1.7.100 scattering; 1.7.8c anisotropy; 1.7.8c, 1.7.14 of colloidal particles; see particles (colloidal), shape annealed (polyelectrolytes); V.1.1 antagonism (in coagulation); IV.3.9k antifoam; V.7.6 antithixotropy; IV.6.14 Antonow's rule; [III.2.1 1.12]. III.5.76 apolar media, double layers; II.3.11
7
8
SUBJECT INDEX
apolar media (continued), electrokinetics; II.4.50 solvation; I.5.3f arabic gum; see gum arabic Archimedes principle (for interaction in a medium); 1.1.30, 1.4.42, 1.4.47, 1.4.50, I.4.69ff, I.fig. 4.15, IV.3.91ff d'Arcy's law (flow in porous media); I.6.4f, IV.2.32ff, IV.4.50-51 area; see surface area association colloids; I.1.6(lntr.), I.1.23ff, IV. 1.5 association colloids, general, esp. modelling; V.chapter 4 bending (of worm-like micelles); V.4.6d bending and vesicles; V.4.7d, V.flg. 4.38, V.fig. 4.47 bilayers (llpid); V.4.37ff, V.fig. 4.7, V.4.7c, V.4.99ff critical micellization concentration; V.4.1c, V.table 4.1. Many examples in V.chapter 4 cylindrical micelles; V.4.6, V.figs. 4.25-30, V.4.115ff disc-like micelles; V.4.7b, V.figs. 4.33-34 end-cap energy; V.4.6c, V.fig. 4.27, V.fig. 4.45 extensive introduction; V.4.1a ionic; V.4.5, V.figs. 4.18-24, V.fig. 4.37 kinetics of micellization; V.4.10 lamellar phases; V.4.7 lamellar phases, interactions; V.4.8 mass action model; V.4.2b micelles, size fluctuations; V.4.2d mixed micelles; V.4.9a, V.figs. 4.42-43 molecular simulations (MD, Monte Carlo); V.4.30, V.4.33ff, V.figs. 4,5-6 non-ionic; V.4.4, V.figs. 4.9-14, V.figs. 4.25-29, V.4.7, V.figs. 4.31-36 pluronics; V.4.4e, V.figs. 4.15-16 profiles; V.fig. 4.1, V.figs. 4.6-7, V.figs. 4.10-12, V.fig. 4.15, V.figs. 4.20-21, V.fig. 4.26, V.fig. 4.32, V.fig. 4.35, V.fig. 4.46 quasi-macroscopic models; V.4.3c, V.fig. 4.8, V.4.5e second c.m.c; V.4.6e self-consistent field theory; V.4.3b, V.fig. 4.6, V.fig. 4.9, V.4.4, V.4.5, V.4.6b, appendix V. 1 solubilization; V.4.9b, V.fig. 4.46 surfactant packing parameter; V.4.1d, [V.4.1.4] thermodynamics (classical); V.4.2 thermodynamics of small systems; V.4.19ff undulation forces; V.4.8a, V.figs. 4.39-40 vesicles; V.figs. 4.35-38
9
SUBJECT INDEX
association constant; I 5.2d association of ions; see ion association association of water; 1.5.3c atomic force microscopy; 1.7.90, II.1.12ff, 11.figs. 1.3-4, II. 1.91, II.fig. 2.17, III.table 3.5, III.3.7d, III.figs. 3.66-68, III.fig. 5.28, V.fig. 1.43, V.fig. 3.14 ATR = attenuated total reflection, attenuated total reflection; 1.7.81, 11.1.18, II.2.54, II.fig. 2.16, V.fig. 3.18 Auger (electron) spectroscopy; 1.7.lla, I.table 7.4 autocorrelation function; Lapp. 11.1 autophobicity; II. 1.80, II. 1.101 averaging; I.3.1a(intr.) azeotrope (in adsorption from solution); II.2.23 a-method (porous surfaces); II. 1.89-90 bacteria, Corynebacterium;
II.fig. 4.39
halophilic; 1.1.27 Nitrosobacter,
Nitrosomonas;
II.3.122
BAM = Brewster angle microscopy Bancroft rule; III. 1.84, III.4.97, V.8.5, V.8.59ff barium sulphate; II.table 1.3 barometric distribution; 1.1.20 barrier crossing; IV.fig. 4.4 barycentric derivative; 1.6.5 Bashforth-Adams tables (for capillarity); III. 1.18ff Batchelor eq. (viscosity); [IV.6.9.9] Baxter model (adhesive hard sphere); IV.5.41, IV.flgs. 5.22-26, IV.fig. 5.30 BBGKY = Bogolubov-Born-Green-Kirkwood-Yvon (recurrency expression); II.3.53ff BDDT = Brunauer-Deming-Deming-Teller, II.(isotherm classification); II. 1.4b beam splitters; 1.7.98 beating (optical) = optical mixing beating (mechanical, foam preparation); V.7.13 bending; III.fig. 1.34 bending moment of interfaces; 1.2.91, V.8.86 bending moduli of interfaces; [III.1.10.2], III.1.55, III.1.15, III.1.79, Ill.tables 1.6 and 1.7 (data); III.4.7, V.4.6d, V.4.7c, V.5.5a, V.fig. 5.34 Bernouilli's law; [V.8.2.2] Berthelot principle (for interaction between different particles): 1.4.3Iff, [III.2.1 1.18] BET = Brunauer-Emmett-Teller; see adsorption isotherm BET-transformed; II. 1.69 Bethe-Guggenheim (approx.) = quasi-chemical biaxiality (anisotropic systems); 1.7.97
10
SUBJECT INDEX
bicontinuity in microemulsions; see there bilayers; V.4.7 Bingham fluid; IV.fig. 6.5, IV.fig. 6.17, IV.fig. 6.21 Bingham viscosity; IV.6.12, 6.40 binodal; 1.2.68, II.5.12, II.fig. 5.4, III.2.26, IV.5.64ff, IV.fig. 5.62 binominal; [IV.A. 1.4] biological activity; V.fig. 3.29 biomineralization; IV.2.38 biopolymers; 1.1.2, 1.1.27, Il.fig. 5.5, V.chapter 3 birefringence; 1.7.14 Bjerrum length; [1.5.2.30a] for polyelectrolytes; [II.5.2.23], V.2.9 Bjerrum theory (ion association); I 5.2d black body (radiation); 1.7.22 black film; see film, liquid blob; II.5.11, V.fig. 1.31 blood clotting; V.3.52ff body (or volume) forces; I.1.8ff(intr.), 1.4.2 Bohr magneton; 1.7.95 boiling point elevation; 1.2.74 Boltzmann equation, II.Boltzmann factor; I.3.10ff, [II.3.5.4], II.3.216 dynamic; II.3.217 Boltzmann's law (for entropy); [1.2.8.2], [1.3.3.7] (also see: Poisson-Boltzmann equation, theory) Booth eq. (prim, electroviscous effect); [IV.6.9.16] Born-Bjerrum equation (solvation); [1.5.3.4] Born equations (solvation); 1.5.3b, II.3.123 Born repulsion; 1.4.5 Bose-Einstein statistics; 1.3.12 Boyle point; 1.2.51, 1.2.64, II.5.6 Boyle-Gay Lussac law; 1.2.51 Bragg-Williams approximation; 1.2.62, I.3.8d, I.3.49ff, II.1.56ff, V.2.17 Bredig sols; IV.2.37 Brewster's angle; 1.7.74, II.2.51, V.fig. 6.4 Brewster angle microscopy; III.table 3.5, III.fig. 3.56 bridging; V.chapter 1, especially V.8.1.11 V.8.73 bridging (foam destruction); V.7.32 Brillouin lines; 1.7.44 Bronsted (acids, bases); 1.5.65, II.2.7, IV.4.2b Brownian motion; 1.3.34, 1.4.2, I.6.3a. I.6.3d, Lapp, l i e , IV.2.7, IV.4.2b in a force field; 1.6.3b, IV.4.3b
SUBJECT INDEX
Brownian motion (continued), rotational; 1.6.73 BSA = bovine serum albumine; see albumine brushes (at interfaces); III.3.4J, IV.4.5, V.I.11, V.fig. 1.28, 29, V . l . l l g , V.2.3c bubbles; see air bubbles Burgers element; III.3.129 Cabannes factor; 1.7.53 Cabosil; see silica Cahn electrobalance; III. 1.44 Cahn-Hilliard theory (for interfacial tension); III.2.6, V. 1.7 calcium carbonate (structure factor); IV.fig. 5.33 calomel reference electrode; 1.5.85 calorimetry (adsorption); II.1.3c, Il.flg. 1.7, II. 1.29, II.2.5b, II.5.60 canal surface viscometer; III.3.183-184 capacitance, electric; 1.4.51, 1.5.13, II.3.7c, II.3.94, II.3.106 differential; I.5.13(def.), 1.5.15, 1.5.100, II.3.10, II.3.21, II.fig. 3.5, II.3.29, Il.flg. 3.9, II.3.33, II.3.36, II.3.6c, Il.flg. 3.22, Il.figs. 3.42-43, Il.flg. 3.49, Il.figs. 3.50-51, Il.flg. 3.53, II.3.149, II.5.60-61 integral; I.5.13(def.), 1.5.15, 1.5.59, II.3.10, II.3.6c capillaries (electrokinetics in), electrokinetic velocity profile; Il.figs. 4.15-16 electro-osmosis; II.4.2Iff electrophoresis; II.4.132 streaming current; II.4.3d streaming potential; II.4.3d capillary bridges; III.1.49, III.1.84, III.5.11d capillary condensation; II.1.42, II.1.6, Il.figs. 1.32-33, Il.flg. 1.35, Il.flg. 1.39 capillary depression; I.1.8ff, I.fig. 1.1, Hl.flg. 1.4b, III.5.4e capillary electrometer; II.3.139ff, Il.flg. 3.47, III.1.20 capillary length; [III. 1.3.3], Ill.table 1.1 capillary number; [III.5.8.1], also see: [V.8.2.3] capillary osmosis; II.4.9 capillary phenomena (general); 1.1.3, 1.2.23, II.1.6d, II.1.6e, III.1.1, III.1.1, III.1.2 capillary pressure; I.1.8ff, I.fig. 1.9, I.fig. 1.10, 1.2.23, II.1.6, V.6.25, V.7.5, V.8.2 Young and Laplace's law; I.1.9(lntr.), 1.1.12, 1.1.15, I 2.23b. especially [1.2.23.19], II.1.85ff, II.1.99, III.1.1, [III. 1.1.2] capillary rise; 1.1.2, I.1.8ff, I.fig. 1.1, II.l.Be, III. 1.3, Ill.flg. 1.4a, III. 1.83, III.5.4e capillary waves; III.2.9c, III.3.6g, III.3.10 capture efficiency (orthokinetic); IV.fig. 4.18
11
12
SUBJECT INDEX
carrier wave; 1.7,38 carbon, graphite, AFM image; II.fig. 1.3 adsorption of: benzene, n-hexane; II.fig. 1.8 carbon tetrachloride; Il.figs. 1.22-23 hexane + hexadecane; II.fig. 2.20 krypton; Il.fig. 1.29 long alkanes from n-heptane; Il.figs. 2.28-29 n-heptane-cyclohexane mixture; Il.fig. 2.18 octadecanol; Il.fig. 2.17 pentane + decane; Il.fig. 2.20 rubber; Il.fig. 5.31 water vapour; Il.fig. 1.11, Il.fig. 1.28 soot; IV. 1.3 immersion (= wetting) enthalpy; II.table 1.3 carboxymethyl cellulose solutions, viscosity; IV.fig. 6.35, V.fig. 2.33 Carnahan-Starling equation [1.3.9.31], [IV.5.4.14] casein ((3 , K ); IV.fig. 5.18, V.figs. 3.24-27, V.fig. 8.19, V.fig. 8.20 Casimir-Polder equation (for retarded Van der Waals forces); [1.4.6.35], [1.4.7.9] Cassie equation; [III.5.5.2] caterpillar trough; Ill.flg. 3.74 cation exchange capacity; 1.5.99, II.3.165ff cationic surfactants, see surfactants CBF = common black film; see films, liquid c.c.c. = coagulation, critical concentration, see colloid stability CD = circular dichroism; see dichroism c.e.c. = cation exchange capacity cell (galvanic); see galvanic cells centrifugation potential (gradient); Il.table 4.4, II.4.6-7, IV.2.54 ceramics; IV. 1.6, IV.3.185 Hamaker constants, IV.app. 3 chain crystallization; V.8.6 chain statistics; II.5; V.I, V.2.3 Chandrasekhar equation; [1.6.3.20] Chapman-Kolmogorov equation; [1.6.3.13], [IV.4.2.5] characteristic curve (in potential theory for gas adsorption); II. 1.74 characteristic functions (in statistical thermodynamics); 1.3.3, [1.3.3.8], III.table 3.2 (in Langmuir monolayers), III.fig. 3.14 charge (electric); 1.5.3, 1.5.9, I.fig. 5.1 (also see: double layer, surface charge density, space charge (density))
SUBJECT INDEX
charge-determining ions; I.5.5b, II.3.7, II.3.8, II.3.84ff, II.3.89, II.3.147ff charge reversal; II.3.62, see further overcharging charged (colloidal) particles; II.chapter 3, II.chapter 4 concentration polarization; II.3.206, II.3.13c, II.4.6c, [II.4.6.53], II.4.8, [II.4.8.22] contribution to conductivity; II.figs. 4.37-39 contribution to dielectric permittivity; Il.figs. 4.37-39 far fields; II.3.207, II.3.13b, II.3.211, II.3.217, II.4.18-19, II.4.6, II.4.8 fluxes; II.3.215ff, II.4.6, II.4.8 (in) alternating fields; II.4.8 interaction; see interaction between colloids induced dipole moment; II.3.206, II.3.210, II.3.212ff, II.4.8 local equilibrium; II.3.213, II.4.79 near field; 11.3.211,11.4.6 polarization field; II.3.207, II.3.209, II.3.21 Iff, II.4.18-19, II.4.70, II.4.87, II.4.8 polarization in external field; II.3.13, Il.fig. 3.86, Il.fig. 3.88, II.4.3a. II.fig. 4.2, II.4.18ff, II.4.6, II.4.8 relaxation; II.3.13d, Il.fig. 3.89, II.4.6c, II.4.8 charging of double layers; I.5.17ff, 1.5.7, II.3.5, IV.3.1, IV.3.2, V.2.3, V.3.8ff (also see: under double layer, electric; Gibbs energy; for specific examples see under the chemical name) charging parameter; 1.5.17, 1.5.106 cheese rheology; IV.6.15, IV.6.19 chemical potential (intr.); 1.2.1 Iff, 1.2.35 dependence on curvature; 1.2.23c dependence on pressure; I.2.41ff dependence on temperature; 1.2.40 (of) polymers; II.5.9 chemisorption; II.1.6, II.1.32, II.1.18, Il.fig. 1.9, II.2.85 chirality; 1.7.100, III.3.216 cholesterol monolayers; Ill.fig. 3.13, III.3.8d, Ill.figs. 3.92-93 chromatography; II.2.47ff, II.2.88 eluate; II.2.48 field flow fractionation; IV.2.61ff high performance liquid (HPLC); II.2.47 retention volume; II.2.48 chymotrypsin; V.fig. 3.29 c.i.p. = common intersection point circular dichroism (CD); see dichroism circular polarization; see electromagnetic radiation, polarization
13
14
SUBJECT INDEX
Clapeyron equation chemical equilibrium; [1.2.21.1 1 and 12] gas adsorption; [ 1.3.39] solubility; 11.2.20.6] in pores; II. 1.99 two-dimensional; III.3.38 Clausius-Mosotti equation (for polarization of a gas); [1.4.4.10] clay minerals (general); 1.5.99 cation exchange capacity; 1.5.99, II.3.165ff double layer; Il.fig. 3.1c, II.3.8, II.3.10d electrokinetics; II.3.168 isomorphic substitution; II.3.2, II.3.165 structures; II.3.163ff, Il.figs. 3.66-67 swelling; II.3.163ff cleaving of solid surfaces; 1.2.99 closure relations; 1.3.69, IV.5.3d cloud point; V.4.11 cloud seeding; II.3.130 CLSM = confocal laser scanning microscopy; see CSLM cluster integral; 1.3.65 c.m.c. = critical micellization concentration; see micellization coacervation; IV.5.95ff, IV.fig. 5.64 coagulation; I.1.6(intr.), 1.1.7, 1.1.28, 1.4.7, 1.7.61, IV.3.9 also see; colloid stability coagulation (flocculation) kinetics; IV.2.2d, IV.figs. 3.65-66, IV.chapter 4, V.I.85 critical concentration; II.3.129ff, IV. 1.11 fractal formation; IV.4.5 irregular series; II.3.62; see further, overcharging orthokinetic; IV.5b particle size effects; IV.4.32 perikinetic (def.); IV.4.37 rapid; IV.4.3a slow; IV.4.3b surface roughness effects; IV.4.32 coalescence, (emulsions); V.8.3b, V.8.53ff, V.8.64 partial; V.8.73 coalescence (foams); V.7.10 coherence (of radiation, electromagnetic waves); 1.7.15, I.7.22ff, 1.7.69 coherence time; 1.7.22 coherent neutron scattering; 1.7.70 cohesion (in liquids); 1.4.5c
SUBJECT INDEX
cohesion (work of); 1.4.47, 111.5.2, III.fig. 5.9 cohesion pressure; 1.3.69 cohesion (or cohesive) energy; 1.4.46 coil (of polymer molecules); 1.1.26, I.fig. 1.17, 11.5.2 co-injection (foaming); V.7.13 co-ions; I.1.21(def.) Cole-Cole diagram; 1.4.35, Il.fig. 3.30b collapse (monolayers) (def.); III.3.23, III.fig. 3.46, III.3.226 collector (in flotation); 1.1.25, III.5.97 colligative properties (intr.); I.2.20f collision broadening (spectral lines); 1.7.22 colloids (general); I.1.5ff(def.), 1.1.2, IV.chapter 1, Volume IV (particulate colloids), Volume V (hydrophilic colloids) characterization; IV.2.3 colour; IV.2.39 light scattering; IV.2.3b microscopies; IV.2.3a, IV.2.41ff sedimentation; IV.2.3d; see further separate entry surface area; III.3.131ff, IV.2.3c concentrated; IV.chapter 5 electron micrographs; Il.fig. 1.1, IV.figs. 2.1, 2.2 and 2.4 emulsions; V.chapter 8 in external fields; II.3.13, Il.chapter 4 fractionation; IV.2.2h history;IV. 1.4 hydrophilic; I.1.7(def.); IV.1.11 (def.), Volume V (general) hydrophobic; I.1.7(def.); IV. 1.11 (def.), Volume IV (general) stability; 1.1.6, I.1.22ff (also see: colloid stability, general) interaction between colloids and macrobodies, interaction curves; I.fig. 6.2, IV.chapter 3, IV.chapter 4, IV.chapter 5 atomic force microscopy; II. 1.13, IV.3.12 constant charge vs. constant potential; 1.5.108, IV.3.8ff, IV.fig. 3.1 density correlation functions; Lapp, l i e , IV.chapter 5 depletion; IV.3.10 Deryagin approximation; I.4.60ff, IV.3.2, IV.3.7c Deryagin-Landau-Verwey-Overbeek (DLVO) theory; IV.chapter 3 Deryagln-Landau-Verwey-Overbeek extended (DLVOE) theory; IV.3.9 in films; V.6.5 disjoining pressure; [V.I. 1.2 and 3], IV.chapter 3 dynamics; IV.chapter 4. IV.4.3 (def.)
