Influence of volume-restriction effect on phase separation phenomena in steric dispersions

Influence of Volume-Restriction Effect on Phase Separation Phenomena in Steric Dispersions D. L. C A T E S AND C. S. H I R T Z E L 1 Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244

Received October 17, 1986; accepted February 10, 1987 Steric systems such as paints, inks, milk and other dairy products, and biocolloids such as tissue and blood cells and protein solutions form a significant class of practically and theoretically important dispersions. Stability and phase separation phenomena in steric systems are complex due, in part, to the role of the polymeric macromolecules which are used to impart steric stability (or, instability). In this study, phase separation phenomena in selected steric systems are examined using a statistical mechanical formalism. Specifically, the volume-restriction effect in steric systems, which arises from the depletion of free polymer from the interparticle region due to the loss of configurational entropy, is investigated. The exclusion of the polymer molecules from the region ofinterparticle separation leads to an attractive force which is described by an interparticle pair potential first developed by Asakura and Oosawa (J. Chem. Phys. 22, 1255 (1954); J. Polym. Sci. 33, 183 (1958)). In the case of aqueous systems in which electrolytes are present, a suitable electrostatic interaction is added to the Asakura-Oosawa potential for the volume-restriction interaction. With or without electrostatic and van der Waals interactions, the volume-restriction interaction potential is used in combination with perturbation theory (one statistical mechanical technique) to obtain osmotic pressures and free energies in fluid-like dispersions. Results for the solid phase are obtained using another statistical mechanical technique, a cell model technique developed by Lennard-Jones and Devonshire (Proc. R. Soc. London Ser. A 163, 53 (1937); 165, 1 (1938)). These results, in turn, are used to construct phase diagrams for nonaqueous and aqueous dispersions in terms of polymer concentration. These results are compared with selected experimental and other theoretical results available in the literature. © 1987AcademicPress,Inc. 1. INTRODUCTION Steric systems such as paints, inks, m i l k a n d o t h e r d a i r y products, a n d b i o c o l l o i d s such as tissues a n d b l o o d cells a n d p r o t e i n s o l u t i o n s f o r m a significant class o f practically, a n d theoretically, i m p o r t a n t dispersions. P o l y m e r s are used in a wide variety o f colloidal systems either to i m p r o v e dispersion stability (as in the m a n u f a c t u r e o f paints a n d inks) o r to p r o m o t e particle separations (as in certain w a t e r a n d w a s t e w a t e r t r e a t m e n t processes). Stability a n d p h a s e s e p a r a t i o n p h e n o m e n a in such steric systems are c o m p l e x due, in part, to the role o f the p o l y m e r i c m a c r o m o l e c u l e s w h i c h are used to p r o m o t e stability (or instability). In ' To whom correspondence should be addressed.

this study, p h a s e s e p a r a t i o n p h e n o m e n a in certain steric systems are e x a m i n e d using a statistical m e c h a n i c a l f o r m a l i s m . T h e precise a c t i o n o f the p o l y m e r i c species d e p e n d s o n t h e c o n c e n t r a t i o n o f the p o l y m e r a n d o n the level o f a d s o r p t i o n (or l a c k o f it) o n the surface o f the colloidal particles. Effects are associated with p o l y m e r s a d s o r b e d o n particle surfaces [see, for e x a m p l e , (1, 2)] as well as with u n a d s o r b e d o r free p o l y m e r (3). Exp e r i m e n t a l studies o f such steric systems d a t e b a c k m a n y years a n d a p p e a r frequently in t h e literature because o f t h e i r p r a c t i c a l i m p a c t ; however, theoretical u n d e r s t a n d i n g a n d interp r e t a t i o n s are o n l y recently b e g i n n i n g to emerge. [ F o r a d d i t i o n a l details, see the recent m o n o g r a p h o n p o l y m e r i c stabilization o f col-

404 0021-9797/87 $3.00 Copyright © 1987 by Academic Press, Inc. All fights of reproduction in any form reserved.

Journal of Colloid and Interface Science. Vol. 120, No. 2, December 1987

INFLUENCE OF VOLUME-RESTRICTIONEFFECT loidal systems by Napper (4); also, see (5, 6) and references cited in those.] The theoretical description and analyses of the effect of polymers on colloidal stability and phase separation are considerably more difficult than similar analyses of electrostatically stabilized dispersions, since the thermodynamics of polymer solutions plays an additional, complicating role in the former case and has a substantial impact on the nature of steric action. Of the many possible effects of the polymeric species on the interparticle interaction, at least two are well recognized. The first is generally known as the volume-restriction effect and arises as a result of the depletion of free polymer from the interparticle region due to the loss of eonfigurational entropy (3, 4, 7, 8). The second effect arises from the interactions between free polymers (in solution) and stabilizers adsorbed on the surface of the particles; this effect is often termed the osmotic effect (3, 9, 10). An extensive discussion of the various possible expressions for pair interaction potentials (between the particles) in the presence of polymers is presented in (2, 4) and in the references cited by Hirtzel and Rajagopalan (5). These interaction potentials can be used (together with electrostatic and van der Waals interactions, if appropriate) in statistical mechanical theories to study the effects of steric interactions on, tbr example, equilibrium properties of the dispersion and phase separation phenomena. In this paper, the volume-restriction effect in steric systems is examined. The basic model used to describe the volume-restriction effect was set forth by Asakura and Oosawa (11, 12); they derived an interparticle pair potential based on the exclusion of the polymer molecules from the region of interparticle separation. The resulting potential, based on the attractive force which arises due to this exclusion of polymer, depends on the interparticle separation, particle diameter, polymer diameter, and volume fraction of polymer. (Additional discussion of the form of the interaction pair potential is presented later, in Section 2.) The

405

primary objective of this present study is to quantify theoretically the influence of this volume-restriction on selected equilibrium properties and phase separations in steric dispersions. In the case of aqueous systems in which electrolytes are present, a suitable electrostatic interaction is added to the volumerestriction pair potential (i.e., the pair potential developed by Asakura and Oosawa above). With or without electrostatic and van der Waals interactions, the volume-restriction pair interaction potential is used in combination with perturbation theory to obtain osmotic pressures and free energies in fluid-like dispersions. The results for the solid-like phase are obtained using a cell model theory developed by Lennard-Jones and Devonshire (13, 14). These results, in turn, are used to construct phase diagrams for both aqueous and nonaqueous dispersions in terms of polymer concentration. Selected results of this study are also compared with certain experimental results (15-17) and other theoretical results (17-19) which are available in the literature. Both of the statistical mechanical techniques used in this study (i.e., perturbation theory and the Lennard-Jones cell model theory) are analytical, approximate techniques as opposed to computer experiment techniques which yield exact results; and both techniques were originally developed for the analysis of molecular fluids. Thus, a secondary objective of the present study is to examine the application of these techniques as adapted to selected steric systems and any associated advantages or limitations. The remainder of this paper is structured as follows. Background material, including a brief review of previous investigations relevant to the problem being studied herein and a description of the theoretical basis of the current investigation, is presented in Section 2. Results for volume-restriction effects on phase separations for nonaqueous and for aqueous dispersions are given in Sections 3.1 and 3.2, respectively. Some selected concluding remarks and comments are summarized in Section 4. Journal of Colloid and Interface Science, Vol. 120, No. 2, December 1987

