Depletion flocculation of aqueous, electrosterically-stabilized latex

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Langmuir 1994,10, 454-463

454

Depletion Flocculation of Aqueous, Electrosterically-Stabilized Latex Dispersions Jill E. Seebergh and John C. Berg* Department of Chemical Engineering, BF-IO, University of Washington, Seattle, Washington 98195 Received November 7 , 1 9 9 P

The depletion interaction may occur when nonadsorbing, or free, polymer is present in a colloidal dispersion. Under appropriate conditions, the interaction can lead to flocculation. The present work examines aggregation stability of aqueous, electrosterically-stabilized polystyrene latices in the presence of free polymer as a function of free polymer concentration,free polymer molar mass, salt concentration, and molar mass of the steric adlayer. The stability ratio, W , is determined from early-stage aggregation kinetics as measured by photon correlationspectroscopy. The effect of salt on polymer solvencyis determined viscometrically. The depletion interaction potential energy is calculated using a modified version of the pragmatic model of Vincent et a1.192 Theoretical stability ratios are calculated from the total potential energy function and compared with experimentalstability ratios. The criticalvolume fractionfor depletion flocculation is found to decrease with increasing molar mass of free polymer. Increases in the salt concentrationdecrease the stability due to the destruction of polymer solvency. In some cases, the stability is observed to increase upon the addition of free polymer; however, this phenomenon is found to be fundamentally differentthan depletion restabilization. The model successfully describes the salientfeatures of the stability behavior.

Introduction The effects of attached polymer on the stability of colloids with respect to aggregation,i.e. steric stabilization, has been the subject of many experimental and theoretical studies. Equally important, though less well understood, are the effectsof unattached, or free, polymer on dispersion stability. Experiments have shown that certain concentrations of free polymer can induce aggregation, while relativelyhigher concentrations can impart stability. These phenomena are known as depletion flocculation and depletion stabilization, respectively. Since many systems which are of practical importance contain both free and attached polymer, it is necessary to consider both steric and depletioneffects. For example, coatingssuch as paints and adhesives include polymers which adsorb to pigment and binder particles to provide steric stabilization, as well as free polymers which act as thickeners, surfactants, etc. Knowledge of depletion phenomena is required in order to formulate the coatings such that the free polymer does not inadvertantly flocculate the pigment or binder particles. The depletion interaction occurs when nonadsorbing polymer is added to a colloidal dispersion. Consider first the case of bare particles in the presence of free polymer coils. (It should be noted that, in practice, it is very difficult to find an experimental system in which added polymer will not adsorb to a truly bare surface.) A coil in solution will maintain, on average, that configuration which is energetically most probable. Under conditions of good solvency, a polymer coil will be in an extended conformation. Assuming that the coil has a Guassian distribution of segments, the segment density is greatest at the center of mass and decreases smoothly with increasing distance from the center of mass. The coil may approach a particle surface to a distance such that its outermost segments just meet the boundary of the surface. If the center of

* Author to whom correspondence should be addressed.

e Abstract published in Advance ACS Abstracts, January 1,1994.

(1) Vincent, B.; Edwards, J.; Emmett, S.; Jones, A. Colloids Suf. 1986,

18, 261. (2) Jones, A.; Vincent, B. Colloid Surf. 1989, 42, 113.

mass of the coil were to approach more closely, either the segments would transgress the boundary, which is obviously forbidden, or the coil would have to adopt a less favorable configuration. The latter situation results in elastic repulsion due to the loss of configurational entropy upon coil deformation. Thus a gradient in the segmental concentration exists in the vicinity of the surface such that the concentration is zero at the surface and increases to the bulk value over a distance known as the depletion layer thickness. The depletion layer thickness of a bare particle, A, is considered to be of the order of either the radius of gyration- or the root mean square end-bend chain length (roughly the “diameter”)6J of a free polymer molecule. Several expressions have been developed to determine A, each of which expresses A as a function of the bulk concentration and the molar mass of the free polymer. As a rule, the depletion layer thickness decreases with increasing bulk free polymer concentration and increases with increasing molar mass. Fleer et a1.3 used a mean field approach to determine the depletion layer thickness, in which the polymer solution was modeled as a lattice. This analysis requires a numerical solution; however, an approximate analytical expression for A which is valid for solvents close to the theta point was provided. De Gennes equated the thickness of the depletion layer with the segment correlation length, E, from scaling the0ry.Q The scaling segment correlation length is a function of the bulk free polymer concentration and is equal to the radius of gyration for dilute polymer solutions. Vincent and coworkers assumed a simple linear form for the depletion layer thickness as a function of the bulk polymer concentration in some of their initial modeling.lP2 In later work, Vincent derived an analytical expression for the (3) Fleer, G. J.; Scheutjens, J. H. M. H.; Vincent, B. ACS Symp. Ser. 1984, No. 240, 245. (4) de Gennes, P. Adu. Colloid Interface Sci. 1987,27, 189. (5) Vincent, B. Colloids Surf. 1990,50, 241. (6) Feigin, R. 1.; Napper, D. H. J. Colloid Interface Sci. 1980,75,525. (7) Hunter, R. J. Foundations of Colloid Science: Oxford University

Press: New York, 1987; Vol. I. (8)de Gennes, P. G. Ann. Chim. (Rome) 1987, 77,389.