15
16
SUBJECT INDEX
colloids, Interaction between colloids and macrobodles, Interaction curves (continued), electric; IV.chapter 3 comparison of models; IV.3.7f diffuse, constant charge; IV.3.4 diffuse, constant potential; IV.3.3 Gouy-Stern layers, regulation; IV.3.5, IV.figs. 3.24-28, IV.3.9d spherical double layers; IV.3.7, IV.fig. 3.30 energy barrier; IV.3.9, IV.fig. 4,4 external field; IV.3.10, IV.4.5, IV.5.30 forced interaction; IV.3.6, IV.3.10, IV.4.3, IV.4.5b, IV.5.3c hetero-interaction; IV.3.4, IV.3.6 hydrodynamics (influence); IV.4.5b kinetics; IV.chapter 4 induction; IV.fig. 3.21 linear superposition approximation (LSA); IV.3.12 Maxwell stress; IV.3.22 measurement; 1.4.8, Il.figs. 2.2-3, II.3.56ff, IV.3.12 nanoparticles; IV.3.81 orthokinetic; IV.4.5b primary minimum; I.fig. 4.2, IV.3.9 regulation; IV.3.4, IV.3.5 Theological consequences; IV.6.13 secondary minimum; I.fig. 4.2, IV.3.9 simplified models; IV.5.2c, IV.5.4, IV.5.5, IV.5.6a, IV.5.81 solvent structure-mediated; 1.5.3, 1.5.4, IV.3.5, IV.3.8c surface force measurements; IV.3.12 surface roughness; IV.3.82ff tabulation of electrical interactions; IV.app. 2 thermodynamics; IV.3.2, V.I.I timescales; IV.4.1ff two-dimensional; HI.3.241 virlal approach; 1.7.8b, IV.chapter 5 irreversible = hydrophic, lyophobic mills; IV.2.2g mixtures; IV.5.7c, IV.5.8c preparation; 1.1.6, 1.2.100, IV.chapter 2 (general) by comminution; IV.2.29 by condensation = by precipitation
SUBJECT INDEX
17
colloids, preparation (continued), by precipitation; V.2.2a (homogeneous), IV.2.2b (kinetics), IV.2.2c, IV.2.2f (heterogeneous) dispersity; IV.2.2d examples of sol preparations; IV.2.4 fractionation; IV.2.2h, IV.2.54ff also see separate entry nucleation and growth; IV.2.2b, IV.2.2f particle growth; IV.2.2d size control; IV.2.2a size distributions; see separate entry sol-gel processing; IV.2.2J reversible = hydrophilic, lyophilic solvent structure contribution; Il.flgs. 2.2-3, II.2.10, III.3.8c solubility; IV.2.2e wetting; III.5.4h (also see: particles, charged (colloidal) particles) colloid stability, stabilization; IV.chapter 3, IV.chapter 4 adsorption versus depletion; IV.3.2, V.I.10 bridging; V.1.6b, V.fig. 1.6 brushes; V.I.11, V.fig. 1.29, V . l . l l g by (bio)polymers (including steric stabilization); I.1.2(intr.), 1.1.7, I.fig. 1.18, I.1.27ff, I.fig. 2.11, II.5.96ff, IV. 1.3, IV. 1.4, IV.fig. 1.2, V.chapter 1 (general), V.6.5c by polyelectrolytes; V.2.7 by surfactants; 1.1.25 case studies; IV.3.13 concentration profiles; V.1.5 critical coagulation concentration (c.c.c); IV.3.98, IV.3.9e, IV.table 3.2 depletion (flocculation); IV. 1.5, V.1.8, V.1.9, V . l . l l h disjoining pressure; IV.chapter 3, V.1.6, V.1.7 DLVO theory; I.1.21ff (intr.), 1.3.59, IV.1.14, IV.chapter 3 DLVOE theory; IV.chapter 3, esp. 3.9 emulsions; V.8.1g equilibrium aspects; V.I, V.I.12 flocculation kinetics; V.1.12e general; I.1.6ff(intr.), I.1.21ff, I.fig. 1.14, 1.2.71, I.fig. 2.11, 1.4.8, IV.chapter 3 gravity influence; IV.3.10a Gibbs energy; IV.chapter 3, V.1.6, V.flgs. 1.7-13, V.1.7, V.1.8b-d, V.1.9c, V . l . l l f grand potential; V.I.I, V.l.3-4, V.1.6 Helmholtz energy; V. 1.1, V. 1.4, V. 1.10, V. 1.57ff, V. 1.65ff
18
SUBJECT INDEX
colloid stability, stabilization (continued), influence electric field; IV.3.10b irregular series; see overcharging lyotropic series; see separate entry magnetic forces; IV.3.10c measurement; IV.3.102ff, IV.3.12 mushroom interaction; V. 1.1 If nonaqueous media; IV.3.11 orthokinetic coagulation, flocculation; IV.4.5b, V. 1.84 Ostwald ripening; IV.2.2e, V.8.3b rheological consequences; IV.6.13 and point of zero charge; II.3.106 Schulze-Hardy rule; IV.3.9e tethered; V.I. 11 (also see: colloids interaction, Van der Waals interaction) colloid titration, calorimetric; II.3.98 conductometric; II.3.88, Il.fig. 3.20 polyelectrolytes; V.2.2d potentiometric; I.5.100ff, I.fig. 5.17, II.3.7, II.3.85, Il.fig. 3.29, II.3.151, Il.figs. 3.57-59, II.5.60, IV.fig. 3.75 proteins; V.3.5ff, V.figs. 3.15-17 colloid vibration potential; Il.table 4.4, II.4.7, II.4.3e, II.4.5d colloidal dispersion; IV. 1.9 (def.) comminution of big particles; IV.2.2g common intersection point (in colloid titration curves); II.3.8a, Il.fig. 3.34, Il.figs. 3.57-59, Il.figs. 3.63-64, Il.fig. 3.77, Il.fig. 3.80, II.3.206 complex coacervate micelles; V.2.6f, V.fig. 2.39 complex coacervation; IV.5.95, V.2.6c, V.figs. 2.34-36 complex quantities; Lapp. 8 compliance; see (interfacial) rheology composition law (polymer adsorption); II.5.40 compositional ripening (emulsions); V.8.71 compressibility; 1.7.46, [III.2.11.4] Ornstein-Zernike equation; [1.3.9.321, [IV.5.2.7] two-dimensional; see interfacial rheology compression; IV.6.2 compression modulus; IV.6.6 concentrated polymer regime; II.5.9, Il.fig. 5.3. IV.6.11, IV.6.12 concentration profiles (ions); Il.fig. 3.8, Il.fig. 3.20, III.3.4h, IV.fig. 3.1 concentration profiles (polym. ads.); V.1.5
SUBJECT INDEX
condensation, counterions; V.2.2a condensation, homogeneous, 1.2.23d condensation (method for preparing colloids); IV. 1.2, IV.2.2 condensation, (two-dimensional); 1.3.43, Il.figs 1.3.5-8, I.3.47ff, I.3.53ff, II.1.59ff, Il.fig. 1.20, Il.figs. 1.31-33, Il.fig. 1.35, Il.fig. 1.39, Il.fig. 1.42, II.2.66 conduction (electrolytes); 1.6.6 (also see: surface conductance, surface conductivity) conduction bond (solids); Il.fig. 3.68, II.3.173 conductivity (electrolytes), limiting; 1.6.6a, I.table 6.5 molar; 1.6.6a (also see: surface conductivity) conductivity of, capillaries and plugs; II.4.55ff, II.4.7, Il.fig. 4.34 colloids: a.c. measurements; II.4.5e, Il.figs. 4.21-22, II.4.8, Il.figs. 4.37-39, IV.fig. 4.16 colloids (non-aqueous); IV.3.134 ions and ionic solutions; 1.5.51, I.6.6a, I.table 6.5, I.6.6b microemulsions; V.5.3f polyelectrolytes; V.2.5b, V.2.5c thin films; V.6.2g, V.fig. 6.37 water; 1.5.43 conductometric titration of colloids and polyelectrolytes; see colloid titration configurations; 1.3.11, I.3.29ff, II.5.2, V.2.3 also see; polymer adsorption configuration integrals; I.3.9a, [1.3.9.6], IV.chapter 5 configurational energy; 1.3.46 configurational entropy; 1.2.52, 1.3.30, V.3.2a confocal laser scanning microscopy; 1.7.91, IV.5.92 conformation; U.S. 1, see under polymers, polyelectrolytes, proteins congruence (adsorption from binary mixtures); II.2.3e charge; II.3.198 electrosorption; II.3.198, Il.fig. 3.81 pH; 11.3.155,11.3.198 temperature; II.2.27, II.3.156 conjugate acid (def.); 1.5.65 conjugate base (def.); 1.5.65 conjugate force; see force conservation (of energy); see energy conservation (of momentum); 1.6.lb, IV.6.4
19
20
SUBJECT INDEX
conservation laws; see hydrodynamics conservative force (def.); 1.4.1 consistency test (adsorption from binary mixtures); II.2.3e, II.fig. 2.19 contact angle; 1.1.3, 1.1.8, I.fig. 1.1, Il.figs. 1.40-41, Ill.fig. 1.1, III.chapter 5 and enthalpy of wetting; II. 1.29, III.5.2 data; Ill.app. 4 heterogeneous precipitation; IV.2.2f hysteresis; Ill.fig. 1.20, III.1.41ff, III.5.5, III.5.4, Ill.fig. 5.4, III.5.9-10, III.5.40 measurement (general); III.5.4, (in films) V.6.2e, V.6.3e captive bubbles; III.5.4b capillary rise/depression; III.5.4e fibers; III.5.4g films, liquid; V.6.2e, V.6.3e, V.flgs. 6.18-19, V.fig. 6.38 individual particles; III.5.4h objects in interface; III.5.4c powders, porous materials; III.5.41 pressure compensation; III.5.50 sessile drops; III.5.4b spinning drop; III. 1.53 tilted plates; III.5.4d contact angle, advancing; see hysteresis contact angle, dynamics; III.5.8 contact angle, interpretation; III.5.7 contact angle, receding; see hysteresis continuity equations; I.6.1a, IV.6.1, IV.6.2 contrast matching (in neutron scattering); 1.7.70 convection; 1.6.37, V.8.75 convective diffusion; 1.6.7c convolution; I.A10.3 co-operativity; II. 1.48 coordination number; 1.3.45, 1.4.46, 1.5.3c copolymers in microemulsions; V.5.6e copper phtalocyanine pigment; IV.fig. 1.5 cordierite; IV.fig. 2.2a core (of micelles) = interior part corona (of micelles) = exterior part correlation coefficient; I.3.9e correlation function; IV.5.2a direct; IV.5.16 pair; 1.3.66, II.3.5Iff
SUBJECT INDEX
correlation function (continued), time (-dependent), 1.6.31, I.7.6c, I.7.6d, 1.7.7, Lapp. 11, II.2.14, IV.2.46ff total; [1.3.9.23], II.3.51ff, IV.5.6ff, IV.5.16 (various examples in chapter IV.5) correlation length; 1.7.46, II.1.94, II.5.11, III.2.27, IV.4.4, V.5.39, V.5.3h correlation time, rotational; 1.5.44 correlator; 1.7.6c, I.7.6d corresponding states; III.2.51, III.2.53, V.5.19ff, V.5.55 corrosion inhibition; II.3.224 Cotton-Mouton effect; 1.7.100 Cottrell equations; [1.6.5.20, 11.21], I.fig. 6.15a Couette viscometers/rheometers; IV.6.7b Coulomb's law, Coulomb interactions; [1.4.3.1], 1.4.38, 1.5.11, 1.5.16, 1.5.17, 1.5.21,11.3.36, II.3.48ff countercharge; I.1.20(def.), II.3.2, II.3.7 see, electric double layers counter ions; I.1.21(def.), see, double layer (ionic components of charge), lyotropic sequences, specific adsorption coupling parameter (Kirkwood); 1.3.68 copper phtalocyanate; II.table 1.3, IV.fig. 1.5 creep flow; 1.6.45, IV.6.6b for interfacial creep, see interfacial rheology critical coagulation concentration (c.c.c); see colloid stability critical micellization concentration; see micellization critical opalescence; 1.3.37, 1.3.69, I.7.7c, IV.2.41 interfacial; 1.7.83 critical point or critical temperature; IV.fig. 2.3, IV.fig. 5.41, IV.fig. 5.43, IV.5.7a, IV.fig. 5.62 for polymer demixing; II.5.12 in pores; II.fig. 1.39 two-dimensional; I.3.49ff, 1.3.53, II. 1.109 critical radius (nucleation); 1.2.101, IV.2.2b cross coefficients (in irreversible thermodynamics); 1.6.12, 1.6.2 cross differentiation (principles); 1.2.14c cross-section (molecular); see surface area cryogenic transmission electron microscopy; IV.2.42-43 crystal defects (in semiconductor); II.3.172 crystal growth; II.5.97, IV.2.2 crystallization (of concentrated colloids); IV.5.8a
21
22
SUBJECT INDEX
CSLM = CLSM = confocal laser scanning microscopy; 1.7.91 Curie temperature; IV.3.124 curvature (of interfaces); I.2.23a,III.1.4ff, III.1.17, III.1.15, V.4.7c, V.5.24ff, V.fig. 5.26, V.fig. 5.27b influence on chemical potential; 1.2.23c mathematical description; III. 1.78 radius of; 1.2.23a, figs. 1.2.14-15, III.1.4ff, III.1.2 spontaneous; III. 1.78, V.5.25, V.fig. 5.14 (also see: bending moment, bending modulus) cut-off length (gel); IV.4.49 CVP = colloid vibration potential Dalton's law; [1.2.17.2] damping (of oscillations); I.4.37ff see further interfacial rheology, wave damping dashpot (and spring); III.fig. 3.50, IV.6.6, see further (interfacial) rheology; Maxwell element and Kelvin (or Voigt) element; Darcy's law; see d'Arcy's law Davies equation (for ionic activity coefficient); [1.5.2.28] Deborah number; 1.2.6, 1.2.86, 1.5.77, 1.6.2, II.1.8, III.1.32, III.1.35, III.3.5, III.3.12, III.3.90, III.4.62, IV.4.33ff, IV.6.16-17, [IV.6.4.3] De Broglie wavelength; 1.3.23,1.7.24 Debye equation (for polarization of gases); [1.4.4.8] Debye-Falkenhagen effect; 1.5.60, I.6.6c, II.4.111 Debye-Huckel approximation (intr.); 1.5.19 Debye-Huckel limiting law; [1.5.2.22] Debye-Huckel theory, for pair interaction; IV.3.3d, IV.3.68 for polyelectrolytes; V.2.2d for strong electrolytes; 1.5.2, 1.6.6b Debye length; [1.5.2.10], I.5.19(def.), I.table 5.2, II.3.19, [II.3.5.7]ff, [II.3.10.22] Debye-Van der Waals forces; see Van der Waals forces Debye relaxation; 1.6.73 decomposition; see demixing deep channel surface shear viscometer; III.fig. 3.70 defoaming; IV.7.6 deformation (in rheology); IV.6.1, IV.6.2 degeneracy; 1.3.4 degree of dissociation; 1.5.30, II.3.76, II.5.56, V.2.2d degrees of freedom (intr.); 1.2.36 de-inking; III.5.102 delayed (elastic) recovery; IV.fig. 6.12
SUBJECT INDEX
23
delta formation (relation to colloid stability); 1.1.1, 1.1.2, 1.1.7, IV. 1.2, IV.3.184 demixing; 1.2.19, II.1.6e, IV.fig. 2.3, IV.5.7a binodal; 1.2.68, Il.fig. 5.4 critical; 1.2.19, II.1.6e spinodal; 1.2.68, Il.fig. 5.4 two-dimensional; III.3.4e density correlation functions; Lapp, l i b , Lapp, l i e density functional; [III.2.5.181, III.2.34, IH.app. 3 density profiles, liquid-fluid interfaces; III.2.4, III.2.5, [HI.2.5.311, Ill.fig. 2.6, Ill.fig. 3.29 see also concentration profiles, distribution functions (of liquids near solids and of fluid interfaces), adsorption of polymers depletion (adsorption) = adsorption, negative layer thickness; V.1.8, V.1.9 depletion interaction; IV.3.10, IV.5.79ff, V.1.8, V.1.9, V.8.72ff depolarization (of polarized interface); II.3.137 deposition; IV.4.3, IV.4.43-44 depolarization ratio; 1.7.54 Deryagin approximation (to compute interactions between non-flat colloids); 1.4.61, I.fig. 4.13, 1.4.64, IV.3.2, IV.3.7c Deryagin-Landau-Verwey-Overbeek (DLVO) theory; see colloids, interaction and colloid stability desalination; 1.1.3 (also see: salt-sieving) Descartes' law = Snell's law desorption; I.1.5(def.) detectors (for radiation); 1.7.lc quadratic; I.7.36ff detergency; III.5.101 dewetting; III.5.4, III.5.10, V.fig. 5.30, V.flg. 7.14 dextrane, adsorption on silver iodide; Il.fig. 5.26b, II.5.80ff, Il.fig. 5.29 DFG = difference frequency generation; III.3.7c.v dialysate; I.5.86ff dialysis; IV.2.32 diamagnetism; IV.3.124 dichroism; 1.7.98, 11.2.56 circular (CD); 1.7.99, V.fig. 3.28, see chapter V.3 (general) dielectric displacement; I.4.5f, 1.7.9, IV.3.124 dielectric dispersion of sols, low and high frequency: 11.3.219. Il.fig. 3.89, II.4.1 10
24
SUBJECT INDEX
dielectric dispersion of sols (continued), measurements; II.4.5e, II.fig. 4.21 theory; II.4.8, IV.4.5a, IV.figs. 4.16-17 dielectric drag; 1.5.51 dielectric increment, of colloids; II.4.8, Il.fig. 4.37-39, IV.figs. 4.16-17 of ions; I.table 5.10 dielectric permittivity (dielectric constant); 1.4.10, I.4.4e, I.4.5a, I.4.5f, 1.5.11,
I.table 5.1,1.5.3e, 1.7.2,1.7.6 complex formalism; I.4.4e, 1.7.2c, Lapp. 8 emulsions; V.8.19ff measurement; 1.4.24, I.4.5f, 11.4.5e, Il.figs. 4.21-22 relation to polarization; I.4.23ff relation to refractive index; 1.7.12 dielectric polarization; see polarization dielectric relaxation; I.4.4e, I.4.5f, II.4.8 dielectric saturation; 1.5.11 dielectrophoresis; II.4.51 differential scanning spectroscopy; V.3.26, V.fig. 3.28 diffraction, principles; 1.7.13, 1.7.24 diffraction colours; IV.2.40 diffuse charge, diffuse double layer; see double layer, diffuse (also see: surface charge) diffuse transmission spectroscopy; V.7.23 diffusing wave spectroscopy, IV.4.9, V.7.23 diffusion (coefficient); 1.6.3, 1.6.5, 1.7.15, Lapp. H e along surface; I.6.5g, II.2.14, II.2.29, III.3.74 and correlation functions; Lapp. 11 and irreversible thermodynamics; 1.6.5a collective; 1.6.55, 1.7.15, Lapp, l i e colloids; IV.4.1ff, IV.4.2 concentrated sols; 1.7.66, 1.7.15, Lapp, l i e convective; 1.6.7c data; I.table 6.4 forced; 1.6.53, 1.6.7 hydrodynamic correction; 1.6.56, Lapp, l i e in condensed media; 1.6.56 in films; V.fig. 6.45 in gases; 1.6.55 in water; I.5.44ff model interpretation; 1.6.5b
SUBJECT INDEX
diffusion (coefficient) (continued). non-linear geometry. I.6.5f non-spherical particles; I.6.69ff, I.fig. 6.19 of colloids from dynamic light scattering; 1.7.8b, 1.7.8c, I.7.8d, 1.7.15 rotational; 1.5.44, 1.6.20, 1.6.53, I.6.70ff, 1.7.8c, 1.7.59 self; 1.5.44, 1.6.53,1.7.15, Lapp, l i e semi-infinite; 1.6.59 thermal; 1.7.44, 1.7.48 to/from (almost) flat surface; I.6.5d, II.2.8, II.4.6c, II.4.8b to growing particles; IV.2.2c diffusion-controlled particle growth; IV.2.2c diffusion equation (theory for polymer adsorption); II.5.32ff diffusion impedance; II.3.96 diffusion layer; 1.6.63, 1.6.68 diffusion-limited aggregation (DLA); IV.2.20ff, IV.4.5 diffusion-limited cluster aggregation (DLCA); IV.4.45, IV.4.48, IV.fig. 4.25 diffusion potentials; I.5.5d, II.4.125 also see: potential difference diffusion relaxation; II.3.13, II.3.219, II.4.6, II.4.8 diffusiophoresis; 1.6.91, II.3.214, II.4.9 dilatant, dilatancy (rheology); IV.6.11 dilatometry (and surface excesses); II.2.7 dilational modulus, interfacial; see interfacial rheology dimple formation (in draining films); V.6.39ff, V.figs. 6.21-23 dipole field; I.4.4b, II.3.13, II.4.6, II.4.8 dipole moment; I.4.4b(def.), 1.7.3b data for molecules; I.table 4.1 of colloids; II.3.13 dipoles; 1.4.4b ideal (point dipole); 1.4.20 induced; 1.4.22, 1.4.27, 1.7.18, I.7.93ff, II.3.13, II.4.6 of colloids; II.3.13, II.fig. 4.1, II.4.8 oscillation; 1.7.3b, II.4.8 permanent; 1.4.22,1.4.27, 1.7.17 Dirac delta function; I.7.40(def.) disc centrifuge; IV.2.60ff discotic fluid (2D); III.3.62 discs (surfactants); V.4.7b
25
26
SUBJECT INDEX
disjoining pressure; I.4.6(def.), II.1.22, Il.fig. 1.37, II.1.95ff, II.1.101, [II.2.2.1], II.5.65, III.3.176, III.5.7, III.5.14-15, III.5.23, Ill.fig. 5.12, 111.fig. 5.15, IV. 1.3, V.I.I, V.6.5 (also see; films, interactions; for interacting colloids, see IV. chapter 3) dispersion, dielectric; see there for preparation of colloids; IV.2.2g of colloids; 1.1.5, IV. 1.2 of refractive index; 1.4.37, 1.7.13 of transverse waves; III.3.116 dispersion forces; see Van der Waals forces dispersity (of colloids); see size distributions displacement , of particles; I.6.18ff, I.6.30ff dielectric; see there dissipation; 1.2.7, 1.2.22, 1.4.3, 1.4.34, 1.6.9, 1.6.13, I.6.35ff, 1.7.2c, 1.7.14, IV.6.3 dissociation constant; I.5.2d (in) double layers; II.3.65ff, II.3.72ff, II.3.76, II.3.82ff (in) polyelectrolytes; V.2.2d (in) proteins; V.3.5ff, V.table 3.1 relation to points of zero charge; II.3.8c, II.table 3.5 dissolution, heat of; 1.2.71 dissymmetry ratio (in scattering); 1.7.58, I.fig. 7.13 distal length (polym. ads.); V.I.18 distributions; 1.3.la, 1.3.7, IV.app. 1 most probable; 1.3.7 Poisson; [IV.2.3.46] also see Gauss distribution distribution (partition) coefficient; 1.2.69 distribution (partition) equilibrium; 1.2.20a distribution function; I.3.9d, II.3.6b, II.3.9 direct/indirect; IV.5.21 higher order; I.3.9e in electrolytes; I.5.28ff, I.fig. 5.9, I.5.57ff in fluid interfaces; Ill.fig. 2.1, III.2.3, III.2.4, III.2.5, III.2.24, [III.2.5.30], III.fig. 2.6, [III.2.5.40] in liquids near surfaces; II. 1.94, Il.fig. 1.38, [II.2.1.2], II.2.6.8, II.figs. 2.2-3, II.2.2b, Il.figs. 2.4-8, II.3.6b, II.3.9 in water; 1.5.3c, I.fig. 5.6 pair; 1.3.71, II.3.5Iff, III.2.30 relation to pair interaction; IV.3.142ff
SUBJECT INDEX