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CATES AND HIRTZEL 2. BACKGROUND

investigate the effects of addition of free polymers to aqueous dispersions of a charge-sta2.1. Stability and Phase Separation bilized latex/hydroxethyl cellulose/water sysPhenomena in Steric Systems: tem. Above a critical concentration of polymer Volume-Restriction Effect (which depends on certain physicochemical As noted in the Introduction, the basic idea parameters), the dispersion was observed to underlying the quantitative description of the separate into a colloid-rich and a colloid-poor volume-restriction effect was first developed phase. The concentration of the particles in by Asakura and Oosawa (11, 12) more than the latter decreases with increasing polymer 30 years ago. However, until only recently, the concentration. Correspondingly, the colloidvolume-restriction effect had received very lit- rich phase becomes progressively more solidfie attention from researchers. [An exception like with addition of polymer. Based upon to this is the study by Sieglaff(20); in that work, some additional experimental studies of this Sieglaff used the volume-restriction effect to same system (i.e., free polymers in an aqueous describe phase separation in systems of mi- dispersion of charge-stabilized latex particles), crogels.] In recent years, the volume-restriction Sperry (16) found that floc phases became effect has received more attention; for exam- more rigid as particle size was increased or as ple, the studies by Vrij (3); Feigin and Napper polymer concentration was increased. The (7, 8), Joanny et al. (9), and Vincent et al. (10) observed floc structures varied from liquid-like cited previously have been concerned with ef- to amorphous to pseudo-crystalline; liquid-like fects of unadsorbed polymer on the stability flocs were observed to occur for the same range of colloidal dispersions. In addition, phase of particle diameter as the liquid-like flocs obseparations have been observed experimen- served by de Hek and Vrij (22). [For additional tally in systems oflyophilic silica particles and discussion or details of these studies, see (5, polystyrene polymers in cyclohexane by de 6, 26).] In a few of the studies mentioned above Hek and Vrij (21, 22). They observed two liquid phases formed during phase separation, [e.g., (15, 22)], the authors have interpreted with one of the liquid phases containing most their (experimental) results--at least, qualiof the silica particles. In a related study, Path- tatively-based on the volume-restriction pair mamanoharan et al. (23) observed similar liq- interaction model of Asakura and Oosawa. [A uid-liquid phase separations, using other recent study by Fleer et al. (27), however, is nonaqueous systems. While these studies sup- concerned with an alternative approach to deport the occurrence of liquid-liquid phase scribe the interaction of hard spheres in the separations, other studies report a phase sep- presence of nonadsorbing polymer. They dearation in which the addition of free polymers rive a particle pair interaction potential based leads to two phases, one rich in particles (solid- on a lattice theory for interacting polymers like) and one relatively poor in particles (fluid- near a surface.] The Asakura-Oosawa pair like) (24). The experimental study of Clarke potential model has also been used by Gast et and Vincent (24) reports a particle-rich phase al. (17-19) in their theoretical and expericonsisting of loosely flocculated particles. This mental investigations of volume-restriction is analogous to the "phase separation" ob- phase separations. Using a hard-sphere fluid served by Sieglaff (20) in his experiments on as the reference structure for the particle-poor dispersions of styrenedivinylbenzene latex phase, and a hard-sphere crystal (a face-cenmicrogels in the presence of unfractionated tered cubic crystal) as the reference structure styrene polymer molecules. (In this context, for the particle-rich phase, these investigators "phase separation" implies "flocculation.") applied perturbation theory to determine osSperry et al. (25) performed experiments to motic pressures and Gibbs free energy values Journal of Colloid and Interface Science, Vol. 120, No. 2, December 1987

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OF VOLUME-RESTRICTION

for both the particle-poor and the particle-rich phases. Although only a liquid-solid phase separation is predicted in their two earlier studies (18, 19) this is--to some extent--a consequence of the assumed structure of the particle-rich phase. In their most recent paper (17), Gast et al. used a hard-sphere fluid as the reference structure for both phases and concluded that their results, based on this assumption, were consistent with the liquid-liquid phase separations observed by Sperry (16).

2.2. Characterization of Particle Interaction Potentials

EFFECT

where R = r/D is the dimensionless center-tocenter distance, r is the center-to-center separation of the particles, D is the particle diameter, E0is the permittivity of a vacuum, Eis the dielectric constant of the medium, ~b0 is the surface potential of the particles, and K is the inverse Debye-Huckel screening length. The above expression for the repulsive potential applies to the case of particle interactions at constant surface potential. The parameter K, the reciprocal Debye-Huckel screening length, is defined (for 1-1 electrolytes) as

K = [8rcCe2NAZ2/%~kBT] 1/2,

The primary objective of this paper is to examine the influence of volume-restriction effects on phase separations of colloid and polymer (nonadsorbing) systems. Both aqueous and nonaqueous systems are considered. The model steric systems considered in this investigation are composed of identical spherical particles and identical spherical macromolecules in a continuous solvent medium. In addition, the particles are assumed to be rigid, with nonadsorbing surfaces. The polymer molecules behave as rigid spheres toward the colloidal particles, but can freely interpenetrate other polymer molecules. The particle interactions can arise due to three sources: electrostatic repulsion, London-van der Waals attraction, and volume-restriction attraction. The expressions used to calculate these are given below; details and discussion on the electrostatic and van der Waals interaction potential energies are available in numerous textbooks on colloid and interfacial science [e.g., (28); also, see the recent monograph by Israelachvili (29)]. The electrostatic forces due to the interaction of the electrical double layers lead to a repulsion between similarly charged particles. The repulsive pair potential, uR(R), used in this study is a logarithmic potential (30): UR(R) = (eoeD~p~/4)ln{ 1 + exp[-KD(R - 1)]}, [2.1]

407

[2.2]

where C is the electrolyte concentration, e is the electronic charge, NA is the Avogadro number, z is the valence of the counterion, ks is the Boltzmann constant, and T is the absolute temperature of the system. The repulsive potential UR(R) given in Eq. [2.1] applies only in the case of thin double layers, i.e., KD >~ 6. [Other expressions for UR(R) for cases other than the above are available; see, for example, (5) and references cited therein.] The attractive potential due to the London-van der Waals forces can be expressed as [e.g., (31)] uA(R) =-(H/12){[1/(R 2 - 1)]