0743-7463/94/2410-0454$04.50/00 1994 American Chemical Society

Depletion Flocculation of Latex Dispersions

depletion layer thickness as a function of bulk polymer concentration, molecular weight of the polymer, and the Flory-Huggins polymer-solvent interaction parameter, x . ~ Depletion phenomena may be understood by considering two particles, each with its own depletion layer, as they approach each other in a solution of free polymer. At separation distances greater than twice the Ydiameter"of the polymer molecule, the segment concentration increases from zero at the surfaces of the particles to the bulk concentration value in the center. As the separation distance decreases, polymer solution at the bulk concentration is displaced into the bulk solution; however, the segment concentration between the particles still rises to the bulk value. When the separation is less than twice the "diameter" of a polymer molecule, there is no longer enough distance for the segment concentration to build up to the bulk value. A t this point, the polymer concentration between the particles is less than that in the bulk. Upon closer approach of the particles, polymer segments from the dilute region between the particles are forced out into the more concentrated bulk region, so the maximum in the segment concentration profile decreases with decreasing separation distance. At separation distances less than the "diameter", essentially no polymer remains between the particles and thus a reservoir of pure solvent exists. The demixing required to create a reservoir of pure solvent is energetically unfavorable in a good solvent, manifested by the existence of a repulsive potential energy barrier which opposes close approach of the particles. Modeling results are in qualitative agreement as to the shape of the depletion interaction potential, but there are some quantitative disagreements. For example, Feigin and Nappers19 found that the height of the energy barrier was sufficient to prevent flocculation, while Fleer et aL3 found that the height of the repulsive barrier was negligible for particles in the colloidal size range. The chemical potential difference between the pure solvent in the reservoir and the solvent in the bulk polymer solution is a driving force which causes pure solvent to flow from between the particles into the bulk polymer solution. In this way, the free energy of the system is lowered via aggregation of the particles. This phenomenon can also be thought of in terms of the osmotic pressure difference. The pressure in the bulk solution is greater than the pressure in the reservoir because of the polymer concentration difference, so the particles are forced together. When particles have thick depletion layers, they do not need to approach each other very closely in order to exclude polymer and create a reservoir of pure solvent. Likewise, when asystem has a relatively high osmotic pressure, there is a large driving force to push the particles together once the polymer has been excluded. Thus a system's tendency to undergo depletion flocculation will increase with increasing osmotic pressure and depletion layer thickness. Unfortunately, these parameters operate in opposite directions as a function of free polymer concentration and molar mass. As the bulk free polymer concentration is increased, the depletion layer thickness will decrease, but the bulk osmotic pressure will increase. Similarly, the depletion layer thickness increases with increasing free polymer molar mass, but the osmotic pressure decreases with increasing molar mass. The depletion interaction is thus governed by the delicate balance between the depletion layer thickness and the bulk osmotic pressure. A t very low concentrations of a given molar mass of free polymer, the depletion layer around each particle is at its maximum thickness, which promotes attraction. However, the pressure exerted on the particles by the bulk solution (9) Feigin, R. I.; Napper, D. H. J. Colloid Interface Sci. 1980,74,567.

Langmuir, Vol. 10, No. 2, 1994 455

is relatively small, so flocculation will not occur. As the polymer concentration is increased, A will decrease and the osmotic pressure will increase until the point at which flocculation occurs. Eventually, at high enough concentrations of free polymer, restabilization can occur. The restabilization is due to the existence of very thin depletion layers which cannot be compensated for by the osmotic pressure at relatively high free polymer concentrations. The depletion interaction is not limited to bare particles. Indeed, the more common practical situation is one in which the particles are sterically stabilized (Le. there is a layer of adsorbed or grafted polymer at the particle surface). The presence of a steric layer complicates the situation considerably, because the interactions of the free polymer with the steric polymer must be accounted for in addition to the interaction of the free polymer with the surface. Consider a system of sterically-stabilizedparticles dispersed i n a solution of nonadsorbing polymer. In a theta solvent, the free polymer can completely penetrate the steric layer, and the depletion layer thickness will be the same as that of a bare particle. In a good solvent, polymer-solvent contacts are energetically preferred over polymer-polymer contacts, so when the free polymer and steric polymer segments mix, osmotic repulsion will occur. Segment interpenetration can also cause elastic repulsion, which arises from the loss of configurational entropy of the free polymer segments due to the presence of the steric polymer segments, and vice versa. (It should be noted that elastic repulsion due to the presence of other polymer segments is likely to be minimal in comparison with elastic repulsion due to the presence of the particle surface.lO) The distance that the free polymer can interpenetrate the steric layer will therefore depend on the magnitude of the mixing and elastic repulsion. The free polymer segment concentration will be effectively zero from the particle surface out to the interpenetration point, whereupon it will rise smoothly up to the bulk value. The distance from the interpenetration point to the point at which the bulk segment concentration is achieved is equal to the depletion layer thickness of a bare particle. The effective depletion layer thickness for a sterically-stabilized particle will be equal to the distance from the particle surface to the point at which the bulk segment concentration is achieved. The correct mechanism for the destabilizing effect of free polymer was first identified by Asakura and Oosawa in 1954.l' They recognized that the attractive force between bare particles (either spheres or plates) suspended in a solution of monodisperse macromolecules was proportional to the osmotic pressure of the medium and that the range of the attraction was of the order of the size of the macromolecules (Le. the depletion layer thickness). The force of attraction between the particles and the interaction potential energy were determined using a statistical thermodynamic approach, the details of which are provided elsewhere.12 For the case of bare spherical particles suspended in a solution of rigid, spherical, nonadsorbing polymer, the potential energy was found to be proportional to the overlap volume of the depletion layers and the osmotic pressure of the bulk medium as given by the van't Hoff expression. SieglafP3 later extended the theory of Asakura and Oosawa by including the second virial coefficient in the osmotic pressure expression. de Hek and VrijI4and Sperry15have developed (10) Napper, D.H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1983. (11) Asakura, S.; Oosawa, F. J. Chem. Phys. 1964, 22, 1255. (12) Asakura, S.;Oosawa, F. J . Polym. Sci. 1968, 33, 183. (13) Sieglaff, C. L. J.Polym. Sci. 1959, 61, 319. (14)de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409.