distribution function (continued). radial (or pair correlation); I.3.9d, 1.7.66, Lapp, l i e , II.3.6b, IV.5.5, IV.5.14, IV.fig. 5.4, IV.fig. 5.8, IV.fig. 5.31 singlet; 1.3.71 dividing plane; see Gibbs dividing plane DLA = diffusion-limited aggregation DLCA = diffusion-limited cluster aggregation DLVO = Deryagin-Landau-Verwey-Overbeek (theory), see colloid stability, colloids interaction DLVOE = Deryagin-Landau-Verwey-Overbeek extended (theory), see colloid stability, colloids interaction DNA, (persistence length); Il.fig. 5.5 Donnan effect; 1.1.21, 1.5.90, 1.5.93, II.3.10, II.3.26, II.3.99, IV.1.6, IV.5.2d, V.2.4 relation to suspension effect; I.5.5f Donnan e.m.f.; 1.5.88 Donnan potential; V.2.38ff donor (in semiconductor); II.3.172ff donor number; 1.5.65 Doppler broadening (spectral lines); 1.7.22 Doppler effect, Doppler shift; 1.7.16, I.fig. 7.6, 1.7.19, 1.7.45, 1.7.94, II.4.46 Dorn effect = sedimentation potential double layer, electric; I.1.20(def.), I.fig. 1.13, I.5.3ff, I.fig. 5.1, Il.chapter 3 in apolar media; II.3.11, IV.3.11 diffuse; 1.1.21, I.fig. 1.13, 1.5.3, I.fig. 5.1, II.3.5 (in) asymmetrical electrolytes; II.3.5c capacitance; II.3.21, Il.fig. 3.5, II.3.29, Il.fig. 3.10, II.3.33, II.3.36 (in) cavity; Il.fig. 3.16 charge; II.3.21, Il.fig. 3.4, II.3.29, Il.fig. 3.9, II.3.32ff, II.3.36, II.3.37, Il.fig. 3.12, Il.table 3.1, H.table 3.2, II.3.40, Il.fig. 3.14, III.4.4 cylindrical; II.3.5f, II.5.14ff, V.2.2b, V.2.2c electrolyte mixtures; II.3.5d field strength; II.3.20, II.3.21, II.3.29, II.3.32 Gibbs energy; II.3.23, Il.fig. 3.6, IV.3.2 Gouy-Chapman theory; see there introduction; II.3.17ff ionic components; II.3.9ff, II.3.5b, Il.fig. 3.8, II.3.33, Il.fig. 3.11, Il.fig. 3.15, II.3.168 negative adsorption of co-ions: II.3.5b, II.3.7e, Il.fig. 3.33 potential distribution; II.3.24ff, Il.fig. 3.7, II.3.35, 11.3.36, Il.fig. 3.12 III.4.4, IV.3.2-3.7, IV.3.11 spherical; II.3.5e, Il.table 3.1. Il.table 3.2
27
28
SUBJECT INDEX
double layer, electric; diffuse (continued), statistical thermodynamics; II.3.6b enthalpy of formation; 1.5.108, II.3.98, II.3.155ff, II.figs. 3.60-61, II.table 3.6 entropy; 1.5.109, Il.fig. 3.44, Il.table 3.6 equivalent circuit; I.fig. 5.11, II.3.7c, Il.fig. 3.31 examples; II.3.1, Il.fig. 3.1 Gibbs energy; 1.5.7, II.3.5, II.3.9, II.3.23, Il.table 3.6, II.3.142, H.3.146 Gouy-Stern model; II.3.6c, Il.figs. 3.20-26, II.3.6f, II.3.133ff, II.3.154, II.3.158 (also see: Gouy-Chapman theory, Stern layer) heterogeneity; II.3.83ff measurements; II.3.7 moment; [II.4.6.50] in monolayers; III.3.4h, III.fig. 3.17 in non-aqueous solvents; 1.5.66, II.3.36, IV.3.11 ionic components of charge; 1.5.2, I.5.90ff, 1.5.6b, 1.6.88, II.3.5b, Il.fig. 3.8, Il.fig. 3.46, Il.figs. 3.53-55, Il.fig. 3.62 (also see: double layer, diffuse) Oosawa model; V.2.2b-c origin; II.3.2, II.3.110, II.3.117, II.3.155ff, II.3.158, V.2.2a overlap; 1.2.72, II.3.24 see colloid stability polarization in external field; II.3.13 (see further: charged (colloidal) particles) polarized vs. relaxed; I.5.5b, II.3.1, II.3.4 polyelectrolytes; V.2.2 polyelectrolytic adsorbates; II.5.5g relaxation; I.5.5b, I.6.6b, I.6.6c, II.3.94, II.3.13d, II.4.6c, II.4.8, IV.4.4 site binding; see Stern layer statistical thermodynamics; II.3.6b Stern layer; 1.5.9,1.5.59, II.3.17, II.3.6c, II.3.6g, II.5.5g, IV.3.9d capacitance; II.3.59ff, Il.fig. 3.43, II.fig. 3.50 charge and potential distribution; II.3.6c, II.3.6d, Il.figs. 3.20-21, II.4.71, Il.figs. 5.17-18, II.5.5g, IV.3.9d condensation; V.2.2a, V.2.2b-c Gibbs energy; II.3.6f, Il.fig. 3.26 site binding; II.3.6e, II.3.6g specific adsorption; II.3.6d zeroth-order; II.3.59, Il.fig. 3.20a, Il.fig. 3.21 surface conduction; see there
SUBJECT INDEX
double layer, electric (continued). thermodynamics; 1.5.6, II.3.4, II.3.1 lOff, II.table 3.6, II.3.138ff, II.3.155ff, II.figs. 3.60-61 triple layer model; II.3.61(def.), II.3.6c two-state models; V.2.2a-b, also see Oosawa model (also see: capacitance, ionic components, surface charge. For specific examples; see under the chemical name of the material.) drainage (of films and foams); I.fig. 1.7, 1.1.15, V.6.4, V.7.10, V.7.3a, V.8.55ff, V.fig. 8.18 drilling fluids; V.7.35 drilling muds; II. 1.80 drop, break-up (emulsification); V.8.34ff, fig. V.8.10 in electric field; III.1.5, III.fig. 1.15 pendant; 1.1.11, I.fig. 1.3 pressure relaxation; III.3.188 rheology; III.3.187, Ill.fig. 3.72 sessile; I.fig. 1.1, Ill.flgs. 5.1-2, III.5.4b, Ill.fig. 5.19 (see interfacial tension, measurement, III.chapter 1) Drude equations; I.7.78ff, II.2.52 dry foam; V. 7.1, V.fig. 7.12 DSC = differential scanning calorimetry Dukhln number; [II.3.13.1], II.3.208ff, II.4.12, II.4.30, II.4.3f, II.4.35ff, II.4.59 Dupre equation; [III.5.2.2b and 2c] DWS = diffusing wave spectroscopy dynamic light scattering; see electromagnetic radiation, scattering dynamics and rheology; IV.6.4 Edwards equation; [V. 1.4.1] Egyptian ink and paints; I.I.Iff, IV. 1.3, IV.2.1 Einstein crystal; I.3.21ff, 1.3.6a Einstein equation (for diffusion); 1.6.20, 1.6.30, [1.7.8.12a], 1.7.64, 1.7.66, 1.7.15 Einstein equation (for viscosity); [IV.3.10.21, IV.6.9a efficiency (thermodynamic); 1.2.9, 1.2.22 elastic (material); IV.fig. 6.10, IV.6.14 elastic aftereffect; IV.fig. 6.12 elastic force, between brushes; V . I . l i e in gels; V.2.3d, V.figs. 2.20-21 elastic recovery; IV.fig. 6.12 elasticity modulus; IV.6.7. IV.6.14
29
30
SUBJECT INDEX
electrical birefringence; 1.7.100 electric double layer; see double layer, electric electric capacitance; see capacitance electric charge; see charge electric current (density); 1.6.6 two-dimensional; I.6.6d electric field; 1.4.3,1.4.5f, 1.5.10, 1.7.6b caused by surface charge; 1.5.11; also see Gauss equation and electromagnetic waves; I.chapter 7 electric potential; see potential electroacoustics; II.4.3e, II.4.5d, II.fig. 4.20 electrocapillary curves; I.5.96ff, 1.5.99,1.fig. 5.16, II.3.139, Il.fig. 3.48, III.1.45 electrocapillary maximum; I.5.99ff, H.3.102, II.3.139ff, Il.fig. 3.48 electrochemical potential; I.5.1c, [1.5.1.18], 1.5.74, II.3.5, II.3.90 electrochemistry (general introduction); I.chapter 5 electrodialysis; II.4.132 electrokinetic charge; II.3.90, II.4.1, Il.fig. 4.13 electrokinetic consistency; II.4.58, II.4.6e, II.table 4.2, II.4.6f, Il.table 4.3 electrokinetic phenomena; II.chapter 4, V.2.5 a.c. phenomena; II.4.8 advanced theory; II.4.6 applications; II.4.6i, II.4.10, II.5.63ff double layer relaxation; II.4.6c elementary theory; II.4.3, II.4.7a irreversible thermodynamics; see Onsager relations polyelectrolytes; V.2.5a survey; Il.table 4.1 techniques; II.4.5 xylene in water; III.fig. 4.21 (see further the specific electrokinetic phenomena) electrokinetic potential; I.5.75ff, 1.6.87, Il.chapter 4, V.2.5 examples; Il.fig. 4.13, II.figs. 4.29-30, IV.fig. 3.64, IV.fig. 3.72 interpretation; II.4.1b, II.4.4 relation to y/d; II.4.41ff, Il.fig. 4.12 electrolytes; 1.5. l b , 1.5.2 electrocratic (colloid stability); IV. 1.2 electromagnetic radiation and waves; I.chapter 7, IV.2.3b absorption; 1.7.2c, 1.7.3, I.7.60ff secondary; 1.7.15 coherence; 1.7.15, 1.7.23 and oscillating dipoles; 1.7.3d
SUBJECT INDEX
electromagnetic radiation and waves (continued), detection; 1.7. l c in a medium; 1.7.2b, 1.7.2c intensity; 1.7.8, 1.7.33 interaction with matter; 1.7.3 in a vacuum; 1.7.la, 1.7.2a irradiance; 1.7.5, 1.7.23 Maxwell equations; 1.7.2 phase shift; 1.7.3 polarization; 1.7.1a, I.fig. 7.2, I.fig. 7.4, 1.7.23, 1.7.26, I.fig. 7.8, 1.7.14 elliptical; 1.7.6, I.fig. 7.4, 1.7.98, III.3.7 circular; I.fig. 7.4 planar; I.fig. 7.4 scattering; 1.7.3 and absorption; I.7.60ff and fluctuations; 1.7.6b dynamic; 1.7.6c, I.fig. 7.10, I.7.6d, 1.7.7, 1.7.8, IV.2.46ff forced Rayleigh; 1.7.103 Guinier; IV.2.44ff inelastic; I.7.16(def.) Mie; I.7.60ff, V.8.22 of colloids; 1.7.8, II.4.46, IV.2.3b, IV.3.142ff, IV.5.21 of emulsions; V.8.17ff, fig V.8.6, V.table.8.1 of foams; V.fig. 7.2 of interfaces; 1.7.10c, III. 1.10 of liquids; I.5.44ff, 1.7.7, 1.7.8a of microemulsions; IV.fig. 5.14, V.5.3d, V.5.3e plane; I.7.27(def.) quasi-elastic, QELS; 1.7.16, 1.7.6, 1.7.7, II.4.46, II.5.62 Raman = inelastic Rayleigh-Brillouin = QELS Rayleigh-Debye; I.7.8d, IV.2.44ff, V.8.17 secondary; 1.7.16 static; 1.7.33, I.7.6d, 1.7.7, 1.7.8 survey; I.table 7.3 wave vector; I.7.27(def.) (also see: neutron scattering, X-ray scattering) sources; 1.7.4 types of; I.fig. 7.1 electromotive force; 1.5.82 electronegative; 1.4.19, 1.4.48
31
32
SUBJECT INDEX
electroneutrality (electrolyte solutions); 1.5.la; 1.5.lb electroneutrality of double layers; 1.5.4, 1.5.6a, II.3.6ff electron microscopies for sols; IV.2.42, also see the photographs in that chapter electron microscopies for microemulsions; V.5.3b electron pair acceptor; 1.5.65 electron pair donor; 1.5.65 electron spin resonance (ESR, principles); 1.7.16, 1.7.13 of aqueous electrolytes; I.5.54ff of interfaces; II.2.8, II.2.55, II.5.59 electro-osmosis; 1.6.12, 1.6.16, II.4.1, Il.table 4.4, II.4.6, II.4.3b, Il.fig. 4.6, II.4.46, Il.fig. 4.15 in plug of arbitrary geometry; II.4.21-22, II.4.5b electro-osmotic dewatering; II.4.132 electro-osmotic flux; II.4.23 electro-osmotic pressure (gradient); Il.table 4.4, II.4.6, II.4.23ff electro-osmotic slip; II.4.19, II.4.2Iff electro-osmotic volume flow; Il.table 4.4, II.4.6, II.4.22ff electrophoresis; II.chapter 4 advanced theory; II.4.6 anticonvectant; II.4.131 applications; II.4.10 elementary theory; II.4.3a experiments; II.4.5a polarization retardation; II.4.3a electrophoretic light scattering; II.4.46 electrophoretic mobility, velocity; II.4.4-5, Il.table 4.1, II.4.3a, Il.fig. 4.41, IV.figs. 3.623.64, IV.fig. 3.68, IV.flg. 3.74 cylindrical particles; Il.fig. 4.4, II.4.16 Dukhin-Semenikhin equation; [II.4.6.45], Il.fig. 4.29 Helmholtz-Smoluchowski equation; [II.4.3.4], II.4.12-14, II.4.17-19, Il.fig. 4.29 Henry; Il.fig. 4.4, II.4.16 Hiickel-Onsager equation; [II.4.3.5] hydrodynamics; II.4.14ff, II.4.6 influence of surface conduction; Il.fig. 4.4, Il.fig. 4.31 irregular particles; II.4.6h, Il.fig. 4.33 measurement; II.4.5a electroacoustics; II.4.5d microelectrophoresis; II.4.45ff, II.figs. 4.14-16 moving boundary; II.4.5Iff, Il.fig. 4.17 Tiselius method; II.4.53
SUBJECT INDEX
electrophoretic mobility, velocity (continued), non-aqueous media; IV.3.135ff O'Brien-Hunter equation; [II.4.6.44] O'Brien-White; Il.figs. 4.26-29 polyelectrolytes; V.2.5a ribonuclease; V.fig. 3.5 sol concentration effect; II.4.6g, II.fig. 4.32 stagnant layer thickness determination; II.4.128ff, II.fig. 4.42 verification of theories; II.4.6e electrophoretic deposition; II.4.132 electrophoretic retardation (in ionic conduction); 1.6.6b, II.4.3a electropositive; 1.4.19, 1.4.48 electrosonic amplitude; II.4.7, II.4.5d electrosorption; II.3.4(def.), II.3.12 electrostriction; 1.5.103 electroviscous effects; II.4.122ff, IV.6.9b, V.2.48ff electrowetting; III.5.103 ellipsometric coefficients; 1.7.75, II.2.51, [H.2.5.7] ellipsometric thickness; [1.7.10.17], II.5.64 ellipsometry; 1.7.10b, II.2.5c, II.5.64ff, III.2.47, Ill.table 3.5, III.3.141ff, V.fig. 6.5 elliptical polarization; see electromagnetic radiation, polarization eluate; see chromatography emission spectrum; 1.7.14 emulsification; 1.1.3, 1.6.45, III.3.237, III.4.97, V.8.2 emulsification failure boundary; V.5.21, V.5.49 emulsifier; V.8.2 emulsifier (biological); 1.1.3 emulsion films; V.6.1 (def), see films, liquid emulsions (general); V.chapter 8 emulsions; I.1.3(intr.), I.1.5(def.), 1.2.98, IV. 1.9 (def. + classif) aggregation; V.8.63, V.8.78ff characterization; V.8.1 coalescence; V.8.3e, V.8.64 creaming; V.fig. 8.25 dielectric properties; V.8.19ff drop size distribution; V.8.1e, V.8.66ff, V.fig. 8.25 formation; V.8.2, V.fig. 8.9, V.fig. 8.10, V.table 8.2 interfacial layers; V.8.2d multiple; IV. 1.9 optical properties; IV.fig. 5.63, V.8.17ff Ostwald ripening; V.8.3b. 8.63
33
34
SUBJECT INDEX
emulsions (continued), phase inversion; V.8.64 Pickering stabilization; III.5.99, V.8.4 preparation; see formation sedimentation; V.8.3d, V.8.63 stability; V.8.3, V.fig. 8.22 type; V.8.3 viscosity; V.8.15, V.8.29 (also see: microemulsions) endothermic; see process energy (principles); 1.2.4, Lapp. 3, Lapp. 4
absorption; I.4.4e, 1.7.3 configurational; 1.3.46 conservation; 1.2.8 interfacial; 1.2.5, 1.2.11, Lapp. 5, II.table 1.2 levels (semiconductors); II.fig. 3.68-69 mixing (polymers); [II.5.2.12] of radiation; 1.7.5 (also see: adsorption, Interaction) engulfment; III.5.lie enhanced oil recovery; 1.1.1,1.1.3, 1.1.11, III. 1.84, V.7.35 ensemble (intr.); 1.3.lc canonical; 1.3. lc grand (canonical); 1.3.lc microcanonical; 1.3. l c entanglements; IV.6.67ff enthalpy (principles); 1.2.6, Lapp. 3, Lapp. 4 interfacial; 1.2.6,1.2.11, Lapp. 5, Il.table 1.2 of chemical reactions; 1.2.21 of dissolution; 1.2.20c, 1.5.3a of electric double layer; 1.5.108, II.3.98, II.3.155ff, Il.figs. 3.60-61 of hydration; I.table 5.4 of transfer; 1.2.69 of wetting; see wetting (also see: adsorption, enthalpy) entropy (principles); I.2.8(intr.), 1.2.9, Lapp. 3, Lapp. 4 absolute; 1.2.24, 1.3.16 configurational; 1.2.52, 1.3.30 interfacial; 1.2.9, 1.2.42, 1.2.83, Lapp. 5, II. 1.2, Il.table 1.2, II.fig. 3.44 intrinsic; 1.2.52 of electric double layers; 1.5.109. II.fig. 3.44
SUBJECT INDEX
entropy (continued), of mixing; 1.2.53, 1.3.28, [II.5.2.1 1 ] of solvation (hydration); 1.5.3, I.table 5.4 production of; 1.6.2a, I.6.2b statistical interpretation; 1.3.16ff (also see: adsorption entropy) environment, double layer effects; II.3.22Off environmental scanning electron microscopy (ESEM); IV.2.42 Eotvos equation (for surface tension); [III.2.11.1] Eotvos number; [III. 1.3c] EPR = electron spin resonance equation of motion; 1.6. l b , IV.6.1, 1V.6.2 equation of state, BET; [II. 1.5.49], H.flg. 1.24b Boyle-Gay Lussac (ideal); [1.1.3.4], 1.2.17a Carnahan-Starling; [1.3.9.31] hard sphere fluid; [1.3.9.26] one-dimensional; [1.3.8.5] Percus-Yevick; [1.3.9.29 and 30] Van der Waals; [1.2.18.26], [1.3.9.28] two-dimensional; 1.1.17, 1.3.42, I.3.8d, II. 1.3, II. 1.3b, II. 1.39, II. 1.45, Il.app. 1, III.3.4, Ill.table 3.3, III.4.2, III.4.3 double layer; II.3.14 electrosorption; II.3.197 Frumkin-Fowler-Guggenheim (FFG); I.3.46ff, [II.A1.5b] Henry; [II.Al.lb] Hill-De Boer = Van der Waals; see there Langmuir; I.3.6d, [1.3.6.23], II. 1.45, [II. 1.5.10], Il.fig. 1.15b, [II.A1.2b] polymer monolayers; III.3.4i, [III.3.4.56] protein monolayers; V.fig. 3.24 quasi-chemical; I.3.8e, [II.A1.6b] sols; [IV.3.12.8] Van der Waals; II. 1.51, II. 1.59, [II. 1.5.28], Il.fig. 1.20b, [II.A1.7b and c], III.3.4e (two dimensional) virial; [II.A1.4b] Volmer [1.5.23], Il.fig. 1.15b, [II.A1.3b] equilibrium (general); 1.2.3, 1.2.8, 1.2.12, 1.3.7, IV.6.3a chemical; 1.2.21 frozen; 1.2.8 local; 1.6.2, 1.6.2a mechanical; 1.2.22, V.6.3b
35
36
SUBJECT INDEX
equilibrium (continued), membrane; I.2.33(def.), I.5.5f, III.3.29 metastable; 1.2.7, 1.2.68 partial; 1.2.7 restricted; V.I. 12c stable; 1.2.7, 1.2.19 equilibrium constant; 1.2.77, II.3.6e equilibrium criteria; 1.2.12 equipartition (of energy); 1.6.18 equipotential planes (in gas adsorption theory); II.fig. 1.25 equivalent circuits (electrochemistry); I.fig. 5.11, II.3.7c, H.fig. 3.31 equivalent circuits (rheology); II1.3.6i, 1V.6.6 also see: interfacial rheology error function; [1.6.5.25], [II.4.6.37] Esin-Markov coefficients; I.5.6d, 1.5.102, II.3.15ff, II.3.26, II.3.103ff, II.3.136, Il.fig. 3.45, II.3.144ff ESA = electrosonic amplitude ESCA (electron spectroscopy for chemical analysis) = XP(E)S ESEM = environmental scanning electron microscopy ESR = electron spin resonance Euler-Lagrange equation; [III.2.5.25], [III.A3.9] Euler's theorem; 1.2.28, 1.2.14a evanescent waves; 1.7.75, II. 1.18, II.2.54, IV.3.157 evaporation; Ill.fig. 2.16 (heat), III.2.55 (entropy) prevention (water conservation); III.3.239 EXAFS = extended X-ray absorption fine structure exchange (adsorption from binary mixtures); II.2.1, II.2.3, II.2.4 constant; [II.2.3.16] excess functions and quantities; I.2.18b(intr.) in regular solutions; 1.2.18c, I.3.46ff (also see: interfacial excesses) excimers; III.3.165 excluded volume; see polymers exothermic; see process expansion coefficient (2D); [III.3.3.2], [III.3.4.5] extended X-ray absorption fine structure (EXAFS); 1.7.88 extraction (of films); V.6.4e extrapolation length (polymer ads.); V. 1.8, V.fig. 1.2 Fabry-Perot interferometer; 1.7.36, I.7.6d falling film (interfacial rheology); III.3.189ff falling meniscus (for measuring interfacial tensions); III. 1.11
SUBJECT INDEX
Faraday's gold sols; IV. 1.13, IV.2.27 fats (metabolism); 1.1.1, 1.1.3, 1.1.7 fatty acids; III.tables 3.7a, 3.7b (overview) fatty acid monolayers; III.fig. 1.32 (interfacial tension, dynamics), III.fig. 3.58, III.fig. 3.60, III.fig. 3.68, lll.fig. 3.76, III.figs. 3.78-87, Ill.fig. 4.26 fatty amine monolayers; III.3.212, fig. III.3.88 FCS = fluorescence correlation specroscopy; III.3.7c.iv FEM = field emission (electron) microscopy; 1.7.lib Fermi-Dirac statistics; 1.3.12, H.3.172 Fermi (energy) level; II.3.172, Il.fig. 3.68, II.3.174 ferrofluids; IV.fig. 2.2c, IV.3.10c preparation; IV.2.4d ferromagnetism; IV.3.124 FFEM = freeze fracture electron microscopy; see transmission electron microscopy FFF (sediment) = (sedimentation) field flow fractionation FFG = Frumkin-Fowler-Guggenheim; see adsorption isotherm fibres, wetting; Ill.fig. 5.2, III.5.4g flbronectin (adsorption); V.fig. 3.14 Fick's first law; I.table 6.1, 1.6.54,1.6.5d, I.6.5e, 1.6.67 two-dimensional; 1.6.69 Fick's second law; 1.6.55, I.6.5d, I.6.5e, II.4.78, II.4.115 field emission techniques; 1.7.lib field strength, electric; I.4.12ff (also see: double layer) film balance; 1.1.16, III.3.3, Ill.figs. 3.4, 3.5 film drainage; V.6.4, V.fig. 8.18, V.8.84 film formation; V. chapter 6, V.8.72 film tension; 1.1.12ff, I.2.5ff, II.1.95ff, V.6.3b, V.fig. 6.17 films, liquid (free); 1.1.6,1.1.12ff, I.fig. 1.4 up to and including fig. 1.11, I.fig. 2.1, 1.2.5, IV. 1.3, V.chapter 6 (general) black or common black; IV. 1.3, V.6.6, V.fig. 6.8, V.fig. 6.10, V.fig. 6.25, V.fig. 6.33, V.fig. 6.37, V.fig. 6.43, V.8.88 Brewster's angle; V.fig. 6.4 colours; 1.7.80 conductivity; V.6.2g, V.fig. 6.37 contact angle; V.6.2e, V.6.3e, V.fig. 6.38 diffusion; V.6.7 disjoining pressure; V.6.2, V.6.2d, V.6.25, V.fig. 6.27, V.6.5, V.figs. 6.31-32, V.6.6a, V.fig. 6.34 elasticity; V.6.2f ellipsometry; V.fig. 6.4
