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Interaction between colloids in solutions containing dissolved polymer
Interaction between Colloids in Solutions Containing Dissolved Polymer ARUN YETHIRAJ, *'1 CAROL K. HALL, *'a AND RONALD D I C ~ A N t Department of Chemical Engineering, * North Carolina State University, Raleigh, North Carolina 27695-7905; and ~Department of Physics and Astronomy, H. H. Lehman College, CUNE Bronx, New York 10468 Received August 26, t991; accepted November 5, t991 The effective interaction between colloids in solutions containing dissolved polymer is investigated using integral equations. The colloidal particles are modeled as hard spheres, the polymer molecules are modeled as freely jointed hard chains, and the solvent is treated as a continuum that doesn!t interact with either the colloidal particles or the polymer molecules. The model therefore concentrates on excluded volume effects in these systems. It is found that at tow polymer volume fractions, the effective intermolecular potential (or potential of mean force) between the colloidal particles is attractive, thus facilitating a phase separation or precipitation of the colloids. As the polymer volume fraction is increased, the strength o f this attraction increases; but a repulsive interaction appears at larger separations, which resembles the double-layer repulsion between charged colloidal particles in an aqueous solution. The effects of varying polymer chain length, colloid particle size, and polymer volume fraction on the effective potential are also studied. © 1992AcademicPress,Inc. 1. INTRODUCTION
The study of colloidal dispersions is of importance in technological systems such as paints, inks, electronic ceramics, oil recovery, and bioseparations and in biological systems such as blood, milk, and protein solutions ( 1). In many of these applications polymers are also present, either to enhance the stability of the dispersion (for example in the manufacture of paints and inks) or to promote separation of the colloids (for example in wastewater treatment and in protein precipitation by nonionic polymer). Consequently, colloidal systems containing polymers have been extensively studied in the last decade, and severn experimental (2-6) and theoretical investigations (7-20) have appeared in the literature. In recent years, there has been a considerable effort to obtain a theoretical understanding of colloidal systems. The increased effort
Present address: Department of Materials Science and Engineering, University of Illinois, Urbana, IL 61801. 2 To whom correspondence should be addressed.
may be attributed in part to the avaitabifity of experimental data on well-characterized simple systems, e.g., polymer latices (5). These systems are easier to model than more complex naturally occurring systems and allow one to concentrate on the physical mechanisms at play in comparisons with experiments. A particularly appealing route to treating colloid-polymer systems is to determine an effective solvent + polymer mediated interaction between the colloidal particles. The dispersion is then treated as a pseudo onecomponent system where the intermolecular potentials are functions &polymer and solvent characteristics. This essentially decouples the problem into two parts: first to determine the effective colloid-colloid interactions as a function of polymer and solvent characteristics and, second, to investigate the behavior of colloids under this interaction. This approach is analogous to the very successful McMiUanMayer theory of solutions (21 ). Theoretical efforts can therefore be classified broadly into two parts: (i) determining the effective interaction potentials or potentials of mean force, and (ii) investigating the prop-
102 0021-9797/92 $5.00 Copyright© 1992by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journalof ColloidandInterfaceScience,VoL 151,No. 1, June 1992
INTERACTION BETWEEN COLLOIDS erties of the colloid fluid, given this effective interaction. These two categories are, of course, far from mutually exclusive, but the primary thrust of most papers lies in one or the other. Effective tools for evaluating phase diagrams have been applied ( 11, 16, 19), but some important tools for estimating the potential of mean force, such as integral equations and Monte Carlo simulation, have been largely neglected. In this paper, we investigate the potential of mean force between colloidal particles immersed in a solution with free (or nonadsorbing) polymer using an integral equation approach. The primary objective of this work is to quantify the effect of excluded volume interactions on the colloidal interactions, using a model that explicitly incorporates both the chain-like nature of polymer molecules and the continuous-space nature of polymeric fluids. We therefore model the colloidal particles as hard spheres and the polymer molecules as a string of freely jointed hard spheres. The solvent is treated as a continuum that interacts with neither the colloids nor the polymers. Simple as the model is, we will show that it displays the qualitative trends that are observed in experiments. The incorporation of more realistic interactions between the molecules and an explicit treatment of the solvent are straightforward generalizations. It has been known for some time that the presence of free polymer can destabilize a colloidal dispersion by inducing an attraction between the colloidal particles (1). The attraction is due to the loss in configurational entropy that the polymer chains experience when they are in the region between the particles, causing them to prefer the region in the "bulk" of the solution. The colloids therefore experience a net attraction because of the pressure that the polymer molecules exert on the exposed surface of the colloidal particles, which pushes the particles together. The attraction has a range that is of the order of the size of the polymer molecules. It has also been suggested that increasing the polymer concentration beyond a certain limit can cause ares-
103
tabilization of the colloidal dispersion (9, 12). A restabilization at high polymer concentrations is intuitively expected, because when a few colloids are dispersed in a dense sea of polymer, their approach toward one another will be hindered by the presence of the chains, thus making flocculation unfavorable. Other reasons that have been proposed for the restabilization will be discussed shortly, after some models for describing polymer-colloid interactions have been reviewed. A model that has been used extensively to quantify the attraction between the colloidal particles due to free polymer is one due to Asakura and Oosawa (7), who modeled the colloids as hard spheres and the polymers as spheres that are impenetrable to the colloids, but fully penetrable to each other. This results in a spherical shell around each colloid particle from which polymer molecules are excluded. Overlap of these "exclusion shells" causes an attraction between the particles that increases with increasing polymer concentration and with increasing polymer molecular weight. The Asakura-Oosawa potential has been widely used to study the destabilization of dispersions due to free polymer (2, 10, 11, 14). The colloid-colloid potential of mean force predicted by this model is always attractive (beyond the hard core separation) and the strength of this attraction increases monotonically with increasing polymer concentration. Therefore the model does not predict a restabilization of the colloidal dispersion (in the absence of electrostatic interactions) at high polymer concentrations. Modifications to the Asakura-Oosawa potential using osmotic virial coefficients for the polymer have been used ( 11 ), but there is reason to believe that these modifications worsen rather than improve the calculated pair potential (18). It is sometimes suggested that the Asakura-Oosawa potential is applicable only when the polymers are much smaller than the colloids ( 11 ). Although the potential was originally derived for the above conditions, the restriction on relative sizes is entirely unnecessary (18). Feigin and Napper (9) derived an effective Journal of Colloid and Inte~aee Science, Vol. 151, No. 1, J une 1992
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YETHIRAJ, HALL, AND DICKMAN
colloid-colloid free energy of interaction using a model that explicitly incorporates the chainlike nature of the polymers and also to some extent the excluded volume of the polymers. They performed Monte Carlo simulations for an isolated rotational isomeric state chain (with no excluded volume) confined between flat plates and monitored the segment density profile perpendicular to the plates for various chain lengths and plate separations. The free energy of mixing was obtained as a function of plate separation by combining the Monte Carlo results with Flory-Huggins solution theory (22) (with an empirical expression for the interaction parameter X). The identification with spherical particles was made using the Derjaguin approximation (23). The free energy change in bringing the colloids from infinitely far apart to a given separation exhibited a minimum at zero separation, as well as a maximum at a separation corresponding to the global size of the chains. Increasing the polymer concentration resulted in a deeper attractive minimum, as well as a higher secondary repulsion. Thus their model accounts for both the destabilization and restabilization of the colloidal dispersion. As the polymer concentration is increased from zero, the attractive minimum causes a destabilization of the dispersion; if the polymer concentration is very high, the secondary maximum is high enough that a metastable state can exist. Physically, the reasons for the restabilization are the same as when electrostatic interactions are present: colloidal particles must overcome a large repulsive barrier before they can approach the attractive minimum. The Feigin and Napper theory is, however, restricted to large colloid/polymer size ratios and does not fully account for excluded volume effects. Joanny et al. (8) determined an effective colloid-colloid interaction via scaling theory. The theory predicts the segment density profile and the free energy of interaction for polymers in semidilute solutions confined between flat plates. The interaction between spheres is again obtained using the Derjaguin approximation (23). This theory predicts an attraction Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
between the spheres; when this attraction is greater than kT, the authors argue, phase separation will occur. We note that scaling theory is applicable only to very long chains. In fact, it is exact only in the limit of infinite chain length. We will argue a little later that in physical systems of interest the equivalent freely jointed chain lengths are of the order of 100 or less; these chains are probably not in the scaling regime. Restabilization of the colloidal dispersion has not been predicted using the Joanny et aL (8) theory although the theory does not rule out this possibility. Fleer et al. ( 12, 17) derived a pair potential between particles in a polymer solution via a generalization of the Scheutjens-Fleer lattice theory (24) for interacting polymers near a flat surface. The attraction between the plates was attributed to the presence of a depletion layer at the surfaces from which polymer chains are excluded. The thickness of this depletion layer decreases as the polymer concentration is increased. Using the Flory-Huggins solution theory (22), the authors calculated an approximate but analytical expression for the depletion layer thickness and the free energy of interaction as a function of chain length, chain concentration, and solvent quality. The interaction between spheres was estimated using the Derjaguin approximation; a geometric factor was included to correct for the fact that the Derjaguin approximation is valid only when the depletion layer thickness is much smaller than the colloidal particle size. They then used the binodal condition to determine the phase behavior of the system. They found that flocculation occurs when the osmotic attraction energy due to the depletion outweighs the loss in configurational entropy of the dispersed particles in going from a dispersion to a floc. At high polymer concentrations, the depletion layer thickness decreases rapidly, resulting in a weakened attraction between the particles. At some (high) polymer concentration, the attraction is weak enough that the particles cannot flocculate, and the model then predicts a restabilization of the dispersion.