456 Langmuir, Vol. 10, No. 2, 1994

expressions which are identical to that of Asakura and Oosawa; however,these investigators used straightforward geometrical considerations to calculate the attractive force and the pair potential energy rather than resorting to statistical thermodynamics. Models of the type discussed above are convenient because they can be applied to spherical particles as well as flat plates, and the depletion potential energy is given by a relatively simple analytical expression. However, these models assume that the polymer molecules act as hard spheres, which is equivalent to assuming that the thickness of the depletion layer is independentof all system parameters, including the free polymer concentration. In reality, polymer molecules are relatively flexible, so the chain dimensions will be altered as a function of the bulk concentration. Fleer e t aL3 used mean field theory to model the interaction between two bare flat plates suspended in a solution of flexible,nonadsorbing polymer. Analytical expressionsbased on the numerical results were given for the concentration-dependent depletion layer thickness and the plate potential energy, from which a sphere potential energy was calculated. Interestingly, the sphere potential expression is equivalent to the models given by Asakura and Oosawa,de Hek and Vrij, and Sperry, although the approach was different. Several other models have allowed for flexible polymers near interfaces, including both mean field and scaling law approaches from de Gennes and cO-workers1C18and a rotational isomeric state Monte Carlo (RISMC) procedure from Feigin and Nappere6 Each of these models is based on the depletion in the concentration of polymer segments near an interface, such that the overlap of depletion layers results in an osmotic attraction. Relatively few theories describe depletion phenomena in dispersions of sterically-stabilized particles. Vincent et al.19 developed a model based on the difference in the free energy of mixing of two steric layers overlapped with each other and two steric layers overlapped with free polymer coils. This theory assumes that the depletion layer is buried within the steric layer, however, this will not be the general case. Feigin and Napper6 allowed for the presence of steric layers in their RJSMC model, but it was assumed that the free polymer could completely interpenetrate the steric layer. This will only be true in the case of a theta solvent, so the model is not generally applicable. Van Lent et a1.20developeda numerical model for the depletion interaction between flat plates with terminally attached steric layers based on the ScheutjensFleer (SF)mean-field theory of polymers at interfaces.21p22 Vincent and his c o - ~ o r k e r have s ~ ~ recently ~~ developed a theory which extends the hard sphere analytical model of Fleer et aL3 to the case of coated spheres. It should be noted that direct comparisons between so-called "pragmatic" models such as that of Vincent and co-workers and models based on Scheutjens-Fleer theory ( e g . Van Lent et al.) are difficult. The parameters used in SF theory, such as the lattice length and the number of statistical segments per chain, must be scaled to reflect the properties (15) Sperry, P. R. J.Colloid Interface Sci. 1982,87,375. (16) Joanny, J. F.; Liebler, L.; de Gennes,P. G. J.Polym. Sci., Polym. Phys. Ed. 1979,17,1073. (17) de Gennes, P. G. Macromolecules 1981, 14, 1637. (18) de Gennes, P. 0 . Macromolecules 1982,15, 492. (19) Vincent, B.; Luckham, P. F.; Waite,F. A. J.Colloid Interface Sci. 1980, 73, 508. (20) Van lent, B.; leraele, R.; Seheutjena, J. M. H. M.; Fleer, G. J. J. Colloid Interface Sci. 1990,137, 380. (21) Scheutjens, J. M. H. M.; Fleer, G. L. J. Phys. Chem. 1979,83, 1619. (22) Scheutjens, J. M. H. M.; Fleer, G. J. J.Phys. Chem. 1980,84,178. (23) Vincent, B.; Clarke, J.; Barnett, K. G. Colloids Surf. 1986,17,51.

