Faraday Discuss. Chem. SOC., 1983, 76, 189-201

Phase Separations Induced in Aqueous Colloidal Suspensions by Dissolved Polymer

BY ALICE P. GAST, CAROL K. HALL AND WILLIAM B. RUSSEL

Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544, U.S.A.

Received 16th May, 1983

Several experimental studies have demonstrated the tendency of non-adsorbing dissolved polymer to cause a phase separation in otherwise homogeneous colloidal suspensions. This separation occurs with hard-sphere systems as well as those stabilized either sterically or electrostatically. A restabilization at higher polymer concentrations has been observed in some electrostatically and sterically stabilized dispersions.

A theory based on statistical mechanics presented here predicts such phase transitions in aqueous colloidal suspensions. The effective interaction potential between two colloidal par- ticles comprises a combination of the volume restriction potential of Asakura and Oosawa and the classical electrostatic repulsion of Derjaguin. The free energy is calculated via second- order perturbation theory with an effective hard-sphere system serving as the reference state. A fluid-solid phase diagram emerges showing the volume fraction of particles in each phase as a function of osmotic pressure. The details of the phase behaviour depend on the polymer concentration and molecular weight, the ionic strength and the electrostatic potential and size of the particles. At high polymer concentrations, a semi-dilute theory of Muthukumar and Edwards is employed to account for the decreasing range of polymer-polymer and polymer-particle interactions with increasing concentration. This behaviour provides a mech- anism for electrostatic restabilization at sufficient polymer concentrations.

The results for the polymer concentration required for destabilization, cF, agree well with experimental observations of Sperry, Hopfenberg and Thomas. Corresponding measurements of the densities of the resulting phases have not been reported. The predictions of restabiliz- ation also remain to be tested quantitatively, but the results conform qualitatively with experi- mental observation.

The flocculation or destabilization of colloidal dispersions by the presence of dissolved polymer has been observed in several systems.'-3 We recently presented a statistical-mechanical treatment of phase transitions induced in non-aqueous hard- sphere suspensions by non-adsorbing dissolved polymer molecules in a theta sol- vent.4 The interaction potential for this system includes an attractive component with magnitude proportional to the polymer osmotic pressure, ascribed to the depl- etion of polymer segments in the region between two approaching colloidal particles. At sufficiently high polymer concentration this leads to a separation of the system into colloid-rich and colloid-poor phases. Similar phase behaviour also has been observed in aqueous colloidal suspensions. Unlike the hard spheres, however, electrostatically or sterically stabilized systems have been observed to restabilize at even higher polymer concentrations.2, In this paper we extend our previous treat- ment of polymer-induced flocculation to describe both destabilization and restabiliz- ation of aqueous systems.

Sperry6 has dealt with the destabilization of aqueous suspensions by non-

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190 POLY MER-INDUCED PHASE SEPARATIONS

adsorbing dissolved polymer by combining a classical description of the electrosta- tic repulsion and van der Waals attraction with the volume restriction mechanism of Asakura and 0 0 s a w a . ~ ~ The original derivation of Asakura and Oosawa assumes that the colloidal particles interact as hard spheres while the polymer molecules behave as hard spheres towards the colloidal particles but freely interpenetrate each other. Sperry retains the same geometrical arguments about depletion of polymer segments from the region of width twice the radius of gyration, 2R,, between two approaching particles but includes polymer-polymer interactions through the sec- ond virial coefficient in the osmotic pressure. He obtains the total effective interac- tion potential from the sum of the repulsive and attractive components. His cal- culations of the depth of the potential well corresponding to the experimentally determined flocculation concentration for a series of polymers indicate an average potential minimum of -2.7kT at the phase boundary. This average minimum is then used as an empirical rule to calculate concentrations required for flocculation.

For semi-dilute polymer solutions in good solvents Joanny et al.9 invoke a similar criterion for phase separation. They use a mean-field theory to predict the segment concentration profile and interaction free energy between two flat plates as a function of polymer concentration and monomer unit length. Their treatment of the polymer solution includes excluded-volume effects and employs the screening length, c, to characterize the range of the monomer-monomer interactions. The interaction potential between two large spheres is derived from that between flat plates through the Derjaguin approximation. The resulting attraction goes to zero at surface-to-surface separations > nt , the distance where the segment concentration between spheres equals the bulk solution concentration. Phase separation is assumed to occur when the minimum free energy of interaction is large compared with kT.