+ [ 1/R 2] + 2 ln[(R e - 1)/R 2] },

[2.3]

where H is the effective Hamaker constant, a material property. Note that, for the steric systems considered in this study, the influence of the London-van der Waals forces is either negligible or canceled by the electrostatic repulsion (if the latter is present). (Selected examples are presented in the next major section, Section 3.) The third source of interaction--and the primary focus of the present work--is that due to the influence of the volume-restriction effect. As noted previously, the volume-restriction effect of free polymer is accounted for using the idea set forth by Asakura and Oosawa (11, 12). The exclusion of polymer molecules from the region of interparticle sepaJournal of Colloid and Interface Science, Vol. 120, No. 2, December 1987

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CATES AND HIRTZEL

ration gives rise to an attractive force which can be described by the following interparticle pair potential, UvR(R),

for properties of the liquid-like phase. A brief description of these is given below. 2.3.1. Cell model theory. The cell model approach used in this study was first developed UvR(R) by Lennard-Jones and Devonshire (13, 14). oo, R < I This cell model approach is characterized by the assumption that the dispersion of volume -(Tr/12)PpolD3[2(1 + o03 V is divided into N identical cells, each of - 3( l + a)2R + R3], l ~ < R ~ < l + a which contains (only) one particle. In addition, each particle is assumed to move about in a O, 1 + a < R , cage or cell formed by its nearest neighbors. [2.41 Additional layers of neighbors surrounding the where a is the ratio of polymer diameter (de- central cell are accounted for by a collection noted d) to particle diameter (i.e., a = d/D) of shells surrounding this central cell; the and Ppo~is the polymer-induced osmotic pres- number of particles in each shell is determined sure. (The above equation can also be ex- by the corresponding coordination number. pressed directly in terms of volume fraction This, in turn, is determined from the assumed of polymer in solution, ~bpol.) Equation [2.4] lattice structure of the model, i.e., the cells are implies that the attractive force associated with chosen so that their centers form a regular latthe volume-restriction effect is proportional to tice (e.g., face-centered cubic crystal). Each of the osmotic pressure of the polymer (at ex- the N particles is assumed to move within its treme dilution). Further, in the absence of any cell independently of the motions of other polymer (q~pol = 0), Eq. [2.4] reveals that a particles. This cell model was first used by hard-sphere interaction between the colloidal Lennard-Jones and Devonshire (LJD) to departicles has been assumed. scribe the behavior of liquids and dense gases; The total pair interaction potential is then the assumption of lattice structure, however, found as the sum of the appropriate individual makes this model more suitable for the study contributions as represented by Eqs. [2.1], of the solid phase. The LJD cell model is de[2.3], and [2.4]. Thus, in nonaqueous systems, scribed in detail in (6) and references therein. Hence, only a brief summary of the relevant the total interaction energy u(R) is equations is presented here. u(R) = uA(R) + uvl~(R), [2.5] The assumption, noted above, of a lattice and for aqueous systems, the electrostatic term structure is needed in order to relate the distances between cell centers to the density. The is also included in the total potential: assumption that each particle moves freely u(R) = uA(R) + uw(R) + urdR). [2.61 within its cell, independently of other particles, is invoked to facilitate the evaluation of the 2.3. Evaluation of Thermodynamic configurational partition function. In particProperties and Phase Separation: ular, this latter assumption implies that the Statistical Mechanical Formalism potential energy of the system can be approxAs noted previously, statistical mechanical imated by a sum of terms, each of which detechniques are used to calculate the thermo- pends on the position of only one particle. dynamic properties (osmotic pressures and Thus, it is assumed that the total potential enfree energies) needed to construct the phase ergy of the system is approximately (32) diagrams in this study. A cell model approach N was used to obtain the properties of the solidU = Uo + ~ [~b(ri)- ~b(0)], [2.7] i=1 like phase and perturbation theory was applied

f

Journal of Colloid and Interface Science, Vol. 120, No. 2, December 1987

INFLUENCE

OF VOLUME-RESTRICTION

where ri is the vector displacement of particle i from the center of its cell, and U0 is the lattice energy, i.e., the potential energy when all the particles are at their centers. The quantity [~(ri) ~(0)] is the change in potential energy when the particle i moves from its center to ri, with all other particles remaining at the centers of their cells. The Helmholtz free energy A is then obtained from the partition function as (33) -

A = - k B T In Q.

[2.8]

In the above expression, Q is the canonical ensemble partition function, Q = x-3Nexp[--/3 U0] VN,

[2.9]

where ~ is the deBroglie thermal wavelength, ~3is the inverse thermal energy [/3 = (kBT)-I], and the factor )-3N is the nonconfigurational part of the partition function. The free volume in Eq. [2.9] above is denoted Vf and is defined as

Vr = ~ell exp{-tX[~(r)- ~(0)]}dr.

[2. I0]

Details of the method of calculation of the values of the cell field due to shell i, ~b,(r), as well as the total cell field, ~(r), are given in (6) together with a discussion of all relevant assumptions. Similarly, final expressions for calculating the free volume Vr and the lattice energy Uo are also presented in that reference. The expression for calculating the Helmholtz free energy A then follows directly from Eq.

[2.81:

EFFECT

409

two components, a reference part and a perturbation part, u(R) = u0(R) + up(R),

[2.121

where u0 denotes the interaction energy of the reference system and up is the perturbation interaction energy. The second step of the procedure is to relate the properties of the reference system to those of a well-known, wellcharacterized system. The reference system used is usually that of a hard-sphere system. The hard-sphere fluid is characterized by a pair potential which is an infinite spike at particle contact and zero otherwise. The perturbation potential Up contains the smoothly varying long-range attractive portion of the potential. This form of the interaction potential, Eq. [2.12], is then used in the statistical mechanical expression for the partition function Q. The free energy, in turn, is calculated from the expression for Q (e.g., see Eq. [2.8]). The free energy is expanded in terms of the perturbation potential up [for details, see (33), or other references cited in (6)]. The result is a series for the Helmholtz free energy, A=Ao+AI +A2+ • • • ,

[2.13]

where A0 is the Helmholtz free energy of the reference (i.e., hard-sphere) system and the terms A j and A2, etc. are the perturbation corrections. The first-order correction term A1 can be written as (AI/NkBT)

(A/NkBT)

= 3 In X + ( U o / N k a T ) - l n

= 12~ Vf.