Seebergh and Berg

of real polymers. Also, SF theory applies to flat surfaces, not spherical surfaces. Each of the models described above has met with some qualitative success, although further experimental investigation is warranted. Only a few studies provide a direct, quantitative comparison between experimental and theoretical r e ~ u l t s . ~ *One ~ J ~of* the ~ drawbacks of many of the theoretical analyses is that interactions other than those due to depletion phenomena are often ignored or assumed negligible.l~3*6J3J4In real systems, electrostatic, van der Waals, steric, and hydration forces may also be present, and these must be accounted for when assessing the overall stability. Observations of depletion phenomena date back to the early part of the twentieth century, although the term "depletion" did not come into usage until the early 198l~.~* Traube= first noted in 1925 that the addition of watersoluble polymers to a dispersion of natural rubber latex resulted in creaming of the latex. Other early investigators, working with systems such as red blood cellsn-B and soils,9o observed aggregation upon the addition of naturally occurringmacromolecules such as gelatin, pectin, dextran, gums, and proteins. These initial experiments are quite interesting from a practical viewpoint; however, studies of model systems are much more useful for probing depletion phenomena. Studies examining the effects of free polymer on model colloidal dispersions can generally be divided into two categories: (1)aggregation studies and (2) phase separation studies.lO The first category includes studies in which the addition of free polymer to a dispersion results in flocculation of the particles. The second category includes studies in which the addition of free polymer results in the formation of two phases separated by a distinct interface, with no aggregates present in either phase. It should be noted that although several investigatorsle931 have chosen to describe depletion flocculation as a "phase separation", in which a "floc phase" coexists with a "dispersed phase", this is merely a construct. In fact, what was observed was aggregation and not a true phase separation. In contrast, de Hek and Vrij observed the formation of a sharp interface between two translucent liquid phases upon the addition of polystyrene to a dispersion of silica in c y c l o h e ~ a n e . ~A* ~ ~ ~ review general of depletion phase separation studies and theories is provided by Napper.lo The focus of this study w i l l be aggregation phenomena. In 1975, Li-in-onet ~ 1 . publishedone 3~ of the f i s t studies to clearly demonstrate that the addition of certain concentrations of free polymer to a model colloidal dispersion caused aggregation. The experimental system in this study was an aqueous polystyrene (PS) latex (average particle radius = 70 nm) with a terminally anchored AB comb stabilizer consisting of PS and poly(ethylene oxide) (PEO). The PS was incorporated into the latex structure and the PEO (molar mass = 750) acted as the stabilizing moiety. Homopolymer PEO ranging in (24) Vincent, B.: Edwards,. J.:. b e t t ., S.:. Croot. R. Colloids Surf. 1988,31,267. (25) Napper,D. H. In Stabilizationofcolloidabyfreepolymer;Tadroe, T. F.. Ed.: Academic Preas: London. 1982. (26) Traube, I. Gummi-Ztg. 1926,39,434. (27) Inokuchi, K. Bull. Chem. Soc. Jpn. 1961,24,78. Ezp. Biol. Med. 1946, (28) Meyer, K.; Hahnel, E.;Feiner, R. Proc. SOC. 58,36. (29) Moneghan, B. R.; White, H. L. J. Gen. Physiol. 19SSM,19, 715. (30) Gwghegan, M. J.; Brian,R C. Biochem. J. 1948,43,5. (31) Cowell, C.;Li-in-on,R.;Vicent,B. J. Chem. Soc.,Faraday Trona 1 1978, 74, 337. (32) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1979, 70, 592. (33) Li-in-on, F. K.; Vincent, B.; Waite, F. A. ACS Symp. Ser. 1976, No. 9, 165.