INTERACTION
BETWEEN
Several other theories for the interaction of colloids in polymer solutions with and without adsorbed polymer have appeared in the literature (7-20). We do not discuss these theories because they do not introduce any physical concepts radically different from the theories that we have already discussed. Our brief review of theoretical progress in the field must therefore be considered representative rather than exhaustive. All of the above theories are in qualitative agreement with experimental data. The physical picture painted by most of these theories, however, is not entirely consistent with the large number of computer simulations (2533) and experiments (34) that address the behavior of chain fluids confined between flat plates. In most of the theories discussed above the relevant length scale (e.g., the polymer "diameter" in the Asakura-Oosawa theory (7), the correlation length in the Joanny et al. theory (8), and the depletion thickness in the Fleer et al. theory (12)) for the interaction between the plates is assumed to be of the order of the global size of the polymer (~Rg). At very low polymer densities, this is generally supported by computer simulations (30) and microscopic theories (35). At moderate polymer densities, however, computer simulations (30) and experiments (34)demonstrate that the relevant length scale for the interaction between plates is not the global size, Rg, of the polymers but rather the bead diameter, ~r, of the polymers (30, 34, 35 ). Therefore, a more correct calculation of the interaction between colloids in polymer solutions must include a more realistic model of the polymer than has been widely used, i.e., a model of polymers that includes both the length scales ( ~Rg and ~ a ) discussed above. Several other facets of the behavior of polymers between plates have been reported in the literature, and these are discussed below since they form an important limiting case for the colloid-colloid interactions we are interested in. In the last five years, there have been several investigations of the behavior of polymers
COLLOIDS
105
confined between surfaces using computer simulation (25-33), integral equation theory (35), density functional theory (36, 37, 38), and in experiments measuring the force between surfaces in polymer fluids (34, 39-42). It has been shown via simulation (30), theory (35), and experiment (34) that the force on surfaces immersed in a polymer fluid is an oscillatory function of plate separation at high chain concentrations, and is repulsive at small plate separations. The behavior of the force as a function of separation between the plates has been obtained via the segment density profile of chains at a surface (30): the segment density profile is easily measured in simulations, is related to the salvation force, and also determines the free energy of the system. It is found that the segment density profile is governed by a competition between so-called packing and entropic effects (25, 26, 30). At low chain concentrations a single-chain molecule experiences a loss in configurational entropy as it approaches the wall; this causes the chains to prefer the region far away from the wall and results in a depletion of chain sites near the wall (entropic effect), causing an attractive force between the walls. At high chain concentrations, the chains near the wall suffer collisions with other chains and are pushed against the wall; this causes an enhancement of chain sites near the wall (packing effect) and results in a repulsive force between the walls. At high chain concentrations, the density profile is qualitatively similar to that of a hard sphere fluid at a wall or to the radial distribution function, g(r), in hard spheres, and is oscillatory with a period of about one bead diameter. The "high" density referred to above is not necessarily a very large number. In fact, using the generalized Flory equation of state, (43) one can show that an enhancement of chain sites at the wall will occur for bulk volume fractions greater than about only 0.2 and is essentially independent of chain length. The oscillatory force as a function of plate separation is, in fact, a consequence of the packing effect, since at small plate separations the Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
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YETHIRAJ,
HALL,
polymers have to be pushed out from between the plates in order to bring the plates close together. Also, for small plate separations and high chain concentrations (bulk volume fraction "~ 0.3) the polymer chains actually prefer the region between the plates to the bulk (33) because the packing effect wins out over the entropic effect. In these cases the chains are severely flattened and are almost two-dimensional (28, 30, 31 ). We note that in colloidal systems, polymer fractions greater than 0.4 are often considered, (3, 14) and packing effects are therefore important. Packing effects are not observed in traditional lattice models for polymer where each chain bead occupies one lattice site. Lattice models are able, however, to display the transition from depletion effects to enhancement effects. In this paper we use integral equations to obtain the potential of mean force between two hard spheres immersed in a hard chain fluid. We apply the polymer-RISM (Reference Interaction Site Model, not to be confused with the rotational isomeric state model) theory of Curro and Schweizer (44-46) to a mixture of hard spheres and hard chains. The colloid-colloid potential of mean force is obtained by calculating the colloid-colloid pair distribution function in the limit as the colloids become infinitely dilute. The approach is similar to that used by two of us earlier to study the adsorption of chains between flat plates (35). The effect of chain length, chain volume fraction, and colloid size on the potential of mean force between colloids is investigated. We find that at low chain densities, the potential of mean force is attractive and is qualitatively similar to the Asakura-Oosawa attraction (7). The range of the potential is of the order Rg of the chains. At high chain densities, the potential is attractive at very short colloid separations and becomes repulsive at a distance of approximately the bead diameter before decaying to zero. In some cases there is a second repulsive peak or shoulder. The depth of the attractive well in the effective potential, which is always present, increases with chain Journal of Colloid and Interface Science, Vol. 15l, No. l, June 1992
AND
DICKMAN
density and with chain length; the height of the repulsion increases with chain density but decreases with chain length. With increasing particle size, both the attraction and the repulsion are accentuated. Our results are consistent with the calculations of Feigin and Napper (9) and suggest the possibility of flocculation at moderate polymer concentrations caused by the attractive m i n i m u m and a restabilization at higher polymer concentrations caused by the repulsive maximum. We study chain lengths ranging from 8 to 100, polymer volume fractions ranging from 0.05 to 0.35, and particle diameters ranging from 5a to 20a where a is diameter of the chain beads. The range of parameters considered is relevant to the experimental data we hope to describe. For example the ratio of partide diameter, dp, to the chain size, R (where R is the root mean square end-to-end distance), ranges from 0.3 to 5, and the range of volume fractions are similar to those considered in the literature (3, 14). The chain lengths considered may seem short, but are actually quite realistic. The equivalent freely jointed chain lengths of the polymers (the Kuhn length) used in experiments may be estimated from the mean square end-to-end distance, R 2, and fully extended chain length, Nl (22). If the equivalent Gaussian chain has a chain length t7 and a segment length {, then if/-= Nl, and ~1-2 = R 2" Therefore, ff ~ ( N I ) Z / R 2 . For the highest molecular weight polymer used by Sperry et al. ( Table II of Ref ( 5 )), the authors report R 2 = 0.0484 #m 2 and Nl = 1.985 um, which gives ff ~ 81. For the smallest molecular weight polymer, r7 ~ 0 . 1 5 3 2 / 0 . 0 7 1 2 = 4.6! The equivalent chain length of the freely jointed hard chain model will be larger than those calculated above for the Gaussian chain, but are of the same order of magnitude. The rest of this paper is organized as follows. In Section 2 we describe polymer-RISM theory and the method for the calculation of the colloid-colloid potential of mean force, in Section 3 we present predictions of the theory, and in Section 4 we present our conclusions.
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INTERACTION BETWEEN COLLOIDS 2. THEORY
sure, which, for a fluid interacting with a pair potential u(r), is given by,
A. Integral Equation Theory In this section we present equations governing our calculation of the potential of mean force between two hard spheres in a hard chain solvent. The approach is to write down integral equations for a mixture of chains and hard spheres and solve these in the limit as the hard spheres become infinitely dilute. In this limit the potential of mean force is simply related to the radial distribution function of the hard spheres. An important relationship in statistical mechanics is the Ornstein-Zernike (OZ) equation (47), which relates the total correlation function, h(r) (=-g(r) - 1, where g(r) is the radial distribution function), to the direct correlation function, c(r). For fluids interacting via spherically symmetric potentials, the OZ equation is given by
h(r) = c(r) + p f c ( r ' ) h ( ] r - r'[)dr',
[1]
where p is the number density of the fluid. The OZ equation divides the interaction between two molecules into a "direct" contribution given by the first term in Eq. [1] and an "indirect" contribution given by the second term in Eq. [1]. The indirect contribution is propagated through the other molecules in the fluid and therefore results in the convolution integral in Eq. [1]. Equation [ t ] may be considered as defining c(r). Defining a Fourier transform pair by
f(k) = k do r sin krf(r)dr 1
f(r) = ~
[2]
~
fo k sin krf(k)dk,
[3]
Eq. [1 ] may be written in Fourier space as
h(k) = ~(k) + p~(k),~(k),
c(r) = [1 + h(r)][1 -
e~U~rq,
[5]
where 13 = 1/kBT, kB is Boltzmann's constant and T is the temperature. If u(r) is the hard sphere potential, i.e.,
u(r)=m, =0,
r>cr
[61
r>~,
[7]
where ~ is the hard sphere diameter, then the PY closure may be represented by the pair of equations
h(r)=-l,
r
[8]
c(r) = 0,
r > ~.
[9]
Equation [ 8 ] is the exact hard core condition, and equation [ 9 ] is the PY approximation. The extension of the OZ equation to molecular fluids, due to Chandler and coworkers (48-50), is the RISM theory and is based on the interaction site model for molecular fluids. In this model, each molecule is assumed to consist of a number of sites; the intermolecular potential, u(r), is assumed to be the sum of site-site potentials and is given by
u(r) = Y~ u~j(r),
[10]
i
where uo is the potential between sites i and j on different molecules and r is the separation between the sites. In the freely jointed chain model, for a chain of length n there is an interaction site located at the center of each of the n hard spheres. The generalization of the Ornstein-Zernike (OZ) equation to molecules consisting of n identical interaction sites is the site-site OZ equation (48-50): h0(r) = f
ar' f
dr" ~ w i k ( ] r - r'l) k,/=l
[41
where the carets denote Fourier transforms. To solve the OZ equation for h(r) another relation between h(r) and c(r), known as a "closure" is required. A popular closure to the OZ equation is the Percus-Yevick (PY) clo-
× era(Jr' - r"])wt/(lr"l) +p
fy dr'
dr" ~ wik(Ir-r'[) k,l= 1
× c ~ l ( l r ' - r"l)h0(Ir"l),
[111
Journal of Colloid and Interface Science. Vol. 151, No. 1, June 1992
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YETHIRAJ, HALL, AND DICKMAN
where o is the number density of molecules in the fluid, his(r) is the total site-site correlation function, (his(r) = go(r) - 1 ), 47rr2go( r) dr is the probability that sites i and j on different molecules are at a distance between r and r + dr apart, 47rrZwij(r)dr is the probability that sites i and j on the same molecule are at a distance between r and r + dr apart, and cij(r) is the direct correlation function, defined by Eq. [ 11 ]. For hard sphere site-site potentials, i.e.,
uis(r)=
{~ 0
if r < ifr>/~,
[12]
Eq. [11 ] may be closed with the PY relations
hij(r) = - 1 ,
r< a
[13]
cifir) = 0,
r > o-,
[14]
where a is the distance of closest approach of any sites i and j. Equation [13] is an exact statement of the hard core condition [12]. Equation [14] is an approximation drawn from an analogy with the PY theory of hard spheres. The accuracy of this approximation may be judged by comparison with simulation data. The wo(r) must be supplied as input to the theory. Equations [ 11 ]- [ 14 ] are often referred to as the RISM equations. This system of order n 2 nonlinear coupled integral equations must be solved to obtain the local structure of a polyatomic fluid. For large n this becomes a difficult numerical problem. Curro and Schweizer (44), using a perturbative scheme, reduced the set of n 2 equations given by Eq. [11 ] to a single equation,
h(r)= f dr' f dr"w(Ir-r'l)c(Ir'-r"l) × [w(r") + pmh(r")], 1
U
Journal of Colloid and Inter/ace Science, Vol. 15 l, No. 1, J une 1992
[171
U
w(r) = 1 E wij(r),
[18]
?l ij
and P m is the monomer (or site) density of the fluid, Pm= np. For molecules in which all the correlation functions are identical, for example in homonuclear diatomics and in ring-like chains, Eq. [ 15 ] is identical to Eq. [ 11 ]. Equation [ 15 ] is referred to as the "polymer-RISM" equation. Polymer-RISM theory predicts the average g(r), while RISM predicts each of the go(r) individually. This is not a limitation, however, because g(r) is the quantity of interest for thermodynamic properties such as the compressibility, and also for use in perturbation theories for chain fluids (47). For chain fluids, Eq. [ 15 ] can be viewed as an approximation where end effects are not explicitly incorporated, though they are included in w(r) via Eq. [18] (44). For hard chains, Curro and Schweizer (44) used a closure similar to Eqs. [13] and [14]:
h(r)= - l ,
r < a,
[19]
c(r) = 0,
r > a.
[20]
Thus, given an expression for w(r), g(r) may be calculated from Equations [ 15 ], [ 19 ], and [20]. A number of approximations for w(r) have been suggested; in this work we use the approximation from the semiflexible chain model of Honnell et al. ( 51 ), which has been shown to be very accurate (52, 53). For a binary mixture of monomers (model colloidal particles, denoted as 1 ) and chains (model polymer molecules, denoted as 2) the polymer-RISM equations may be written (in Fourier space) as (46) h l l ( k ) = Cll(k) + p l C l l ( k ) h l l ( k )
[15]
where
h(r) = -~ Z h°(r)'
1
c(r) = ~5 ~ cij(r),
[16]
+ np2~12(k)h12(k),
[21]
h12(k) = vv(k)ct2(k) + pxc11(k)hzl(k)
+ nozYlz(k)hz2(k),
[22]
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INTERACTION BETWEEN COLLOIDS
h21(k) = ff(k)~2~(k) + p1~(k)~21(k)hll(k)
+ np2~(k)C22(k)l112(k),
[23]
h22(k) = if(k) c22(k)w(k)
+ old(k) Ql(k)h21(k) + nozv~(k)C22(k)hzz(k),
[24]
where the carets denote three-dimensional Fourier transforms defined by Eq. [2]. The chain-chain and c h a i n - m o n o m e r correlation functions are averaged over all the sites on the chain. By symmetry, h12 = h21 and clz = c21. In the limit as 01 -+ 0, Eqs. [ 2 2 ] - [ 2 4 ] reduce to hll(k) = c11(k) + np2cl2(k)hl2(k.)