These two theoretical approaches concentrate on the form of the interaction potential and choose a value for the depth of the potential well as a criteria for phase separation. This implies no dependence on the volume fraction of particles and provides no information about the nature of the resulting phases. Our goal is to employ a more rigorous statistical-mechanical approach with a similar potential, comprising an electrostatic repulsion and an osmotic attraction, to predict the complete phase diagrams for aqueous suspensions over a range of experimental conditions.

We combine the classical electrostatic interaction potential for short-range repul- sions developed by Derjaguin * with the volume-exclusion potential for the effective colloid-colloid interaction potential. Thermodynamic properties are calculated via the perturbation theory developed by Barker and Henderson for Lennard-Jones fluids. This approach determines the volume fraction of colloidal particles in each phase as well as the phase boundaries as functions of the colloidal concentration, polymer osmotic pressure and electrostatic potential.

In our derivation of the attractive potential due to dissolved polymer, we retain the geometric derivation of Asakura and 00sawa.~. The resulting potential depends on the osmotic pressure of the polymer solution and the thickness of the excluded shell surrounding each particle, that is the layer depleted of polymer due to the polymer-particle interaction. Polymer-polymer interactions enter solely through the dependence of the osmotic pressure and depleted-layer thickness on concentration. In the dilute-solution regime we relate the osmotic pressure to the polymer con- centration through a virial expansion and take the range of the interaction, i.e. twice the shell thickness, to be twice the radius of gyration. At higher polymer concent- rations we employ the theory developed by Muthukumar and Edwards l 2 for both the osmotic pressure and the screening length. In this regime the screening length,

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A. P. GAST, C. K. HALL AND W. B. RUSSEL 191

which governs the depleted-layer thickness and hence the range of the interaction, decreases with increasing concentration.

Whenever the range of the attraction and of the electrostatic repulsion be- come comparable, the suspension should remain a single phase due to the greater magnitude of the latter. In dilute solutions this occurs at low ionic strengths or low polymer molecular weights. But in semi-dilute solutions increasing the polymer con- centration produces a similar effect by reducing the screening length, thereby re- stabilizing the suspensions.

Our predictions of the polymer concentration required for destabilization com- pare favourably with the experimental observations of Sperry et al.3 for polystyrene particles in aqueous solutions of hydroxyethylcellulose and ammonium nitrate. No experiments to date have recorded the volume fractions in the two phases for electrostatically stabilized colloids. Future experiments clearly should include measurements of phase densities and structure to provide a rigorous test of the proposed interaction mechanism.

THEORY

This section describes the statistical mechanics necessary to predict the phase behaviour of electrostatically stabilized aqueous suspensions in the presence of non- adsorbing dissolved polymer. The technique, perturbation theory, involves an expansion of the Helmholtz free energy as calculated from a pairwise additive inter- action potential; hence we first describe the effective colloid+olloid interaction potential.

Two contributions to the pair potential between charged colloidal particles sus- pended in an electrolyte solution containing dissolved polymer are considered: an electrostatic repulsion and an osmotic attraction. The suspensions are assumed to be kinetically stable due to electrostatic repulsions. Hence we ignore the van der Waals attraction which, due to its short range, generally contributes insignificantly to the attraction caused by polymer molecules at separations outside the primary maximum.6

ELECTROSTATIC REPULSION

Here we model systems wherein the Debye length (ic-l) is small relative to the particle radius, viz. d2rc/2 >> 1. The interaction potential for spheres at infinite di- lution separated by a centre-to-centre distance Y developed by Derjaguin O

u&) = ~ ln(l + exp[ - ~ ( r - d 2 ) ] ) 4

is a good approximation for d 2 4 2 > 10, where

E is the dielectric constant of the solvent, e is the charge of an electron, I is the ionic strength, $o is the surface potential and d2 is the sphere diameter.

At finite particle volume fractions and low ionic strengths the effective Debye length K - differs from the value at infinite dilution noted above due to the contri- bution of counterions to the total ionic strength as well as the displacement of solvent by the particles. This effect has been quantified for concentrated suspensions in osmotic equilibrium with the fluid containing electrolyte at ionic strength I as l 3

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192 POLY MER-INDUCED PHASE SEPARATIONS

d i K 2 = 12Q- (3)

with F(x) = 1 + 1/x2[1 + ( 1 + x - ~ ) ” ~ ] . The ratio of the dimensionless surface charge, Q G ed2zq/2ckT, where z is the ionic valence, to the dimensionless ionic strength, N = e2d$,1/4&kT, determines the effect of the colloidal volume fraction, cp,, on d 2 ~ . This expression for K reduces to the simple expression given by eqn ( 1 ) at low colloid volume fractions or high electrolyte concentrations.