Depletion Flocculation of Latex Dispersions

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molar mass from 200 to 4000 was added to the dispersion land, OR) were used as the model monodisperse colloidal over a broad range of concentrations. Aggregation of the system. The latices were prepared by emulsion polymerization of styrene and were received suspended in latex upon addition of free polymer was monitored using distilled water at a concentration of 10.6f 0.1 w t % solids. turbidimetry, in which the optical density was monitored as a function of time. The results showed that for each According to the manufacturer, the dispersion was free of excess surfactant, residual monomer, preservatives, and molar mass of free polymer, a certain critical concentration other additives, and was thus used without further existed below which the dispersion was stable and above cleaning. The mean diameter as determined from transwhich aggregation was observed. The critical concentramission electron microscopy was given as 0.086 pm f tion necessary to induce aggregation was found to decrease with increasing molar mass of free polymer. In addition, 11.6% ,and the parking area was given as 2571A per charge group. The surface charge density is 0.62 pC/cm2. a second, higher critical concentration was observed above which the dispersion became increasinglymore stable with Two commercial ABA poly(ethy1ene oxide)/poly(prorespect to aggregation. The critical concentration necpylene oxide) triblock copolymers, L43 and L35, were used essary to induce restabilization was not a function of the as steric stabilizers. These nonionic, water-soluble comolar mass of the free polymer. These results were the polymers are manufactured by BASF Wyandotte Corp. first reported observations of depletion restabilization. (Wyandotte, MI) under the tradename Pluronic. L43 has a total molar mass of 1850 g/mol and a PEO molar mass Numerous studies of the parameters affecting depletion of 650 g/mol, while L35 has a total molar mass of 1900 phenomena have been published since the early work of g/mol and a PEO molar mass of 950 g/mol. It should be Li-in-on et al. In particular, Vincent and his co-workers noted that the purity and polydispersity of the Pluronic have contributed a large body of experimentalwork. Some samples are unknown, so the molar masses reported above of the parameters which have been studied include free are only approximate. polymer concentration and free polymer molecular weight,2J9*31*34-37 particle size,35*37particle volume fracUpon addition of the triblock copolymer to a dispersion of polystyrene latices, the poly(propy1ene oxide) (PPO) tion,1131*34*38 particle composition,34presence or absence of segment adsorbsto the latex surface, acting as an anchoring a steric physisorbed versus chemicallygrafted steric extent layers,3' molecular weight of the steric layer,2*31t36g37 moiety. Under conditioesof good solvency the PEO chains of steric layer ~overage,1f~J6*~9 and temperature.1g~31~37p3s extend out into the solvent, acting as a stabilizing moiety. Adsorption isotherms for L43 and L35 adsorbing on Although the role of solvency in the depletion interaction polystyrene at 24 "C were determined using differential has been considered by several investigator^^^^^^^ there refractometry (C.N. Wood Mfg., Newtown, PA). The has not been a systematic study of the effect of salts on plateau adsorption for both polymers was approximately the depletion interaction via alteration of polymer sol1.1mg/m2,and the plateau adsorption concentrationswere vency. Sperry et a1.a4 studied the effect of ammonium 0.25 and 0.2 g/L for L43 and L35, respectively, in good sulfate concentration on the concentration of hydroxyethyl agreement with values reported in the literature for cellulose necessary to flocculate an aqueous acrylic coPluronic/polystyrene systems.40-42 In a typical sample polymer latex dispersion. They found that the flocculation preparation, L35 or L43 was added to a polystyrene concentration decreased with increasingsalt concentration; dispersion a t the plateau adsorption concentration and however, no attempt was made to model this behavior in stirred for 24 h to achieve equilibrium. The dispersions terms of the change in polymer solvency upon the addition were refrigerated when not in use and discarded after 4 of salt. Likewise, Vincent and co-workers have studied the depletion interaction in systems with salt p r e ~ e n t , ~ ~days. ~~~ Four different molar mass homopolymer PEOs were yet no quantitative consideration was given to the effects used as free polymer. PEO lo00 was obtained from Fluka of salt on polymer solvency. (Switzerland), PEO 8000 was obtained from Sigma The purpose of this study was to examine systematically Chemicals (St. Louis, MO), and PEO 18 500 and PEO the effect of electrolyte concentration on the stability of 100 OOO were obtained from Polysciences, Inc. (Warrington, an electrosterically-stabilized latex in the presence of free PA). Free polymer solutions ranging in volume fraction polymer. Polymer solvency as a function of salt concenfrom 101-6 to 0.1 were used in aggregation experiments. tration was tracked through the second virial coefficient, Experiments at higher volume fractions of polymer were which was determined from viscosity measurements. not performed because it appeared as if gel structures Photon correlation spectroscopy was used to monitor the were forming in solution. The kinematic viscosity of each early-stage aggregation kinetics (i.e. the rate of doublet polymer solution was measured using a Cannon-Fenske formation) of the dispersions. A modified version of the capillary viscometer, and the density was measured using pragmatic theory of Vincent and co-workers was used to a Mettler-Paar densitometer. All measurements were calculate the depletion interaction potential energy. Sterperformed at 24 "C. ic, van der Waals, and electrostatic interactions were also Polymer Solvency Experiments. Theory predicts accounted for. Theoretical stability ratios were calculated that polymer solvency,as described by the Flory-Huggins and compared with experimental stability ratios. polymer/solvent interaction parameter, x , or the second virial coefficient,B2, is independent of polymer molar mass, Experimental Work polymer concentration, and electrolyte concentration. In Materials. Polystyrene latices with sulfate surface fact, experiments have shown that solvency depends on groups obtained from Interfacial Dynamics Corp. (Portall of these parameters to some extent.10*41143-rsIn this work, reagent grade sodium chloride (J.T.Baker Chemical (341 Sww,P.R.:Hopfenben,H.B.;Thomas,N.L. J.Colloidlnterface Sci. 198i,82;62. (35)Clarke, J.; Vincent, B. J. Colloid Interface Sci. 1981,82,208. (36)Clarke.. J.:. Vincent, B. J. Chem. Soc. Faraday Tram. I 1981.77, 1831. . (37)Cowell, C.; Vincent, B. In The stability ofpolystyrene latices in thepresenceofpoly(ethyleneoride);Tadros,T.F., Ed.; Academic Press: London, 1982. (38) Cowell, C.; Vincent, B. J. Colloid Interface Sci. 1982,87,518. (39)Milling, A.;Vincent,B.;Emmett,S.;Jones, A. Colloids Surf.1991, 57, 185.