37
38
SUBJECT INDEX
films, liquid (free) (continued), emulsion; V.6.2c, V.fig. 6.9 experiments; V.6.2 flow in them, drainage; 1.1.15, I.6.4d, IV. 1.3, V.6.4 FTIR; V.fig. 6.6 interferometry; V.6.2a macroscopic; V.6.2b, V.6.4e microscopic; V.6.2c multiple reflections; 1.7.80 Newton (black); III.5.39, IV. 1.3, V.6.2, V.fig. 6.33, V.6.6b, V.6.6c, V.figs. 6.43-44 oscillatory forces; see stratification reflectometry; V.fig. 6.4-6.5 permeability; V.6.2b, V.6.7a-b, V.figs. 6.43-44 phospholipid; V.6.7c pinch-off (in foam breaking); V.7.32 rupture; V.6.4c, V.6.4d, V.6.3b, V.6.6c, V.7.3b, V.7.6, also see; stability critical thickness; V.6.4c, V.fig. 6.23, V.fig. 6.25 stability; 1.6.45, V.7.3, V.fig. 7.14, V.8.81ff, V.8.85ff, V.fig. 8.26 Kabalnov-Wennerstrom theory; V.8.86 Vrij-Scheludko theory; V.6.4c, V.8.87 de Vries theory; V.8.85 stratification; V.6.2, V.6.5d thermodynamics; V.6.3 thickness; V.6.2a (def.), V.6.3a,b, V.6.4, V.fig. 6.28 thinning; III.5.3d Van der Waals forces; 1.4.72, III.5.3 X-ray; V.6.9 also see; foams films, liquid (on solid supports), disjoining pressure; II.1.22, [II.1.3.15], II.1.6d, [II.1.6.18], Ill.fig. 1.14, III.5.3 (also see; wetting films) filtration; IV.2.32ff filtration law; [IV.2.2.66] FIM = field-ion microscopy; 1.7.lib fire fighting; V.7.35-36 first curvature (of interfaces), see mean curvature First Law of thermodynamics; see thermodynamics First Postulate of statistical thermodynamics; see statistical thermodynamics fish diagrams; see microemulsions flatbands (semiconductors); II.3.174, H.fig. 3.69
SUBJECT INDEX
flickering clusters; 1.5.42 FLIM = fluorescence lifetime imaging microscopy floating (objects on liquids); I.l.lOff, I.fig. 1.2, I.fig. 1.4 flocculation, flocculants; 1.1.28, II.2.88, II.5.97, IV. 1.5 see colloid stability, especially by polymers flocculation kinetics; V. 1.12e orthokinetic; V.I.84 Flory-Huggins interaction parameter (x); 1.3.45, II.1.56, II.2.34, II.5.5ff, V.1.4, V.I.8 flotation; 1.1.25, II.2.88, III.4.4d, III.5.lib flow; see viscous flow flow birefringence = streaming birefringence fluctuations; 1.2.68, 1.2.102, 1.3.1, 1.3.7, 1.4.26, 1.7.26 in micelles; V.4.2d; more Illustrations in V.chapter 4 in surfaces; I.7.81ff, II. 1.45 of dielectric permittivity; 1.7.6 fluctuation potential; II.3.52 fluidity; I.6.52(def.) fluorescence; 1.7.14 in surfaces and adsorbates; 1.7.11a, II.2.54, II.2.80, IH.table 3.5 fluorescence correlation spectroscopy; III.3.7c.iv fluorescence lifetime imaging microscopy; III.3.7c.iv fluorescence recovery (after photobleaching) (FRAP); 1.7.104, III.3.7c.iv fluorescence resonance energy transfer; III.3.7c.iv fluorophores; Ill.fig. 3.63 flux; I.6.5ff, 1.6.11, 1.6.2b, 1.6.32 around polarized double layers; II.3.13c, IV.4.2-4.4 radial; 1.6.68 foam; I.fig. 1.7, 1.2.98, IV.1.11 (classif.), breaking; III.5.99, V.7.6 characterization; V.table 7.1, V.7.4 coalescence; V.7.3b, V.7.6 definition; V.7.1 drainage; V.7.3a dry; V.7.1 foam film; V.6.1 (def.), see films, liquid foam fractionation; V.7.35 flotation; V.7.35 (also see: films, liquid) formation; V.7.2, V.figs. 7.6-7 general; V.chapter 7 old; V.7.2
39
40
SUBJECT INDEX
foam (continued), optical properties; V.7.5b Ostwald ripening; V.7.3c polyhedral; V.figs. 7.1-3, V.7.1b, V.fig. 7.13 rheology; V.7.5a stability; V.7.2, V.7.3 structure; V.7.1a, V.7.1b, V.fig. 7.1 wet; V.7.2 young; V.7.2 foaming agents; V.7.1c fog; 1.1.6 Fokker-Planck equation; I.6.3c, 1.6.73, IV.4.7, IV.4.9 forces (principles); conjugate; 1.6.12, I.6.2b, conservative; 1.4.2 directional; 1.6.1, I.6.3b, electrostatic; 1.4.2 fields; 1.4.3 generalized interpretation; 1.6.1 Iff, 1.6.54 hydrophobic; see hydrophobic interaction and bonding internal vs. external; 1.6.13 mechanical vs. thermodynamic; 1.2.27, 1.4.2, 1.4.49 steric; 1.4.2 stochastic; 1.4.2, 1.6.1, 1.6.3 Van der Waals; see Van der Waals forces vectorial interpretation; 1.4.3a, Lapp. 7 (also see: interaction and interaction force) force constant; 1.4.44 force (effect) microscope; 1.7.90 forced interaction; IV.3.10, IV.4.5, IV.5.3a forced Rayleigh scattering; 1.7.103 forced wetting; III.5.4, Hl.fig. 5.5 form drag; 1.6.47 form factor (in scattering); IV.2.44, IV.3.143, IV.5.21 Fourier transform infrared (spectroscopy) (FTIR); II.2.53, V.fig. 6.6 Fourier transforms; Lapp. 10 fractal dimensionality; IV.2.21ff, IV.4.6, IV.6.75 fractal growth; IV.2.21ff, IV.4.6 fractal structures; IV.4.6, IV.6.75ff, IV.fig. 6.40 fractionation; IV.2.7, IV.2.2h adsorptives of different sizes; II.2.45
SUBJECT INDEX
FRAP = fluorescence recovery after photobleaching free energy; see Helmholtz energy free enthalpy; see Gibbs energy free path (mean); 1.6.55 freezing point depression; 1.2.75 Fredholm integrals; II. 1.108 Frenkel defects; II.3.173 Fresnel equations; 1.7.10a Fresnel Interfaces; 1.7.73, II.2.5c FRET = fluorescence resonance energy transfer Freundlich adsorption isotherm; see adsorption isotherm friction; 1.4.3, 1.6.10 friction coefficient (intr.); I.6.21ff, I.6.30ff, 1.6.56, 1.7.101, Lapp. 11 frictional drag (intr.); 1.6.47 froth = foam FRS = forced Rayleigh scattering FTIR = Fourier transform infrared (spectroscopy) function of state (def.); 1.2.9, 1.2.14c functional adsorption; see adsorption, functional; III.2.22, III.2.34, Ill.app. 3 fundamental constants (table); Lapp. 1 funnel technique (interfacial rheology); III.3.185 Fuoss-Onsager equation for limiting conductivity; [1.6.6.29] Fuoss theory for ion association; I.5.2d Galvani potential; see potential galvanic cells; I.5.5e gas pockets (and nucleation); V.7.9, V.fig. 7.5 gases, adsorption on solids; chapter 1 ideal; 1.1.17, 1.2.40, 1.2.17a, 1.3.If, I.3.6b, I.3.6c, 1.3.58 non-ideal; 1.3.9 Gauss distribution; 1.3.39, 1.6.21, 1.6.3c, 1.6.63, [IV.2.3.41] for surface heterogeneity; II. 1.106, Il.fig. 1.43 Gauss' law (surface charge and field strength); I.4.53ff, Lapp. 7e, II.3.59, IV.3.11, IV.3.54 Gauss(ian) beams; 1.7.23 Gauss(ian) curvature; I.2.90(def.), III.1.4ff, III.1.15, V.4.94ff, V.5.24 (also see: bending moduus) gel, gelation; IV.4.48ff, IV.fig. 4.26, IV.5.86ff, IV.6.13. IV.6.14, V.2.3d, V.5.86 gel electrophoresis; 11.4.131 gel, thermoreversible; IV. 1.6
41
42
SUBJECT INDEX
generic phenomena, properties; I.5.67(def.), II.3.6 Gibbs adsorption law; I.1.5(intr.), 1.1.16, 1.1.25, 1.2.13, 1.2.22, II. 1.2, II.2.27 Gibbs adsorption law for; charged species; 1.5.3, 11.3.12a curved interfaces; 1.2.94 dissociated monolayer; 1.5.94 electrosorption; II.3.12d liquid films; V.6.3d, [V.6.3.34] narrow pores; II. 1.96 polarized charged interfaces; I.5.6c, II.3.4, II.3.138 relaxed (reversible) charged interfaces; I.5.6b, II.3.4, II.3.138 water-air interfaces; II.3.178 see III.chapter 4 for many examples Gibbs convention (for locating dividing plane); 1.2.5, I.fig. 2.3, V.6.3 Gibbs dividing plane; 1.2.5 (intr.), 1.2.22, II. 1.2, II.2.4, II.2.64, III.4.4, V.6.2a, V.6.3 for curved interfaces; 1.2.23b Gibbs-Duhem relation; 1.2.10, 1.2.13, 1.2.78, 1.2.84, 1.5.92, [II. 1.3.35-36], V.6.3c, [V.6.3.20], [V.6.3.36, 37] Gibbs elasticity; III.3.30, V.6.2f Gibbs energy; 1.2.10, Lapp. 3, Lapp. 4 interfacial; 1.1.4,1.2.10, 1.2.11, Lapp. 5, Il.table 1.2, V.1.3, V.8.2a, V.8.64 self; I.4.5c, 1.5.17,1.5.3a, 1.5.3b statistical interpretation; 1.3.18, [1.3.3.14] Gibbs energy of; adsorption; see adsorption, II.Gibbs energy chemical substances; 1.2.77 colloid interaction; IV.chapter 3, V.chapter 1 double layer; 1.5.7, II.3.5, II.3.9, II.3.23, Il.fig. 3.6, Il.table 3.6 Il.fig. 3.26, II.3.142, II.3.146, II.3.155ff, III.3.4h, IV.chapter 3 ions;I.5.17ff, (also see: solvation, hydration) liquid films; V.6.3 nucleation; I.2.23d; IV.2.2b, IV.2.2f polarization; 1.4.55 proteins; V.3.2 solvation (hydration); 1.4.45, 1.5.3a, I.table 5.4, I.5.3f transfer; I.5.3f, I.table 5.11, 1.5.5a see further the pertaining system Gibbs free energy = Gibbs energy Gibbs triangle = phase diagram, ternary Gibbs-Helmholtz relations; 1.2.41, 1.2.15, 1.2.61, 1.2.78, II.3.156, III.3.34
SUBJECT INDEX
Gibbs-Kelvin equation; see Kelvin equation Gibbs-Thomson equation; see Kelvin equation Girifalco-Good-Fowkes theory (for interfacial tensions); III.2.lib, III.table 2.3 glass, double layer; II.fig. 3.64 glass electrodes; II.3.224 glass transition point; IV.5.90 goethite; see iron oxide gold sols; IV.1.2, IV. 1.3, IV.1.13; IV.2.27, IV.3.185 Gouy-Chapman length; [V.2.2.13] Gouy-Chapman theory (diffuse double layers); 1.5.16, 1.5.18, II.3.5 (in) cavities; II.3.5g cylindrical surfaces; II.3.5f, V.2.2 defects; II.3.6a flat surfaces; II.3.5a-d, Il.fig. 5.18 improvements; II.3.6b, Il.figs. 3.18-19 spherical surfaces; II.3.5e (also see: double layer, Gouy-Stern model) grafting (macromolecules); V.I.10, V.I.11, V.3.9, V.fig. 3.30 grand potential; III.2.7, V.I.I, V.l.3-4, V.1.8c, association colloids; V.chapter 4 translationally restricted; V.4.25ff also see interfacial grand potential graphite; see carbon Graphon, graphite; see carbon gravity (influence on colloid stability); IV.3.10a, see further sedimentation grazing incidence; II.2.56, III.3.147, HI.3.152, Ill.fig. 59 ground state approximation (Edwards eq.); V. 1.11 ground water table; 1.1.1 growth (particles, drops); V.8.31 GSA = ground state approximation guar solution; IV.fig. 6.34 Guggenheim convention (for treating interfacial excesses); 1.2.15,1.fig. 2.4 Guinier radius; IV.2.45 gum arable; 1.1.2,1.1.7, IV. 1.3 Gurvitsch's rule; II. 1.94 gyration radius; see radius of gyration haematite (a-Fe2O3); see iron oxides Hageman factor; V.3.52 Hagen-Poiseuille law; 1.6.42, II.4.47, II.5.62, IV.6.7a
43
44
SUBJECT INDEX
Hamaker constant, Hamaker function; 1.3.45, 1.4.59, [1.4.7.7], 1.4.79, 11.2.5, 11.3.129, (tables) Lapp. 9, IV.app. 3 and interfacial tensions; III.2.5c Hamiltonian; I.3.57ff, II.3.47ff hard core or hard sphere interaction (molecules or colloids); 1.3.65, 1.4.5, 1.4.42, IV.5.4 hard sphere liquid; IV.5.3 heat, statistical interpretation; 1.3.16 (also see: enthalpy) heat capacity; 1.2.7,1.3.36, 1.5.42 interfacial; 1.2.7, III.2.74 of ions; I.table 5.6 heavy metal pollution; II.3.221 Helfrich equation (bending); [III. 1.78], [III. 1.15.1 and 2], [V.5.5.1] helix ( a ) ; V.fig. 3.2 Helmholtz energy; 1.2.10, Lapp. 3, Lapp. 4 interfacial; 1.2.10, 1.2.11, Lapp. 5, Il.table 1.2, V.1.3b, V.1.4 liquid films, V.6.3 statistical interpretation; 1.3.17, II.especially [1.3.3.10] see further the pertaining system Helmholtz free energy = Helmholtz energy Helmholtz planes; see inner Helmholtz plane and outer Helmholtz plane Helmholtz-Smoluchowski equation; see electrophoretic mobility hematite = haematite, see iron oxides Henderson equation, for liquid junction potential; [1.6.7.11 ] for solvent structure contribution to disjoining pressure; [ 1.6.13] for-^-potential; [II.3.9.9] Henderson-Hasselbalch equation, plot; [1.5.2.34], [II.3.6.52, 11.53], II.3.88ff, V.2.19ff, V.fig. 2.12 Henry adsorption isotherm; see adsorption isotherm Henry constant, for adsorption; 1.1.19, 1.2.71 for solubility of a gas; 1 2.20b Henry's law for gas solubility; 1.2.20b Herschel-Bulkley (rheological model); [IV.6.3.4] Hess' law; 1.2.16 heterodispersity (of colloids); I.7.8e, IV.5.3 Iff, IV.fig. 6.27 heterodyne beating; see optical mixing heterogeneity of surfaces; see surface, heterogeneity
SUBJECT INDEX
hetero-interaction; 1.4.72, I.fig. 4.17, IV.3.4, IV.3.12, IV.fig. 3.58. IV.fig. 3.61, 1V.5.3Iff, IV.fig. 5.64 hexadecylpyridinium chloride (adsorption); II.fig. 2.22 higher-order Tyndall spectra (HOTS); 1.7.61 Hill plot; II. 1.48 HLB = hydrophile-lipophile balance HNC = hypernetted chain Hofmeister series = lyotropic series holes (in semiconductors); II.3.171 homodisperse colloids; 1.1.14, 1.1.28, 1.7.53, IV.1.6, IV.fig. 1.3 homodyne beating; see optical mixing homogeneous condensation, see condensation homogenizer; V.fig. 8.17 homointeraction; 1.4.72 homopolyelectrolytes; II.5.1 homopolymers; II.5.1 honeycomb symmetry; 1.1.14 Hooke (material, law); IV.6.9, IV.6.13 HOTS = higher order Tyndall spectra HPLC = high performance liquid chromatography; see chromatography HSA = human serum albumin; see albumin Hiickel-Onsager equation; see electrophoretic mobility Huggins constant; IV.6.61, V.2.48 Huggins equation (rheol.); [IV.6.11.9], [V.2.4.3] Huygens oscillator; V.fig. 1.26 hyaluronic acid (charge); V.fig. 2.7 hydration; 1.2.58, 1.5.3, II.3.121, Il.table 3.7 (also see: solvation) hydration number; 1.5.50 hydraulic radius; 1.6.50, II. 1.84 hydrodynamic radius, layer thickness; 1.7.50, II.5.61 hydrodynamics; 1.6.1, conservation laws; I.6.1a, I.6.1b, II.4.6, II.4.8, IV.6.1 in colloid interaction; IV.4.5b in electrophoresis; II.4.3, II.4.6 in emulsification; V.8.2b in polarized double layers; II.3.215ff, II.4.6 hydrogen bonding, hydrogen bridges; 1 4.5d. 1.5.3c hydrophile-lipophile balance (HLB); V.4.1b. V.8.5 hydrophilic; 1.1.7, 1.1.23, Il.table 1.3 (also see: colloids)
45
46
SUBJECT INDEX
hydrophilicity/phobicity; Il.table 1.3, III.5.5, III.5.11, III.5.11a hydrophobic; 1.1.2, 1.1.7, 1.1.23, Il.table 1.3 (also see: colloids) hydrophobic interaction and -bonding, hydrophobic effect; 1.1.30, I.4.5e, 1.5.3c, 1.5.4, I.table 5.12, II.2.7d, II.3.12d, V.3.2b, V.3.6b (also see: lyotropic sequences) hyperbolic functions; Lapp. 1.2 hypernetted chain (HNC); 1.3.69, [IV.5.3.41] hysteresis; 1.2.7 in rheology; IV.6.3b, IV.fig. 6.7 also see, (hysteresis of) adsorption, contact angles, monolayers ideal dilute (polymer solution); II.5.9 i.e.p. = isoelectric point IgG = immunoglobulin iHp = inner Helmholtz plane illites; II.3.165 image charges; II.3.48ff, Il.fig. 3.17, Il.table 3.3 imaginary quantities; Lapp. 8 imaging techniques; 1.7.11b (also see: transmission electron microscopy, atomic force microscopy, surface force microscopy, etc.) immersion method (to determine points of zero charge); II.3.105 immersion heats or enthalpies; II. 1.29, Il.table 1.3, II.2.5, II.2.6, II.2.7, II.2.3d, Il.fig. 2.10, Il.fig. 2.20, 0.3.98, II.3.114, III.5.2 (also see: wetting, immersional) immunoglobulins, adsorption; V.fig. 3.13, V.fig. 3.18 impedance (spectrum); II.3.92, Il.fig. 3.30, II.3.149, II.4.8 incident angle (for radiation); 1.7.10a incident plane (for radiation); 1.7.10a index of refraction; see refractive index indicator electrode; 1.5.82 indifferent (ions, electrolytes); II.3.6, II.3.103, IV. 1.11 inertia; II.4.2 infrared spectroscopy; 1.7.12, II.2.8, II.2.7Iff, II.5.57; III.3.7c.i, Ill.fig. 3.62 infrared reflection-absorption spectroscopy; III.3.7c.i injection (emulsification); V.8.31 injection (foaming); V.7.8ff ink (Egyptian); 1.1.1, 1.1.2, 1.1.7, 1.1.27, IV. 1.3, IV.2.1 ink jet printing; III. 1.84 inner Helmholtz plane (intr.); II.3.61ff insoluble monolayers, see monolayers, Langmuir
SUBJECT INDEX
interaction (principles), energy, force; 1.4.2, 1.4.8, Il.figs. 2.2-3 multiparticle, Born-Green-Yvon; 1.3.69 Carnahan-Starling; I.3.69ff hypernetted chain; 1.3.69 Percus-Yevick; 1.3.69 relation to distribution functions; I.3.9d, 1.3.9e relation to vlrial coefficients; 1.3.9c pairwlse; 1.3.8, 1.3.9, 1.4.1, 1.4.2, 1.4.3, 1.4.4 tabulation for electric repulsion; IV.app. 2 potential; see interaction energy sign; 1.4.4 solvent structure-originated; 1.5.15, 1.5.3, 1.5.4, Il.table 1.5.12, II.1.95-96, Il.fig. 2.2, II.3.184ff, IV.3.8c see further, colloids, Interaction Interaction between colloids and macrobodies, see colloid stability, Van der Waals forces and colloids, interaction interaction between Ions; 1.5.2 interaction between molecules and surfaces; [1.4.6.1 ], II. 1.5, II.chapters 1, 2 and 5 interaction curves; I.fig. 3.4, I.fig. 4.1, figs. 1.4.2-3 interaction energy parameter; I.3.40ff, 1.3.43-45, [1.3.8.9], II. 1.56, II.2.34, II.5.5ff excess; 1.3.45, especially [1.3.8.9] interaction forces, general introduction; I.chapter 4 Interactions inside proteins; V.3.3 interaction in solution (excess nature of); 1.1.29, 1.4.5, 1.4.6b, 1.4.7 interface; I.1.3(intr.) curved; see capillary phenomena of tension; 1.2.94 optical study; 1.7.10, 1.7.11, II.2.5c reflection of light; 1.7.10a, II.2.5c refraction of light; 1.7.10a scattering; 1.7.10c (also see: purity criteria) Interfacial area; see surface area interfacial charge; see surface charge interfacial concentration (intr.); 1.1.5, see further, Interfacial excess Interfacial energy; III.2.9a, III.fig. 2.14, Ill.fig. 2.16 (relation to heat of evaporation)
47
48
SUBJECT INDEX
interfacial entropy; III.2.9a, Ill.flg. 2.14-1S5, III.2.55 (relation to entropy of vaporization), III.4.2d, Ill.flg. 4.19 interfacial excess; 1.2.5, 1.2.42, 1.2.22, III. 1.2, II.2.2, Il.flg. 2.1, [II.2.1.2], II.2.3, III.2.2, V.6.3, [V.6.3.24] isotherm; II.2.3, III.4.2 interfacial Gibbs energy; III.2.2, V.I.3 interfacial grand potential; III.2.2, III.5.18ff, V.I.I, V.l.3-4, V.1.6, V.1.8, V.1.9c, V.I.53 (also see: adsorption, Gibbs adsorption law, surface excess) interfacial polarization; see potential difference, % interfacial potential jump (x); see potential difference, % interfacial potentials; 1.5.5, II.3.9, III.4.4 also see: electrokinetic potential; Gouy-Chapman theory; monolayers, ionized; Poisson's law; potential difference between adjacent phases interfacial pressure; see surface pressure interfacial rheology; III.3.6, Ill.table 3.5 Burgers element; III.3.129 compliance; Ill.table 3.4, III.3.105 compressibility; [III.3.3.1], [III.3.4.3], III.3.93 compression; III.3.83, III.3.91 creep; Ill.flg. 3.40, III.3.61 deformation types; Ill.flg. 3.34 dilation; III.3.81, III.3.83, III.3.91, Ill.flg. 3.38 dilational elasticity (modulus); [III.3.4.4], III.3.40, III.3.81, Ill.table 3.4ff, [III.6.18-19], [III.3.6.34-39], III.3.6g, Ill.flg. 3.48, Ill.flg. 3.87, Ill.flg. 3.91, III.4.5, Ill.figs. 4.26-27, Ill.table 4.3, Ill.flg. 4.38, V.fig. 3.27, V.7.6, V.7.16, V.8.1c, V.fig. 8.4, V.8.69, V.8.83, V.8.88 distance coefficient; III.3.113, Ill.flg. 3.44, III.3.117ff distance damping; III.3.112 elasticity; III. 1.55, III.3.82, Ill.table 3.4 emulsification; V.8.2b, V.fig. 8.11, V.8.47ff equivalent mechanical circuits; III.3.6i experimental methods; III.3.6f, III.3.7e Fourier transform method; III.4.59ff Gibbs monolayers; Ill.table 4.2 Kelvin element = Voigt element Kelvin equation (damping); [III.3.6.63] kinetics; III.4.5 loss angle; [III.3.6.12], [III.3.6.41a], [III.3.6.66], [III.3.6.74]
SUBJECT INDEX
49