[25]
hlz(k) = v~(k)~12(k) + np2C12(k)h22(k)
Cl2(r) = 0,
r > al2
[32]
h22(r) = - l ,
r < a
[33]
c22(r) = 0,
r > a,
[34]
where ff12 ---- (dp q- 0-)/2. ( W e have assumed that the hard sphere diameters are additive.) Once the polymer-RISM equations have been solved for h ll (r), the dimensionless colloid-colloid potential of mean force, ueer(r), is obtained from the relation
uefr(r) = - l n ( 1 + hll(r)),
[35]
and the dimensionless force between the colloid particles, Fs is obtained by differentiation of u ~ ( r ) , i.e.,
Fs = - ~
OUe~(r ) Or
[36]
[261 = l # ( k ) c12(k)
B. Numerical Procedure
+ np2~(k)42(k)ll12(k ) h22(k) = if(k)
[27]
Q2(k)~(k)
+ npgw(k)42(k)l122(k),
[281
where we have used c12 = c21 and h12 = h21. Equation [28] is the polymer-RISM equation for pure chains and may be solved independently of the other equations. Equations [26 ] and [27] are equivalent, and either may be used to solve for the colloid-polymer correlation functions (we use equation [27]). Once c22(k) is known, Eq. [27] is used to calculate h12(k) and c12(k), and then hll (k) is obtained via equation [25]. In addition to equations [ 25 ]- [ 28 ] we require closure relations between the total and direct correlation functions. We use the PY closure throughout. For colloids of diameter dp and for polymer chains with bead diameters 0-, the PY closures are given by h11(r) = - 1 ,
r < dp
[29]
c11(r) = O,
r > dp
[30]
h12(r) = - 1 ,
r < 0-12
[31]
The first step is to obtain an expression for the intramolecular structure factor ~ ( k ) for the polymers, for which, as mentioned earlier, we use the semiflexible chain model of Honnell et al. (51 ). This theory (51 ) requires, as input, the mean-square end-to-end distance R E of the polymer chains. In earlier work (52, 54), we have obtained R 2 from Monte Carlo simulations for several chain lengths and at several densities. These numbers are reproduced in Table I, where ~ is the volume fraction of the chains (n = npzrra3/3). The equations for the calculation of ~ ( k ) are somewhat lengthy and are not reported here. The model allows some unphysical overlaps between nonbonded segments on the same chain. We introduce an overlap correction for the density as suggested by Honnell et al. ( 51 ). We refer the reader to the original paper (51 ) for details of the calculation of if(k) and of the overlap correction. Once ~ ( k ) is known, the polymer-RISM equation (Eq. [28]) may be solved for the functions h2:(k) and c22(k). For numerical convenience, we rewrite Eq. [28] as JournalofColloidand[nterjaceScience,Vol. 151,No. I, June 1992
110
YETHIRAJ, HALL, AND DICKMAN
TABLE I S i m u l a t i o n Values for R 2 (52, 54) R2/¢ 2
r/
n=20
n=50
n=100
t3.01 _+ 0.01 --
-44.75 __+0.35
---
---
0.10
12.64 + 0.08
43.01 -+ 1.79
--
--
0.15 0.20
12.20 --_ 0.09 12.10 _+ 0.30
38.48 + 0.69 37.37 + 1.32
-118.70 + 4.83
-242.51 + 6.53
0.25 0.30
10.78 _+ 0.30 10.61 --+_0.31
35.70 -+ t.05 34.94 _+ 2.81
-106.84 + 6.05
-220.09 -+ 5.08
0.35
10.22 -_+ 0.28
32.23 -+ 4.30
--
--
0.05 0.06
n=8
"~22(k) -= hz2(k) - d22(k) [37] [ (v~(k))2 ] 1 -- np2~(k)&z(k) - 1 c22(k) [381
profiles of chain fluids at hard walls (35); we therefore expect the theory to be accurate for both h22(r) and hi2 (r). For very large colloid = diameters (dp > 20), however, the PY theory produces unphysical results, i.e., negative valand rewrite the PY closure as ues for the colloid-colloid pair correlation C22(r) = --I -- "Y22(r), r < a [391 function, gll (r), when Eq. [ 25 ] is used for the calculation, although reasonable results are c22(r) = 0, r > o'. [40] always obtained for g12(r). This is consistent Equations [38]-[40] form a set of two with the well known fact (56) that, even for equations for the two unknowns c22 and "Y22 hard sphere mixtures, the PY theory is inapand may be solved using standard iterative propriate for the correlation function between techniques. We solve Eqs. [ 38] with closures very large spheres at infinite dilution. In this given by Eqs. [39] and [40] using the very paper, we do not consider colloids with dp > efficient algorithm proposed by Labik et aL 20 but note that larger colloids may also be (55) for solving the OZ equation, modified to studied in the general framework by obtaining solve the polymer-RISM equation. This is an the force between the colloids approximately iterative procedure but does not require ac- from a knowledge of gl2(r), as described curate initial guesses (we use initial guesses of below. 3,(r) = 0 for all r for all the cases studied here). The force between two colloids (located at Generally less than 20 iterations were required the origin and at R) at a separation R (~ IR I) to solve the integral equation. may be calculated from the exact relation Equations [25] and [27] are solved using a procedure similar to that used for solving Eq. flFd R ) [28 ]. The entire procedure, starting with the = -2~rn~2p2 dO sin 0 cos Ogl2(s, R) calculation of v~(k) and ending with the calculation of uefc(r)generally takes less than 50[411 60 seconds on a DecStation 3100 workstation. The polymer-RISM theory has been shown where s is defined so that [s - RI = a~2, and to be very accurate (when compared to Monte 0 is the anne between s and R, i.e., Carlo simulations) for the site-site correlations in bulk chain fluids (52) and for the density s. R = Is]R cos 0. [421 Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
INTERACTION BETWEEN COLLOIDS Equation [ 41 ] is a straightforward generalization to hard chain solvents of the expression for hard sphere solvents presented in reference (18). To calculate the solvation force via equation [41] one requires the angle dependent colloid-polymer pair correlation function, gl2(S, R), for two colloids immersed in a polymer fluid, which is not available from the polymer-RISM theory presented here. Attard (18) presented a superposition approximation for calculating g12(s, R) from the colloid-solvent pair correlation function for a single colloid immersed in the solvent. For colloids located at positions R1 and R2, the superposition approximation is gl2(S, R1, R2) ~ g 1 2 ( l s - a l 1)g12 ( I s -
R21),
[43]
where gl2(S, R1, R2) is the value of the colloidpolymer pair correlation function between polymer sites at s and colloids at R~ and R2 and g~2(Irl) is the value of the colloid-polymer pair correlation function between polymer sites at r and a single colloid at the origin. The function g~2(Irl ) is calculated using the polymer-RISM theory presented. The resulting expression for the force is
111
packing and entropic effects similar to those discussed earlier in the context of density profiles of polymers at hard walls. This is depicted in Fig. l, which compares the polymer-colloid correlation function g12(r) in 8-mer and 20mer fluids, for a colloid diameter dp= 10, and for polymer volume fractions of ~ -- 0.05 and 0.35. The abscissa in Figs. 1 and 2 is the distance from the colloids measured from contact in units of the chain bead diameter. At low polymer densities, the local polymer density is depleted in the region near the colloid, while at high polymer densities the local polymer density is enhanced near the colloid when compared to the bulk. In the bulk region, g12(r) = l, this is achieved within a few bead diameters. The depletion at low densities is more severe for longer chains, as evidenced by the lower value ofgl2(r) near r 0 for 20 mers when compared to 8 mers. The enhancement is stronger for the shorter chains as evidenced by the higher value ofgl2 (r) near r - a12 = 0 for 8 mers when compared to 20 mers. Both of these trends are observed be-
o-12
z
2.4 ] ,,,~TI= 0,35
flFs( R ) =-2~rn0-212p2g12(0-12)
f:
2"0/ !~,~ n=8
dO sin 0 cos Og~2(s), [44]
where s = !/R 2 + ~ 2 - 2R0-12cos 0. By comparing predictions using the superposition approximation to a more rigorous theory for the density profile of a fluid at a single sphere, Attard found ( 18) that the superposition approximation was very accurate for the solvation force between colloids immersed in a hard sphere solvent when one of the colloids was the same size as the solvent spheres. 3. RESULTS AND DISCUSSION
A. Colloid-Polymer Correlations The colloid-polymer site distribution function is governed by a competition between
}
1.6 1.2 1.0
0.8
0.4
0
1
2
3
(r - o 12)/o
FIG. 1. Polymer-colloid pair distribution function, glz(r), for n = 8 (--) and 20 (---), for ~ = 0.05 and 0.35, and for dp= 10. Journal of Colloid and Inleffhce Sciemz~, Vol. 151, No. 1, June 1992
112
YETHIRAJ,
HALL, AND
cause the restrictions on chain configurational entropy are smaller for 8 mers than for 20 mers. At the higher density, the profiles are similar for 8 mers and 20 mers. At a distance of r - o,2 = a the correlation functions show a discontinuity in slope; at the higher density this takes the form of a cusp. This cusp is a consequence of the fixed bond lengths employed in the model for the chains and has been observed before in simulations of chain molecules (52). The packing/entropic effects discussed above become more severe as the colloid diameter is increased. Figure 2 compares the polymer-colloid correlation function in 20 mers, for colloid diameters of 1 and 20 and for polymer volume fractions of 0.05 and 0.35. At low density, the depletion of the chains near the colloid surface is greater for the larger colloid since the restrictions on the configurational entropy of a single chain are greater near a surface with less curvature. At high density the enhancement of the chains near the colloid surface is greater for the larger colloid since
2.4
2.0
!~!i!
~q = 0 . 3 5
1.6
~
1.2 1,0
J
0.8
.~J
" ~=0.05
"np
20
0.4
o
~
2
3
(r - %2)/c~ FIG. 2. P o l y m e r - c o l l o i d p a i r d i s t r i b u t i o n f u n c t i o n ,
gl2(r), for n = 20, for ;7 = 0.05 and 0.35, and for dp = 1 (--) and 20 (---). Journal~?fColloidandInterfaceScience,Vol. 151, No. 1, June 1992
DICKMAN
the chains can pack more efficiently at a surface with less curvature. B. Colloid-Colloid Potential o f Mean Force At low polymer volume fractions, where entropic effects dominate, one expects the potential of mean force to be attractive. At high polymer volume fractions, where packing effects dominate, one expects a repulsive potential of mean force at some separations (because the polymer chains have to be pushed out from between the colloids for them to be brought closer), but an attractive mean force at separations smaller than the size of the polymer beads (because all the polymer chains have already been pushed out). This is in fact observed. In Fig. 3 we plot the potential of m e a n force between colloids (d o = 10) in an 8-mer fluid for polymer volume fractions of 0.05, 0.2, and 0.3. In Fig. 3 and the figures that follow the abscissa is the separation of the colloids measured from contact in units of the polymer bead diameter. At the lowest density, the potential of m e a n force is always attractive; as the density is increased, the potential minim u m at small separations deepens, but a potential m a x i m u m develops at a separation of the order of the polymer bead diameter. At the highest density depicted, the repulsion is quite marked. An intuitive picture of the repulsive force is that it results from squeezing polymer beads out from the region between the colloids. At the highest density the potential of mean force resembles that observed for charged colloids in aqueous solutions, where electrostatic repulsion causes a peak in the potential of mean force. In the limit as the colloids touch each other, the strength of the attraction increases with increasing polymer density. This simply reflects an increased osmotic pressure. In Fig. 4 we plot the potential of mean force between colloids ( d p = 15 ) immersed in a 20-mer fluid at three different densities. The features of the potential of mean force are qualitatively similar to those discussed above for colloids in an 8-mer fluid.
INTERACTION
BETWEEN
113
COLLOIDS
:I
1.5
4
1.0 =0.3
./I]=0.3
0.5 0.0 \ "~-0.5
11=0.2 = 0,05
-1.0
-1.5
-2.0
-2.5
0
1
2
3
(r - dp)/G
0
1
2 (r - dp)/(~
F~G. 3. Potential of mean force between colloids for n = 8, for dp= 10, and for 7/= 0.05, 0.2, and 0.3.
FIG. 5. Force between colloids for n = 20, for dp= 10, and for v/= 0.1 and 0.3.
Further insight into the potential of m e a n force m a y be obtained by looking at the force between the particles. In Fig. 5 we depict the force between two colloid particles ( d p = 10)
immersed in a 20-mer fluid at volume fractions of 0.1 and 0.3. (The curves are obtained from a crude numerical differentiation using a two-point formula and are therefore not smooth.) At the lower density the force is always attractive and is similar to that reported for the force between two walls immersed in a polymer fluid (35). At the higher density and at large separations, the force is an oscillatory function with a period of approximately one polymer bead diameter. A sharp repulsion is present at a separation of exactly one bead diameter; this marks the expulsion of the last polymer bead from the region between the colloids. At smaller separations the force becomes attractive since polymer beads are now largely excluded from this region. The strength of the attraction in the potential of m e a n force increases with increasing polymer chain length (at the same polymer volume fraction) at low polymer volume fractions. On the other hand, the strength of the repulsion in the potential of mean force decreases with increasing polymer chain length. Figure 6 depicts the potential of m e a n force
251 1.25
~
/
~"~" 11= 0.3
0.0
TI = 0.05 -1.25
-2.5 0
1
2
3
(r - dp)/~
F)G. 4. Potential of mean force between colloids for n = 20, for dp= 15, and for ~/= 0.05, 0.2, and 0.3.
Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
114
YETHIRAJ,
HALL, AND
0.5 n=8 0.0
-0.5
-1,0
-1.5
-2.0
o
;
2
3
(r dp)/C~ -
FIG. 6. Potential of mean forcebetween colloidsfor dp = 10, n = 0.2, and n = 8, 50, and 100. between colloids (dv = 10) immersed in 8mer, 50-mer, and 100-mer fluids for a polymer volume fraction of 0.2. As the polymer chain length is increased, the strength of the attraction increases, but it is possible that it becomes independent of chain length for large n. Notice that the difference between the potential of mean force in 8-mer and 50-mer fluids is quite large, but the corresponding difference between the potential of mean force in 50-mer and 100-mer fluids is much smaller. Figure 7 depicts the potential of mean force between colloids (dp = 10) immersed in 8-mer, 50-mer, and 100-mer fluids for a polymer volume fraction of 0.3. In this case the strength of the repulsion decreases with increasing chain length as mentioned earlier. Again, the figure shows that the potential of mean force becomes less sensitive to the actual value of n for large n. Both the decrease in the strength of the repulsion and the increase in the strength of the attraction are caused by the increased restrictions on the chain configurational entropy as the chain length is increased. The qualitative features of the potential of mean force discussed above are the same for Journal of Colloid and Interface Science, Vol. ! 51, No. I, June 1992
DICKMAN
various colloid diameters, but the attraction and the repulsion become more marked as the colloid diameter is increased. Figure 8 depicts the potential of mean force between colloids immersed in a 20-mer fluid for a polymer volume fraction of 0.1 and for colloid diameters of 1, 5, 10, and 20. As noted above, the strength of the attraction increases with increasing colloid diameter. This is consistent with the observation that the packing and entropic effects are more important for large colloids (relative to the polymer beads) than for small ones. Figure 9 depicts the potential of mean force between colloids immersed in a 20-mer fluid for a polymer volume fraction of 0.3 and for colloid diameters of 1, 5, 10, and 15. (We could not get meaningful results for gll(r) for dp = 20). Here we see that the strength of both the attractive and repulsive potentials increase with increasing colloid diameter, since packing effects are more important for polymers between large colloids. For the smallest colloid diameter, the repulsion is small enough that it is almost negligible. In this case, packing effects are negligible, consistent with the conclusions of Fig. 2.
1.0
n=8
0.5
0.0
~
=0.5
-1.0 -1.5 =2.0 -2.5 ,
(r
- dp)/O
FIG. 7. P o t e n t i a l o f m e a n force b e t w e e n colloids for dp
= 10, ~ = 0.3, and n = 8, 50, and I00.
INTERACTION
o.ol---
dp=l
BETWEEN
dp=5
,o1 / -1.5
0
1
2
(r- dp)/(~ FIG. 8. Potential of mean force betweencolloids for n = 20, n = 0.1, and dp= 1, 5, 10, and 20. 4. C O N C L U S I O N S
We have presented an integral equation theory to determine the potential of mean force between two colloidal particles immersed in a polymer solution. The colloid particles are modeled as hard spheres, the polymer molecules are modeled as freely jointed chains of hard spheres (beads), and the solvent is modeled as a continuum that does not interact with either the colloids or the polymers. The model therefore focuses on excluded volume effects in the system. We find that at low polymer densities, the potential of mean force is attractive. This is because entropic restrictions on the chain molecules cause them to prefer the region in the bulk to that between the colloids. This results in a mean force, due to the osmotic pressure of the polymer solution, that tends to push the colloids together. At colloid separations smaller than the bead diameter, the strength of the attraction increases with increasing chain length and increasing polymer density. At high polymer densities, however, the packing of the chains at the colloid surfaces causes
1 15
COLLOIDS
a repulsion to appear in the potential of mean force at a colloid separation of the order of the bead diameter. The strength of this repulsion increases with increasing polymer density and decreases with increasing chain length. The molecular model, in itself, is fairly crude, and we have therefore not attempted to determine phase behavior or compare with experimental results. The model could be made more realistic by including the effect of the solvent and by including other interactions between the colloids, between the colloids and the polymers, and between the polymers. The solvent could be treated either as a continuum that modifies the colloid-colloid, colloid-polymer, and polymer-polymer intermolecular potentials, i.e., in the McMillanMayer approach (21 ), or as a third component in the mixture RISM format that interacts with the colloids and polymers. The McMillanMayer approach is probably more appearing. Solving the RISM equations for three-component systems is not difficult, but treating the solvent explicitly introduces six interaction potentials that must be determined from a comparison with experiment; treating the solvent
2.5 I 2.0-
,01 A
~
0.0 .
.
.
.
.
.
.
.
.
.
.
.
.
.
-0.5
"1"0 i -1.5 -2.0
-2.5 -3.0 .
o
i
i (r - dp)/~
FIG. 9. P o t e n t i a l o f m e a n force b e t w e e n colloids for n = 20, n = 0.3, a n d d~ = 1, 5, 10, a n d 15.
Journal oJCot[oid and Interface Science, Vol. 151, No. 1, June 1992
1 16
YETHIRAJ, HALL, AND DICKMAN
as a continuum reduces the number of unknown interaction potentials from six to three. If the McMillan-Mayer approach is to be used, effective colloid-colloid, colloid-polymer, and polymer-polymer interactions must be incorporated into the RISM formalism. Other interactions between the colloids (such as electrostatic interactions), and between the colloids and the polymer are easy to include, with the appropriate changes in the closure relations. Various closures, such as the hypernetted-chain or the mean spherical (47), are known to be fairly accurate. Incorporating other interactions between the polymer beads are not as simple. Recall that the RISM theory used in this paper requires as input the intramolecular pair correlation function, which is obtained here in an approximate (though accurate) manner from the semiflexible chain model. When bead-bead interactions other than the hard sphere interaction are considered it may be difficult to obtain approximations for the intramolecular pair correlation function. In fact, the intramolecular and intermolecular correlations ought to be treated in a self-consistent manner (46); to date this problem has not been solved. A simple way to get around this problem, however, is to model the polymers as hard chains towards each other and to use an experimentally determined value for the mean square end-to-end distance, R 2, in the RISM theory. Alternatively, the intramolecular correlation function could be determined via Monte Carlo simulations for R 2. One disadvantage of our approach is that very large colloidal particles (relative to the size of the chain) cannot be directly treated. This is because the PY theory fails for large spheres at infinite dilution. The superposition approximation, however, has been found to yield an accurate prediction for the force between the colloids (18) and so is a viable alternative to the direct calculation of the potential of mean force. Also, if the phase behavior is of primary importance, it might be useful to solve the RISM equations for a mixJournal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
ture of colloids and polymer directly for the thermodynamic properties; at higher colloid densities, it is likely that the numerical problems associated with large colloids might not be present. In conclusion, we have presented an integral equation theory that promises to be very useful in the study of polymer-induced colloid-colloid interactions. The theory is simple to implement, and its extension to more complex systems here is straightforward. ACKNOWLEDGMENTS We thank S. Labik and A. Malijevskyfor providingus with a copy of their FORTRAN code for the numerical solution of the OZ equation for hard spheres. This study was supported by the Gas Research Institute under Grant 5082-260-724 and by the Department of Energy under Grant DE-FG05-91ER14181. Acknowledgementis made to the donors of the Petroleum Research Fund, administered by the AmericanChemicalSociety,for partial support of this research. REFERENCES 1. Napper, D. H., "PolymericStabilization of Colloidal Dispersions," AcademicPress, (1983). 2. Vrij, A., PureAppL Chem. 48, 471 (1976). 3. Vincent, B., Luckham, P. F., and Waite, F. A., J. Colloid Interface Sci. 73, 508 (1980). 4. De Hek, H., and Vrij, A., J. Colloid Interface Sci. 84, 409 ( 1981). 5. Sperry,P. R., Hopfenberg,H. B., and Thomas, N. L., J. Colloidlnterface Sci. 82, 62 ( 1981). 6. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 109, 161 (1986). 7. Asakura,S., and Oosawa,F., J. Chem. Phys. 22, 1255 (1954); J. Polym. Sci. 33, 183 (1958). 8. Joanny, J. F., Leibler, L., and de Gennes, P. G., J. Polym. Sci. Polym. Phys. Ed. 17, 1073 (1979). 9. Feigin, R. I., and Napper, D. H., J. Colloid Interface Sci. 75, 525 (1980). 10. Sperry,P. R., J. Colloid Interface Sci. 87, 375 (1982). 11. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 96, 251 ( 1983); Faraday Discuss. Chem. Soc. 76, 189 (1983). 12. Fleer, G. J., Scheutjens, J. H. M. H., and Vincent, B., in "PolymerAdsorptionand Dispersion Stability" (E. D. Goddard and B. Vincent, Eds.). ACS Symposium Series 240, 245, Washington,DC, (1984). 13. Vincent, B., in "PolymerAdsorption and Dispersion Stability" (E. D. Goddard and B. Vincent, Eds.). ACS SymposiumSeries 240, 1, Washington, DC, (1984).
INTERACTION BETWEEN COLLOIDS 14. Cates, D. L., and Hirtzel, C. S., J. Colloid Interface Sci. 120, 404 (1987). 15. Vincent, B., Colloids Surf 24, 269 (1987). 16. Vincent, B., Edwards, J., Emmett, S., and Croot, R., Colloids Sur. 31, 267 (1988). 17. Fleer, G. J., Scheutjens, J. M. H. M., and Cohen Stuart, M. A., Colloids Surf. 31, 1 (1988). 18. Attard, P., J. Chem. Phys. 91, 3083 (1989). 19. Canessa, E., Grimson, M. J., and Silbert, M., Molec. Phys. 67, 1153 (1989). 20. Santore, M. M., Russel, W. B., and Prud'homme, R. K., Macromolecules 23, 3821 (1990). 21. McMillan, W. G., and Mayer, J. E., J. Chem. Phys. 13, 276 (1945). 22. Flory, P. J., "Principles of Polymer Chemistry," Cornell University Press, Ithaca, NY, 1953. 23. Derjaguin, B., Koll. Z. 69, 155 (1934). 24. Scheutjens, J. M. H. M., and Fleer, G. J., J. Phys. Chem. 83, 1619 (1979); J. Phys. Chem. 84, 178 (1980). 25. Dickman, R., and Hall, C. K., Y. Chem. Phys. 89, 3168 (1988). 26. Yethiraj, A., and Hall, C. K., Y. Chem. Phys. 91, 4827 (1989). 27. Madden, W. G., J. Chem. Phys. 88, 3934 (1988). 28. Kumar, S. K., Vacatello, M., and Yoon, D. Y., J. Chem. Phys. 89, 5206 (1988). 29. Mansfield, K. F., and Theodorou, D. N., Macromolecules 22, 3143 (1989). 30. Yethiraj, A., and Hall, C. K., Macromolecules 23, 1865 (1990). 31. Bitsanis, I., and Hadziioannou, G., £ Chem. Phys. 92, 3827 (1990). 32. Yethiraj, A., and Hall, C. K., Macromolecules 24, 709 (1991). 33. Yethirja, A., and Hall, C. K., Mol. Phys. 73, 503 (1991). 34. Gee, M. L., and Israelachvili, J. N., Y. Chem. Soc. Faraday Trans. 86, 4049 (1990). 35. Yethiraj, A., and Hall, C. K., J. Chem. Phys. 95, 3749 (1991).
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36. Chandler, D., McCoy, J. D., and Singer, S. J., J. Chem. Phys. 85, 5971 (1986). 37. McMullen, W. E., and Freed, K. F., J. Chem. Phys. 92, 1413 (1990). 38. Woodward, C. E., J. Chem. Phys. 94, 3183 (1991). 39. Christenson, H. K., Gruen, D. W. R., and Israelachvili, J. N., J. Chem. Phys. 87, 1834 (1987). 40. Horn, R. G., and Israelachvili, J. N., J. Chem. Phys. 75, 1400 (1981). 41. Israelachvili, J. N., and Kott, S. J., J. Chem. Phys. 88, 7162 (1988). 42. Patel, S., and Tirrell, M., Ann. Rev. Phys. Chem. 40, 597 (1989). 43. Dickman, R., and Hall, C. K., J. Chem. Phys. 85, 4108 (1986). 44. Curro, J. G., and Sehweizer, K. S., J. Chem. Phys. 87, 1842 (1987). 45. Schweizer, K. S., and Curro, J. G., Macromoleeules 21, 3070 ( 1988); J. Chem. Phys. 89, 3342 ( 1988); J. Chem. Phys. 89, 3350 (1988). 46. Schweizer K. S., and Curro, J. G., J. Chem. Phys. 91, 5059 (1989). 47. Hansen, J.-P., and McDonald, I. R., "Theory of Simple Liquids," Academic Press, San Diego, 1986. 48. Chandler, D., and Andersen, H. C., J. Chem. Phys. 57, 1930 (1972). 49. Chandler, D., J. Chem. Phys. 59, 2742 (1973). 50. Chandler, D., in "Studies in Statistical Mechanics VIII," North-Holland, Amsterdam, 1982. 51. Honnell, K. G., Curro, J. G., and Schweizer, K. S., Macromolecules 23, 3496 (1990). 52. Yethiraj, A., Hall, C. K., and Honnell, K. G., J. Chem. Phys. 93, 4453 (1990). 53. Yethiraj, A., and Hall, C. K., J. Chem. Phys. 93, 5315 (1990). 54. Yethiraj, A., and Hall, C. K., J. Chem. Phys. 96, 797 (1992). 55. Labik, S., Malijevsky, A., and Vonka, P., Molec. Phys. 56, 709 (1985). 56. Torrie, G. M., Kusalik, P. G., and Patey, G. N., J. Chem. Phys. 88, 7826 (1988).
Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
The study of colloidal dispersions is of importance in technological systems such as paints, inks, electronic ceramics, oil recovery, and bioseparations and in biological systems such as blood, milk, and protein solutions ( 1). In many of these applications polymers are also present, either to enhance the stability of the dispersion (for example in the manufacture of paints and inks) or to promote separation of the colloids (for example in wastewater treatment and in protein precipitation by nonionic polymer). Consequently, colloidal systems containing polymers have been extensively studied in the last decade, and severn experimental (2-6) and theoretical investigations (7-20) have appeared in the literature. In recent years, there has been a considerable effort to obtain a theoretical understanding of colloidal systems. The increased effort
Present address: Department of Materials Science and Engineering, University of Illinois, Urbana, IL 61801. 2 To whom correspondence should be addressed.
may be attributed in part to the avaitabifity of experimental data on well-characterized simple systems, e.g., polymer latices (5). These systems are easier to model than more complex naturally occurring systems and allow one to concentrate on the physical mechanisms at play in comparisons with experiments. A particularly appealing route to treating colloid-polymer systems is to determine an effective solvent + polymer mediated interaction between the colloidal particles. The dispersion is then treated as a pseudo onecomponent system where the intermolecular potentials are functions &polymer and solvent characteristics. This essentially decouples the problem into two parts: first to determine the effective colloid-colloid interactions as a function of polymer and solvent characteristics and, second, to investigate the behavior of colloids under this interaction. This approach is analogous to the very successful McMiUanMayer theory of solutions (21 ). Theoretical efforts can therefore be classified broadly into two parts: (i) determining the effective interaction potentials or potentials of mean force, and (ii) investigating the prop-
102 0021-9797/92 $5.00 Copyright© 1992by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journalof ColloidandInterfaceScience,VoL 151,No. 1, June 1992
INTERACTION BETWEEN COLLOIDS erties of the colloid fluid, given this effective interaction. These two categories are, of course, far from mutually exclusive, but the primary thrust of most papers lies in one or the other. Effective tools for evaluating phase diagrams have been applied ( 11, 16, 19), but some important tools for estimating the potential of mean force, such as integral equations and Monte Carlo simulation, have been largely neglected. In this paper, we investigate the potential of mean force between colloidal particles immersed in a solution with free (or nonadsorbing) polymer using an integral equation approach. The primary objective of this work is to quantify the effect of excluded volume interactions on the colloidal interactions, using a model that explicitly incorporates both the chain-like nature of polymer molecules and the continuous-space nature of polymeric fluids. We therefore model the colloidal particles as hard spheres and the polymer molecules as a string of freely jointed hard spheres. The solvent is treated as a continuum that interacts with neither the colloids nor the polymers. Simple as the model is, we will show that it displays the qualitative trends that are observed in experiments. The incorporation of more realistic interactions between the molecules and an explicit treatment of the solvent are straightforward generalizations. It has been known for some time that the presence of free polymer can destabilize a colloidal dispersion by inducing an attraction between the colloidal particles (1). The attraction is due to the loss in configurational entropy that the polymer chains experience when they are in the region between the particles, causing them to prefer the region in the "bulk" of the solution. The colloids therefore experience a net attraction because of the pressure that the polymer molecules exert on the exposed surface of the colloidal particles, which pushes the particles together. The attraction has a range that is of the order of the size of the polymer molecules. It has also been suggested that increasing the polymer concentration beyond a certain limit can cause ares-
103
tabilization of the colloidal dispersion (9, 12). A restabilization at high polymer concentrations is intuitively expected, because when a few colloids are dispersed in a dense sea of polymer, their approach toward one another will be hindered by the presence of the chains, thus making flocculation unfavorable. Other reasons that have been proposed for the restabilization will be discussed shortly, after some models for describing polymer-colloid interactions have been reviewed. A model that has been used extensively to quantify the attraction between the colloidal particles due to free polymer is one due to Asakura and Oosawa (7), who modeled the colloids as hard spheres and the polymers as spheres that are impenetrable to the colloids, but fully penetrable to each other. This results in a spherical shell around each colloid particle from which polymer molecules are excluded. Overlap of these "exclusion shells" causes an attraction between the particles that increases with increasing polymer concentration and with increasing polymer molecular weight. The Asakura-Oosawa potential has been widely used to study the destabilization of dispersions due to free polymer (2, 10, 11, 14). The colloid-colloid potential of mean force predicted by this model is always attractive (beyond the hard core separation) and the strength of this attraction increases monotonically with increasing polymer concentration. Therefore the model does not predict a restabilization of the colloidal dispersion (in the absence of electrostatic interactions) at high polymer concentrations. Modifications to the Asakura-Oosawa potential using osmotic virial coefficients for the polymer have been used ( 11 ), but there is reason to believe that these modifications worsen rather than improve the calculated pair potential (18). It is sometimes suggested that the Asakura-Oosawa potential is applicable only when the polymers are much smaller than the colloids ( 11 ). Although the potential was originally derived for the above conditions, the restriction on relative sizes is entirely unnecessary (18). Feigin and Napper (9) derived an effective Journal of Colloid and Inte~aee Science, Vol. 151, No. 1, J une 1992
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YETHIRAJ, HALL, AND DICKMAN
colloid-colloid free energy of interaction using a model that explicitly incorporates the chainlike nature of the polymers and also to some extent the excluded volume of the polymers. They performed Monte Carlo simulations for an isolated rotational isomeric state chain (with no excluded volume) confined between flat plates and monitored the segment density profile perpendicular to the plates for various chain lengths and plate separations. The free energy of mixing was obtained as a function of plate separation by combining the Monte Carlo results with Flory-Huggins solution theory (22) (with an empirical expression for the interaction parameter X). The identification with spherical particles was made using the Derjaguin approximation (23). The free energy change in bringing the colloids from infinitely far apart to a given separation exhibited a minimum at zero separation, as well as a maximum at a separation corresponding to the global size of the chains. Increasing the polymer concentration resulted in a deeper attractive minimum, as well as a higher secondary repulsion. Thus their model accounts for both the destabilization and restabilization of the colloidal dispersion. As the polymer concentration is increased from zero, the attractive minimum causes a destabilization of the dispersion; if the polymer concentration is very high, the secondary maximum is high enough that a metastable state can exist. Physically, the reasons for the restabilization are the same as when electrostatic interactions are present: colloidal particles must overcome a large repulsive barrier before they can approach the attractive minimum. The Feigin and Napper theory is, however, restricted to large colloid/polymer size ratios and does not fully account for excluded volume effects. Joanny et al. (8) determined an effective colloid-colloid interaction via scaling theory. The theory predicts the segment density profile and the free energy of interaction for polymers in semidilute solutions confined between flat plates. The interaction between spheres is again obtained using the Derjaguin approximation (23). This theory predicts an attraction Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
between the spheres; when this attraction is greater than kT, the authors argue, phase separation will occur. We note that scaling theory is applicable only to very long chains. In fact, it is exact only in the limit of infinite chain length. We will argue a little later that in physical systems of interest the equivalent freely jointed chain lengths are of the order of 100 or less; these chains are probably not in the scaling regime. Restabilization of the colloidal dispersion has not been predicted using the Joanny et aL (8) theory although the theory does not rule out this possibility. Fleer et al. ( 12, 17) derived a pair potential between particles in a polymer solution via a generalization of the Scheutjens-Fleer lattice theory (24) for interacting polymers near a flat surface. The attraction between the plates was attributed to the presence of a depletion layer at the surfaces from which polymer chains are excluded. The thickness of this depletion layer decreases as the polymer concentration is increased. Using the Flory-Huggins solution theory (22), the authors calculated an approximate but analytical expression for the depletion layer thickness and the free energy of interaction as a function of chain length, chain concentration, and solvent quality. The interaction between spheres was estimated using the Derjaguin approximation; a geometric factor was included to correct for the fact that the Derjaguin approximation is valid only when the depletion layer thickness is much smaller than the colloidal particle size. They then used the binodal condition to determine the phase behavior of the system. They found that flocculation occurs when the osmotic attraction energy due to the depletion outweighs the loss in configurational entropy of the dispersed particles in going from a dispersion to a floc. At high polymer concentrations, the depletion layer thickness decreases rapidly, resulting in a weakened attraction between the particles. At some (high) polymer concentration, the attraction is weak enough that the particles cannot flocculate, and the model then predicts a restabilization of the dispersion.
INTERACTION
BETWEEN
Several other theories for the interaction of colloids in polymer solutions with and without adsorbed polymer have appeared in the literature (7-20). We do not discuss these theories because they do not introduce any physical concepts radically different from the theories that we have already discussed. Our brief review of theoretical progress in the field must therefore be considered representative rather than exhaustive. All of the above theories are in qualitative agreement with experimental data. The physical picture painted by most of these theories, however, is not entirely consistent with the large number of computer simulations (2533) and experiments (34) that address the behavior of chain fluids confined between flat plates. In most of the theories discussed above the relevant length scale (e.g., the polymer "diameter" in the Asakura-Oosawa theory (7), the correlation length in the Joanny et al. theory (8), and the depletion thickness in the Fleer et al. theory (12)) for the interaction between the plates is assumed to be of the order of the global size of the polymer (~Rg). At very low polymer densities, this is generally supported by computer simulations (30) and microscopic theories (35). At moderate polymer densities, however, computer simulations (30) and experiments (34)demonstrate that the relevant length scale for the interaction between plates is not the global size, Rg, of the polymers but rather the bead diameter, ~r, of the polymers (30, 34, 35 ). Therefore, a more correct calculation of the interaction between colloids in polymer solutions must include a more realistic model of the polymer than has been widely used, i.e., a model of polymers that includes both the length scales ( ~Rg and ~ a ) discussed above. Several other facets of the behavior of polymers between plates have been reported in the literature, and these are discussed below since they form an important limiting case for the colloid-colloid interactions we are interested in. In the last five years, there have been several investigations of the behavior of polymers
COLLOIDS
105
confined between surfaces using computer simulation (25-33), integral equation theory (35), density functional theory (36, 37, 38), and in experiments measuring the force between surfaces in polymer fluids (34, 39-42). It has been shown via simulation (30), theory (35), and experiment (34) that the force on surfaces immersed in a polymer fluid is an oscillatory function of plate separation at high chain concentrations, and is repulsive at small plate separations. The behavior of the force as a function of separation between the plates has been obtained via the segment density profile of chains at a surface (30): the segment density profile is easily measured in simulations, is related to the salvation force, and also determines the free energy of the system. It is found that the segment density profile is governed by a competition between so-called packing and entropic effects (25, 26, 30). At low chain concentrations a single-chain molecule experiences a loss in configurational entropy as it approaches the wall; this causes the chains to prefer the region far away from the wall and results in a depletion of chain sites near the wall (entropic effect), causing an attractive force between the walls. At high chain concentrations, the chains near the wall suffer collisions with other chains and are pushed against the wall; this causes an enhancement of chain sites near the wall (packing effect) and results in a repulsive force between the walls. At high chain concentrations, the density profile is qualitatively similar to that of a hard sphere fluid at a wall or to the radial distribution function, g(r), in hard spheres, and is oscillatory with a period of about one bead diameter. The "high" density referred to above is not necessarily a very large number. In fact, using the generalized Flory equation of state, (43) one can show that an enhancement of chain sites at the wall will occur for bulk volume fractions greater than about only 0.2 and is essentially independent of chain length. The oscillatory force as a function of plate separation is, in fact, a consequence of the packing effect, since at small plate separations the Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
106
YETHIRAJ,
HALL,
polymers have to be pushed out from between the plates in order to bring the plates close together. Also, for small plate separations and high chain concentrations (bulk volume fraction "~ 0.3) the polymer chains actually prefer the region between the plates to the bulk (33) because the packing effect wins out over the entropic effect. In these cases the chains are severely flattened and are almost two-dimensional (28, 30, 31 ). We note that in colloidal systems, polymer fractions greater than 0.4 are often considered, (3, 14) and packing effects are therefore important. Packing effects are not observed in traditional lattice models for polymer where each chain bead occupies one lattice site. Lattice models are able, however, to display the transition from depletion effects to enhancement effects. In this paper we use integral equations to obtain the potential of mean force between two hard spheres immersed in a hard chain fluid. We apply the polymer-RISM (Reference Interaction Site Model, not to be confused with the rotational isomeric state model) theory of Curro and Schweizer (44-46) to a mixture of hard spheres and hard chains. The colloid-colloid potential of mean force is obtained by calculating the colloid-colloid pair distribution function in the limit as the colloids become infinitely dilute. The approach is similar to that used by two of us earlier to study the adsorption of chains between flat plates (35). The effect of chain length, chain volume fraction, and colloid size on the potential of mean force between colloids is investigated. We find that at low chain densities, the potential of mean force is attractive and is qualitatively similar to the Asakura-Oosawa attraction (7). The range of the potential is of the order Rg of the chains. At high chain densities, the potential is attractive at very short colloid separations and becomes repulsive at a distance of approximately the bead diameter before decaying to zero. In some cases there is a second repulsive peak or shoulder. The depth of the attractive well in the effective potential, which is always present, increases with chain Journal of Colloid and Interface Science, Vol. 15l, No. l, June 1992
AND
DICKMAN
density and with chain length; the height of the repulsion increases with chain density but decreases with chain length. With increasing particle size, both the attraction and the repulsion are accentuated. Our results are consistent with the calculations of Feigin and Napper (9) and suggest the possibility of flocculation at moderate polymer concentrations caused by the attractive m i n i m u m and a restabilization at higher polymer concentrations caused by the repulsive maximum. We study chain lengths ranging from 8 to 100, polymer volume fractions ranging from 0.05 to 0.35, and particle diameters ranging from 5a to 20a where a is diameter of the chain beads. The range of parameters considered is relevant to the experimental data we hope to describe. For example the ratio of partide diameter, dp, to the chain size, R (where R is the root mean square end-to-end distance), ranges from 0.3 to 5, and the range of volume fractions are similar to those considered in the literature (3, 14). The chain lengths considered may seem short, but are actually quite realistic. The equivalent freely jointed chain lengths of the polymers (the Kuhn length) used in experiments may be estimated from the mean square end-to-end distance, R 2, and fully extended chain length, Nl (22). If the equivalent Gaussian chain has a chain length t7 and a segment length {, then if/-= Nl, and ~1-2 = R 2" Therefore, ff ~ ( N I ) Z / R 2 . For the highest molecular weight polymer used by Sperry et al. ( Table II of Ref ( 5 )), the authors report R 2 = 0.0484 #m 2 and Nl = 1.985 um, which gives ff ~ 81. For the smallest molecular weight polymer, r7 ~ 0 . 1 5 3 2 / 0 . 0 7 1 2 = 4.6! The equivalent chain length of the freely jointed hard chain model will be larger than those calculated above for the Gaussian chain, but are of the same order of magnitude. The rest of this paper is organized as follows. In Section 2 we describe polymer-RISM theory and the method for the calculation of the colloid-colloid potential of mean force, in Section 3 we present predictions of the theory, and in Section 4 we present our conclusions.