OSMOTIC ATTRACTION

The model originated by Asakura and Oosawa7p8 and developed by Vrij1,14 treats the solvent (species 1) as a continuous medium and the colloidal particles (species 2) as hard spheres of diameter d2. In the dilute limit, the polymer molecules (species 3) behave as hard spheres of diameter d3 with respect to the colloidal par- ticles, forming a sheath of excluded volume of thickness d3/2 = R, around each particle. In this dilute regime we account for polymer-polymer interactions by an osmotic pressure described by a conventional virial expansion. As the polymer con- centration increases beyond the overlap concentration the solution behaves as a sea of segments with a screening length 5 < R, which decreases with increasing con- centration. In this regime the mean-field treatment of Joanny et al.9 indicates that the polymer segments are depleted from a layer of thickness g around each colloidal particle.

Two colloidal particles approaching each other within d2 + d3 (where d3 = 2R, in the dilute regime and d3 = 25 in the semi-dilute regime) exclude polymer molecules from the region between them, resulting in a deficit in the osmotic force in the gap and hence a net attraction. The attractive force equals the osmotic pressure Po,, times the area inaccessible to polymer molecules so that

7L F(r) = { -4(4dg3 - r2)Posm d2 < r < 2d23 (4)

C O 2 d 2 3 < r

where dZ3 = (d2 + d3)/2 (and d3 = 2R, or 25). Integration of this force determines the pair potential as

r < d2

where Po,, is determined from the polymer solution thermodynamics.

expansion in polymer number density n At low polymer concentrations the osmotic pressure is described by a virial

+ . - . ) A2M2,n kT

where A 2 is the second virial coefficient, M , is the molecular weight and N is Avogadro’s number.

At higher polymer concentrations, in the semi-dilute regime, Muthukumar and Edwards have developed general expressions for chain size, osmotic pressure and

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A. P. GAST, C. K . HALL AND W. B.

screening length as functions of polymer concentration The screening length

RUSSEL 193

and excluded-volume effects.

where a is a constant, w is the excluded volume parameter, f is the Kuhn step length and L is the contour length, and the mean-square end-to-end distance determine the polymer dimensions and interaction range. The osmotic pressure

varies as n9I4 in accord with scaling results13 rather than as n2 as in the dilute regime.

We determine the crossover point between the dilute and semi-dilute regimes by requiring that the range of the attractive interaction between colloidal particles be continuous at the crossover concentration n,. Thus, setting the dilute solution range d3 equal to 25 and using eqn (7) yields the crossover concentration applicable to polymer solutions interacting with colloidal particles as

Tn order that the magnitude of the attractive potential be continuous at the crossover, the osmotic pressure from the virial expansion must match that of Muthu- kumar and Edwards;I4 hence

40n ~ 16nna3 1i4w3/413N5/4n:'4

NA - 243( 9 ) relates A2 to w. Rearranging and substitution into eqn (9) provides

ylc = ( 5NA )"'(chains/volurne) 12nA2M,2d;

or in concentration units

thereby enabling the calculation of b, or c, from dilute solution data for A 2 . Hence, above n, the interaction range follows the scaling law

5 = "(">'I4 2 n

from eqn (7) and (9) and the osmotic pressure follows eqn (8).

PERTURBATION THEORY

In order to determine the phase equilibria for a one-component system with a given interparticle potential, it is necessary to calculate the Gibbs free energy and pressure as a function of the system temperature and density. The perturbation theory based on the hard-sphere reference state was outlined previously for inter-

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194 POLYMER-INDUCED PHASE SEPARATIONS

actions between hard spheres experiencing the osmotic attraction. In order to use the same approach with the electrostatic interactions the steep repulsive potential must be converted to an effective hard-sphere potential of diameter, i.e. range, d’ .

we separate the potential Following the approach of Barker and Henderson into the repulsive and attractive parts as

u(d2 + 6s) s =

Aii(r) r > u(r) =

with u(a) = 0. The steepness of the repulsion

r - d , a - d z 6 6

<-

0

1/6 is gauged by

-KEdz$i - kT 1 - -- dul = - dr r = dz 8 d2 6’

For electrostatically stabilized systems at moderate ionic strengths 6 << l / d 2 ~ << 1 . The magnitude of the dimensionless parameter 3, characterizing the attraction de- pends on the details of the electrostatic and osmotic potentials but cannot exceed (P,,,u3/kT)(dz/d3) (where v3 = 7cd$/6), the magnitude of the latter at contact. Hence for 6,3, + 0 the potential reduces to that for hard spheres.