interfacial rheology (continued), Marangoni effect; 1.1.2 (intr.), 1.1.17, I.6.4.43ff, III.1.35, III.1.72, III.3.81 III.3.6e, Ill.figs. 3.35-37, III.3.239, V.6.2, V.6.37ff, V.7.6, V.8.1c, V.fig. 8.3, V.8.48, V.8.52ff Maxwell element; Ill.figs. 3.51-52 proteins; V.3.7, V.figs. 3.25-27 relation to interfacial tension; III.3.92, III.3.6d recovery; Ill.fig. 3.40, Ill.fig. 3.52 relaxation (Gibbs monolayers); Ill.fig. 4.20 relaxation (Langmuir monolayer); III.3.6h, III.3.61 respiratory stress syndrome; III.3.238-239 shear modulus; III.3.84, HI.3.92 shear viscosity; III.3.84, III.3.92, [III.3.6.20], III.3.6g, Ill.fig. 3.91, V.8.89 stress relaxation; Ill.fig. 3.39, III.3.6i stress tensor; III.3.91ff, IH.table 3.4, V.figs. 3.25-27 time damping; HI.3.113 viscosity; III. 1.55, III.3.82, III.table 3.4, V.figs. 3.25-27 Voigt element; IH.3.94, Ill.figs. 3.51-52 wave damping and propagation; III. 1.1, III. 1.58, III.3.6g, Ill.fig. 3.42-44, III.3.183, III.3.185, also see: loss angle interfacial science (first review); 1.1.2, 1.1.3, Volumes II and III interfacial tension, surface tension; I.1.4(intr.), 1.1.25, I.fig. 1.16 binary mixtures; III.4.2 data; Ill.app. 1,111.1.12 dynamic conditions; III.1.14b, Ill.fig. 1.31, Ill.fig. 1.32, V.fig. 8.3, V.8.1c, V.8.48ff interpretation; Ill.chapter 2, III.3.6d Cahn-Hilliard; III.2.6 capillary waves; III.2.9c distribution function; III.2.4, [III.2.4.6-8], III.2.24 empirical; III.2.11 and geometric means; III.2.lib and grand potential; III.2.7 Hamaker-de Boer approximation; III.2.5c lattice theory; III.2.10 mechanical; III. 1.3, V.6.3b pressure tensor (interfacial); III.2.3, [III.2.3.5], III.3.6d, V.6.3b scaling; [III.2.5.35], V.5.74ff simulations; III.2.7, Ill.figs. 2.9-10, Ill.table 2.2 statistical thermodynamics; III.2.4, IH.2.30-31, III.2.5Iff
50
SUBJECT INDEX
interfacial tension, surface tension, interpretation (continued), thermodynamic or quasithermodynamic; I.2.10ff, I.2.26ff, 1.2.11, 1.2.91, III.2.2, III.2.9 van der Waals; III.2.5 measurement; 1.1.11, 1.2.5, 1.2.96, II.3.139, Ill.chapter 1 'Bugler method'; III. 1.49 capillary rise; III.1.3, (differential) III. 1.19 (in a wedge), Ill.fig. 1.10 captive drops; Ill.fig. 1.11, further see sessile and hanging drops drop oscillations; III. 1.58 drop weight; III.1.6, III.1.72ff drops in a gradient; III.1.5 (also see: growing drops, sessile drops, spinning drops, etc.) du Noiiy ring; III. 1.8b, III.1.72ff dynamic (conditions); III.1.14, III.1.3-4, III.1.53ff (also see; interfacial tension, relaxation) falling drop; Ill.fig. 1.17 (see further, drop weight) falling meniscus; III.1.11, Ill.fig. 1.26 growing drops; III. 1.74 hanging drops; see sessile drops maximum bubble pressure; III.1.7, III.1.72ff micropipette; III. 1.57 'Padday's pencil'; III. 1.48 pendant drop = hanging drop; see sessile and hanging drops rheology; III. 1.57 sessile and hanging drops; III.1.4, III.1.72ff sphere tensiometry; III. 1.48 spinning bubbles; see spinning drops spinning drops and bubbles; III.1.9, Ill.fig. 1.24 surface light scattering; III.1.10 tensiometers; III. 1.8 wave damping; Wilhelmy plate; III.1.8a, III.1.72ff of curved interfaces; 1.2.23, II.1.6d, III.1.1, III.1.15, V.5.4 of electrolyte solutions; III.4.4 of films; II.1.95ff of microemulsions; V.5.4 of solid surfaces; 1.2.24, III. 1.5 measurement; III.1.13 pressure dependendence; III.2.9b relation to adsorption from binary mixtures; II.2.4f
SUBJECT INDEX
interfacial tension, surface tension, measurement (continued), relation to compressibility; III.2.11a, [III.2.11.6-7] relation to interfacial Helmholtz, Gibbs or internal energy; 1.2.11, Lapp. 5, [IIM.la.b] relation to molar volume; III.2.11a relation to surface light scattering; 1.7.10c, III.1.10 relation to work of cohesion; 1.4.47 relaxation; III.1.14, Ill.fig. 1.29 temperature dependence; 1.2.42, [II. 1.3.42], III.2.9a, V.fig. 8.1, V.5.4 (also see: capillary phenomena, monolayers, wetting, interfacial rheology) interfacial turbulence; V.8.52ff interfacial viscometers; III.3.180ff, Ill.figs. 3.69-71 interfacial work; 1.2.10,1.2.3, 1.3.17 interference (intr.); 1.7.8, 1.7.15 interferometry (contact angle); III.5.43ff ion association (in solution); I 5.2d ion binding; I.5.3ff, II.3.6d-e ion condensation; II.4.43, V.2.5 ion correlations; II.3.6b ion exchange; II.3.35, II.3.168ff ion mass spectroscopy (SIMS); 1.7.1 la, I.table 7.4 ion pairs; 1.5.3 ion scattering spectroscopy (ISS); 1.7.11a, I.table 7.4, II.1.15 ion specificity; see lyotropic series ion transfer (resistance); II.3.95, IV.4.19-20 ion vibration potential; II.4.29 ionic atmosphere; 1.5.16, see double layer, diffuse ionic components of charge; see double layer, electric ionization, ionization (Gibbs) energy; 1.4.29, I.5.34ff, 1.7.86 ionomer; V.2.70 ions, activity coefficient; see there bound vs. free; 1.5.la, II.3.6, III.3.4h, V.2.2, V.2.5a hydration; 1.5.3 hydrophilic; 1.5.47 hydrophobic; 1.5.46 radii; I.table 5.4, I.fig. 5.7 solvation; 1.5.3 structure-breaking; 1.5.47 structure-forming; 1.5.47 transfer; I.5.3f, II.3.9
51
52
SUBJECT INDEX
ions (continued), volumes; I.tables 5.7 and 8 ion-solvent interaction; see hydration, solvation ionic surfactants; see surfactants IRAS = infrared reflection-absorption spectroscopy iron oxides goethite (oe-FeOOH), electrokinetic charge; Il.fig. 4.13 point of zero charge; II.table 3.5, Il.app. 3b haematite (a-Fe2O3); Il.fig. 1.1, IV. fig. 2.1b adsorption of fatty acids from heptane; Il.fig. 2.26 conductivity of sols; Il.fig. 4.38 dielectric relaxation of sols; Il.fig. 4.38 double layer; II.3.94, Il.table 3.6, Il.figs. 3.59-62, II.table 3.8 electrokinetic properties; Il.table 4.3 point of zero charge; Il.table 3.5, Il.fig. 3.37, Il.app. 3b immersion, wetting; Il.table 1.1 irradiance (intr.); 1.7.5 IRRAS = infrared reflection-absorption spectroscopy irregular coagulation series; see coagulation irreversible thermodynamics; see thermodynamics irreversible colloids = colloids, lyophobic irreversible process; see process, natural Ising problem; 1.3.40,1.3.43, V.2.20 isobaric (process, def.); 1.2.3 isochoric (process, def.); 1.2.3 isoconduction; II.3.215 isoelectric point; II.3.8, II.3.103, II.3.106, Il.fig. 3.78, Il.fig. 4.41, II.4.127ff relation to point of zero charge; II.3.8b, Il.fig. 3.35 isoelectric focusing; II.4.131ff isodisperse = homodisperse isomorphic substitution (in clay minerals); II.3.2, II.3.165 isosteric (process); 1.2.3 isosteric heat of adsorption; see adsorption, isosteric enthalpy isotachophoresis; II.4.131 isothermal (process); 1.2.3 destination; see Ostwald ripening reversible work; 1.2.27 ISS = ion scattering spectroscopy Jones-Dole equation, coefficients (viscosity of electrolytes); 1.5.52. I.table 5.9, I.6.78ff, IV.6.53-54
SUBJECT INDEX
kaolinite; II.3.164, II.fig. 3.66, IV.fig. 2.2b wetting; Il.table 1.34, Keesom-Van der Waals forces; see Van der Waals forces Kelvin cells; V.7.5 Kelvin element (rheol.) = Voigt element; see interfacial rheology Kelvin equation; [1.2.23.24], [II.1.64, [II.1.6.17], [III.1.13.3], [IV.2.2.50], IV.2.2e Kelvin equation (wave damping); [III.3.6.63] Kerr effect; 1.7.100 Kiessing fringes; III.3.150, III.fig. 3.58 kinetics (coagulation); IV.4.3 kinetics (micellization); V.4.10 Kirkwood equation (electric polarization); [1.4.5.22] Kirkwood-Buff equation (interfacial tension); [III.2.4.6-9] Kirkwood-Frbhlich equation (electric polarization); [1.4.5.23] Kolmogorov theory (for emulsification); V.8.41ff Kozeny equation; [1.6.4.39], II.4.55 Kozeny-Carman equation; [1.6.4.41], II.4.55, [IV.2.2.67] Krafft temperature; V.4.13, V.8.5 Kramers-Kronig relations; [1.4.4.31, 11.32], 1.4.36, 1.4.77, 1.7.13, II.3.93 Krieger-Dougherty eq. (viscosity); [IV.6.9.10], V.8.15 Kugelschaum; V.7.2 Kuhn segment = statistical chain element lactalbumin ( a ) , adsorption; V.figs. 3.15-18 lactoglobulin (P ); V.figs. 3.25-26, V.fig. 8.20 Lambert-Beer's law; 1.7.13,1.7.41 Landau-Ginzburg analysis (microemulsions); V.5.38ff Langevin equation (for forced stochastic processes); [1.6.3.4], 1.6.3d, Lapp. 11.2, IV.4.2 Langmuir adsorption isotherm; see adsorption isotherm Langmuir-Blodgett layers; II.2.56, III. 1.42, III.3.7a Langmuir monolayers, see monolayers Langmuir trough; see film balance Laplace's law = Young and Laplace's law, see capillary pressure Laplace pressure = capillary pressure Laplace transformations; Lapp. 10 lasers; 1.7.4c laser-Doppler microscopy = QELS; see electromagnetic radiation latex, (pi. latices or latexes); 1.1.6, 1.5.99, II.3.87, II.fig. 3.29, IV. 1.9 compressibility and scattering; IV.fig. 5.32 conductivity; Il.table 4.2, Il.fig. 4.34 crystallization; IV.5.8a, IV.fig. 5.58
53
54
SUBJECT INDEX
latex (continued), distribution function; IV.fig. 5.4 electro-osmosis; II.table 4.2 electrophoresis; Il.fig. 4.29, Il.table 4.2 rheology; IV.fig.6.28-29 sedimentation; IV.fig. 5.60 stability; IV.3.13c, IV.fig. 4.8b, IV.figs. 4.19-21 streaming potential; Il.fig. 4.30, Il.fig. 4.35 structure factor; IV.fig. 5.19, IV.fig. 5.34, IV.fig. 5.36 surface charge; Il.fig. 3.29, IV.fig. 3.75 lattice statistics; I 3.6d. I.3.6e, 1.3.8b, Il.chapter 5, V.chapter 1 polymer adsorption; II.5.30ff random walk; 1.6.3d LC = liquid condensed (2D phase); III.3.3b LE = liquid expanded (2D phase); III.3.3b lead oxides; IV. 1.3 LEED = low-energy electron diffraction Lennard-Jones pair interaction energy; I.fig. 4.9, 1.4.5b, [1.4.5.1] in adsorbates; [II. 1.1.14], II. 1.74 in liquids near solids; II.figs. 2.4-5 structure factor; IV.fig. 5.5 leucocytes; III.5.100 leukemia; III.5.100 Levich equations (for convective diffusion); I.6.92ff levitation; IV.3.12d Lewis acids, bases; 1.5.65, II.3.185 Lifshits theory; see Van der Waals interaction between colloids Lifshits-Slezov-Wagner (LSW) theory (Ostwald ripening); IV.2.26, V.8.66ff light scattering; see electromagnetic radiation line tension; III. 1.6, III.5.5, III.5.6 lipase; 1.1.3 lipids (phospho-); Ill.table 3.8 lipids (phospho-) films; V.6.67ff, V.fig. 6.35, V.fig. 6.41, V.fig. 6.45 lipids (phospho-) monolayers; Ill.fig. 3.8, Ill.fig. 3.12, Ill.fig. 3.14, Ill.fig. 3.29, III.3.140, Ill.fig. 3.55, Ill.figs. 3.62-62, Ill.fig. 3.67, III.3.8c, Ill.figs. 3.89-91, III.3.238 Lippmann capillary electrometer; see capillary electrometer Lippmann equation (for electrocapillary curves); [1.5.6.17], 1.5.100, 1.5.108, II.3.138 liquid bridges, see capillary bridges liquid junction potentials; see potential difference
SUBJECT INDEX
liquid-fluid interface, general; III.chapter 1.2, III.2.8 (thickness), III.2.9b density profile; Ill.flg. 2.1, III.2.3, [III.2.5.31], III.2.8, Ill.figs. 2.11-13, Ill.fig. 2.19 double layer; II.3.10g thermodynamics; III.2.2 liquids, apolar, double layers; II.3.11, II.4.50 solvation; I.5.3f in pores; II 1.6d near surfaces; II.1.6d, II.3.123ff, II.4.38, Il.fig. 4.11 London-Van der Waals forces; see Van der Waals forces longitudinal waves; III.3.110, see interfacial rheology loops; see adsorption of polymers Lorentzian peaks; 1.7.50 Lorenz-Lorentz equation (electric polarization); [1.4.5.21], 1.7.43 loss angle (rheology), III.3.90, [III.4.5.44] see interfacial rheology low-energy electron diffraction (LEED); 1.7.25, 1.7.86, Il.fig. 1.2, II. 1.1 Iff LSA = linear superposition approximation, see colloids, interaction LSW (theory) = Lifshits-Slezov-Wagner (theory) lunar soil, spherules in; 1.1.1, 1.1.2 sorption of methanol; Il.fig. 1.35 lung surfactant; 1.1.1, 1.1.2, III.3.219, HI.3.238, V.6.8 lyotropic series; I.5.66ff(intr.) in coagulation; 1.5.67, IV.3.9i, IV.table 3.3 in ionic binding, double layer charge or double layer capacitance; I.5.66ff, II.3.15, II.3.109, II.3.132, Il.fig. 3.41, II.3.135, Il.fig. 3.53, Il.fig. 3.55, II.3.147, Il.fig. 3.75, II.3.10h, Il.table 3.8, II.3.203, III.3.207ff, Ill.figs. 3.85-86, III.4.89ff, IV.3.149, IV.table 3.6, IV.3.162ff, IV.3.170, IV.3.173, IV.figs. 3.72-73, V.2.22, V.2.53, V.2.68 lysozyme; IV.fig. 5.27, V.fig. 3.24-25, V.fig. 8.20 macromolecules; see polymers, polyelectrolytes, proteins macropores; see pores magnetic birefringe; 1.7.100 magnetic colloids; IV.3.10c magnetic fields; I.7.1a, 1.7.2, 1.7.16, 1.7.13, IV.3.10c magnetic induction; 1.7.2, IV.3.10c magnetic permeability; 1.7.9, IV.3.10c magnetite; IV.fig. 2.2, IV.2.4d, IV.fig. Al.l
55
56
SUBJECT INDEX
magnetization; 1.7.9, IV.3.10c Mandelstam equation (for surface scattering); [1.7.10.24], [III. 1.10. II, V.6.43 manganese dioxide, double layer; Il.table 3.8, Il.app. 3b Marangoni effect; see interfacial rheology Marangoni number; V.6.38, V.7.15ff, [V.8.1.4], V.8.57, V.8.83, V.8.89 marginal regime (polymer concentration); II.5.9, II.fig. 5.3 marginal regeneration (in films); V.6.4e Mark-Houwink eq. (rheology); [IV.6.11.8] Markov chain; 1.6.24, V.A1.7 Markov process, first order; 1.6.24, II.5.30 Martin eq. (rheology); [IV.6.11.14] masers; 1.7.4c mass action (micelle formation); V.4.2b mass conservation (in hydrodynamics); 1.6.la, IV.6.1 mass, reduced; 1.4.44 maximum term method (statistical thermodynamics); 1.3.37 Maxwell-Boltzmann statistics, distribution; 1.3.12,1.6.26ff, II.3.172, IV.4.6ff Maxwell element, see rheology and interfacial rheology Maxwell equations (for electromagnetic waves); 1.7.2 Maxwell (-Wagner) relaxation; 1.6.84, II.3.219, Il.fig. 3.89, IV.4.23 Mayer function; I.3.60ff, I.3.64ff mayonnaise; 1.1.6 mean curvature (of interfaces); 1.2.23a, III.1.1, III.1.15 mean field theories; II.5.7, II.5.29, III.2.5 membrane emulsification; V.8.32, V.fig. 8.8 membrane equilibrium; see equilibrium mercury-solution interface, double layer; 1.5.6c, II.3.10b, II.figs. 3.48-55, Il.table 3.8 interfacial tension; II.3.138ff, Il.fig. 3.48 mercury sulphide; IV. 1.3 mesopores; see pores mesoscopic = colloidal metabolism; see fats metal(s), contact angles on ...; III.table A4.1 Hamaker constants; Lapp. 9.4, IV.app. 3 points of zero charge; Il.app. 3a sol preparation; IV.2.37 methylviologen; [II.3.14.1], II.3.224
SUBJECT INDEX
mica; II.3.165, IV.fig. 3.56 interaction; IV.3.12b, IV.fig. 3.57 micelle; I.1.6(def.), 1.1.24, I.fig. 1.15, III.4.6 complex coacervate; V.2.6f reverse (or inverted); 1.1.25 for general discussion, see V.chapter 4 micellization, critical concentration of (am.a); I.1.24ff(intr.), III.4.6a, Ill.table 4.4, IV. 1.5 determination; I.1.25ff, V.4.1c critical temperature; V.4.11 for general discussion, see V.chapter 4 microelectrophoresis; II.4.45ff, Il.figs. 4.14-16, IV.fig. 5.14 microemulsions; 1.1.3, 1.1.7, 1.2.68, IV. 1.6, V.chapter 5 (mostly non-ionic) applications; V.5.6 bending; V.5.5a, V.fig. 5.34, V.fig. 5.36, V.5.93 bicontinuity; V.5.1 Iff, V.5.30 conductometry; V.5.3f, V.fig. 5.21 correlation lengths; V.5.39, V.5.3h, V.fig. 5.25, V.fig. 5.27a, V.5.4e, V.5.5 curvatures; see microstructure diffusion; V.5.3e, V.fig. 5.20 efficiency boosting; V.5.6e emulsiflcation failure boundary; V.5.21, V.5.49 experimental methods; V.5.3 fish diagrams; V.flgs. 5.6, 5.13, 5.22, 5.38-39, 5.41, 5.43, 5.47 Gibbs triangle = ternary phase diagram interfacial tensions; V.5.4, V.flgs. 5.28-33, V.5.5, V.fig. 5.37 Landau-Ginzburg approach; V.5.38ff microstructures; V.5.3 middle phase (= one of the Winsor states); V.5.2g optimalization; V.5.2d, V.5.2e phase behaviour (-diagrams); V.5.2, V.5.4b binary systems; V.5.2a quaternary systems; V.5.84, V.fig. 5.39, V.fig. 5.43 quinary systems; V.5.87, V.fig. 5.43 ternary systems; V.5.2b phase inversion (intr.); V.5.3 (many examples in chapter V.5) phase inversion temperature; V.5.6ff phase trajectories; V.5.2g and elsewhere in V.chapter 5 plumbers nightmare = bicontinuity scaling; V.5.2h, V.fig. 5.27, V.5.4e. V.5.73ff, V.fig. 5.37 surfactant solubility; V.5.2f, and elsewhere in V.chapter 5
57
58
SUBJECT INDEX
microemulsions (continued), theory; V.5.5 wetting; V.5.4c, V.fig. 5.30 Winsor states (def.); V.5.1, V.fig. 5.1 micropores; see pores microscopies of sols; IV.2.41ff middle phase (microemulsions); V.5.2g Mie theory (light scattering); see electromagnetic radiation milk; V.8.1 mixtures, athermal; 1.2.55 colloidal; IV.5.7c, IV.5.8c homogeneous, thermodynamics; 1.2.16 ideal; 1.2.17 non-ideal; 1.2.18 mobile films; V.6.48 mobility (of ions); 1.6.6a molality; I.2.45(def.) molarity; I.2.45(def.) mole fraction; I.2.44(def.) molecular condensor; I.fig. 5.1, II.3.59 molecular dynamics; I.3.1e(intr.), association colloids; V.4.3a, V.fig. 4.7 electrolytes; I.5.57ff, I.fig. 5.9 liquids in pores; Il.fig. 1.38 liquids near surfaces; Il.figs. 2.5-7, II.3.55, Il.fig. 3.39 wetting; Ill.fig. 5.36 molecular mass (of colloids); see polymers, particles molecular sieve, adsorption of krypton; Il.fig. 1.19 adsorption of methane; Il.fig. 1.36 molecular state; see state molecular thermodynamics; see statistical thermodynamics moment, of a distribution; 1.3.7b of a double layer; [II.4.6.50] moment expansion; IV.2.45, IV.app. 1 momentum; I.3.57(def.) momentum conservation (in hydrodynamics); 1.6.lb, IV.6.4 (also see: transport of momentum) monatomic crystal: see Einstein crystal
SUBJECT INDEX
monochromatic (waves, radiation); 1.7.2 monochromator; 1.7.3 monodisperse = homodisperse monolayers (at liquid-fluid interfaces); I.fig. 1.15b adsorbed = Gibbs monolayers bending moduli; Ill.tables 1.6 and 7 binary mixtures; see Gibbs monolayers characterization; III.3.7 cholesterol; III.3.8d curved; see Gibbs monolayers; diffraction; III.3.7b dilute solution; see Gibbs monolayers electrolytes; see Gibbs monolayers in emulsions; V.8.1f film balances; III.3.3a fatty acids; alcohols; III.3.8b Gibbs (monolayers); III.chapter 4 binary mixtures; III.4.2, V.8.7 curved; III.4.7 dilute solutions; III.4.3 distinction from Langmuir monolayers; III.3.1 dynamics; III.1.14b, III.4.5 electrolytes; III.4.4 Gibbs equation; 1.5.94 ionized; II.3.2, Il.fig. 3.1b, V.6.5b proteins; V.3.7 rheology and kinetics; see interfacial rheology surfactants; III.1.14b, III.4.6 temperature dependence; V.fig. 8.1 Langmuir (monolayers); III.chapter 3 Brewster angle microscopy; III.table 3.5 characteristic functions; III.table 3.2 cholesterol; III.3.8d collapse; III.fig. 3.46 diffraction; III.3.7b distinction from Gibbs monolayers; III.3.1 ellipsometry; Ill.table 3.5, III.3.7b energy-entropy compensation; III.3.37 fluorescence; Ill.table 3.5 hysteresis; III.3.13, III.3.8a, III.fig. 3.79 ionized; III.3.4h
59
60
SUBJECT INDEX