(40)Baker, J. Ph.D. Thesis, University of Washington, 1986. (41)Einamon, M. B. Ph.D. Thesis, University of Washington, 1992. (42) Kayes, J. B.; Rawlins, D. A. Colloid Polym. Sci. 1979,257,622. (43)Ataman, M.Colloid Polym. Sci. 1987,265,19. (44)Boucher, E. A.; Hines, P. M. J.Polym. Sci., Polym. Phys. 1976, 14,2241. (45)Napper, D. H.; Netachey, A. J. Colloid Interface Sei. 1971,37, 528. (46)Napper, D.H. J. Colloid Interface Sci. 1970,33, 384.

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Seebergh and Berg

Co.) was used to induce changes in the solvency of 1000 molar mass homopolymer PEO. Polymer solvency is dependent on molar mass in the oligomer range, but above about 1O00, the dependency disappears.'O Molar mass of 1000was chosen for these experiments because the results can be applied to the high molar mass free polymer as well as the low molar mass ( 1OOO) steric polymer. The salt concentrations investigated were 0.03,0.1,0.4, and 1.0 M. The change in solvency as a function of sodium chloride concentration was evaluated through the second virial coefficient, which is given by the Flory equation4I

0.0025

3

N

B, =

(ff5

0

- ff3)v,2

where ( r2)O1/, is the unperturbed root mean square endto-end length of the polymer. The intramolecular expansion factor is evaluated as the ratio of intrinsic viscosities (3)

where 191is the intrinsicviscosity of the polymer in solution at the conditions of interest and [ q ] e is the intrinsic viscosity of the polymer in solution at theta conditions. a is equal to unity for a theta solvent, greater than 1for a good solvent, and less than unity for a poor solvent. The intrinsic viscosity at theta conditions was calculated using the Mark-Houwink-Sakurada equation4 = K@"

(4)

where Ke is a constant determined at theta conditions for the polymer of interest. The intrinsic viscosity is equivalent to the reduced viscosity in the limit of infinite dilution [VI

=p..+o'("-l) 90 c2

0.4

0.6

0.8

1

1.2

NaCl Concentration (M)

2C,VlM1/2

where a is the intramolecular expansion factor for the polymer in solution, Y, is the partial specific volume of the polymer, VI is the molar volume of the solvent, M is the molar mass of the polymer, and C, is a molecular constant. C, is defined as follows

[VIS

0.2

(5)

where c2 is the mass concentration of polymer in solution, Q is the viscosity of the polymer solution, and 90 is the viscosity of the solvent. For a given salt concentration, the viscosities of five or six solutions with varying polymer concentrations were measured a t 24 OC using a CannonFenske viscometer immersed in a thermostated water bath. Values of Bz were then calculated using the above relationships. Reference 48 gives the following values for poly(ethy1ene oxide): Ke = 0.115 f 0.015mL/g, v2 = 0.833 cm3/g, and (r2)01/2= Mo.b(O.0775 f 0.0030) nm. Aggregation Experiments. All latex dispersions and salt and polymer solutions were prepared using deionized, double distilled, filtered water with a pH in the range of 5.5 to 6.0. All glassware was acid washed and rinsed with tap, deionized, distilled, and filtered distilled water, successively. Great care was taken to avoid contamination and to remove dust during sample preparation and handling. (47) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1963. (48) Brandup, J.; Immergut, E. H. Polymer Handbook, 1975.

Figure 1. Effect of sodium chloride on poly(ethy1ene oxide) (PEO)solvency as measured by the second virial coefficient: M = 1000 g/mol, T = 24 "C.

In a typical flocculation experiment, NaCl solution was added to a sterically-stabilized polystyrene dispersion in a test tube and gently stirred for 1min. At time zero, free polymer solution was added to the dispersion. The initial number concentration,No, of the latices in all experiments was 7 X log particles/mL and the final adsorbed polymer concentration was 2.5 X 1V g/mL. The change in latex apparent diameter with time following mixing was monitored using the technique of photon correlation spectroscopy (PCS).49 The PCS setup consisted of a Brookhaven Instruments (Holtsville, NY) Model BI-2030 automated laser light scattering system with a 72-channel digital correlator and a Physics Stabilite 15-mW He-Ne laser. All experiments were performed at a temperature of 24 "C,a scattering angle of 90°, and a wavelength of 632.8 nm. Fluctuations in the intensity of scattered light due to Brownian motion of the particles were measured as a function of time, and the mean diffusion coefficient of the particles was determined from the autocorrelation function using the method of ~ u m u l a n t s .The ~ StokesEinstein equation was used to relate the mean diffusion coefficient to the mean hydrodynamic diameter.5l The rate constant of doublet formation, k,, was determined from the initial rate of change of hydrodynamic radius with time52