The Helmholtz free energy

A = -kT In Q (1 5 ) follows from the partition function Q which depends on the configurational integral ZN as

Q = ZN/NA3N (16) for a system of N particles. The configurational integral is related to the interaction potential by

ZN = J . . . J exp[ - u,(r)/kr]dr, . . . drN (17) where ri is the position of the ith sphere and uT equals the sum of the pair potentials.

This relationship between the partition function and the interaction permits a Taylor-series expansion for Helmholtz free energy in terms of 6

Substitution of eqn (1 6 t ( 18) leads to

- N k T - A - -- NkT A H S 627cpd.ipHs(dr){d’ - I:[ 1 - exp( - %)Id.)

potential and A

(18)

(19)

where gHS(r) and AHS are the radial distribution function and Helmholtz free energy for a system of hard spheres of diameter d’. Choice of the effective hard-sphere diameter as

d’ = Ji[I - exp( - $)Id.

then suppresses the (6 (6) term in the expansion for all densities.

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A. P. GAST, C. K . HALL AND W. B. RUSSEL 195

The final expression for the Helmholtz free energy, retaining the A 2 term in- dicated formally above, is

where ( d ~ / d P ) ~ ~ is given by eqn (20). systems of interest second-order term

the bulk compressibility function for the system of hard spheres The 0 (d2) and C (61) terms are negligible since 6 << 1 for all the here. The use of the macroscopic compressibi!ity ( d ~ / a P ) ~ ~ in the will be discussed in the next section.

This expansion, along with the thermodynamic identities

and

determines the pressure and Gibbs free energy of the system.

HARD-SPHERE REFERENCE SYSTEM

Calculation of the free energy from eqn (21) requires the free energy, compressi- bility and radial distribution function of the hard-sphere reference system. Molecular-dynamics studies have demonstrated a hard-sphere fluid-solid phase transition in the absence of any attraction; hence both fluid and solid reference states are required for our calculations. The former exists for volume fractions (p2 = p d 3 ~ / 6 up to 0.497, at which point a solid phase of (p2 = 0.551 appears and persists up to closest packing cpz = 0.740. The equations of state and radial distri- bution functions employed in our calculations are briefly discussed here and are described in more detail el~ewhere.~

The free energy, pressure and bulk compressibility are extracted from the appro- priate equations of state zHS((p2) = PHs/pkT. Carnahan and Starling derived an expression for the fluid phase which agrees with exact virial expansions and molecular-dynamics calculations. The equation of state for the solid phase has been derived by Hall l8 in terms of Pad6 approximants fitted to Monte Carlo data.

The radial distribution function for pairs of particles is defined by

but exact calculations from eqn (24) are very difficult even for hard spheres. For the fluid phase several approximate integral equations are available for the radial distri- bution function. We adopt the Percus-Yevick approximation in the form of an explicit analytical solution derived by Smith and Henderson. This result agrees quite well with molecular-dynamics calculations at low densities; however, several defects emerge at higher densities. Verlet and Weis 2o developed an iterative correc- tion to the Percus-Yevick gHS(r) requiring that gas(d) conform with the virial theorem

g H S ( d ) = [zHS(V2) - 11/4%- (25)

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196 POLYMER-INDUCED PHASE SEPARATIONS

The Verlet-Weis gHS(r ) agrees with molecular dynamics to within & 0.03. The solid-phase radial distribution function, generated by Monte Carlo calcu-

lations, was fitted with a simple approximate analytical equation by Kincaid and Weis.21 Their equations provide values for gHS(r) in the range 0.52 < ( p 2 < 0.73 with mean-square deviations from the Monte Carlo data of < 0.026.

PROCEDURE

Our calculations of the phase transitions in aqueous colloidal suspensions via second-order perturbation theory follow the work of Barker and Henderson with a Lennard-Jones fluid. We verified our fluid-phase routine by recalculating their tabulated values for the free energy and pressure for a Lennard-Jones fluid.22 Use of the macroscopic, instead of the local, compressibility approximation introduced an error no greater than that due to the discrepancy of kO.03 in the Verlet-Weis g(r). Since the local compressibility approximation requires an additional differentiation of gHS(r ) with respect to density, thereby magnifying the error in gHS(r), we retained the simpler macroscopic approximation.