107
INTERACTION BETWEEN COLLOIDS 2. THEORY
sure, which, for a fluid interacting with a pair potential u(r), is given by,
A. Integral Equation Theory In this section we present equations governing our calculation of the potential of mean force between two hard spheres in a hard chain solvent. The approach is to write down integral equations for a mixture of chains and hard spheres and solve these in the limit as the hard spheres become infinitely dilute. In this limit the potential of mean force is simply related to the radial distribution function of the hard spheres. An important relationship in statistical mechanics is the Ornstein-Zernike (OZ) equation (47), which relates the total correlation function, h(r) (=-g(r) - 1, where g(r) is the radial distribution function), to the direct correlation function, c(r). For fluids interacting via spherically symmetric potentials, the OZ equation is given by
h(r) = c(r) + p f c ( r ' ) h ( ] r - r'[)dr',
[1]
where p is the number density of the fluid. The OZ equation divides the interaction between two molecules into a "direct" contribution given by the first term in Eq. [1] and an "indirect" contribution given by the second term in Eq. [1]. The indirect contribution is propagated through the other molecules in the fluid and therefore results in the convolution integral in Eq. [1]. Equation [ t ] may be considered as defining c(r). Defining a Fourier transform pair by
f(k) = k do r sin krf(r)dr 1
f(r) = ~
[2]
~
fo k sin krf(k)dk,
[3]
Eq. [1 ] may be written in Fourier space as
h(k) = ~(k) + p~(k),~(k),
c(r) = [1 + h(r)][1 -
e~U~rq,
[5]
where 13 = 1/kBT, kB is Boltzmann's constant and T is the temperature. If u(r) is the hard sphere potential, i.e.,
u(r)=m, =0,
r>cr
[61
r>~,
[7]
where ~ is the hard sphere diameter, then the PY closure may be represented by the pair of equations
h(r)=-l,
r
[8]
c(r) = 0,
r > ~.
[9]
Equation [ 8 ] is the exact hard core condition, and equation [ 9 ] is the PY approximation. The extension of the OZ equation to molecular fluids, due to Chandler and coworkers (48-50), is the RISM theory and is based on the interaction site model for molecular fluids. In this model, each molecule is assumed to consist of a number of sites; the intermolecular potential, u(r), is assumed to be the sum of site-site potentials and is given by
u(r) = Y~ u~j(r),
[10]
i
where uo is the potential between sites i and j on different molecules and r is the separation between the sites. In the freely jointed chain model, for a chain of length n there is an interaction site located at the center of each of the n hard spheres. The generalization of the Ornstein-Zernike (OZ) equation to molecules consisting of n identical interaction sites is the site-site OZ equation (48-50): h0(r) = f
ar' f
dr" ~ w i k ( ] r - r'l) k,/=l
[41
where the carets denote Fourier transforms. To solve the OZ equation for h(r) another relation between h(r) and c(r), known as a "closure" is required. A popular closure to the OZ equation is the Percus-Yevick (PY) clo-
× era(Jr' - r"])wt/(lr"l) +p
fy dr'
dr" ~ wik(Ir-r'[) k,l= 1
× c ~ l ( l r ' - r"l)h0(Ir"l),
[111
Journal of Colloid and Interface Science. Vol. 151, No. 1, June 1992
108
YETHIRAJ, HALL, AND DICKMAN
where o is the number density of molecules in the fluid, his(r) is the total site-site correlation function, (his(r) = go(r) - 1 ), 47rr2go( r) dr is the probability that sites i and j on different molecules are at a distance between r and r + dr apart, 47rrZwij(r)dr is the probability that sites i and j on the same molecule are at a distance between r and r + dr apart, and cij(r) is the direct correlation function, defined by Eq. [ 11 ]. For hard sphere site-site potentials, i.e.,
uis(r)=
{~ 0
if r < ifr>/~,
[12]
Eq. [11 ] may be closed with the PY relations
hij(r) = - 1 ,
r< a
[13]
cifir) = 0,
r > o-,
[14]
where a is the distance of closest approach of any sites i and j. Equation [13] is an exact statement of the hard core condition [12]. Equation [14] is an approximation drawn from an analogy with the PY theory of hard spheres. The accuracy of this approximation may be judged by comparison with simulation data. The wo(r) must be supplied as input to the theory. Equations [ 11 ]- [ 14 ] are often referred to as the RISM equations. This system of order n 2 nonlinear coupled integral equations must be solved to obtain the local structure of a polyatomic fluid. For large n this becomes a difficult numerical problem. Curro and Schweizer (44), using a perturbative scheme, reduced the set of n 2 equations given by Eq. [11 ] to a single equation,
h(r)= f dr' f dr"w(Ir-r'l)c(Ir'-r"l) × [w(r") + pmh(r")], 1
U
Journal of Colloid and Inter/ace Science, Vol. 15 l, No. 1, J une 1992
[171
U
w(r) = 1 E wij(r),
[18]
?l ij
and P m is the monomer (or site) density of the fluid, Pm= np. For molecules in which all the correlation functions are identical, for example in homonuclear diatomics and in ring-like chains, Eq. [ 15 ] is identical to Eq. [ 11 ]. Equation [ 15 ] is referred to as the "polymer-RISM" equation. Polymer-RISM theory predicts the average g(r), while RISM predicts each of the go(r) individually. This is not a limitation, however, because g(r) is the quantity of interest for thermodynamic properties such as the compressibility, and also for use in perturbation theories for chain fluids (47). For chain fluids, Eq. [ 15 ] can be viewed as an approximation where end effects are not explicitly incorporated, though they are included in w(r) via Eq. [18] (44). For hard chains, Curro and Schweizer (44) used a closure similar to Eqs. [13] and [14]:
h(r)= - l ,
r < a,
[19]
c(r) = 0,
r > a.
[20]
Thus, given an expression for w(r), g(r) may be calculated from Equations [ 15 ], [ 19 ], and [20]. A number of approximations for w(r) have been suggested; in this work we use the approximation from the semiflexible chain model of Honnell et al. ( 51 ), which has been shown to be very accurate (52, 53). For a binary mixture of monomers (model colloidal particles, denoted as 1 ) and chains (model polymer molecules, denoted as 2) the polymer-RISM equations may be written (in Fourier space) as (46) h l l ( k ) = Cll(k) + p l C l l ( k ) h l l ( k )
[15]
where
h(r) = -~ Z h°(r)'
1
c(r) = ~5 ~ cij(r),
[16]
+ np2~12(k)h12(k),
[21]
h12(k) = vv(k)ct2(k) + pxc11(k)hzl(k)
+ nozYlz(k)hz2(k),
[22]
109
INTERACTION BETWEEN COLLOIDS
h21(k) = ff(k)~2~(k) + p1~(k)~21(k)hll(k)
+ np2~(k)C22(k)l112(k),
[23]
h22(k) = if(k) c22(k)w(k)
+ old(k) Ql(k)h21(k) + nozv~(k)C22(k)hzz(k),
[24]
where the carets denote three-dimensional Fourier transforms defined by Eq. [2]. The chain-chain and c h a i n - m o n o m e r correlation functions are averaged over all the sites on the chain. By symmetry, h12 = h21 and clz = c21. In the limit as 01 -+ 0, Eqs. [ 2 2 ] - [ 2 4 ] reduce to hll(k) = c11(k) + np2cl2(k)hl2(k.)