where Rh,l is the initial hydrodynamic radius at time zero and Cr is an optical factor that is a function of particle radius, solvent refractive index, scattering angle, and wavelength of light in the medium . The stability of a dispersion can be quantified through the stability ratio, W, defined as the ratio of rate constants for doublet formation: W = kdk, (7) where kf is the Smoluchowski diffusion-limited rate constant for fast aggregation (Le. every collision is effective) and k,,as defiied above, is the rate constant for aggregation during which every collision may not be effective. Results. The change in the second virial coefficient, B2, as a function of sodium chloride concentration is shown in Figure 1. With no added salt, B2 equals 0.00215 cm3 (49) Dahneke, B. E. Measurement of Suspended Particles by Q u a i Elastic Light Scattering, 1983. (50)Berne, B. J.; Pecora, R. Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics; John Wiley & Sone, Inc.: New York, 1976. (51) Pecora, R. In Measurement of Supended Particles by Quaielastic Light Scattering; Dahneke, B. E., Ed.;Wiley: New York, 1983. (52) Virden, J. W.; Berg, J. C. J. Colloid Interface Sci. 1992,149,528.

Depletion Flocculation of Latex Dispersions 10)

,

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' ")''''I

, ,'"'"I

' """'I

,

' ~ ' " " I '

Langmuir, Vol. 10, No. 2,1994 459

+

. I . . .

A A L

M=lW.NoSalt M=lMM;O.lM M=1000;0.4M M=lMX),l.OM

0 0

M=8000;1.OM M = 185oO.l.OM.

c M = 100,ooO. 1.0 M

W+

10 @Ib

Figure 2. Measured effectof free polymer molecular weight on the stability of L35-coated polystyrene latices. mol/g2. As the salt concentration increases to 1.0 M, B2 decreases smoothly to a value of 0.0001 cm3mol/g2. This trend in polymer solvency is the same as the trends observed previously for PEO with BaC12 and AlC13as the electrolytes.53 The results of the aggregation studies are presented in terms of the normalized stability ratio, W*, which is the absolute stability ratio for the system of interest divided by the stability ratio for the case of bare latex particles undergoing fast aggregation. The results for bare latex particles aggregating in the presence of NaCl were reported previously." Figure 2 shows the experimentally observed effect of free polymer molecular weight in terms of stability plots for polystyrene latices with an L35 adlayer ( M zz 950 g/mol) in the presence of four different free polymers. No sodium chloride was added to these systems. In all cases, there is a plateau at a normalized stability ratio of approximately 8OOO. The relativelyhigh values of W* indicate quite stable dispersions. This is expected because stability is imparted by electrostatic as well as stericmechanisms in the absence of salt. The onset of depletion flocculation occurs between the followingvolume fractions: (1)0.004 and 0.01 for PEO 1000, (2)0.001 and 0.004 for PEO 8000 and PEO 18 500, (3) 10-5 and 10-4 for PEO 100 000. (It should be noted that the data for the three lowest molar masses lie on top of each other at volume fractions of 10-4 and W3). The stability ratios decrease monotonically with increased free polymer volume fraction above the criticalvolume fraction for the onset of depletion flocculation. Depletion stabilization was not observed for any of the systems. The decrease in critical depletion flocculation volume fraction with increased free polymer molar mass has been widely observed by other i n v e ~ t i g a t o r s . ~ J ~ ~ ~ ~ * ~ ~ ~ 7 Figure 3 shows the effect of salt addition. Two types of data are plotted: (1)changes in the stability ratio as a function of salt concentration for a constant free polymer molar mass (1000 g/mol) and (2) changes in the stability ratio as a function of free polymer molar mass at aconstant salt concentration (1.0 M). Consider first the effect of salt on the stability. As the NaCl concentration is increased from no salt td 1.0 M, the plateau value of the stability ratio decreases over an order of magnitude from BOO0 to 350. This decrease can be explained in terms of the effect of salt on the electrostatic double layer and polymer solvency. The critical coagulation concentration (ccc) for the bare polystyrene latices is 0.095 M NaCl," so as the salt concentration is increased from no salt to 0.1 (53) Einarson, M. B.; Berg, J. C.Langmir 1992,8, 2611. (54) Einareon, M.; Berg, J. C. J. Colloid Interface Sci. 1993, 155.

' '',,..J

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'

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'',,.,.'

'

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Seebergh and Berg

460 Langmuir, Vol. 10, No. 2, 1994

aggregationwhich the system undergoes, whether it is weak flocculation or coagulation, is treated as a loss of stability due to the addition of free polymer. Jar testing, on the other hand, is the commonly-usedexperimental technique, based on the visual determination of aggregation. By default, the critical volume fraction for the onset of depletion flocculation is defined as that concentration of free polymer which induces visually-detectable aggregation. Flocculation which is unobservable by the human eye will therefore be disregarded. Clearly, it will take higher concentrations of free polymer to induce visuallydetectable aggregation than to induce PCS-detectable aggregation.