[~up(R)]go(R)R2dR,

[2.14]

[2.11]

The term 3 In X represents the nonconfigurational contribution to the Helmholtz free energy and the remaining two right-hand side terms represent the configurational part of the free energy. 2.3.2. Perturbation theory. In the general scheme of perturbation theories, two main steps can be recognized. First, the interaction energy u(R) of the actual fluid is separated into

where q~denotes volume fraction and go(R) is the radial distribution function of the reference system and, as before, R = riD. The rigorous form of the second-order correction term includes terms to account for three-body and four-body interactions and is, therefore, difficult to evaluate. However, superposition principles can be applied to approximate the multibody interaction effects. Specifically, based upon the macroscopic compressibility Journal of Colloid and Interface Science, Vol. 120, No. 2, December 1987

410

CATES AND HIRTZEL

approximation, the second-order term A2 is written as [see, for example, (33, 34)] (A2/NkBT) = -6kBT~b(~)o fo~° [flup(R)]Zgo(R)R2dR, [2.15] where p is the number density of particles, and Pis the thermodynamic pressure of the system. As mentioned above, the first step of a perturbation theory procedure is concerned with the separation of the potential into reference and perturbation parts. Two separation schemes are well known; one is due to Barker and Henderson (35) and the other is due to Weeks et al. (36). [For a detailed discussion and comparison of these two schemes, see (6).] In this study, the reference system proposed by Barker and Henderson has been used. The reference and perturbation potentials for the Barker-Henderson scheme are, respectively, u(r); u°=

r~

O;

r > a

0;

r~
and Up=

u(r);

r>cr.

[2.16]

In the above potentials, a denotes the distance at which u(r) becomes zero near the core. This separation scheme assumes that the region where the repulsive forces dominate is in the range of distances r ~< a. The correspondence between this reference system and the hardsphere system is obtained through the definition of an effective hard-sphere diameter dE as dE = do +

L

1 - exp[-flu(r)]dr,

[2.17]

That is, the radial distribution function and Helmholtz free energy of the reference system are, respectively, go(R) = gr~s(R; q~E)

[2.18]

A o = AHS ((~E).

[2.19]

and The hard-sphere equation of state used is the well-known equation due to Carnahan and Starling (37); the radial distribution function of hard-sphere fluids [from Throop and Bearman (38) and Verlet and Weis (39)] is also well known and available in the literature [see (33)]. Additional details on the development of the calculations and relevant equations are available in (26) and references cited therein. Summary tables describing the computational schemes for both the LJD cell model and the perturbation theory are also given in (26). 2.3.3. Construction of phase diagrams. Once the expressions for the Helmholtz free energy A = A(T, ~b)are available (Sections 2.3.1 and 2.3.2), other properties follow from standard thermodynamic relationships. For example, the pressure P and Gibbs free energy are calculated as [see, for example, (33)] P= --( OA/ OV)T

[2.20]

G=A+PV.

[2.21]

and The equilibria values needed to construct the phase diagrams are obtained from plots of Helmholtz free energy versus volume and Gibbs free energy versus pressure, based on the Maxwell construction procedure [see (40)]. Several such Maxwell constructions are made, for a range of polymer concentrations, to determine the boundaries of the regions of the stability diagram. Phase diagrams are expressed in terms of volume fraction of colloid (¢) versus volume fraction of polymer (q~po0, for fixed values of ~ (a = d/D). Results are presented in the next section.

where dE is the effective hard-sphere diameter of particles and the point do is chosen as that point where the term exp[-fiu(r)] is negligible (e.g., less than about 10-6). The effective vol3. RESULTS AND DISCUSSION ume fraction, ~bE, and the properties of the The primary objective of this paper is to reference system are taken to be those of a examine the influence of volume-restriction hard-sphere system with volume fraction q~E. Journal of Colloid and Interface Science, Vol. 120, No. 2, December 1987

INFLUENCE

OF VOLUME-RESTRICTION

effects on phase separations in dispersions of colloidal particles and unadsorbed (free) polymer using statistical mechanical techniques (specifically, a perturbation theory and the LJD cell model), as outlined in Section 2. The volume-restriction attraction is characterized by the Asakura-Oosawa potential, as shown in Eq. [2.4]. Results are given for both nonaqueous systems [Section 3.1] and aqueous systems, in which electrostatic repulsion and L o n d o n - v a n der Waals attraction also are present [Section 3.2]. For both cases, comparisons with experimental and theoretical resuRs available from other studies are also presented. However, before these are given, a few general comments related to the computations and methods herein are appropriate. The assumptions underlying the pair potential used in this study to describe the volume-restriction effect (i.e., based on the idea of Asakura and Oosawa) include the assumption that the concentration is sufficiently low that polymer-polymer interactions can be neglected. [Note that in the Asakura-Oosawa pair potential (Eq. [2.4]), polymer-polymer interactions enter only through the concentration dependence of the osmotic pressure and of the thickness of the layer, depleted of polymer, which surrounds each particle.] The form of the potential implies that the volumerestriction attraction is proportional to the osmotic pressure at extreme dilution; this is given by the van't Hoff ideal solution expression. Hence, in this study, systems were considered for conditions such that the ratio of the actual polymer-induced osmotic pressure (as represented by a two-term virial equation of state) to the ideal osmotic pressure o f the polymer (as given by the van't Hoff expression) is less than or equal to about 1.1. Second, the geometric reasoning underlying the AsakuraOosawa characterization strictly applies for systems in which the polymer molecules are much smaller than the particles. This constraint is dealt with by considering values of c~ (=d/D) less than or equal to approximately 0.1.

EFFECT

411

3.1. Influence of Volume-Restriction Effect: Phase Separation in Nonaqueous Dispersions The nonaqueous systems considered here are characterized by particles which interact under the influence of volume-restriction (only), in "theta-state" polymer solutions. [A polymer in solution in the theta-state, i.e., a polymer in its theta solvent and at the theta temperature, is unaffected by other polymers, at low concentrations of polymers; e.g., see (2).] Phase separation calculations based on the statistical mechanical methodology used in this study (see Section 2) were generally possible as long as the magnitude of the minimum in the pair potential was less than approximately 1.5 times the thermal energy. However, for deeper minima, the cell model yielded Helmholtz free energy values which were found to be physically unreasonable [in brief, such physically unreasonable results were determined to be so during use of the Maxwell construction to obtain the phase equilibria conditions; such values are probably due to the fact that the cell model becomes increasingly less accurate at lower volume fractions of particles; see (26) for details and specific examples]. For those conditions for which phase separations were predicted, the flocculated phase always formed into (solid) face-centered cubic (FCC) type structures. The possibility of noncrystalline floes was also considered, based on the use of data for the radial distribution function of a simple Bernal liquid (41), but such floc structures were not predicted in this study. Figure 1 presents a dispersion stability diagram which is based upon the FCC flocculation resuits; results for two values of a -- d/D are shown, a = 0.05 and 0.1. This stability diagram reveals an enhanced volume-restriction effect in the case of the smaller value of a (for fixed particle diameter D, smaller values of cr correspond to smaller values of polymer diameter). Phase boundaries for both cases are truncated at the value of polymer volume fraction or polymer concentration at which Journal of Colloid andlnterface Science, Vol. 120, No. 2, December 1987