monolayers (at liquid-fluid interfaces), Langmuir (continued), Langmuir trough; III.3.3a Langmuir-Blodgett; see there lattice theory; III.3.5e mixed; III.3.4f molecular dynamics; III.3.5d molecular thermodynamics; III.3.5 Monte Carlo; III.3.5c neutron reflection; III.table 3.5, III.3.7b optical techniques; III.3.7b-c permeation; III.3.238 phase behaviour; III.3.3 phospholipids; III.3.8c polymer brushes; III.3.4J, III.3.8f polymers; III.3.4i, III.3.8e, V.8.8, V.flg. 8.2, V.I.77 preparation; III.3.2 proteins; V.3.7 reflection; III.III.3.7b relaxation; III.3.6h, V.I. 12b reproducibility; III.3.8a, III.fig. 3.79 rheology; see interfacial rheology; scanning probe; III.3.7d, III.table 3.5 simulations; III.3.5c, III.3.5d spectroscopy; 111.3.7c, Ill.table 3.5 thermodymamics; III.3.4 thermodynamics; III.3.4 transfer; III.3.7a, III.fig. 3.5,, Ill.figs. 3.53-54 Volta potential; see there, For the optical techniques see the entry in question X-ray diffraction; Ill.table 3.5, III.3.7b X-ray reflection; Ill.table 3.5, III.3.7b monolayer formers; III.3.200 monolayer spreading; III.3.2 montmorillonite; II.3.165 adsorption of alcohols + benzene; II.fig. 2.21 adsorption of methane + benzene; II.fig. 2.22 adsorption of poly(acryl amide); II.fig. 5.39b adsorption of water vapour; II.fig. 1.30 disjoining pressure; IV.fig. 3.55 Monte Carlo simulations; I.3.1e(intr.), I.fig. 5.4, 1.5.30, IV.fig. 5.16, IV.fig. 5.30 adsorbed liquids; II.fig. 2.4 adsorbed polymers; II.5.30
SUBJECT INDEX
Monte Carlo simulations (continued), association colloids; V.4.3a, V.figs. 4.5-4.6 electric double layer; II.fig. 3.18 Mountain lines; 1.7.45 moving boundary electrophoresis; II.4.5Iff, II.fig. 4.17 mushrooms (polymer ads.); V . I . l i d , V.fig. 1.27 muscovite; II.3.165 myoglobin (ads.); V.fig. 3.19 natural; see process nanoparticles = small colloids nanoscience = science of small colloids Navier-Stokes equation; [1.6.1.15], 1.6.51, II.4.18, [II.4.6.4]ff NBF = Newton black film; see films, liquid negative adsorption; see adsorption, negative Nernst-Einstein equation; [1.6.6.15], [II.3.13.14], [II.4.3.55] Nernst-Planck equation; I.6.7a, 1.6.89, II.2.85, (11.3.13.12], [II.4.6.2] Nernst's heat theorem; 1.2.24 Nernst's law for distribution equilibrium; 1.2.20a. 1.2.81 Nernst's law for electrode potential; 1.2.34, 1.5.5c, I.5.5e, II.3.8, II.3.91, II.3.147ff, II.3.150 networks; IV.6.14 also see: gels, percolation Neumann triangle; [III.5.1.3], Ill.fig. 5.6 neutron reflection (by surfaces); II.2.7, II.5.66ff neutron scattering (by colloids); 1.7.9, 1.7.102, IV.fig. 5.25, IV.fig. 5.33, IV.fig. 5.36 Newton films; see films, liquid Newton(ian) fluids; 1.6.8,1.table 6.1,1.6.4a, III.3.6b, IV.6.1, IV.6.2, IV.fig. 6.5, IV.table 6.3 Newton's second law; [1.6.1.12], 1.6.4 NMR = nuclear magnetic resonance non-ionic surfactants; see surfactants non-linear optical techniques; III.table 3.5 non-Newton(ian) flow; 1.6.36, III.3.6b, IV.6.7a, IV.fig. 6.17 non-solvent; 1.1.27 normal stress; see stress nuclear magnetic resonance (NMR); 1.7.16, 1.7.13, 1.7.102 chemical shift; I.5.54ff, I.fig. 5.8 of emulsions; V.8.23 of interfaces; II.2.8, II.2.55, II.5.58ff, II.5.71 of microemulsions; V.5.3e, V.fig. 5.20 of pores; II. 1.90
61
62
SUBJECT INDEX
nuclear magnetic resonance (NMR) (continued), of water; I.5.54ff, I.fig. 5.8 (also see: spin, etc.) nucleation, in colloid preparation; IV.2.9ff, IV.2.2b, IV.2.2c in emulsification; V.8.31 heterogeneous; 1.2.100, II.1.42 homogeneous; 1.2.23d, IV.2.9ff in pores; see capillary condensation number of realizations; 1.3.4 (def.) octupole moment; 1.4.19 odd-even parity; III.3.302, Ill.fig. 4.31, Ill.fig. 4.36 oHp = outer Helmholtz plane Odijk-Skolnick-Fixman theory (polyelectr.); V.2.27ff oil recovery, tertiary; see enhanced oil recovery ointments; 1.1.6, 1.1.28 Onsager formula (for limiting conductivity); [1.6.6.26] Onsager formula (for polarization); [1.4.5.20] Onsager relations (irreversible thermodynamics); 1.6.2b application to electrokinetics; 1.6.2c, II.4.2, II.4.7, II.4.21, II.4.27, II.4.61, II.4.106 Onsager theorem (for approach to equilibrium); 1.7.44, 1.7.48, Lapp. 11.3, Lapp. 11.5, Lapp. 11.7-8 optical activity; I.7.99ff optical axes; 1.7.14 homodyne; 1.7.37, I.7.6d optical levitation; IV.3.157ff optical mixing (beating); 1.7.37 heterodyne; 1.7.37, I.7.6d optical trapping; IV.3.12d optical tweezers; IV.3.158 ordering parameter; 1.6.73, III.3.71ff, Ill.fig. 3.61, III.3.166, [III.3.7.13] open circuit potential; II.3.149 ore benification; 1.1.25, II.5.97 orientation of adsorbed molecules; II.2.55 Ornstein-Zernike equation (for compressibility); [1.3.9.32] Ornstein-Zernike equation (for correlation functions); [IV.5.3.19], [IV.5.3.33b] Ornstein-Zernike equation (for critical opalescence); [1.7.7.10] orthokinetic coagulation or flocculation; IV.2.41, IV.4.5b, IV.fig. 4.18, IV.4.47-48, V.1.84, V.flg. 1.50 oscillating drop; Ill.fig. 3.72
SUBJECT INDEX
oscillating liquid jet; Hl.figs. 1.28 and 29, III. 1.84 oscillation, harmonic; 1.4.38, 1.4.44, I.7.3d, III.3.80ff, III.3.105ff, Ill.fig. 3.41, Ill.fig. 3.45 oscillator; 1.7.3b oscillator, harmonic; 1.3.5a, 1.4.37 oscillator strength; 1.4.38 osmotic coefficient; 1.2.18a osmotic compressibility; IV.5.3d osmotic equilibrium; 1.2.34, IV.5.2 osmotic pressure; 1.2.34, 1.2.64, I.2.20d, IV.3.144, IV.5.46ff, V.I.5, V.I. 10, V.2.11ff, V.figs. 2.2-3, V.2.39 osmotic repulsion; 1.2.72, I.fig. 2.11 Ostwald equation; [1.2.23.25], II.1.19, [III.1.13.2] Ostwald ripening; 1.2.97, II.1.103, II.3.110, IV.2.2e, V7.14ff, V.7.3c, V.8.3b Ostwald viscometer; IV.fig, 6.18 outer Helmholtz plane; II.3.17, II.3.59ff Overbeek equations (for retarded London-Van der Waals forces); [1.4.4.23a, b], 1.4.74 overcharging; Il.fig. 3.20c, IV.3.9J, IV.3.164ff, IV.figs. 3.62-64, IV.fig. 3.67, IV.3.74 overflowing cylinder (in rheology); Ill.fig. 3.73, V.fig. 3.25 overpotential; 1.5.79 overrun (foams); V.7.7 oxides (in general), contact angles on ...; III.table A4.3 double layer; 1.5.6a, 1.5.6b, II.3.71ff, Il.table 3.5, II.3.8, II.3.71ff, 11.3.10c point of zero charge; II.3.112, Il.app. 3b, Il.table 3.5 (for specific oxides, see under the chemical name) paints; 1.1.22, 1.1.28, IV.2.1ff, IV.3.185 pair correlation function; 1.3.66, II.3.51ff, IV.5.3 pair interaction; see interaction pair potential; see interaction Pallmann effect = suspension effect pancake (polymer ads.); V.I.11, V.fig. 1.27 paper electrophoresis; II.4.131 papermaking; IV.2.1 papyrus; IV. 1.3, fig. IV. 1.1 parachor; III.2.67 paramagnetism; IV. 3.124
63
64
SUBJECT INDEX
parameter (def.), extensive; 1.2.10 intensive; 1.2.10 mechanical; 1.3.40 thermodynamic; 1.3.40 paraquat; [II.3.14.1] partial molar quantities; 1.2.46 particle-in-a-box problem; 1.3.23 particles (colloidal), form factor; 1.7.56,1.7.70ff interaction and rheology; IV.6.9, IV.fig. 6.26 networks; IV.6.14a shape; I.6.5g (2), 11.(3), 1.7.8c, I.7.8d, 1.7.69 size; 1.7.8, 1.7.26, 1.7.63, 1.7.67, IV.4.32 size distributions; see separate entry structure factor; I.7.64ff (see separate entry) also see: charged (colloidal) particles particle-wave duality; 1.7.5 partition; see distribution partition coefficients (micelles); V.4.9b partition function; I.3.2(intr.), 1.3.3,1.3.4,1.3.5, Lapp. 6 canonical; I.3.2(intr.), 1.3.3,1.3.4,1.3.5,1.3.51,1.3.54, 1.3.59,1.3.63, Lapp. 6 for ideal gas; I.3.6b, I.3.6c for localized adsorbate; I.3.6d for subsystem; 1.3.5 grand (canonical); I.3.2(intr.), 1.3.3,1.3.4,1.3.31,1.3.33, I.3.54ff, 1.3.63, Lapp. 6, 11.1.95,11.1.99 isobaric-isothermal; 1.3.13, 1.3.18,1.3.19 microcanonical; I.3.2(intr.), 1.3.3, 1.3.4 separable; 1.3.20 Pascal's law; 1.2.90, III.2.9 Pauli principle; 1.4.5,1.4.42, II.3.172 PCS = photon correlation spectroscopy = QELS; see electromagnetic radiation Pearson's rule; II.3.185 Peclet number; 1.7.97, [IV.4.5.11], [V.8.1.18], [V.8.3.5], [V.8.3.21] pendant drop; see drop, pendant penetration depth (evanescent waves); [1.7.10.12] Percus-Yevick (PY) equations [1.3.9.29 and 30], [IV.5.3.40], IV.5.4b, IV.fig. 5.16 percolation (threshold); IV.5.86ff, IV.6.83 perikinetic coagulation (def.); IV.4.37
SUBJECT INDEX
65
period (of a wave); 1.7.4 permeability, of liquid films; V.6.2h of monolayers; III.3.239ff also see: porous plugs perpetual motion; 1.2.8 of second kind; 1.2.23 persistence length; see polymers/polyelectrolytes in solution for worm-like micelles; V.4.6d persistence parameter; see polymers/polyelectrolytes in solution persistence time; 1.5.45 PFM = polarized fluorescence microscopy phagocytosis; III.5.2, III.5.100 phase angle; Lapp. 8, see further loss angle phase diagrams of microemulsions; see V.chapter 5 phase diagrams of surfactants; V.4.1e, V.fig. 4.4 phase diagrams (2D); Ill.flg. 3.15, Ill.fig. 3.19, (see the /r(A) curves in III.chapter 3) phase diagrams and nucleation; IV.fig. 2.3, IV.figs. 5.37-41, IV.fig. 5.43, IV.figs. 5.445.45, IV.figs. 5.49-53, IV.fig. 5.56, IV.fig. 5.62, IV.fig. 5.63c. phase integral; 1.3.57 phase inversion; V.5.2c, V.8.64 phase inversion temperature; V.5.6ff, V.8.5, V.8.49ff, V.fig. 8.14, V.8.89 phase rule (Gibbs); 1.2.13 phase separation, transitions and coexistence; 1.2.19, II.5.2e IV.5.7 in capillaries; II.1.6e in interfaces; III.2.18, III.3.3b, Ill.table 3.1, III.3.4d, III.3.217ff (also see: demixing, critical point, condensation: two-dimensional, polymers in solution and microemulsions) phase space; I.3.57(def.) phase stability (cone, colloids); IV.5.7 phase transitions (cone, colloids); IV.5.8 phenomenological approach; I.1.29(def.), 1.2.2 phosphate binding (soils); II.3.222 phospholipids, see lipids photobleaching; 1.7.103 photocatalysis; II.3.222 photochromic probes; 1.7.103 photoconduction; II.3.173 photocorrelation spectroscopy = QELS; see electromagnetic radiation photoelectric effect; 1.7.85 photographic emulsion; IV. 1.9
66
SUBJECT INDEX
photolysis of water; II.2.87, II.3.223 photons; 1.7.5, III.3.168 (counting) physisorption; II. 1.5, II. 1.18, II.1.30ff Pickering (emulsion stabilization); III.5.99, V.8.4 pipettes, emptying; III.fig. 1.8 p.i.t. = phase inversion temperature plant growth in arid regions; 1.1.1, 1.1.2 plastic behaviour (rheology); IV.6.3a Plateau border; 1.1.16, I.fig. 1.11, V.6.2, V.6.3, V.6.48, V.fig. 7.2 Plateau rules (foam structure); V.7.4 plumber's nightmare; V.4.18, see further microemulsions, bicontinuous pluronics (in micelles); V.4.4e point of zero charge; 1.5.90, 1.5.6e, II.3.8, II.3.11, II.3.17, II.3.74, II.3.8, Il.app. 3, II.3.118, II.3.120, Il.fig. 3.61, II.3.162 experimental determination; II.3.8a influence of organic additives; II.3.12d, Il.figs. 3.77-80, Il.fig. 3.82, II.3.223 influence of specific adsorption; II.3.68ff, II.3.103ff, Il.fig. 3.34 interpretation; II.3.8c pristine; 1.5.102, II.3.8, II.3.103ff, Il.fig. 3.34, II.3.140, II.3.152 relation to isoelectric point; II.3.8b, Il.fig. 3.35 tabulation; app. II.3 temperature dependence; 3.75ff, II.3.115ff, Il.figs. 3.36-37 Poiseuille's law; see Hagen-Poiseuille's law Poisson-Boltzmann equation; 1.4.16, 1.5.18, [II.3.5.6], [II.3.5.44], [II.3.5.57ff] Poisson-Boltzmann theory; II.3.6a for flat interfaces = Gouy-Chapman theory for low potentials = Debye-Huckel theory improvements; 1.5.2c, II.3.6b, Il.figs. 3.18-19 Poisson distribution; [IV.2.3.46] Poisson's law (electrostatics); 1.4.53, 1.5.10, [I.5.1.20-20a], II.3.19, II.3.35, [II.3.5.43], [II.3.6.14], II.3.211, II.4.18, II.4.70, [II.4.6.12]ff, II.4.115, II.5.55 Poisson ratio; IV.6.9 polarimeters; 1.7.99 polarizability (intr.); I.4.22ff, I.4.4d, I.4.4e, 1.7.18, 1.7.53, 1.7.94 data for molecules; I.table 4.2 molar; 1.4.24 polarization, dielectric (phenomenon); 1.4.4b, I.4.4e, I.4.5f of colloids; see dielectric dispersion of sols of interfaces; 1.5.5b, II.3.9, (see potential difference, z) polarization, dielectric (physical quantity); I.4.5f, 1.7.2, 1.7.6, IV.3.10 polarization (of radiation, etc.); see electromagnetic radiation
SUBJECT INDEX
polarized fluorescence microscopy; 111.3.7c.iv polarizer; I.fig. 7.7, 1.7.99 polarizer-sample-analyzer; 111. 3. 7b .i polar molecules; 1.4.4b poly(phenylene), V.fig.2.3 polyampholytes; II.5.13, V.2.1, V.2.6d polydispersity; 1.1.13, IV.2.2d, IV.figs. 5.15-17, IV.fig. 6.27 relative; IV.2.61, IV.app. 1 also see, size distributions polyelectrolytes in solution; 1.1.6, II.5.2f, V.chaper 2 (general) annealed; V.2.2 brushes; V.2.3c chain statistics; V.2.3 charge; II.5.14ff, V.2.2, V.fig. 2.4 chemical composition; V.2.1b, V.table 1 colloid flocculation; V.2.7a colloid stabilization; V.2.7b complex coacervation; V.2.6e, V.2.6f (compl. coac. micelles) complexation; V.2.6 concentration regimes; V.2.3, V.fig. 2.15 conductivity; V.2.5 configurations; V.2.3 dielectrics; V.2.5d dissociation; V.2.2d electrokinetics; V.2.5a gels; V.2.3d grafts, see adsorption of polyelectrolytes interaction; V.2.15, V.fig. 2.5 multilayers, see adsorption of polyelectrolytes persistence length; II.5.14, II.fig. 5.5 phase separation; V.2.6 quenched; V.2.2 solubility; V.2.6a viscosity; V.2.4 polyelectrolyte adsorption; see adsorption of polyelectrolyte effect; II.3.71, II.3.76, V.2.2d polyelectrolyte gels; V.2.3d polyhedral (foams); see foams polymer brush, monolayers; III.3.8f, III.figs. 3.96-98, III.fig. 3.100. V.I.11 polymer colloids; see latex, latices
67
68
SUBJECT INDEX
polymer melt near a wall; II.5.45ff polymer surfactants; V. 1.51 polymers in solution; I.1.2(intr.), I.flg. 1.17, 1.3.34, IV.6.11-12; V.I.2 concentration regimes; II.5.9ff, Il.fig. 5.3, IV.6.11-12 conformation; Il.figs. 5.1-2, II.5.1ff, IV.fig. 6.30 end-to-end distance; II.5.4, Il.fig. 5.2, IV.6.61 entanglements; IV.6.67ff excluded volume (parameter); II.5.3, II.5.2b, II.5.5b, IV.6.62, [IV.6.11.5] expansion coefficient; II.5.6ff, IV.6.62 Flory-Fox constant; IV.6.63, V.2.51 ideal chain; II.5.3, II.5.2a intrinsic viscosity; IV.6.11 light scattering; I.7.56ff, I.7.62ff molecular mass; 1.7.26,1.7.62ff, I.7.68ff, IV.6.11, also see, size distributions networks; IV.6.14a overlap; II.5.2c, V.chapter 1 persistence (stiffness) parameter; II.5.4 phase separation; Il.fig. 5.3, II.5.2e, Il.fig. 5.4, V.2.3a reptation; IV.figs. 6.36-37 solvent quality; II.5.2b swollen chain; II.5.3, II.5.2b, II.5.9 thermodynamics; II.5.2c viscous flow; IV.6.12 (also see: radius of gyration, adsorption of polymers, colloid stability) polymers, contact angles on ...; III.table A4.2 poly(acrylic acid), adsorption; V.fig. 2.19 charge; V.fig. 2.12 transference number; V.fig. 2.30 viscosity; V.fig. 2.24 poly(acryl amide), adsorption on montmorillonite; Il.fig. 5.39b polyfdiallyl dimethylamm. chloride), conductivity; V.fig. 2.31 poly(ethylene imine) (charge); V.fig. 2.8 poly(ethylene), AFM image; Il.fig. 1.3 poly(ethylene oxide or oxyethylene), adsorption on latex and SiO2; Il.fig. 5.25 adsorption of polyfstyrene sulfonate); Il.fig. 5.35
SUBJECT INDEX
poly(maleic acid) (charge); V.fig. 2.9 polyfmethacrylic acid), adsorption on silver iodide; II.fig. 5.36-37 charge; V.figs. 2.11, 2.12 polyfmethacrylic ester), monolayer; III.3.8e, Ill.figs. 3.94-95 poly(oxyethylene) (PEO), influence on interaction; V.figs. 1.43-45 surface pressue; V. 1.77 poly(phenylene); V.fig. 2.3 poly(styrene), adsorption on silica; II.fig. 5.28, II.fig. 5.30 mixtures with coated silica; IV.fig. 5.64 poly(stvrene) latex, adsorption of C9<|)P^E(m^ (non-ionic); II.fig. 2.32 adsorption of C g ^ ^ E ^ ^ (non-ionic); II.fig. 2.33b adsorption of lactalbumin; V.figs. 3.15-18 adsorption of polyfethylene oxide or oxyethylene); II.fig. 5.25a coagulation; IV.sec.3.13c, V.figs. 1.47-50 rheology; IV.figs. 6.28-29 poly(styrene sulfonate), adsorption on poly(oxymethylene); II.fig. 5.35 brushes and stability; V.fig. 1.36 conductivity; V.fig. 2.32 diffusion; V.fig. 2.17 overlap cone; V.fig. 2.16 stabilization of mica; V.fig. 1.42 viscosity; V.fig. 2.23, V.fig. 2.25-28 poly(styrene co 2-vinyl pyridine), AFM image; II.fig. 1.4 poly(vinyl alcohol) (PVA); V.fig. 3.24 poly(vinyl chloride), flow behaviour; IV.fig. 6.31 poly(vinylpyrrolidone), adsorption on SiO2; Il.fig. 5.22 influence on flocculation; V.fig. 1.46 pores (in surfaces); II.1.6 ad- and desorption of gases; II.figs. 1.32-35 classification into macro-, Il.meso- and micropores; II.1.6a connectivity; II. 1.82 mesopore filling; II. 1.6b micropore filling; 11.1.82, II.1.6c molecular dynamics: II.fig. 1.38
69
70
SUBJECT INDEX
pores (In surfaces) (continued), radius (effective); II. 1.84 size distribution; II. 1.85, II. 1.88 volume; II.1.84ff (also see: porosity, hysteresis, capillary condensation) porosity, mesoporosity; II. 1.6b microporosity; II. 1.6c of plugs; I.6.50ff of surfaces; II.1.6, II.2.67, II.3.161 by mercury penetration; II.1.90, II.1.100 classification; II. 1.6a (also see: adsorption hysteresis, capillary condensation, pores (in surfaces)) porous plugs, permeability; I.6.4f, II. 1.90, II.4.55ff, Il.flg. 4.18, IV.2.32ff, IV.fig. 4.26 electro-osmosis; II.4.3b, Il.flg. 4.6, II.4.5b, Il.flg. 4.18 other electrokinetic phenomena; II.4.5b, Il.flg. 4.18-19, II.4.7, Il.figs. 4.34-35 wetting; HI.5.41, III.5.9 porous surfaces; see pores (in surfaces) potential-determining ions; see surface ions or charge-determining ions potential difference (between adjacent phases); I.table 5.13, II.3.7b, II.3.138 X, 1.5.73-74, II.2.19, II.3.91, II.3.102, II.3.115, II.3.9, Il.figs. 3.38-39, Il.table 3.7, II.3.179, Il.flg. 3.75, Il.table 3.9, II.3.200ff, Il.flg. 3.79, III.2.47, [III.3.7.22], III.4.4a,b, Ill.fig. 4.20, Ill.fig. 4.24 electrokinetic (£"); see electrokinetic potential Galvani; 1.5.5a, I.5.5c, II.3.14, II.3.90, II.3.119ff, Il.flg. 3.38, II.3.138 liquid junction (diffusion); I.5.5d, I.fig. 5.12, 1.6.7b real; 1.5.75, II.3.121ff, Il.table 3.7 Volta; 1.5.5a, II.3.119ff, Il.flg. 3.38, Il.figs. 3.74-75, II.3.179, III.3.7f, Ill.fig. 3.75, Ill.fig. 3.76, Ill.fig. 3.85, Ill.fig. 3.88, Ill.fig. 4.14 (also see: suspension effect) potential of a force; 1.4.3b electric; 1.4.12, 1.5.3, I.5.7ff, 1.5.10, I.fig. 5.1, II.3.3 interfacial; 1.5.5, II.3.6b in diffuse layer, Stern layer etc.; see there of mean force vs. mean potential; I.4.3c, 1.5.18, I.5.24ff, I.5.27ff, II.3.51ff, IV.5.2b potentiometric titration (of colloids); see colloid titration pouring (foaming); V.7.13 powder technology; II.5.97 powders (wetting); III.5.4b, III.5.9 Poynting vector; 1.7.5, 1.7.10, 1.7.97
SUBJECT INDEX