[25]
hlz(k) = v~(k)~12(k) + np2C12(k)h22(k)
Cl2(r) = 0,
r > al2
[32]
h22(r) = - l ,
r < a
[33]
c22(r) = 0,
r > a,
[34]
where ff12 ---- (dp q- 0-)/2. ( W e have assumed that the hard sphere diameters are additive.) Once the polymer-RISM equations have been solved for h ll (r), the dimensionless colloid-colloid potential of mean force, ueer(r), is obtained from the relation
uefr(r) = - l n ( 1 + hll(r)),
[35]
and the dimensionless force between the colloid particles, Fs is obtained by differentiation of u ~ ( r ) , i.e.,
Fs = - ~
OUe~(r ) Or
[36]
[261 = l # ( k ) c12(k)
B. Numerical Procedure
+ np2~(k)42(k)ll12(k ) h22(k) = if(k)
[27]
Q2(k)~(k)
+ npgw(k)42(k)l122(k),
[281
where we have used c12 = c21 and h12 = h21. Equation [28] is the polymer-RISM equation for pure chains and may be solved independently of the other equations. Equations [26 ] and [27] are equivalent, and either may be used to solve for the colloid-polymer correlation functions (we use equation [27]). Once c22(k) is known, Eq. [27] is used to calculate h12(k) and c12(k), and then hll (k) is obtained via equation [25]. In addition to equations [ 25 ]- [ 28 ] we require closure relations between the total and direct correlation functions. We use the PY closure throughout. For colloids of diameter dp and for polymer chains with bead diameters 0-, the PY closures are given by h11(r) = - 1 ,
r < dp
[29]
c11(r) = O,
r > dp
[30]
h12(r) = - 1 ,
r < 0-12
[31]
The first step is to obtain an expression for the intramolecular structure factor ~ ( k ) for the polymers, for which, as mentioned earlier, we use the semiflexible chain model of Honnell et al. (51 ). This theory (51 ) requires, as input, the mean-square end-to-end distance R E of the polymer chains. In earlier work (52, 54), we have obtained R 2 from Monte Carlo simulations for several chain lengths and at several densities. These numbers are reproduced in Table I, where ~ is the volume fraction of the chains (n = npzrra3/3). The equations for the calculation of ~ ( k ) are somewhat lengthy and are not reported here. The model allows some unphysical overlaps between nonbonded segments on the same chain. We introduce an overlap correction for the density as suggested by Honnell et al. ( 51 ). We refer the reader to the original paper (51 ) for details of the calculation of if(k) and of the overlap correction. Once ~ ( k ) is known, the polymer-RISM equation (Eq. [28]) may be solved for the functions h2:(k) and c22(k). For numerical convenience, we rewrite Eq. [28] as JournalofColloidand[nterjaceScience,Vol. 151,No. I, June 1992
110
YETHIRAJ, HALL, AND DICKMAN
TABLE I S i m u l a t i o n Values for R 2 (52, 54) R2/¢ 2
r/
n=20
n=50
n=100
t3.01 _+ 0.01 --
-44.75 __+0.35
---
---
0.10
12.64 + 0.08
43.01 -+ 1.79
--
--
0.15 0.20
12.20 --_ 0.09 12.10 _+ 0.30
38.48 + 0.69 37.37 + 1.32
-118.70 + 4.83
-242.51 + 6.53
0.25 0.30
10.78 _+ 0.30 10.61 --+_0.31
35.70 -+ t.05 34.94 _+ 2.81
-106.84 + 6.05
-220.09 -+ 5.08
0.35
10.22 -_+ 0.28
32.23 -+ 4.30
--
--
0.05 0.06
n=8
"~22(k) -= hz2(k) - d22(k) [37] [ (v~(k))2 ] 1 -- np2~(k)&z(k) - 1 c22(k) [381
profiles of chain fluids at hard walls (35); we therefore expect the theory to be accurate for both h22(r) and hi2 (r). For very large colloid = diameters (dp > 20), however, the PY theory produces unphysical results, i.e., negative valand rewrite the PY closure as ues for the colloid-colloid pair correlation C22(r) = --I -- "Y22(r), r < a [391 function, gll (r), when Eq. [ 25 ] is used for the calculation, although reasonable results are c22(r) = 0, r > o'. [40] always obtained for g12(r). This is consistent Equations [38]-[40] form a set of two with the well known fact (56) that, even for equations for the two unknowns c22 and "Y22 hard sphere mixtures, the PY theory is inapand may be solved using standard iterative propriate for the correlation function between techniques. We solve Eqs. [ 38] with closures very large spheres at infinite dilution. In this given by Eqs. [39] and [40] using the very paper, we do not consider colloids with dp > efficient algorithm proposed by Labik et aL 20 but note that larger colloids may also be (55) for solving the OZ equation, modified to studied in the general framework by obtaining solve the polymer-RISM equation. This is an the force between the colloids approximately iterative procedure but does not require ac- from a knowledge of gl2(r), as described curate initial guesses (we use initial guesses of below. 3,(r) = 0 for all r for all the cases studied here). The force between two colloids (located at Generally less than 20 iterations were required the origin and at R) at a separation R (~ IR I) to solve the integral equation. may be calculated from the exact relation Equations [25] and [27] are solved using a procedure similar to that used for solving Eq. flFd R ) [28 ]. The entire procedure, starting with the = -2~rn~2p2 dO sin 0 cos Ogl2(s, R) calculation of v~(k) and ending with the calculation of uefc(r)generally takes less than 50[411 60 seconds on a DecStation 3100 workstation. The polymer-RISM theory has been shown where s is defined so that [s - RI = a~2, and to be very accurate (when compared to Monte 0 is the anne between s and R, i.e., Carlo simulations) for the site-site correlations in bulk chain fluids (52) and for the density s. R = Is]R cos 0. [421 Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
INTERACTION BETWEEN COLLOIDS Equation [ 41 ] is a straightforward generalization to hard chain solvents of the expression for hard sphere solvents presented in reference (18). To calculate the solvation force via equation [41] one requires the angle dependent colloid-polymer pair correlation function, gl2(S, R), for two colloids immersed in a polymer fluid, which is not available from the polymer-RISM theory presented here. Attard (18) presented a superposition approximation for calculating g12(s, R) from the colloid-solvent pair correlation function for a single colloid immersed in the solvent. For colloids located at positions R1 and R2, the superposition approximation is gl2(S, R1, R2) ~ g 1 2 ( l s - a l 1)g12 ( I s -
R21),
[43]
where gl2(S, R1, R2) is the value of the colloidpolymer pair correlation function between polymer sites at s and colloids at R~ and R2 and g~2(Irl) is the value of the colloid-polymer pair correlation function between polymer sites at r and a single colloid at the origin. The function g~2(Irl ) is calculated using the polymer-RISM theory presented. The resulting expression for the force is
111
packing and entropic effects similar to those discussed earlier in the context of density profiles of polymers at hard walls. This is depicted in Fig. l, which compares the polymer-colloid correlation function g12(r) in 8-mer and 20mer fluids, for a colloid diameter dp= 10, and for polymer volume fractions of ~ -- 0.05 and 0.35. The abscissa in Figs. 1 and 2 is the distance from the colloids measured from contact in units of the chain bead diameter. At low polymer densities, the local polymer density is depleted in the region near the colloid, while at high polymer densities the local polymer density is enhanced near the colloid when compared to the bulk. In the bulk region, g12(r) = l, this is achieved within a few bead diameters. The depletion at low densities is more severe for longer chains, as evidenced by the lower value ofgl2(r) near r 0 for 20 mers when compared to 8 mers. The enhancement is stronger for the shorter chains as evidenced by the higher value ofgl2 (r) near r - a12 = 0 for 8 mers when compared to 20 mers. Both of these trends are observed be-
o-12
z
2.4 ] ,,,~TI= 0,35
flFs( R ) =-2~rn0-212p2g12(0-12)
f:
2"0/ !~,~ n=8
dO sin 0 cos Og~2(s), [44]
where s = !/R 2 + ~ 2 - 2R0-12cos 0. By comparing predictions using the superposition approximation to a more rigorous theory for the density profile of a fluid at a single sphere, Attard found ( 18) that the superposition approximation was very accurate for the solvation force between colloids immersed in a hard sphere solvent when one of the colloids was the same size as the solvent spheres. 3. RESULTS AND DISCUSSION
A. Colloid-Polymer Correlations The colloid-polymer site distribution function is governed by a competition between
}
1.6 1.2 1.0
0.8
0.4
0
1
2
3
(r - o 12)/o
FIG. 1. Polymer-colloid pair distribution function, glz(r), for n = 8 (--) and 20 (---), for ~ = 0.05 and 0.35, and for dp= 10. Journal of Colloid and Inleffhce Sciemz~, Vol. 151, No. 1, June 1992
112
YETHIRAJ,
HALL, AND
cause the restrictions on chain configurational entropy are smaller for 8 mers than for 20 mers. At the higher density, the profiles are similar for 8 mers and 20 mers. At a distance of r - o,2 = a the correlation functions show a discontinuity in slope; at the higher density this takes the form of a cusp. This cusp is a consequence of the fixed bond lengths employed in the model for the chains and has been observed before in simulations of chain molecules (52). The packing/entropic effects discussed above become more severe as the colloid diameter is increased. Figure 2 compares the polymer-colloid correlation function in 20 mers, for colloid diameters of 1 and 20 and for polymer volume fractions of 0.05 and 0.35. At low density, the depletion of the chains near the colloid surface is greater for the larger colloid since the restrictions on the configurational entropy of a single chain are greater near a surface with less curvature. At high density the enhancement of the chains near the colloid surface is greater for the larger colloid since
2.4
2.0
!~!i!
~q = 0 . 3 5
1.6
~
1.2 1,0
J
0.8
.~J
" ~=0.05
"np
20
0.4
o
~
2
3
(r - %2)/c~ FIG. 2. P o l y m e r - c o l l o i d p a i r d i s t r i b u t i o n f u n c t i o n ,
gl2(r), for n = 20, for ;7 = 0.05 and 0.35, and for dp = 1 (--) and 20 (---). Journal~?fColloidandInterfaceScience,Vol. 151, No. 1, June 1992
DICKMAN
the chains can pack more efficiently at a surface with less curvature. B. Colloid-Colloid Potential o f Mean Force At low polymer volume fractions, where entropic effects dominate, one expects the potential of mean force to be attractive. At high polymer volume fractions, where packing effects dominate, one expects a repulsive potential of mean force at some separations (because the polymer chains have to be pushed out from between the colloids for them to be brought closer), but an attractive mean force at separations smaller than the size of the polymer beads (because all the polymer chains have already been pushed out). This is in fact observed. In Fig. 3 we plot the potential of m e a n force between colloids (d o = 10) in an 8-mer fluid for polymer volume fractions of 0.05, 0.2, and 0.3. In Fig. 3 and the figures that follow the abscissa is the separation of the colloids measured from contact in units of the polymer bead diameter. At the lowest density, the potential of m e a n force is always attractive; as the density is increased, the potential minim u m at small separations deepens, but a potential m a x i m u m develops at a separation of the order of the polymer bead diameter. At the highest density depicted, the repulsion is quite marked. An intuitive picture of the repulsive force is that it results from squeezing polymer beads out from the region between the colloids. At the highest density the potential of mean force resembles that observed for charged colloids in aqueous solutions, where electrostatic repulsion causes a peak in the potential of mean force. In the limit as the colloids touch each other, the strength of the attraction increases with increasing polymer density. This simply reflects an increased osmotic pressure. In Fig. 4 we plot the potential of mean force between colloids ( d p = 15 ) immersed in a 20-mer fluid at three different densities. The features of the potential of mean force are qualitatively similar to those discussed above for colloids in an 8-mer fluid.
INTERACTION
BETWEEN
113
COLLOIDS
:I
1.5
4
1.0 =0.3
./I]=0.3
0.5 0.0 \ "~-0.5
11=0.2 = 0,05
-1.0
-1.5
-2.0
-2.5
0
1
2
3
(r - dp)/G
0
1
2 (r - dp)/(~
F~G. 3. Potential of mean force between colloids for n = 8, for dp= 10, and for 7/= 0.05, 0.2, and 0.3.
FIG. 5. Force between colloids for n = 20, for dp= 10, and for v/= 0.1 and 0.3.
Further insight into the potential of m e a n force m a y be obtained by looking at the force between the particles. In Fig. 5 we depict the force between two colloid particles ( d p = 10)
immersed in a 20-mer fluid at volume fractions of 0.1 and 0.3. (The curves are obtained from a crude numerical differentiation using a two-point formula and are therefore not smooth.) At the lower density the force is always attractive and is similar to that reported for the force between two walls immersed in a polymer fluid (35). At the higher density and at large separations, the force is an oscillatory function with a period of approximately one polymer bead diameter. A sharp repulsion is present at a separation of exactly one bead diameter; this marks the expulsion of the last polymer bead from the region between the colloids. At smaller separations the force becomes attractive since polymer beads are now largely excluded from this region. The strength of the attraction in the potential of m e a n force increases with increasing polymer chain length (at the same polymer volume fraction) at low polymer volume fractions. On the other hand, the strength of the repulsion in the potential of mean force decreases with increasing polymer chain length. Figure 6 depicts the potential of m e a n force
251 1.25
~
/
~"~" 11= 0.3
0.0
TI = 0.05 -1.25
-2.5 0
1
2
3
(r - dp)/~
F)G. 4. Potential of mean force between colloids for n = 20, for dp= 15, and for ~/= 0.05, 0.2, and 0.3.
Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
114
YETHIRAJ,
HALL, AND
0.5 n=8 0.0
-0.5
-1,0
-1.5
-2.0
o
;
2
3
(r dp)/C~ -
FIG. 6. Potential of mean forcebetween colloidsfor dp = 10, n = 0.2, and n = 8, 50, and 100. between colloids (dv = 10) immersed in 8mer, 50-mer, and 100-mer fluids for a polymer volume fraction of 0.2. As the polymer chain length is increased, the strength of the attraction increases, but it is possible that it becomes independent of chain length for large n. Notice that the difference between the potential of mean force in 8-mer and 50-mer fluids is quite large, but the corresponding difference between the potential of mean force in 50-mer and 100-mer fluids is much smaller. Figure 7 depicts the potential of mean force between colloids (dp = 10) immersed in 8-mer, 50-mer, and 100-mer fluids for a polymer volume fraction of 0.3. In this case the strength of the repulsion decreases with increasing chain length as mentioned earlier. Again, the figure shows that the potential of mean force becomes less sensitive to the actual value of n for large n. Both the decrease in the strength of the repulsion and the increase in the strength of the attraction are caused by the increased restrictions on the chain configurational entropy as the chain length is increased. The qualitative features of the potential of mean force discussed above are the same for Journal of Colloid and Interface Science, Vol. ! 51, No. I, June 1992
DICKMAN
various colloid diameters, but the attraction and the repulsion become more marked as the colloid diameter is increased. Figure 8 depicts the potential of mean force between colloids immersed in a 20-mer fluid for a polymer volume fraction of 0.1 and for colloid diameters of 1, 5, 10, and 20. As noted above, the strength of the attraction increases with increasing colloid diameter. This is consistent with the observation that the packing and entropic effects are more important for large colloids (relative to the polymer beads) than for small ones. Figure 9 depicts the potential of mean force between colloids immersed in a 20-mer fluid for a polymer volume fraction of 0.3 and for colloid diameters of 1, 5, 10, and 15. (We could not get meaningful results for gll(r) for dp = 20). Here we see that the strength of both the attractive and repulsive potentials increase with increasing colloid diameter, since packing effects are more important for polymers between large colloids. For the smallest colloid diameter, the repulsion is small enough that it is almost negligible. In this case, packing effects are negligible, consistent with the conclusions of Fig. 2.