412

CATES AND HIRTZEL

0.7 Stable Particle-Rich Phase

0.6

a

T w o - Phase Region = 0.05

a=O,I

0"5I Stable

O4

Particle-Poor

I

o

O.Ol

Phase

, [

o.o2

[

I

0o3

0.04

0.05

~pol

FIG. 1. Dispersion stability diagram for nonaqueous system with volume-restriction effect [a = d / D = 0.05 and 0. I]. The asymptotic limits of the coexistence region for q~poJ = 0 differ slightly from the appropriate hard-sphere limits because of numerical approximations in the calculations.

the cell model results became questionable (see discussion above). Points which lie within the two-phase region correspond to phase separation conditions, as noted in the figure. Some additional phase separation calculations for nonaqueous systems, for different values of a, are shown in Table I. The results of this study were also examined in comparison with the results reported by de Hek and Vrij (22) (see Section 2.1). Values of parameters a = d/D, 4~pol,and overall value of volume fraction, 4)T, corresponding to the experimental systems studied by de Hek and Vrij (22) are summarized in Table II. (The overall volume fraction ~bTshown in Table II is based on the total number of particles in the coexisting phases.) However, based on the methodology used in this current study, we were unable to predict the phase separations which de Hek and Vrij observed. This may be due to the fact that the perturbation theory

TABLE I Phase Separation Predictions for Nonaqueous Systems a a =d/D

~bpad

0.001

0.005

0.01

0.05

0.1

0.001

-BUminb ~bpoc 4~nc

1.501 0.51 (7) 0.57 (9)

O.301 0:51 (7) 0.57 (8)

O. 151 0.5I (7) 0.58 (0)

0.031 0.51 (9) 0.57 (8)

0.016 0.51 (5) 0.58 (1)

0.005

-flUmi,

7.505 ---

1.505 0.52 (4) 0:58 (4)

0.755 0.52 (1) 0.58 (5)

0.155 0.52 (1) 0.58 (6)

0.080 0.51 (4) 0.58 (2)

15.010 ---

3.010 ---

1.510 0.53 (2) 0.59 (4)

0.310 0.52 (3) 0.59 (6)

0.160 0.51 (6) 0.58 (2)

75.010 ---

15.050 ---

7.550 ---

1.550 0.50 (6) 0.64 (8)

0.800 0.51 (7) 0.58 (6)

3.100

1.600 0.46 (5) 0.69 (8)

~bpo q~, 0.01

-flUrnin

4~w 4)~

0.05

--/~Umin ~bpo ~n

O. 1

-flUmi,~

q~po ~b,

150.100 . .

. .

30.100 . .

15.100 .

. .

a Properties of the particle-poor phase and the particle-rich phase were obtained based on the perturbation method and cell model, respectively. b For cases with excessively deep pair potential minima(i.e., --~Umitl > 3), calculations were not made. c 4~po, particle volume fraction in particle-poor phase; q~d, particle volume fraction in particle-rich phase. d 4)po~,volume fraction of polymer. Journal of Colloidand InterfaceScience, Vol. 120,No. 2, December1987

413

INFLUENCE OF VOLUME-RESTRICTION EFFECT TABLE II Characterization of the Experimental Systems of de Hek and Vrij (22) Systemcodea

a

~r~l

-3u,,,:~

&rb

SBI, 1% PS1 SBI, 5% PS1 SB1, 1% PS2 SB1, 5% PS2 $6, 1% PS1 SB1, 1% PS3 SB1, 5% PS3 SB1, 5% PS5

0.059 0.059 0.12 0,12 0.13 0.19 0.19 0.26

0.26 0.23 0.32 0.28 0.50 0.39 0.33 0.38

6.8 6.2 4.4 3.8 6.3 3.5 2.9 2.5

0.0054 0.027 0.0054 0.027 0.0060 0.0054 0.027 0.027

-~12X 100%c 190% 150% 120% 90% 180% 97% 69% 60%

"All systems are at their theta temperature; the code is that used by de Hek and Vrij (22). bThe overallvolume fraction (of particles), 4~T,is determined based on the total volume of the destabilizeddispersion and the total number of particles in the coexistingphases. c The terms A1 and A2 denote the first-order and second-order perturbation correction terms, respectively, for the Helmholtz free energy A of a liquid-phase system with volume fraction q~T. does n o t converge for particle c o n c e n t r a t i o n s associated with those systems, as indicated by the values shown i n the last c o l u m n o f T a b l e II. T h e values there (i.e., ] A 2 / A I [ × 100%) represent the m a g n i t u d e of the ratio of the secondorder to first-order p e r t u r b a t i o n correction terms, based o n the overall particle v o l u m e fractions i n the systems studied b y de H e k a n d Vrij (22). Clearly, the p e r t u r b a t i o n theory does n o t p r o d u c e c o n v e r g e n t results for those conditions shown. Gast et a l . (18), i n their theoretical study of n o n a q u e o u s systems, applied p e r t u r b a t i o n theory to b o t h the liquid-like phase a n d the solid-like phase. [Their predictions indicate solid-phase flocs rather t h a n liquid-phase flocs for c o n d i t i o n s which corres p o n d to the l i q u i d - l i q u i d separations reported by de H e k a n d Vfij (22).] Table III presents a c o m p a r i s o n of selected theoretical results from the study by Gast e t a l . (18) wYth results of this study. As c a n be seen, for c o n d i t i o n s for which the p e r t u r b a t i o n calculations (for the particle-poor phase) converge, the two sets of results are very similar. However, o u r results indicate that p e r t u r b a t i o n theory applied to the particle-poor phase calculations does n o t give convergent results for all values o f v o l u m e fraction (e.g., see the entries c o r r e s p o n d i n g to q~pol = 0.3 i n T a b l e III). A n d , i n the case o f the

particle-rich phase, the results herein are based o n the L J D cell m o d e l theory; those o f Gast e t a l . (18) are based o n use of p e r t u r b a t i o n

TABLE III Comparison of Nonaqueous Phase Separation Predictions of This Study to Corresponding Predictions of Gast et al. (18)

a - d/D = 0.2

O.1

0.2

0.3

-3umi. q~n(this study) 49d(GHR)a ~po (this study) 0oo (GHR)~ IA2/A~I × 100%b

0.85 0.63 0.61 0.51 0.50 0.50%

1.70 0.66 0.66 0.44 0.42 2.0%

2.55 c 0.68 -0.026 61%

Values for equilibrium volume fractions of particles were obtained from Fig. 5 ofGast et al. (18). Comparisons of first-order and second-order perturbation correction terms are for liquid-phasesystemsat the 4~ values of Gast et al. (18). Cell model resultsbecame physicallyunreasonable for the case of -3umin = 2.55; therefore, equilibrium phase compositionswere not predictedfor that case by this study. (See text for additional discussion.) a 4~vol,volume fraction of polymer; q~n,volume fraction of particle-rich phase; ~bpQ,volume fraction of particlepoor phase. Journal of Colloid and Interface Science, VoL120,No. 2, December1987

414

CATES AND HIRTZEL

theory for the particle-rich (as well as the particle-poor) phase.