precursor film; III.5.8, Ill.fig. 3.35 preparation of colloids; IV.chapter 2 kinetics; IV.2.2C preparation of emulsions; V.8.2 pressure, two-dimensional; see surface pressure pressure tensor; III.2.3, IV.6.1, V.6.3b prefixes (table); Lapp. 2 primary minimum; see colloids, interaction primary structure (proteins); V.3.3 primitive (liquid model); I.5.1(def.) in conduction; 1.6.79 in diffusion; 1.6.56 in hydrodynamics; 1.6. Iff in solvation; 1.5.3b principal axes (in optics); 1.7.98 principal radii of curvature; see curvature, radius of probability; 1.3.1,1.3.2d, 1.3.3 probability distributions; I.3.2d, I.3.7b, IV.app. 1 process; I.2.3(def.) endothermic; 1.2.8 exothermic; 1.2.8 isobaric; 1.2.3 isochoric; 1.2.3 isosteric; 1.2.3 isothermal; 1.2.3 natural (or irreversible); 1.2.4,1.2.8 reversible; 1.2.3,1.2.21 spontaneous; 1.2.4, 1.2.8 stochastic; 1.6.3 (also see: transport) protection, of colloids against aggregation; 1.1.2(intr.), 1.1.27 of colloids against aggregation by (bio-) polymers; V.chapter 1 see further; colloid stability proteins; 1.1.23 conformation; V.3.2a, 2b relaxation at interfaces; V.3.3b structure changes upon adsorption; V.3.4 structures in solution; V.3.2, V.table 3.3 proton acceptor; 1.5.65 proton donor; 1.5.65
71
72
SUBJECT INDEX
Prussian blue; IV.2.2 PSA = polarizer-sample-analyzer pseudoplastic behaviour (rheology); IV.6.3a pullulan (radius of gyration); V.fig. 2.14 pulmonary surfactant = lung surfactant purity criteria (of interfaces); III. 1.7, 111.1.14c PY = Percus-Yevick p.z.c. = point of zero charge QELS = quasi-elastic (light) scattering; see electromagnetic radiation, scattering quadrupole moment; 1.4.19,1.5.42 quartz; see silica, etc. quasi-chemical approximation (in statistical thermodynamics); I.3.8e quasi-elastic (light) scattering (QELS); see electromagnetic radiation, scattering quaternary structure (proteins); V.3.3ff quenched (polyelectrolytes); V.2.2 quenchers; III.3.165 radial distribution functions; see distribution function radiant intensity; 1.7.5 radiation; see electromagnetic radiation radius, Guinier; IV.2.45, IV.A 1.4 hydraulic; 1.6.50 hydrodynamic (viscometric); 1.7.51; IV.6.9, [IV.6.9.2], IV.6.13, IV.A1.4 (of) gyration; 1.7.57, II.5.4, II.5.6ff, IV.6.11, V.2.3, V.fig. 2.14 ionic; I.table 5.4,1.fig. 5.7 Raman scattering; see electromagnetic radiation scattering Raman spectroscopy; 1.7.12, III.3.7c.ii Randies circuit; Il.fig. 3.31 random coil (intr.); II.5.3 random flight or random walk; 1.3.34, 1.6.3, II.5.3ff, Il.fig. 5.2, II.5.24, IV.4.2 random phase approximation (interaction); IV.5.50ff, IV.fig. 5.30 random sequential adsorption; V.3.16ff Raoult's law for vapour pressure lowering; 1.2.74, II. 1.70 rate of coagulation; IV.4.3 rate of strain; see strain tensor, [IV.6.1.5] Rayleigh-Brillouin scattering; see electromagnetic radiation scattering Rayleigh-Debye (-Gans) scattering; I.7.8d, 1.7.67 Rayleigh instability; III.5.11d, HI.fig. 5.47 Rayleigh line; 1.7.44 Rayleigh ratio; [I.7.7.6](def.), 1.7.3 (table) recipes for sol preparation; IV.2.4
SUBJECT INDEX
red shift (of spectra); 1.7.19 reference electrode; 1.5.82 reference state; see state, standard reflection, multiple; 1.7.80, II. 1.18 total; 1.7.74 reflection angle; 1.7.72 reflection at interfaces; 1.7.10a, Ill.fig. 3.57 reflection coefficient; 1.7.73, II.2.50 reflection electron spectroscopy; see scanning electron spectroscopy reflectometry; 1.7.10b, II.2.5c, Il.figs. 2.15-16, II.5.64, III.2.47, V.figs. 6.4-6.5 refraction angle = transmission angle refraction by interfaces; 1.7.10a refractive index; 1.7.12, 1.7.14 complex; 1.7.2c, 1.7.61,1.7.98 regular solutions; 1.2.18c, II.2.30, II.5.8 regulation (of double layers), see colloids, interactions relaxation (time); I.4.4e, 1.6.6c, II.3.13, II.4. lOff, II.4.6c, II.4.8, IV.4.4, IV.table 4.3 adsorbed proteins; V.3.3b colloid Interaction; IV.4.4 Debye; 1.6.73 dielectric; see dielectric relaxation double layers; see there (in) external fields; IV.4.5 Maxwell (-Wagner); 1.6.84, II.3.219, II.fig. 3.89, II.4.111, IV.4.23 mechanical; IV.6.4 of interfaces (electric); 1.5.5b in Langmuir monolayers; III.3.6h, III.figs. 3.46-47 retardation (in ionic conduction); I.6.6b, 1.6.6c thermodynamic; 1.2.3 (also see: diffusion, rotational, double layer, relaxation) reptation; IV.6.69ff, IV.flgs. 6.36-37 repulsion, electric; 1.1.2Iff osmotic; 1.2.72 (see further: colloids, interaction) residence time, and adsorption; II.1.46ff, [II.2.4.1] and hydration; 1.5.53 resonance band = absorption band resonance (electric); I.4.4e
73
74
SUBJECT INDEX
resonance frequency; 1.4.34 resonators; 1.7.4a respiratory distress syndrome (RDS); III.3.238, V.6.87-88 retardation (of dispersion forces); see Van der Waals forces retention volume; see chromatography reversible, reversibility (in thermodynamic sense); 1.2.3,1.2.9,1.2.8 colloids = colloids, lyophilic interfaces (in electrical sense); 1.5.5b (also see: process; for adsorption reversibility, see (adsorption) hysteresis) Reynolds eq. (film thinning); see Stephan-Reynolds eq. Reynolds limit (wave damping); Ill.fig. 3.44, III.3.117ff Reynolds number; I.6.4b, I.table 6.2 rheology (general); IV.chapter 6 rheology; III.3.6b, IV. 1.3 (and) colloid interaction; IV.2.41, IV.3.144, IV.6.13 compliance; IV.6.23ff concentrated dispersions; IV.6.10 creep compliance; IV.6.24 creep; IV.6.6b, IV.flg. 6.12 definition; IV.6.1 descriptive; IV.6.3 dilute sols; IV.6.9 distribution functions; IV.3.144 electroviscous effects; IV.6.9b emulsions; V.8.15ff foams; V.7.5a (and) fractals; IV.6.13 fracture; IV.6.5, IV.flgs. 6.8-6.9 (of) gels; IV.6.14 instrumentation; see measurements Kelvin element = Voigt element Maxwell element; IV.6.16, IV.fig. 6.11, IV.6.20-21 Maxwell modulus; IV.6.16 measurements; IV.6.6 constant strain rate; IV.6.6c creep; IV.6.6b dynamic; see oscillatory oscillatory; IV.6.6d stress relaxation; IV.6.6a also see, viscometers overshoot; IV.6.6c
SUBJECT INDEX
rheology (continued), particle networks; IV.6.14b polymer networks; IV.6.14a polymer solutions; IV.6.11, IV.6.12 principles; IV.6.1 quantities; IV.6.2 structure; IV.6.8 time scale effects; IV.6.4 Voigt element; IV.flg. 6.13 yield; IV.6.5 yield stress; IV.flg. 3.73, IV.6.3a yield value = yield stress also see: interfacial rheology, stress, strain rheopexy = antithixothropy ribonuclease; V.fig. 3.3, V.fig. 3.5 adsorption; V.fig. 3.23 rigid (films); V.6.48 ring trough (interfacial rheology); Ill.fig. 3.71 ringing gels; IV.2.41 RIS = rotational isomeric state RNase = ribonuclease Ross Miles test (foams); V.7.13ff rotating molecule; I.3.5e rotational correlation time; see correlation time rotational isomeric states (RIS); V.4.38 rotational diffusion (coefficient); see diffusion Rouse-Zimm theory (polym.); IV.6.64-65. Rowlinson-Widom equation (for surface tension); [III.2.5.40] RSA = random sequential adsorption RSD = respiratory distress syndrome rubber, adsorption on carbon black; Il.fig. 5.31 rupture (of films); see films, stability ruthenium dioxide, double layer; Il.fig. 3.56, Il.fig. 3.59 point of zero charge; Il.app. 3b XPS (= ESCA) spectrum; Il.fig. 1.5 rutile; see titanium dioxide saddle splay modulus = Gauss modulus Sackur-Tetrode equation; [1.3.1.9], 1.5.35, [II1.2.9.12] (for surfaces), III.3.37 salt-sieving; 1.1.1. 1.1.3, 1.1.21 ff, II.3.28. 11.3.223, 11.4.56. IV. 1.6
75
76
SUBJECT INDEX
salting-out; 1.5.71 sal ting-in; 1.5.71 Sand equation; [1.6.5.23], I.fig. 6.15b SAMS = self assembled monolayers; III.3.240 SANS = small angle neutron scattering saponine (films); V.fig. 6.27 Saxen's rule (electrokinetics); 1.6.17, II.4.2 SAXS = small angle X-ray scattering SCAF = self-consistent anisotropic field scaling theory; II.5.11, II.5.4c, III.3.8e,f scanning electron microscopy (SEM); 1.7.11b, II.fig. 1.1 scanning, optical; III.3.7c.iv, III.3.7d scanning probe microscopy (SPM); Ill.table 3.5 (includes STM and AFM), III.3.7d scanning transmission electron microscopy (STEM); 1.7.11b scanning tunnelling microscope (STM); 1.7.90, II. 1.12, III.3.7d Scatchard plot; II. 1.48 scattering, length density; 1.7.70, II.5.66 length; 1.7.70, II.5.66 plane; I.7.27(def.) from surfaces; III.1.10 wave vector; I.7.27(def.), III. 1.54 scattering of, neutrons; see neutron scattering, small angle neutron scattering (SANS) radiation; see electromagnetic radiation X-rays; see X-ray scattering Schiller layers; IV.2.40 Schottky defects; II.3.173 Schulze-Hardy rule (for coagulation of colloids); 1.5.67, 1.6.83, II.3.129ff, IV. 1.11, IV.3.9f SCF = self-consistent field Schrodinger equation; 1.3.1, 1.3.11,1.3.20ff, V.I.7 screening (of charges); 1.5.11, also see: double layer charge, capacitance, etc. Searle viscometers/rheometers; IV.6.7b second harmonics, generation; II.2.55, III.3.7c.v, Ill.figs. 3.64-65 second central moment; 1.3.35, IV.app. 1 Second Law of thermodynamics; see thermodynamics Second Postulate of statistical thermodynamics; see statistical thermodynamics second virial coefficient; see virial coefficient secondary ion mass spectroscopy (SIMS): 1.7.lla. I.table 7.4. II.1.15, II.fig. 1.6
SUBJECT INDEX
secondary minimum; see colloids, interaction secondary structure (proteins); V.3.3 sediment, sedimentation; 1.1.2, 1.1.22, I.fig. 1.14, 1.6.48, IV. 1.2, IV.2.40ff, IV.2.3d, V.8.3d, V.8.23, V.fig. 8.24 equilibrium; IV.fig. 5.11, V.8.75 groupwise; V.8.77 hindered; V.8.76 sedimentation coefficient; IV.2.51 sedimentation current; II.4.24 sedimentation-diffusion equilibrium; IV.2.52ff sedimentation field flow fractionatin (FFF); IV.2.61ff sedimentation potential (gradient); Il.table 4.1, 11.4.6-7, II.4.3c sedimentation profiles; IV.2.3d, V.8.80 seeding, seeds (in nucleation); 1.2.100, IV.2.2f segment weighting factors; II.5.37ff selection rules, infrared; I.table 7.5 Raman; I.table 7.6 self-assembly; see III.chapter 3, V.chapter 4 self-avoiding walk; II.5.6 self-consistent anisotropic field (SCAF); V.4.39 self-consistent field (SCF) theory; II.5.29, II.5.5, V.I.4, V.appendix 1 association colloids; V.chapter 4 polymers; II.5.29, II.5.5, V.I.4 self-diffusion; see diffusion self Gibbs energy; see Gibbs energy self-similarity, in Brownian motion; 1.6.18 in fractal structures; IV.6.7Iff in scaling theory; II.5.34 SEM = scanning electron microscopy semiconductors; 11.3. lOe double layer; II.3.10e, II.figs. 3.60-72 intrinsic; II.3.170 n-type and p-type; II.3.173 semidilute (polymer) solution; II.fig. 5.3, II.5.2d settling ~ sedimentation SER = surface enhanced Raman spectroscopy SF = Scheutjens-Fleer (polymer adsorption theory) SFG = sum frequency generation
77
78
SUBJECT INDEX
SFM = scanning force microscopy, (including scanning, probe microscopy, scanning tunnelling microscopy); IV.3.12c, IV.table 3.5, IV.3.58 also see AFM = atomic force microscopy shaking (foam formation); V.7.13 shear rate; 1.6.32, IV.fig. 6.2, IV.fig. 6.3, IV.6.2, IV.table 6.1 shear stress; see stress shear thickening, thinning; III.3.87, IV.fig. 6.5, IV.6.3a SHG = second harmonic generation Shinoda cut (microemulsions); V.5.42, V.flg. 5.22b), V.5.46 silica, silicium dioxide, adsorption of poly(oxyethylene); II.fig. 5.25b adsorption of poly(sryrene); Il.fig. 5.28, II.fig. 5.30 adsorption of poly(vinyl pyrrolidone); Il.fig. 5.22 crystals; fig. IV. 14 point of zero charge; Il.app. 3b Aerosil, adsorption of Cg^P^g^E^y) (non-ionic); Il.fig. 2.33a adsorption of various organic substances from carbon tetrachloride; Il.fig. 2.27 adsorption of water vapour; Il.fig. 1.28 Cab-o-Sil, adsorption of water vapour; Il.fig. 1.26 adsorption of nitrogen, Il.fig. 1.26 double layer; II.table 3.5 hydrophilic, adsorption of C12E5 (non-ionic); Il.figs. 2.30-31 immersion, wetting; II.table 1.3 Ludox, adsorption of C12E6 (non-ionic); Il.fig. 2.31 oxidized wafers, adsorption of C12E6 (non-ionic); Il.fig. 2.31 precipitated, adsorption of nitrogen; Il.fig. 1.34d double layer; Il.figs. 3.64-65, Il.table 3.8 pyrogenic, double layer; Il.fig. 3.65 quartz, double layer; Il.fig. 3.65 wetting by water; III.5.3c
SUBJECT INDEX
79
silica sols; IV.fig. 2.1a, IV.2.4, IV.fig. 2.8, IV.3.13b, IV.fig. Al.l Ludox; IV.figs. 3.71-72 mobility; IV.fig. 3.68, IV.fig. 3.72, IV.fig. 3.74 preparation; IV.2.4a stability and structure; II.3.161-2, IV.3.13b, IV.fig. 5.13, IV.fig. 5.17, IV.figs. 5.24-25, IV.fig. 5.35, IV.fig. 5.62 Stober; IV.2.63 silicium dioxide-zirconium dioxide catalyst, SIMS spectrum; Il.fig. 1.6 silver bromide, point of zero charge; Il.app. 3c silver iodide, adsorption of alcohols; Il.flgs. 3.77-79, Il.table 3.9 adsorption of dextrane; Il.fig. 5.26b, II.5.80ff, Il.fig. 5.29 adsorption of polyfmethacrylic acid); II.figs. 5.36-37 adsorption of tetraalkylammonium salts; II.figs. 3.80-81 double layer; II.3.8, Il.fig. 3.28, Il.fig. 3.32, Il.table 3.6, II.3.10a, Il.flgs. 3.40-46, II.3.112, Il.flgs. 3.52-53, Il.fig. 3.56, Il.table 3.8, II.3.202ff, Il.flgs. 3.77-81, Il.table 3.9 electrokinetic charge; Il.fig. 4.13 electrosorption; II 3.12d. Il.fig. 3.77-81, Il.table 3.9 negative adsorption of ions; Il.fig. 3.40 point of zero charge; II.3.1 lOff, Il.fig. 3.36, Il.flgs. 3.41-43, Il.flgs. 3.77-80, Il.app. 3c, Il.fig. 5.37 relaxation of double layers; IV.fig. 4.9 sols;IV.2.16 site binding (adsorption); II. 1.47-48, II.3.6e, II.3.159, Il.fig. 3.63 SIMS = secondary ion mass spectroscopy single ionic activities; 1.5. l b (also see: activity coefficient) size distributions and averages; IV. 1.9, IV.2.2d, IV.2.45ff, IV.2.3f, IV.2.61ff, IV.app. 1, V.8.4, V.8.1e, V.table 8.1, V.8.66 relative dispersity; IV.app. 1 self-sharpening; IV.2.17ff viscometric; IV.6.63 sky (blue colour); 1.3.34, 1.7.25 slip plane, slip process; 1.5.75, II.4.1b, Il.fig. 4.3 interpretation; II.4.4, V.2.5a sludges; 1.1.23 small-angle neutron scattering (SANS); 1.7.9b. IV.fig. 5.25, IV.fig. 5.31. IV.fig. 5.33. IV.fig. 5.36, V.5.3d, V.fig. 5.18, V.fig. 5.19 small-angle X-ray scattering (SAXS); 1.7.9a
80
SUBJECT INDEX
smectite, wetting; II.table 1.3, II.3.165 smoke; 1.1.6 Smoluchowski eq. (electroviscous effect); [IV.6.9.15] Smoluchowski's theorem (electrokinetics); II.4.21-22 Smoluchowski's theory (coagulation); IV.4.3a Snell's law; 1.7.11, 1.7.72 soap bubbles; I.fig. 1.4, I.figs. 1.9-10 soap films; see films, liquid sodium dodecylsulphate; see surfactants, anionic sodium laurylsulphate = sodium dodecylsulphate soft depletion; V . l . l l h soils (permeation in); 1.1.2, 1.1.3, I.1.22ff, 1.1.28 soil structure; IV.3.184 sol; I.1.5(def.) ageing; 1.2.99, IV.fig. 2.8 colour; I.7.60ff, IV.2.39ff preparations; IV.2.4 sol-gel processing; IV.2.37 solar energy conversion; II.3.223 solid surfaces and interfaces, characterization; II. 1.2 solid-liquid; II.2.2, Il.figs. 2.4-7 thermodynamics; 1.2.24 (for adsorption, double layers etc. see there; also see, interfacial tension of solid surfaces) solubility, of colloids; IV.2.2e, IV.fig. 2.7 of gas in liquid; 1.2.20b of liquid in liquid; 1.2.20c of monomers in microemulsions; V.5.2f of small drops and particles; 1.2.23c of solid in liquid; 1.2.20c of surfactants in microemlusions; V.chapter 5 solubilization; IV. 1.5, V.4.9b solubility parameter (Hildebrand); 1.4.47 solutions, principles, (ideally) dilute; 1.2.17c, 1.2.20 non-ideal; 1.2.18 also see regular solutions solvation; 1.2.58, 1.4.42, I.4.5c, 1.5.3
SUBJECT INDEX
solvent, quality of; 1.1.7, I.1.25ff, I.fig. 1.17, 1.1.30, II.5.2, Il.fig. 5.24 solvent structure; 1.5.3, 1.5.4 near surfaces; see distribution functions of liquids near surfaces (see; interactions, solvent structure-mediated) space charge (density); 1.5.9, Il.fig. 3.70 speciation; I.5.2(def.) specific binding, specifically bound charge; 1.5.3; see further, double layer, Stern specific vs. generic (properties, phenomena); 1.5.67, II.3.6 specific adsorption; I.5.6d, I.5.104ff, II.3.6, Il.fig. 3.20b, II.c, II.3.6d, II.3.6e, IV.3.9i criteria for absence or presence; 1.5.102, II.3.108ff, II.3.132 first and second kind; II.3.64 site binding models; II.3.6e speckle pattern; IV.2.46 spectral density; 1.7.34 (intr.) spectroscopy, of adsorbed proteins; V.3.4a of surfaces; 1.7.11,1.table 7.4, II.2.7ff, Ill.table 3.5, III.3.7c also see, microscopies spectrum analyzer; 1.7.37 (intr.) spin-echo techniques; 1.7.102 spin (electronic); 1.7.16,1.7.13 spin labels; 1.7.13 spin-lattice relaxation; 1.7.96 spin (nuclear); 1.7.16,1.7.13 spinning drop; see interfacial tension, measurement spinodal; 1.2.68, II.5.12, Il.fig. 5.4 decomposition; IV.2.8, IV.figs. 2.3-4, IV.5.65ff, IV.fig. 5.41, IV.fig. 5.62 SPM = scanning probe microscopy spin quantum number; 1.7.16 spin-spin relaxation; 1.7.96 spontaneous; see process spontaneous curvature; III.1.15, HI.4.7 spreading; 1.1.8, III.3.2 rate of; III.3.12 spreading coefficient = spreading tension spreading parameter = spreading tension spreading tension; III.3.8, III.5.4, III.5.6, [III.5.1.1 ], III.5.15ff spring (and dashpot); III.fig.3.50, IV.6.20ff see further; (interfacial) rheology, Maxwell element and Kelvin (Voigt) element sputtering; II. 1.110
81
82
SUBJECT INDEX
square gradient (across interfaces); [III.2.5.28], [III.2.5.30], III.2.28ff, III.2.36, V.I.7, V.1.4e square gradient method (polymer adsorption); II.5.33ff, V.1.4e stability, stabilization, thermodynamics; 1.2.7, 1.2.19 (see colloids, stability of, emulsions, stability; steric stabilization, state) stability ratio; IV.figs. 3.65-67, IV.fig. 3.71, IV.4.17, IV.fig. 4.14, IV.fig. 4.21 stagnant layer (electrokinetics); II.2.15, II.4.1b, II.4.4, II.4.128ff, V.2.5a standard deviations; I.3.7a, I.6.19ff, [II. 1.5.11], IV.5.33 starch; 1.1.2 Stark effect (for spectral lines); 1.7.16 state (def.), metastable; 1.2.7 molecular; 1.2.3, 1.3.2 stable; 1.2.7 standard; 1.2.4 thermodynamic; 1.2.3 (also see: equation of, function of) state variables; 1.2.3 stationary state; 1.6.8, 1.6.13, 1.6.15, II.4.2, II.4.6 statistical chain element; I.3.5f, II.5.5 statistical mechanics; see statistical thermodynamics statistical thermodynamics; I chapter 3 classical; 1.3.9 postulates; 1.3.2,1.3.Id (also see: adsorption isotherm, Fermi-Dirac, Maxwell-Boltzmann, interfacial tension: interpretation, self-consistent field theories and the various applications) Stefan-Osrwald rule (for surface tensions); [III.2.11.25] Stephan-Reynolds eq. (for film thinning); [V.6.4.2 and 3] STEM = scanning transmission electron microscopy step-weighted lattice walk; 1.6.28, II.5.5 steric stabilization; see colloid stability (by polymers) Stern layer; see double layer stiffness parameter = persistence parameter, see polymers in solution Stirling approximation; [1.3.6.5] STM = scanning tunnelling microscope Stober sols (silica); IV.2.63ff stochastic; see processes, forces Stockmayer-Fixman equation (vise); [V.2.4.1 1] Stokes' law; [1.6.4.30], 1.6.56, 11.4.18, II.5.62, V.8.74ff Stokes limit (wave damping); III.fig. 3.44, III.3.117ff