1.0
n=8
0.5
0.0
~
=0.5
-1.0 -1.5 =2.0 -2.5 ,
(r
- dp)/O
FIG. 7. P o t e n t i a l o f m e a n force b e t w e e n colloids for dp
= 10, ~ = 0.3, and n = 8, 50, and I00.
INTERACTION
o.ol---
dp=l
BETWEEN
dp=5
,o1 / -1.5
0
1
2
(r- dp)/(~ FIG. 8. Potential of mean force betweencolloids for n = 20, n = 0.1, and dp= 1, 5, 10, and 20. 4. C O N C L U S I O N S
We have presented an integral equation theory to determine the potential of mean force between two colloidal particles immersed in a polymer solution. The colloid particles are modeled as hard spheres, the polymer molecules are modeled as freely jointed chains of hard spheres (beads), and the solvent is modeled as a continuum that does not interact with either the colloids or the polymers. The model therefore focuses on excluded volume effects in the system. We find that at low polymer densities, the potential of mean force is attractive. This is because entropic restrictions on the chain molecules cause them to prefer the region in the bulk to that between the colloids. This results in a mean force, due to the osmotic pressure of the polymer solution, that tends to push the colloids together. At colloid separations smaller than the bead diameter, the strength of the attraction increases with increasing chain length and increasing polymer density. At high polymer densities, however, the packing of the chains at the colloid surfaces causes
1 15
COLLOIDS
a repulsion to appear in the potential of mean force at a colloid separation of the order of the bead diameter. The strength of this repulsion increases with increasing polymer density and decreases with increasing chain length. The molecular model, in itself, is fairly crude, and we have therefore not attempted to determine phase behavior or compare with experimental results. The model could be made more realistic by including the effect of the solvent and by including other interactions between the colloids, between the colloids and the polymers, and between the polymers. The solvent could be treated either as a continuum that modifies the colloid-colloid, colloid-polymer, and polymer-polymer intermolecular potentials, i.e., in the McMillanMayer approach (21 ), or as a third component in the mixture RISM format that interacts with the colloids and polymers. The McMillanMayer approach is probably more appearing. Solving the RISM equations for three-component systems is not difficult, but treating the solvent explicitly introduces six interaction potentials that must be determined from a comparison with experiment; treating the solvent
2.5 I 2.0-
,01 A
~
0.0 .
.
.
.
.
.
.
.
.
.
.
.
.
.
-0.5
"1"0 i -1.5 -2.0
-2.5 -3.0 .
o
i
i (r - dp)/~
FIG. 9. P o t e n t i a l o f m e a n force b e t w e e n colloids for n = 20, n = 0.3, a n d d~ = 1, 5, 10, a n d 15.
Journal oJCot[oid and Interface Science, Vol. 151, No. 1, June 1992
1 16
YETHIRAJ, HALL, AND DICKMAN
as a continuum reduces the number of unknown interaction potentials from six to three. If the McMillan-Mayer approach is to be used, effective colloid-colloid, colloid-polymer, and polymer-polymer interactions must be incorporated into the RISM formalism. Other interactions between the colloids (such as electrostatic interactions), and between the colloids and the polymer are easy to include, with the appropriate changes in the closure relations. Various closures, such as the hypernetted-chain or the mean spherical (47), are known to be fairly accurate. Incorporating other interactions between the polymer beads are not as simple. Recall that the RISM theory used in this paper requires as input the intramolecular pair correlation function, which is obtained here in an approximate (though accurate) manner from the semiflexible chain model. When bead-bead interactions other than the hard sphere interaction are considered it may be difficult to obtain approximations for the intramolecular pair correlation function. In fact, the intramolecular and intermolecular correlations ought to be treated in a self-consistent manner (46); to date this problem has not been solved. A simple way to get around this problem, however, is to model the polymers as hard chains towards each other and to use an experimentally determined value for the mean square end-to-end distance, R 2, in the RISM theory. Alternatively, the intramolecular correlation function could be determined via Monte Carlo simulations for R 2. One disadvantage of our approach is that very large colloidal particles (relative to the size of the chain) cannot be directly treated. This is because the PY theory fails for large spheres at infinite dilution. The superposition approximation, however, has been found to yield an accurate prediction for the force between the colloids (18) and so is a viable alternative to the direct calculation of the potential of mean force. Also, if the phase behavior is of primary importance, it might be useful to solve the RISM equations for a mixJournal of Colloid and Interface Science, Vol. 151, No. 1, June 1992
ture of colloids and polymer directly for the thermodynamic properties; at higher colloid densities, it is likely that the numerical problems associated with large colloids might not be present. In conclusion, we have presented an integral equation theory that promises to be very useful in the study of polymer-induced colloid-colloid interactions. The theory is simple to implement, and its extension to more complex systems here is straightforward. ACKNOWLEDGMENTS We thank S. Labik and A. Malijevskyfor providingus with a copy of their FORTRAN code for the numerical solution of the OZ equation for hard spheres. This study was supported by the Gas Research Institute under Grant 5082-260-724 and by the Department of Energy under Grant DE-FG05-91ER14181. Acknowledgementis made to the donors of the Petroleum Research Fund, administered by the AmericanChemicalSociety,for partial support of this research. REFERENCES 1. Napper, D. H., "PolymericStabilization of Colloidal Dispersions," AcademicPress, (1983). 2. Vrij, A., PureAppL Chem. 48, 471 (1976). 3. Vincent, B., Luckham, P. F., and Waite, F. A., J. Colloid Interface Sci. 73, 508 (1980). 4. De Hek, H., and Vrij, A., J. Colloid Interface Sci. 84, 409 ( 1981). 5. Sperry,P. R., Hopfenberg,H. B., and Thomas, N. L., J. Colloidlnterface Sci. 82, 62 ( 1981). 6. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 109, 161 (1986). 7. Asakura,S., and Oosawa,F., J. Chem. Phys. 22, 1255 (1954); J. Polym. Sci. 33, 183 (1958). 8. Joanny, J. F., Leibler, L., and de Gennes, P. G., J. Polym. Sci. Polym. Phys. Ed. 17, 1073 (1979). 9. Feigin, R. I., and Napper, D. H., J. Colloid Interface Sci. 75, 525 (1980). 10. Sperry,P. R., J. Colloid Interface Sci. 87, 375 (1982). 11. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 96, 251 ( 1983); Faraday Discuss. Chem. Soc. 76, 189 (1983). 12. Fleer, G. J., Scheutjens, J. H. M. H., and Vincent, B., in "PolymerAdsorptionand Dispersion Stability" (E. D. Goddard and B. Vincent, Eds.). ACS Symposium Series 240, 245, Washington,DC, (1984). 13. Vincent, B., in "PolymerAdsorption and Dispersion Stability" (E. D. Goddard and B. Vincent, Eds.). ACS SymposiumSeries 240, 1, Washington, DC, (1984).
INTERACTION BETWEEN COLLOIDS 14. Cates, D. L., and Hirtzel, C. S., J. Colloid Interface Sci. 120, 404 (1987). 15. Vincent, B., Colloids Surf 24, 269 (1987). 16. Vincent, B., Edwards, J., Emmett, S., and Croot, R., Colloids Sur. 31, 267 (1988). 17. Fleer, G. J., Scheutjens, J. M. H. M., and Cohen Stuart, M. A., Colloids Surf. 31, 1 (1988). 18. Attard, P., J. Chem. Phys. 91, 3083 (1989). 19. Canessa, E., Grimson, M. J., and Silbert, M., Molec. Phys. 67, 1153 (1989). 20. Santore, M. M., Russel, W. B., and Prud'homme, R. K., Macromolecules 23, 3821 (1990). 21. McMillan, W. G., and Mayer, J. E., J. Chem. Phys. 13, 276 (1945). 22. Flory, P. J., "Principles of Polymer Chemistry," Cornell University Press, Ithaca, NY, 1953. 23. Derjaguin, B., Koll. Z. 69, 155 (1934). 24. Scheutjens, J. M. H. M., and Fleer, G. J., J. Phys. Chem. 83, 1619 (1979); J. Phys. Chem. 84, 178 (1980). 25. Dickman, R., and Hall, C. K., Y. Chem. Phys. 89, 3168 (1988). 26. Yethiraj, A., and Hall, C. K., Y. Chem. Phys. 91, 4827 (1989). 27. Madden, W. G., J. Chem. Phys. 88, 3934 (1988). 28. Kumar, S. K., Vacatello, M., and Yoon, D. Y., J. Chem. Phys. 89, 5206 (1988). 29. Mansfield, K. F., and Theodorou, D. N., Macromolecules 22, 3143 (1989). 30. Yethiraj, A., and Hall, C. K., Macromolecules 23, 1865 (1990). 31. Bitsanis, I., and Hadziioannou, G., £ Chem. Phys. 92, 3827 (1990). 32. Yethiraj, A., and Hall, C. K., Macromolecules 24, 709 (1991). 33. Yethirja, A., and Hall, C. K., Mol. Phys. 73, 503 (1991). 34. Gee, M. L., and Israelachvili, J. N., Y. Chem. Soc. Faraday Trans. 86, 4049 (1990). 35. Yethiraj, A., and Hall, C. K., J. Chem. Phys. 95, 3749 (1991).
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36. Chandler, D., McCoy, J. D., and Singer, S. J., J. Chem. Phys. 85, 5971 (1986). 37. McMullen, W. E., and Freed, K. F., J. Chem. Phys. 92, 1413 (1990). 38. Woodward, C. E., J. Chem. Phys. 94, 3183 (1991). 39. Christenson, H. K., Gruen, D. W. R., and Israelachvili, J. N., J. Chem. Phys. 87, 1834 (1987). 40. Horn, R. G., and Israelachvili, J. N., J. Chem. Phys. 75, 1400 (1981). 41. Israelachvili, J. N., and Kott, S. J., J. Chem. Phys. 88, 7162 (1988). 42. Patel, S., and Tirrell, M., Ann. Rev. Phys. Chem. 40, 597 (1989). 43. Dickman, R., and Hall, C. K., J. Chem. Phys. 85, 4108 (1986). 44. Curro, J. G., and Sehweizer, K. S., J. Chem. Phys. 87, 1842 (1987). 45. Schweizer, K. S., and Curro, J. G., Macromoleeules 21, 3070 ( 1988); J. Chem. Phys. 89, 3342 ( 1988); J. Chem. Phys. 89, 3350 (1988). 46. Schweizer K. S., and Curro, J. G., J. Chem. Phys. 91, 5059 (1989). 47. Hansen, J.-P., and McDonald, I. R., "Theory of Simple Liquids," Academic Press, San Diego, 1986. 48. Chandler, D., and Andersen, H. C., J. Chem. Phys. 57, 1930 (1972). 49. Chandler, D., J. Chem. Phys. 59, 2742 (1973). 50. Chandler, D., in "Studies in Statistical Mechanics VIII," North-Holland, Amsterdam, 1982. 51. Honnell, K. G., Curro, J. G., and Schweizer, K. S., Macromolecules 23, 3496 (1990). 52. Yethiraj, A., Hall, C. K., and Honnell, K. G., J. Chem. Phys. 93, 4453 (1990). 53. Yethiraj, A., and Hall, C. K., J. Chem. Phys. 93, 5315 (1990). 54. Yethiraj, A., and Hall, C. K., J. Chem. Phys. 96, 797 (1992). 55. Labik, S., Malijevsky, A., and Vonka, P., Molec. Phys. 56, 709 (1985). 56. Torrie, G. M., Kusalik, P. G., and Patey, G. N., J. Chem. Phys. 88, 7826 (1988).
Journal of Colloid and Interface Science, Vol. 151, No. 1, June 1992