I

I

I

Stable Particle-Rich Phase

0.6

3.2. Influence of Volume-Restriction Effect: Phase Separation in Aqueous Dispersions Aqueous dispersions in which particles interact through electrostatic repulsion and London-van der Waals attraction as well as volume-restriction attraction were also examined. As in the case of the results for the nonaqueous systems studied (Section 3.1), the cell model calculations yielded results which were physically unreasonable at higher concentrations of polymer and/or deep minima. (However, there appeared to be no single, clear-cut value of limiting minima depth which would guarantee physically reasonable cell model predictions; this was true for both the nonaqueous and the aqueous dispersions cases.) The phase separations predicted always formed (solid-like phase) FCC floc structures; as in the case of nonaqueous systems, the possibility of Bernal liquid-type floc structures was considered, but these structures were never predicted for the range of conditions and techniques used herein. Figure 2 is an example of a stability diagram of an aqueous system with volume-restriction attraction; results for two values o f ~ are shown. As can be seen by comparing Figs. 1 and 2, the stability diagrams for the aqueous and for the nonaqueous system are qualitatively similar. In the aqueous dispersion with volume-restriction attraction, smaller values of cr (i.e., smaller polymer diameter for fixed particle diameter) correspond to a stronger volume-restriction attraction than that for larger a, and the influence of the volume-restriction potential is restricted to a smaller range (since particle surfaces must be within one polymer diameter of each other to experience the volume-restriction effect; see Eq. [2.4]). Thus, the total particle interaction potential [i.e., u(R) = UR(R) + UA(R) + UvR(R)] is influenced to a lesser extent by the volumerestriction contribution for smaller values of Journal of Colloid and Interface Science, Vol. 120, No. 2, December 1987

a=o.I a = 0.05 0.5

Two-Phase Region a = 0 . 0 5 and 0.1 0.4

Stable Particle-Poor Phase 0

0.02

0.04

0.06

0.08

oJ

~pol

FIG. 2. Dispersion stability diagram for aqueous system with volume-restriction effect [a = 0.05 and 0.1; T = 298.15 K, H = 0.02 picoerg, e = 78.54, ¢0 = 50 mV, D = 0.5 urn, C = I × 10-z mole/liter; and Zpol = I, Cz = 7.1 × 10-4 cm3-mole/g2, M , = 115600.0 g/mole in the two-term virial equation for pressure, Pool = poojk~T[1 + (C2M~Pool/NA)], where C2 is the virial coetficient for polymer/solvent system and Mn is the molecular weight of polymers. The value of Zoo1is Zooj = Pool/Pvol,ia~,where Pooud~ is given by the van't Hoffequation, Pideal = ppolkBT].

a (see Fig. 2). (Values of the relevant physicochemical variables in the interaction potentials used in the calculations shown and values of phase separation calculations for a = 0.1 are shown in Table IV.) Again, it is of interest to compare the results of this study's methodology with available results from other investigators. Tables V and VI list the characteristics and parameter values corresponding to the experimental systems studied by Sperry et al. (15, 25) (in Table V), and Sperry (16) (in Table VI). Based on the values shown in those tables, the present study did not predict the experimentally observed phase separations. However, this is not surprising since examination of the values shown in these two tables indicates that none of the systems studied in those experiments satisfy both the polymer concentration constraint and the polymer size constraint which apply to the theoretical methodology used in this study (see

415

INFLUENCE OF VOLUME-RESTRICTION EFFECT TABLE IV Phase Separation Predictions for Aqueous Systems

--/~ Urnin

Zpold ~b[,,b ~bnb'c

0

0.001

0.oo5

0.Ol

0.05

0.1

0.151 0 0.44 (0) 0.50 (5)

0.154 1.000 0.44 (2) 0.50 (5)

0.164 1.001 0.44 (2) 0.50 (6)

0.176 1.002 0.44 (2) 0.50 (7)

0.297 1.012 0.44 (4) 0.51 (8)

0.481 1.024 0.44 (3) 0.54 (2)

Note. T = 298.15 K, H = 0.02 picoerg, e = 78.54, fro = 50 mV, D = 0.5 um, C = 1 × 10-2 mole/liter, Zpo~= 1, d = 0.05 t~m, C2 = 7.1 × 10-4 cm3-mole/gZdM, = 115600.0 g/mole d, a = d/D = 0.1. 4)po~denotes volume fraction of polymer. b Properties of the particle-poor phase and the particle-rich phase were obtained in terms of the perturbation method and cell model, respectively; 4¥o denotes volume fraction of particle-poor phase; 4)~denotes volume fraetion of particlerich phase. Face-centered cubic structures are predicted for the particle-rich phases. d The pressure Ppol is calculated from a two-term virial equation Ppo~ = ppo~kBT[1 + ( C z M 2 Ppot]NA)] where C2 is the second virial coefficient for a polymer/solvent system and M, is the molecular weight of polymers. The value of Zpo~ is Zpol =

Ppol/Ppol,ideal, w h e r e epol,id~ = OpolkBT.

d i s c u s s i o n i n S e c t i o n s 2 a n d 3). T h e e x p e r i ments reported in the studies by Sperry and co-workers demonstrate volume-restriction p h a s e s e p a r a t i o n s , b u t w o u l d a p p e a r t o d o so for conditions which the Asakura-Oosawa interaction potential cannot adequately account for. l i t s h o u l d a l s o b e n o t e d t h a t t h e r e is s o m e a m b i g u i t y i n t h e r e p o r t e d v a l u e s o f c e r t a i n ex-

p e r i m e n t a l p a r a m e t e r s (e.g., s e c o n d v i r i a l c o e f ficient and inverse Debye-Huckel screening l e n g t h ) u s e d i n t h e s y s t e m s o f S p e r r y et al. (15, 16, 25). A l s o , t h e e x p e r i m e n t a l v a l u e s o f i o n i c s t r e n g t h a r e n o t g i v e n i n (16).] In their recent study of volume-restriction effects, G a s t et al. (17) p r e d i c t o n e o f t h e liquid-liquid phase separations observed by