SUBJECT INDEX
stopping mechanism (in micelle formation); V.4.8ff strain; III.3.6b, IV.6.2, IV.6.1 strain rate; III.3.85ff strain tensor; [IV.6.1.4) strain energy release; IV.fig. 6.9 strain hardening; IV.6.12 strain (rate) thinning; IV.6.12 stratification (in films); see films, liquid streaming, or flow birefringence; 1.7.97,1.7.100 streaming current; I.6.16ff, Il.table 4.4, II.4.7, II.4.3d, Il.fig. 4.8, II.4.55 streaming potential; I.6.16ff, Il.flg. 3.78, Il.table 4.4, II.4.7, II.4.3d, Il.fig. 4.8, II.4.5b, Il.fig. 4.30, Il.fig. 4.35, II.5.63 stress; IV.fig. 6.4 normal; 1.6.7, III.3.6b, IV.6.1, IV.fig. 6.1 shear; 1.6.7, 1.6.1 Iff, I.6.4a, III.3.6b, IV.fig. 6.1, V.8.2 stress overshoot; V.fig. 7.11 stress relaxation; IV.figs. 6.9-10, IV.6.6a modulus; IV.6.20-21 spectrum; IV.6.20-21 stress tensor; I.6.6ff, [III.3.6.1-2], [IV.6.1.1] also see: interfacial rheology stretching of solid surfaces; 1.2.103 structural forces; 1.4.2, II. 1.95 structure breaking; 1.5.38, I.5.3d structure factor; 1.3.67, 1.7.64, Lapp, l i e , IV.3.143, IV.5.3, IV.5.21, IV.figs. 5.5-5.6, IV.figs 5.16-18, IV.figs. 5.21-23, IV.5.49, [IV.5.6.8], IV.fig. 5.30, IV.figs. 5.33-36, V.2.26, structure of colloids; IV.chapter 5 rheology; IV.5.1, IV.6.8-10 structure of water; see water, structure structure promotion; 1.5.38, 1.5.3d substantial derivative; 1.6.5 subsystems (statistical thermodynamics); 1.3.5, 1.3.6 dependent; 1.3.5, 1.3.20, 1.3.8 independent; 1.3.5,1.3.20,1.3.6 sulphur sol (preparation); IV.2.4b sum frequency generation; III.3.7c.v superadditivity (in coagulation); IV.3.9k supercooling; 1.2.23d supermolecular fluids; 1.7.63 supersaturation: 1.2.23d. IV.2.9ff, V.7.8
83
84
SUBJECT INDEX
supersaturatlon ratio; [IV.2.2.14], IV.2.15 surface; I.1.3(intr.), acidity/basicity (dry surfaces); II. 1.18 characterization in general; II.1.2, Il.table 1.1, II.2.2a external vs. internal; II.1.6a heterogeneity; I.1.18(intr.), 1.5.106, II. 1.5, II.1.7, II.2.29, II.3.83 patchwise vs. random; II.1.103ff 'high' vs. 'low' energy; II. 1.35 hydrophilicity/hydrophobicity; II.1.19, II.1.35, II.2.7, II.2.87, II.3.130 imaging techniques; 1.7.lib modulus; see interfacial rheology porosity; see porosity of surfaces reconstruction; II. 1.8 scattering by; 1.7.10c spectroscopic characterization; 1.7.11, II.1.9ff, Il.figs. 1.1-6 (also see: interface, especially for 'wet' surfaces, equations of state) surface (or interfacial) area, (molecular, in monolayers), III.3.15, III.3.24, see further the ;r(A) isotherms in Ill.chapter 3, Ill.fig. 3.16, Ill.fig. 3.82, III.3.84, V.5.3h(ii), V.fig. 5.24 specific; I.1.18(def.), 1.1.20, I.6.4f, IH.Sf, II.2.67, II.2.73, II.3.7e, II.3.127, II.3.131ff, IV.fig. 2.8, IV.fig. 2.10, IV.2.33ff, IV.2.3c surface charge (density); 1.1.20, 1.5.3, 1.5.9, 1.5.6, II.3.3, II.3.21, Il.fig. 3.18, Il.fig. 3.28, Il.fig. 3.41, Il.fig. 3.52, Il.figs. 3.56-59, Il.figs. 3.63-65, Il.figs. 3.69-70, Il.fig. 3.77, Il.fig. 3,.80, Il.fig. 3.82-83, IV.fig. 3.75 determination; I.5.6e, II.3.7a dipolar contribution; II.3.126 discrete nature; II.3.46, II.3.6e for clay mineral-type particles; 11.3.lOd for polarized interfaces; I.5.6c, II.3.10b, II.3.163 for relaxed (reversible) interfaces; I.5.6a, II.3.10a, II.3,10c, II.3.163 formation thermodynamics; II.3.1 lOff Gouy-Stern layer; Il.figs. 3.23-25 relation to D and E; I.4.53ff site-binding models; II.3.6e (from) statistical theories; Il.fig. 3.18 (see double layer, diffuse, charge) surface concentration; see interfacial concentration surface conduction and conductivity; 1.5.4, I.6.6d, II.3.208, II.4.1, II.4.28, II.4.3f, Il.fig. 4.9, 11,4.91, Il.table 4.3 behind slip plane; II.4.32ff, II.4.37ff, II.4.67, II.4.94, II.4.6f, Il.table 4.3
SUBJECT INDEX
surface conduction and conductivity (continued], in diffuse double layer; 11.4.32ff, II.fig. 4.10 Bikerman equations; [II.4.3.59]ff influence on ^-potential; II.4.6e, II.figs. 4.29-31, II.4.6f, II.table 4.3 measurements; II.4.5c surface correlation length; II.2.10 surface diffusion (coefficient); I.6.69ff, II.2.14, II.2.29 surface excess; see interfacial excess surface energy; see energy surface enhanced Raman spectroscopy; III.3.7c.ii surface equation of state; see : equation of state, two-dimensional surface force apparatus; 1.4.8, IV.3.12b surface forces versus body forces; I.1.8ff, 1.4.2 surface ions; 1.5.89 surface modification; II. 1.110, II.2.88, II.5.97, IV.2.2i surface porosity; see pores (in surfaces), porosity of surfaces surface potential; 1.5.5 ('surface potentials' of monolayers = Volta potentials; see under potentials) surface pressure; I.1.16(def.), 1.3.17, I.3.32ff, I.3.47ff, I.3.51ff, II.1.3, II.1.3b, II.1.28, II.1.51, Il.fig. 1.15, II.1.59ff, Il.app. 1, II.3.14, II.3.140, III.chapters 3, 4, V.fig. 8.2, V.8.8ff (also see: equation of state, two-dimensional; for measurement, see film balance) surface pressure isotherms; III.chapter 3 surface rheology; see interfacial rheology Marangoni effect; I.1.2(intr.), 1.1.17,1.6.43ff surface roughness, and Van der Waals forces; 1.4.68 in colloid interaction; IV.3.82ff in electrokinetics; II.4.39 in optics; 1.7.10 surface states (semiconductors); II.3.172ff, II.3.176 surface tension; see interfacial tension surface of tension; 1.2.93, V.6.5 surface undulations; 1.7.77, 1.7.10c surface wave; 1.7.75 surface work; see interfacial work surfactants; I.1.4(def.), I.1.6(intr.), I.1.23ff, II.figs. 1.1.15-16, II1.4.6a, III.table 4.4 anionic; I.1.23ff, I.fig. 1.15, III.4.6d adsorption of; II.fig. 3.Id, also see monolayers
85
86
SUBJECT INDEX
surfactants, anionic (continued), in films; V. figs. 6.23-24, V.fig. 6.28, V.fig. 6.32, V.flg. 6.34, V.6.64ff, V.fig. 6.36, V.fig. 6.44 in mlcroemulsions; V.5.60 monolayers; Ill.fig. 1.30, III.4.6d, Ill.fig. 4.36 association behaviour etc.; see V.chapter 4 bending moduli of monolayers; III.tables 1.6 and 7 cationic; 1.1.23, III.4.6d adsorption, see monolayers interfaclal tension (dynamic and rheological); Ill.figs. 3.43-44, Ill.fig. 4.17 monolayers; III.4.6d, Ill.fig. 4.35, IH.table 4.6, Ill.fig. 4.38 coalescence; V.8.83ff emulsifiers; V.8.1b, V.8.2c, V.fig. 8.13, V.8.15 interfacial tension (dynamic and rheological); Ill.fig. 1.31, III.4.6 monolayers; Ill.fig. 3.65, III.4.6 non-ionic; 1.1.23 adsorption of; see poly(styrene) latex, silica, and monolayers cloud point; Ill.fig. 4.29 in emulsions; V.fig. 8.13, V.fig. 8.14, V.8.49ff in films; V.fig. 6.26, V.6.66ff, V.fig. 6.40 in microemulsions; V.chapter 5 interfacial and surface tension (static, dynamic and rheological); Ill.fig. 1.31, Ill.table 4.5 monolayers; III.4.6c, Ill.figs. 4.30-34 packing parameter; V.4.1d, [V.4.1.4] surroundings (in thermodynamic sense); 1.2.2, 1.3.2 susceptibility (electric); 1.4.52 (intr.) suspension; 1.1.2, 1.1.22, I.fig. 1.14 ageing; 1.2.99 suspension effect; I.5.5f, I.fig. 5.15, II.3.105 Svedberg equation (sedimentation); [IV.2.3.23] swelling; V.2.23, V.2.39, V.2.3d, V.4.115ff swollen dilute (polymer solution); II.5.9, II.fig. 5.3 synergism (in coagulation); IV.3.9k system, (in statistical sense); 1.3.la (in thermodynamic sense); 1.2.2 Szyzskowski Isotherm; [III.4.3.14] tactoids; II. 1.80 tails; see adsorption of polymers t-plot; see adsorption
SUBJECT INDEX
tangential stress = shear stress; see stress Tate's law; (III. 1.6.1] Taylor number; 1.6.36 Taylor vortices; fig 1.6.8 Teflon, wetting; II.table 1.3 tensiometers; III. 1.8 tensors; Lapp. 7f TEM = transmission electron microscopy ternary phase diagrams; V.chapter 5 tertiary oil recovery; see enhanced oil recovery tertiary structure (proteins); V.3.3ff tethered chains; V.I. 11 thermal diffusion; 1.6.12, 1.7.44,1.7.48 thermal neutrons; 1.7.25 thermal wavelength; [1.3.5.14] thermodynamic state; 1.2.3 thermodynamics (general); I.chapter 2 First Law; 1.2.4,1.4.3 irreversible; 1.6.2,1.6.5a, I.6.6a, 1.6.7 Second Law; 1.2.8 'Third Law1; 1.2.24 (also see: statistical thermodynamics) thermodynamics of small systems; V.4.2a theta (0) solvent; 1.6.28 thickness of adsorbed layers; see adsorbate thin liquid films; see films thixotropy, thixotropic; 1.1.23, III.3.87, IV.1.3, IV.6.14, IV.fig. 6.7 tilt angle; Ill.flg. 3.60 tilted plate; III.5.4d, Ill.flg. 5.24 time correlation functions; see correlation functions TIRF = total internal reflection fluorescence TIRM = total internal reflection microscopy titanium dioxide; Il.table 1.3, II.table 3.6, IV.3.13a anatase, double layer; II.3.94 point of zero charge; Il.table 3.5, Il.app. 3b rutile, adsorption of anionic surfactants; II.figs. 3.82-83 adsorption of water vapour; II.fig. 1.9 double layer; II.figs. 3.58-60, II.fig. 3.63, Il.table 3.8, Il.figs. 3.82-83 electrokinetic charge; II.fig. 4.13
87
88
SUBJECT INDEX
titanium dioxide, rutile (continued), mobility; IV.figs. 3.62-64 point of zero charge; II.table 3.5, Il.fig. 3.37, II.fig. 3.82, Il.app. 3b stability ratios; IV.figs. 3.65-67 £ -potential; Il.fig. 3.63 topology (vesicle formation); V.4.7d torque; lV.6.7b total internal reflection fluorescence (TIRF); II.2.54, III.3.7c.iv total internal reflection microscopy (TIRM); IV.3.157 total reflection; 1.7.74, II.2.51, II.2.54 trains; see adsorption of polymers trajectories (of particles); IV.flg. 4.2, IV.flg. 4.20, V.fig. 8.7 transfer in galvanic cells; I.5.5e transfer, of molecules; 1.2.18a (ions to other phases); I.5.3f, 1.5.5 (molecules to other phases); 1.2.20, 1.4.47, 1.6.44 (also see: transport, Il.viscous flow) transference numbers; see transport numbers transmission angle; 1.7.72 transmission coefficient; 1.7.73, II.2.50 transmission electron microscopy (TEM); 1.7.lib, Il.fig. 1.1, V.5.3b cryo-direct imaging; V.5.3b freeze fracture direct imaging; V.5.3b, V.figs. 5.15-17 transport processes; I.chapter 6 (also see: hydrodynamics, diffusion, conduction) transport, linear laws; 1.6.lc, I.table 6.1 of charge; I.table6.1 in double layers; IV.4.4a of heat; I.table 6.1 of mass; 1.6.1a, I.table 6.1 through interfaces; 1.6.44 of momentum; I.6.1b, I.table 6.1,1.6.4a transport numbers; I.6.76ff, [1.6.6.14](def), V.2.5b, V.fig. 2.30 transverse waves; III.3.110, III.fig. 3.43, see further: interfacial rheology trapping (optical); IV.3.157 Traube's rule; 1.4.51 triboelectricity; II.3.187 Trouton number; V.8.37 Trouton ratio; IV.6.10
SUBJECT INDEX
Trouton's rule; III.2.54 (also for surfaces) tunnelling (of electrons); 1.7.90 turbidity; 1.7.41, 1.7.47, V.flg. 8.6 turbulence; I.6.4b, V.8.40ff Tyndall effect; 1.7.26, II.4.45, IV. 1.3 Ubbelohde viscometer; IV.flg. 6.18 ultracentrifuge; IV. 1.13 ultracentrifugatlon, see sedimentation; IV.2.3d ultramlcroscope; 1.7.26, II.fig. 4.14 ultrasonic emulslfication; V.8.33 ultrasonic vibration potential; II.4.7, II.4.3e Ultra Turrax; V.fig. 8.17 uncertainty principle (Heisenberg); 1.3.4, 1.3.58 undulations (of fluid interfaces); III. 1.78 undulation forces (membranes, etc.); V.4.8 unlaxlality; 1.7.97 UPES = UPS = ultraviolet photo-electron speetroscopy; 1.7. l l a UVP = ultrasonic vibration potential vacancy (In semiconductor); II.3.171 valence band (solids); Il.fig. 3.68, H.3.173 Van der Waals interactions; I.chapter 4, IV.3.8a between molecules (general); I.4.10ff, 1.4.4 additivity; I.4.18ff London (or dispersion); 1.4.17, I.4.4d, I.table 4.3, I.4.4e, 1.4.5, 1.4.6, 1.4.7 Debye; 1.4.17, I.4.4c, I.table 4.3, 1.4.41 Keesom; 1.4.17, I.4.4c, I.table 4.3, 1.4.41 retardation; 1.4.17, 1.4.31, I.4.78ff between colloids and macrobodies; 1.4.6, IV.3.9 Hamaker-De Boer; 1.4.6 Lifshits; 1.4.7 measurement (direct); 1.4.8, I.fig. 4.19 macroscopic; see Lifshits microscopic; see Hamaker-De Boer repulsive; 1.4.72, 1.4.78 retardation; 1.4.6c in thin films; II. 1.101 Van der Waals equation of state; [1.2.18.26], [1.3.9.28], [1.4.4.1], III.2.17, [III.2.9.3] (reduced), IV.5.7a Van der Waals loops; 1.3.47, 1.4.17, Il.fig. 1.20, II.1.101, Il.fig. 1.42, IV.5.7a, IV.flg. 5.37
89
90
SUBJECT INDEX
Van der Waals' theory (interfacial tensions); III.2.5, III.2.3.1 (generalized ) Van 't Hoffs law, for boiling point elevation; 1.2.74 for freezing point depression; 1.2.74 for osmotic pressure; l.2.20d. 1.7.50 vapour pressure, lowering; 1.2.74 of small drops; 1.2.23c variance; 1.3.35 vector, vector field etc.; Lapp. 7 velocity correlation function; Lapp, l l a , II.2.14 velocity distribution; 1.6.2Iff, I.6.3c, I.fig. 6.4 velocity persistence length; IV.4.5 vermiculite, wetting; Il.table 1.3 vertical plate, wetting of; III.1.3b end effect correction; III. 1.22 vibration; 1.3.5a, 1.4.44 vibrational spectroscopy; III.table 3.5 virial coefficients; I 2.18d. I.3.8f, I.3.9c second; I.2.18d, I.3.8f, I.3.9b, [1.3.9.12], [1.4.2.8-11], 1.7.51, 1.7.57, IV.table5.1, IV.fig. 5.24, [IV.5.6.2], IV.fig. 5.29, IV.flg. 5.55 two-dimensional; [II. 1.5.24], II. 1.60 virial expansions; I 2.18d. I.3.9b, I.3.9c, I.5.27ff, 1.7.51, [II.1.5.30], IV.5.2e viscoelasticity; I.2.7(intr.), III.3.88, IV.chapter 6, IV.6.1, IV.6.11, IV.6.14, IV.fig. 6.10, IV.flg. 6.12, IV.6.6, IV.fig. 6.15 viscoelectric coefficient; II.4.40 viscoelectric effect; II.4.40 viscometers, capillary; IV.6.7a Couette; IV.6.7b Ostwald; IV.fig. 6.18 rotation(al); I.6.36ff, IV.6.7b Searly; IV.6.7b Ubbelohde; IV.fig. 6.18 viscosity; 1.5.43, I.6.10ff, IV.6.1, IV.6.2 apparent; IV.6.11 definitions; IV.table 6.3 dispersity effect; IV.fig. 6.27 dynamic vs. kinematic; 1.6.11 Einstein; IV.6.9a extensions; IV.table 6.3, IV.table 6.4
SUBJECT INDEX
viscosity (continued), (in) electrokinetics; II.4.4 elongational; IV.6.9 emulsions; V.8.15ff, V.fig. 8.5 examples: I.table 6.3, I.6.4g, IV.figs. 6.28-29, IV.figs. 6.31-32, IV.figs. 6.34-35, IV.fig. 6.38, V.2.4 intrinsic; IV.6.47, IV.fig. 6.24, V.2.4c Newton, definition; IV.table 6.4 polyelectrolytes; V.2.4 polymer solutions; IV.6.11, 6.12 shear vs. elongational; 1.6.8 viscosity-averaged molecular mass; IV.6.63 viscous flow; 1.6.1, 1.6.4, IV.6.1, IV.6.2, IV.fig. 6.5, IV.6.8-13 around spheres; I.6.4e, II.4.6, II.4.8 between parallel plates; I.6.40ff, IV.fig. 6.3 dilational vs. rotational; I.fig. 6.7 due to Marangoni effects; 1.1.17,1.6.44 due to temperature gradients; 1.6.4c fluid-fluid interfaces; I.6.42ff in cylindrical tube; I.6.41ff in porous media; I.6.4f laminar linear; 1.6.4a, I.6.4d turbulent; I.6.4b Volta potential; see potential Volmer adsorption isotherm; see adsorption isotherm volumes, of ions (partial); I.table 5.7 Vonnegut equation (for spinning drops); [III. 1.9.6] vortices; 1.6.4b Vroman effect (protein ads.); V.3.52 Warburg coefficient; II.3.96 Ward-Tordai equation; [1.6.5.36], [II.1.1.15] Washburn equation; [III.5.4.4] III.5.57ff, III.5.84ff waste water treatment; IV.3.184 water, interactions in; I 4.5d. I.4.5e structure; 1.5.3c, 1.5.4, II.2.16 near surfaces; II.2.2c, Il.figs. 2.6-7, II.3.122ff, II.fig. 3.39, II.4.38ff water-air interface, double layer; II.3.10f, Il.fig. 3.78, III.4.4, III.fig. 4.20 reflectivity; III.fig. 3.57
91
92
SUBJECT INDEX
water-air interface (continued), surface relaxation; III.fig. 1.29 surface tension; III.1.12, III.table 1.2 influence of electrolytes; II.3.180, II.fig. 3.73 influence of temperature; III.1.12b, Ill.table 1.3, Ill.table 1.4, Ill.fig. 1.27 simulation; Ill.table 2.2, Ill.figs. 2.12-13 waterglass; IV.2.2 wave damping; see interfacial rheology waves, electromagnetic; I.chapter 7 evanescent; 1.7.75 in a vacuum; 1.7.1 plane; 1.7. l a films; V.8.87 polarization; 1.7. l a spherical; 1.7. l b surface; see surface wave (also see: electromagnetic radiation) wave vector; I.7.4(def.) wave vector transfer = wave vector Weber number; [V.8.2.3], V.fig. 8.12, V.flg. 8.15, [V.8.3.25], V.table 8.4, V.8.89 Wenzel equation; [III.5.5.1] wet foam; V.7.2, V.fig. 7.12 wetting (general); III.chapter 5 wetting; 1.1.2, 1.1.8, 1.1.3, I.fig. 1.13, Il.table 1.3, Ill.fig. 5.7 adhesional; II.2.5, III.5.2 and gas adsorption; II. 1.19, III.5.3, Ill.fig. 5.16 and Van der Waals forces; 1.4.72 complete; III. 1.5, III.5.1, III.5.4 dynamics; III.5.8 enthalpy or heat; [II. 1.3.43], Ill.table 1.3, II.fig. 1.10b, II.2.6, II.2.3d, Il.fig. 2.10, Il.fig. 2.20, III.5.2, III.5.20 entropy; II.2.7 immersional; II.2.5, Il.fig. 2.10, III.5.2 liquids by liquids; III.5.3 microemulsions; V.5.4c molecular dynamics; Ill.fig. 5.36 (and) nucleation; IV.fig. 2.9, IV.2.2f partial; III.1.5, III.5.1, Ill.fig. 5.1, Ill.fig. 5.13. porosity; III.5.9 scales; III.5.5 selective; II.2.88
SUBJECT INDEX
93
wetting (continued), silica by water; III.5.3c surfactant influence; III.5.10 thermodynamics; III.5.2 wetting agents; III.5.86, III.5.10 wetting films; III.5.3 wetting transition; II.l.lOl, Il.fig. 1.41, III.5.14, III.5.30, Ill.fig. 5.14, V.5.60, V.fig. 5.30 Wiegner effect; see suspension effect Wien effect (in electrolytic conductance); 1.6.6c Wilson chamber; 1.2.100 wine tears; 1.1.1, 1.1.2, 1.1.17 wolfram surface, covered with palladium; Il.fig. 1.2 work; 1.2.4 of adhesion; III.2.34, III.2.7 Iff isothermal reversible; 1.2.27 statistical interpretation; 1.3.15, [1.3.3.11] (also see: interfacial work, transfer, potentials) work function; 1.5.75, II.3.114, II.3.174 work hardening or softening; III.3.88 worm-like chain; V.2.27 Wulff relations; IV.2.23 XAFS = X-ray absorption and fluorescence spectroscopy; 1.7.lla XPS = XPES = X-ray photoelectron spectroscopy; 1.7.1 la, I.table 7.4, II. 1.15, Il.fig. 1.5 X-ray reflection and diffraction; III.2.47, Ill.table 3.5, III.3.7b, IV.2.40 X-ray scattering; I.fig. 5.6,1.7.9a of films; V.6.9 yield stress; see rheology (of) emulsions; V.8.17 young foam; V.72 Young and Laplace's law; see capillary pressure Young's law for capillary pressure; see capillary pressure, Young and Laplace Young's law for contact angle; [III. 1.1.7], III.5.1b, [III.5.1.2], III.5.2 Yukawa pair interaction; IV.5.6a, IV.fig. 5.30 Z-average; 1.7.63 Zeeman effect (for spectral lines); 1.7.16 zeolithe; see molecular sieve zero point of charge; see point of zero charge zero point vibrations; 1.3.22, 1.4.29 Zimmplot; I.7.57ff. I.fig. 7.12, I.fig. 7.16 zinc oxide, SEM: Il.fig. 1.1
94
SUBJECT INDEX
Zisman plot; III.fig. 5.42 zone electrophoresis; II.4.131 zwitterionic surfaces; II.3.74 a-helix; V.chapter 3 P -sheet; V.chapter 3 X-parameter (polymers etc.); see Flory-Huggins interaction parameter X&-parameter (polymer ads.); [II.5.4.1], II.chapter 5 ^ r i t ; II.5.40, Il.fig. 5.22 X-potential; see potential f -potential; see electrokinetic potential <9-point, ©-temperature; II.5.6ff, II.5.2b