The calculations began with integration of eqn (20) with Simpson’s rule to deter- mine the effective hard-sphere diameter d’ as a function of polymer osmotic pressure for each set of the parameters d2/d3, d 2 ~ and &$;d2/4kT. Subsequent calculation of A through integration of eqn (21) via Simpson’s rule required 120 steps for conver- gence to within 0.05%. Integrations for the solid phase were limited to the non-zero portions of gHS(r), minimizing the number of steps required for convergence.

The first-order perturbation term is always negative, as expected for an attractive potential, and usually less than the reference term, while the second-order term varies in sign but remains < 1/20th of the first-order term. Since the 0 ( A 2 ) term in the expansion is much less than the 0 (A) term, the perturbation theory is clearly valid for the conditions considered.

To determine the phase transition at a particular polymer osmotic pressure, we plot the Gibbs free energy against the pressure for colloid volume fractions in both the fluid and solid phases. The intersection of the curves from plots of G/kT against P/pokT, where po is the density at closest packing, for the individual phases deter- mines the coexistence pressure, p’, for the phase transition. The densities of the fluid and solid phases follow from the corresponding plot of pressure P/p,kT against density p/po . Repeating these calculations for different polymer densities maps out the phase diagrams which indicate the onset of the phase transition, and the possi- bility of restabilization, for a range of polymer concentrations and colloidal volume fractions.

RESULTS

We have calculated the phase diagrams for the destabilization at dilute polymer concentrations for a range of values of the colloid-to-polymer diameter ratio, $2/d3, at a fixed d21c (fig. 1) as well as for a range of d2rc at fixed d2/d3 (fig. 2). These phase diagrams are presented as plots of the volume fraction of colloidal particles against a dimensionless osmotic pressure, (P,, ,u~/k T)(d2/d3), which reflects the magnitude of the osmotic force at contact. Conversion to the polymer concentration requires knowledge of the second virial coefficient and molecular weight. Note the transition without polymer corresponding to the phase transition for hard spheres of effective diameter given by eqn (20).

As d2/d3 increases the attractive part of the interaction potential [eqn (4)]

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A. P. GAST, C. K. HALL AND W. B. RUSSEL 197

sol id

(P2

I V . " ,

/ two-phase

0 2 5 6 8 10 12 14 Posm u3 dz ~-

kT d3

Fig. 1. Phase diagrams showing plot of volume fraction of colloid, c p 2 , against dimensionless polymer osmotic pressure, ( P o s m u3/kT)(d2/d3), for d,/d3 = (a) 3.72, (b) 4.67, (c) 8.67 and (d) 12.54 with dzlc = 141.4 and €d2$8/4kT = 1459.8. The lower line is the fluid-phase boundary;

the upper line is the solid boundary.

solid 0.8-

t w o - p h a s e

J 0 2.0 4.0 6 .O 8 .O

Posm ~3 dz ___- kT d3

Fig. 2. Phase diagrams showing plots of cp, against (POsm u3/kT)(d2/d3) for ionic strengths I/mol dmP3 = (a) 0.076 (dZK = 388.9), (b) 0.038 (d2x = 275.0), (c) 0.010 ( d 2 ~ = 141.4) and

(d) 0.0076 (d2x = 122.8) with d2/d3 = 8.67 and &d2$;/4kT = 1459.8.

becomes stronger but shorter in range. In fig. 1, dzlc is held fixed, thereby maintain- ing the range of the electrostatic repulsion constant. As the range of the attraction decreases, a larger fraction of the total attraction falls inside dzrc suppressing the

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198 POLYMER-INDUCED PHASE SEPARATIONS

kT

Fig. 3. Total interaction potential plotted against reduced distance at the phase boundaries for cp2 = 0.10 in fig. 1 [(a)-(d) as in fig. 11.

secondary well. Hence as d2/d3 increases, a larger osmotic pressure is required to induce a phase transition at a given (p2. This is illustrated by a plot of the total interaction potential at the phase boundary for a range of d2/d3, as shown in fig. 3 for the phase transition at (p2 = 0.10. Note that the depth of the minimum required to cause a phase transition increases slightly as the range of the attraction, i.e. d3/d2, decreases, invalidating the simple criterion used by Sperry .

Increasing the ionic strength screens the electrostatic repulsion more efficiently, thereby diminishing its range. Hence at higher electrolyte concentrations less poly- mer is required to induce the phase transition (fig. 2).