TABLE V Characterization of the Experimental Systems of Sperry (15) Polymer code ~

d (~m) C2 (cmLmole/g 2) Mn (g/mole) cqb 4~pol

-flUmi~ Zpolc

LR

ER

GR

MR

HR

0.0239 12 X 10 -4 36,000 0.055 0.43 0.44 1.15

0.0343 9.4 X 10 -4 63,900 0.079 0.54 1.3 1.16

0.0496 7.1 X 10-4 115,000 0.11 0.50 2.3 1.12

0.0921 4.4 X 10 - 4 307,000 0.21 0.60 3.2 1.10

0.116 3.7 X 10 4 440,000 0.27 0.66 3.3 1.10

Note. T = 298 K, H = 0.07 picoerg, ~ = 78.5, fro = 80 mV, D = 0.43 ~zm, C = 0.01 mole/liter, Zpot = 1, 6 = 0.0022 ~m b, 49T = 0.95. a Code used by Sperry (15). b b is the thickness of adsorbed layer of surfactant on particle surfaces; c~ = d/(D + 26). c The pressure Ppol is calculated from a two-term virial equation Ppo~ = p~olkBT[1 + (C2M~p~ol/NA)]where C2 is the second virial coefficient of the polymer/solvent system and Mn is the molecular weight of polymers; Z~o~= PpoJPpo~,i~e~, where Ppol,ideal = ppolkBT. Journal of Colloid andlnterface Science, Vol. 120, No. 2, December 1987

416

CATES A N D H I R T Z E L TABLE VI Characterization of the Experimental Systems of Sperry (16) Particle diameter, D(/Lm)

a~a q~pol --~Umin Zoo~b Type of floc

0.050

0.089

0.14

0.31

0.60

1.7 1.7 2.5 1.28 Liquid

0.98 1.5 3.0 1.26 Liquid

0.64 1.2 3.0 1.20 Liquid

0.29 0.68 3.0 1.12 Solid

0.15 0.52 4.5 1.09 Solid

Note. T = 298 K, H = 0.07 picoerg, E = 78.5, ~0 = 75 mV, C = 0.0135 mole/liter, Zpol = 1, 6 = 0.0025 # m a, q~T = 0.09, d = 0.0921 m, C2 = 4.4 × 10 -4 cm3-mole/g 2 b, Mn = 307,000 g/mole b. a 6 is the thickness o f adsorbed layer of surfactant on particle surfaces; c~ = d/(D + 26). b The pressure Pool is calculated from a two-term virial equation, Pool = ppolkBT[1 + (C2M~pool/NA)] where C2 is the second virial coefficient for polymer/solvent system and Mn is the molecular weight of polymers. The value of Zoo~ is Zpol = Ppol/Pool,i~leal, where Ppol,ideal = ,Ooolkl3T.

Sperry (16) (specifically, that one corresponding to the case where particle diameter D = 0.14 t~m; see Table VI). This prediction was obtained by representing both the particle-rich phase and the particle-poor phase in terms of liquid-phase perturbation theory. Similar calculations performed in this present study led to a similar prediction of a liquid-liquid phase separation; however, an examination of the values of the first- and second-order perturbation corrections A1 and A2 to the free energy indicated that the perturbation results do not converge for a system with the associated parameter values for that case. A closer examination of this particular system revealed that, for the system in question (i.e., that one described in Table VI corresponding to D = 0.14 #m), the contributions to the total pair interaction potential from London-van der Waals attraction and from electrostatic repulsion are approximately of the same magnitude. In this study, when the London-van der Waals attractions are included in the phase separations calculations, the perturbation theory results for the particle-rich phase lead to physically unreasonable results from the Maxwell construction (similar to the case of physically unreasonable results obtained for certain cell model calculations, as discussed previously). Journal of Colloidand InterfaceScience, Vol. 120, No. 2, December1987

The results obtained by Gast et al. (17), however, were based on phase separation calculations which did not include the Londonvan der Waals effects. And, although Gast et al. (17) predict one of the three cases of liquidlike flocs reported by Sperry (16), they were not able to predict [in (18)] the liquid-liquid separations for conditions corresponding to the liquid-liquid separations observed and reported by de Hek and Vrij (22). Thus, at this point, the meaning and interpretation of predictions of liquid-liquid phase separations using only liquid-phase perturbation theory for both particle-rich and particle-poor phases seem unclear. 4. CONCLUDING REMARKS The primary objective of this study has been to investigate theoretically the volume-restriction effect of free (unadsorbed) polymers on the equilibrium properties and phase separation behavior of selected steric dispersions. The volume-restriction effect on particle interactions was characterized by the interaction pair potential developed by Asakura and Oosawa (11, 12). Conditions for coexisting phases, in both nonaqueous and aqueous systems with volume-restriction interactions, were determined. The necessary thermodynamic prop-

INFLUENCE OF VOLUME-RESTRICTION EFFECT

erties were calculated based on two statistical mechanical techniques, a cell model theory (for the solid-like, particle-rich phase) and a perturbation theory (for the liquid-like, particle-poor phase). For the systems considered, phase separations which lead to the formation of solidphase flocs of a face-centered cubic structure are predicted. The non-face-centered cubic flocs (i.e., liquid-phase and amorphous-type flocs) observed experimentally by other investigators [de Hek and Vrij (22) and Sperry (16)] were not predicted by the methodology herein. However, these experimental systems have system values (e.g., pair potential minimum depth, volume fraction of polymer) which are outside the range of values considered appropriate for the Asakura-Oosawa volume-restriction potential (the assumptions underlying it, etc.) used herein. Moreover, attempts in this study to predict such liquid-liquid transitions using perturbation theory for both the particlerich and the particle-poor phases did not succeed in predicting theoretically the observed experimental results. This failure may be due--at least, in part--to a lack of consistency between the system conditions and the assumptions underlying the Asakura-Oosawa potential. Certain limitations of the techniques used herein also were examined, both with respect to the use of the Asakura-Oosawa potential and with respect to limitations of the statistical mechanical techniques which were used. Additional work is needed in order to resolve some of these questions. Although some experimental results on volume-restriction phase separation are available in the literature (see Sections 2 and 3), the conditions characteristic of those systems do not, in general, appear to be consistent with use of the Asakura-Oosawa potential alone, and, hence, interpretation of comparisons is difficult. Many research problems related to the steric systems studied herein require additional attention. Additional studies are needed with respect to the liquid-liquid transitions observed in experimental studies, the interpre-

417

tation of experimental observations, and theoretical explanations of these. A key outstanding problem in steric systems is that of the thermodynamics of interaction between steric barriers. There is, to date, only a partial and inadequate understanding of such interactions and experimental and theoretical investigations are necessary. [See (5) for discussion on other research needs related to steric dispersions.] The theoretical approaches considered in this study treat the steric dispersion as a system composed of colloidal particles (spheres) and nonadsorbing polymer macromolecules (also spheres) in a continuous solvent medium. The phase separations and instabilities observed in some of the experimental studies of such systems discussed herein (see Section 2) are similar to phase separation phenomena observed in solutions containing more than one polymeric species [see, for example, (42)]. In view of this similar behavior, the colloidal dispersions containing polymer macromolecules may be considered as multicomponent mixtures. Thus, the steric systems considered here (such as inks, paints, biocolloids, and protein solutions, to name a few) fall under this much broader classification. Experimental studies of certain multicomponent mixtures have been available for some time; however, theoretical interpretation and understanding are only recently receiving attention and much work needs to be done. ACKNOWLEDGMENTS One of the authors (C.S.H.) would like to express appreciation to B. Boa, M.S., and S.D. for their continued support and encouragement throughout the research program and the preparation of this work. We also acknowledge and thank Ms. Ruth Dewey for her expert and skilled assistance in the preparation of this manuscript. In addition, we thank Mr. Rolf Ziemer for his excellent graphics. REFERENCES 1. Lyklema, J., Adv. C o l l o i d l n t e r f a c e Sci. 2, 66 (1968). 2. Sato, T., and Ruch, R., "Stabilization of Colloidal Dispersions by Polymer Adsorption." Dekker, New York, 1980. Journal of Colloidand InterfaceScience. Vol. 120,No. 2, December1987

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CATES AND HIRTZEL

3. Vrij, A., PureAppl. Chem. 48, 471 (1976). 4. Napper, D. H., "Polymeric Stabilization of Colloidal Dispersions." Academic Press, London, 1983. 5. Hirtzel, C. S., and Rajagopalan, R., "Colloidal Phenomena: Advanced Topics." Noyes Press, Park Ridge, NJ, 1985. 6. Castillo, C. A., Rajagopalan, R., and Hirtzel, C. S., Rev. Chem. Eng. 2, 237 (1984). 7. Feigin, R. I., and Napper, D. H., J. Colloidlnterface Sci. 74, 567 (1980a). 8. Feigin, R. I., and Napper, D. H., J. Colloid Interface Sci. 75, 525 (1980b). 9. Joanny, J. F., Liebler, L., and de Gennes, P. G., J. Polym. Sci. 17, 1073 (1979). 10. Vincent, B., Luckham, P. F., and Waite, F. A., J. Colloid Interface Sci. 73, 508 (1980). 11. Asakura, S., and Oosawa, F., J. Chem. Phys. 22, 1255 (1954). 12. Asakura, S., and Oosawa, F., J. Polym. Sci. 33, 183 (1958). 13. Lennard-Jones, J. E., and Devonshire, A. F., Proc. R. Soc. Ser. A 163, 53 (1937). 14. Lennard-Jones, J. E., and Devonshire, A. F., Proc. R. Soc. Ser. A 165, 1 (1938). 15. Sperry, P. R., Z Colloid Interface Sci. 87, 375 (1982). 16. Sperry, P, R., aT.Colloid Interface Sci. 99, 97 (1984). 17. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 109, 161 (1985). 18. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 96, 251 (1983a). 19. Gast, A. P., Hall, C. K., and Russel, W. B., Faraday Discuss. Chem. Soc. 76, 189 (1983b). 20. Sieglaff, C. L., J. Polym: Sci. 41, 319 (1959). 21. de Hek, H., and Vrij, A., J. Colloid Interface Sci. 70, 592 (1979). 22. de Hek, H., and Vrij, A., J. Colloid Interface Sci. 84, 409 (1981). 23. Pathmamanoharan, C., de Hek, H., and Vrij, A., ColloidPolym. Sci. 259, 769 (1981). 24. Clarke, J., and Vincent, B., J. Colloid Interface Sci. 82, 208 (1981).

JournalofColloidandInterfaceScience,Vol.120,No. 2, December1987

25. Sperry, P. R., Hopfenberg, H. B., and Thomas, N. L., J. ColloM Interface Sci. 82, 62 (1981). 26. Cates, D. L., Ph.D. dissertation, Rensselaer Polytechnic Institute, Troy, NY, 1986. 27. Fleer, G. J., Scheutjens, J. M. H. M., and Vincent, B., in "Polymer Adsorption and Dispersion Stability" (E. D. Goddard and B. Vincent, Eds.), pp. 245-263. American Chemical Soc., Washington, DC, 1984. 28. Void, R. D., and Vold, M. J., "Colloid and Interface Chemistry." Addison-Wesley, Reading, MA, 1983. 29. Israelachvili, J. N., "Intermolecular and Surface Forces with Applications to Colloidal and Biological Systems." Academic Press, London, 1985. 30. Hogg, R., Healy, T. W., and Fuerstenau, D. W., Trans. Faraday Soc. 62, 1638 (1966). 31. Overbeek, J. Th. G., in "Colloid and Interface Science" (M. Kerker, R. L. Rowell, and A. C. Zettlemoyer, Eds.), Vol. 1, pp. 431-445. Academic Press, New York, 1977. 32. Barker, J. A., "Lattice Theories of the Liquid State." Pergamon, London, 1963. 33. McQuarrie, D. A., "Statistical Mechanics." Harper & Row, New York, 1976. 34. Barker, J. A., and Henderson, D., Rev. Mod. Phys. 48, 587 (1976). 35. Barker, J. A., and Henderson, D., J. Chem. Phys. 47, 2856 (1967). 36. Weeks, J. D., Chandler, D., and Andersen, H. C., J. Chem. Phys. 54, 5237 (1971). 37. Carnahan, N. F., and Starling, K. E., J. Chem. Phys. 51, 635 (1969). 38. Throop, G. J., and Bearman, R. J., J. Chem. Phys. 42, 2408 (1965). 39. Verlet, L., and Weis, J. J., Phys. Rev. A 5, 939 (1972). 40. Stanley, H. E., "Introduction to Phase Transitions and Critical Phenomena." Oxford Univ. Press, London, 1971. 41. Finney, J. L., Proc. R. Soc. London Ser. A 319, 479 (1970). 42. Edmond, E., and Ogston, A. G., Biochem. J. 109, 569 (1968).