As the polymer concentration increases into the semi-dilute regime, the range of

o+8c s o l i d

0 . € i p = =

0.4 402

0.2

0 0.2 0.4 0.6 0.8 sld1

Fig. 4. Phase diagrams showing plots of c p z against (Po,, v3/kT)(d2/d3) for destabilization and restabilization for two ionic strengths: (a) 0.076 mol dm-3(d2rC = 388.9) and (b) 0.038 mol

d m - 3 ( d 2 ~ = 275.0) for the conditions in fig. 2.

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A. P. GAST, C. K. HALL AND W. B. RUSSEL 199

the attractive potential, 2t, decreases as n-3/4. At fixed dzrc this results in a diminish- ing secondary minimum with increasing polymer concentration, ultimately leaving a purely repulsive potential. In the absence of the attractive well the suspension reverts to a single phase, providing a mechanism for restabilization at high polymer con- centrations. The complete destabilization and restabilization are illustrated as phase diagrams for d2/d3 = 8.673 at two ionic strengths in fig. 4. Note that as the electro- lyte concentration decreases, the size of the unstable region in the colloid-polymer phase diagram diminishes.

Comparison of our predictions with the data of Sperry et al.3 for polystyrene latices in aqueous solutions of hydroxyethylcellulose with added ammonium sul-

Table 1. Comparison with data of Sperry et al. 3 5 6

A2/cm3 g-2 mol-1 polymer d2/d3 M“ [ref. (23)]

~~~ ~

ER 12.547 63,800 9.3 x 10-4 GR 8.673 114,700 7.3 x 10-4 MR 4.673 306,000 4.1 x 10-4 HR 3.723 438,800 4.0 x 10-4

$o = -80 mV; d2 = 4.3 x em; K = 3.28 x lo7 x 11/* (mol dm-3)1’2 em-’; ~ d ~ $ ~ ~ / 4 k T = 1459.8.

0.4

0.3

3 5 0.2 v

0”

0 . I

0 I I 1 0.1 0.2 0.3

4 1 4 Fig. 5. Comparison of the predictions for the onset of destabilization, cF(wt %) as a function of polymer size, d3/dz, with the data of Sperry et al.3 for polystyrene particles in an aqueous solution of hydroxyethylcellulose with properties listed in table 1 at an ionic strength of 0.01

mol dm-3.

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200 POLYMER-INDUCED PHASE SEPARATIONS

phate clearly substantiates the trends predicted for the destabilization. The para- meters used in the calculations are summarized in table 1. The concentration re- quired to induce phase separation, cF(wt %), is shown as a function of polymer molecular weight in fig. 5 and electrolyte concentration in fig. 6. Sperry’s data shows the anticipated drop to cF = 0 at electrolyte concentrations sufficient to induce a primary minimum flocculation due to van der Waals attraction. Our results tend asymptotically instead to a hard-sphere limit at high ionic strengths due to our exclusion of the van der Waals attraction. Over most of the range in electrolyte concentration the van der Waals attraction does not contribute significantly. At low amounts of added salt our theory predicts a monotonically increasing cF as the range of the repulsion increases, while the measured values attain an asymptote, as if the total ionic strength were in fact constant.

0

0

0.011 1 I I I I I I l l 1 1 1 1 I I I I I I I I I I I I I J

0.01 0.1 1 .o 10.0 salt (wt%)

Fig. 6. Comparison of the predictions for the onset of destabilization, cF(wt %), as a function of the ionic strength with the tabulated data of Sperry et al. ( A ) and the graph in ref. (3) (0), for polystyrene particles in aqueous ammonium sulphate with hydroxyethylcellulose

(Ad, = 114 700).

CONCLUSIONS

We have shown that perturbation theory, combined with the volume-exclusion potential, can predict both the destabilization and the restabilization of electrostati- cally stabilized, aqueous colloidal suspensions containing non-adsorbing dissolved polymer. The technique predicts the onset of the phase separations in dilute solu- tions in quantitative agreement with experimental results and also determines the amounts and densities of the resulting phases as functions of the osmotic pressure of the polymer solutions.

The predicted restabilization arises from the screening of polymer-polymer inter- actions in semi-dilute solutions, which reduces the range of the attractive potential. Here our analysis combines the solution thermodynamics of Muthukumar and Edwards l 2 with the mean-field treatment of Joanny et aL9 for polymer-surface

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A . P. GAST, C . K. HALL AND W. B. RUSSEL 20 1

interactions. The mechanism differs fundamentally from that proposed by Feigen and N a ~ p e r . ~ ~ These initial predictions demonstrate the effect; detailed exploration still remains necessary.

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