Id

W’

100

10

1

0

0.0001

0.001

0.01

0.1

Figure 4. Measured effect on sodium chloride on the stability of L43-coated polystyrene latices.

For the three NaCl concentrations shown in Figure 4 , the stability ratios in the absence of free polymer are in good agreement with previously measured values.” The decrease in stability as the salt concentration increases is a direct consequence of the loss of polymer solvency, as explained above. Interestingly, upon the addition of free polymer there is actually an increase in the stability. This phenomenon is fundamentally different than depletion restabilization, which occurs after the system has already flocculated. To the authors’ knowledge, this behavior has not been reported by other investigators. The observations may possibly be explained by solvency considerations. In the presence of electrolyte, the solvency of the free polymer is reduced and this may cause some of the free polymer chains to anchor themselves in the steric layer. The molar mass of the free polymer in this case is lo00g/mol, whereas the steric layer molar mass is 650 g/mol. The presence of longer polymer tails in the steric layer would effectively increase the adlayer thickness, leading to a more stable system. In addition, cations have been observed to associate with PEO in solution.s6 These complexes are analogous to the “crown ether” complexes formed between macrocyclic polyethers and alkali-metal ions, where the ion is bound in the central cavity of the crownlike structure.m If the sodium ion is in fact binding to the PEO, the polymer chain could elongate and/or stiffen. The changes in polymer conformation, along with the postulated presence of free polymer chains in the steric layer, could well be responsible for the observed increases in stability. It should be noted that in general, the critical volume fractionsfor depletion flocculation reported here are lower than those reported elsewhere. The reason for this discrepancy can most likely be attributed to differences in experimental techniques which lead to differences in how “aggregation” is defined. The light scattering technique used in this study allows the measurement of the early-stage aggregation kinetics, i.e. the formation of doublets from singlets. Due to the sensitive nature of PCS, weak flocculation is easily detected even though it may not be macroscopically observable. In this study, the critical concentration for the onset of depletion flocculation is defined as the lowest volume fraction of free polymer at which the stability ratio is less than the stability ratio of the same system in the absence of free polymer. Any (55) Bailey, F. E.; Koleske, J. V. In Configuration and hydrodynamic properties of the polyoxyethylene chain in solution; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Vol. 23. (56) C m , J. InMetalion Complexes of polyoxyalkylene chains;Cross, J., Ed.; Marcel Dekker, Inc.: New York, 1987; Vol. 19.

Modeling Background. The stability of a colloidal system is determined by the total interaction potential energy. According to convention, the total potential energy of interaction for a pair of particles, V,t, can be expressed by summing each of the attractive and repulsive potential energies: VvdW is the attractive potential energy arising from van der Waals interactions (primarily London dispersion forces), vdep is the depletion potential energy arising from the presence of free polymer in solution, Vel& is the repulsive potential energy arising from electrostatic charges on the particle surface, and Vaisthe steric potential energy arising from polymer adsorbed on the particle surface. The hypothesis of independent additivity of the van der Waals, electrostatic, and steric potential energies was shown to be valid in previous work in this laboratory;63154however, the validity of adding in the depletion potential energy was not examined. In order to calculate V,t, theoretical expressions are needed to describe each of the potential energies. (Note that all potential energiesgiven in the following expressions are normalized by kBT.1 The van der Waals interaction energy for spherical particles of equal size is given by the Hamaker expression67

VVdW =

4a2 “-[ + + 1 2 k ~ T h2 + 4ah h2 + 4ah + 4a2 4a2

2

4

)]

h2+4ah (9) h2 4ah 4a2 where A is the Hamaker constant, a is the radius of the particle, and h is the interparticle separation distance. A value of 4 X J was used for the Hamaker constant of polystyrene in water. This was the best-fit modeling value as determined previously for the case of bare particles” and is well within the range of values reported by other investigators.58-61 Exact analytical expressions for the electrostatic potential energy cannot be given, so either numerical solutions or analytical approximations must be used. For the case of low surface potential (ze$o/Pk~T< 11, the Debye-Huckel approximation can be made and an anahave derived lytical expression will result. Healy et al.62*63

+

+

(57) Hiemenz, P. Principles of Colloidand Surface Chemistry; Marcel Dekker: New York, 1986. (58) Ottewill, R. H.; Shaw, J. N. Diacuas. Faraday SOC.1966,42,154. (59) Kotera, A.; Furusawa, K.; Kudo, K. Kolloid 2.2.Polym. 1970, 240, 837. (60) Lichtenbelt, J. W. T.;Pathmamanoharan, C.; Wiemema, P. H. J. Colloid Interface Sci. 1974,49, 281. (61) Ash, S. G.; Clayfield, E. J. J. Colloid Interface Sei. 1976,55,645. (62) Weise, G. R.; Healy, T. W. Trans. Faraday SOC.1970, 66, 490. (63) Hogg, R.; Healy, T. W.; Fuemtenau, D. W. Tram.Faraday SOC. 1966,62, 1638.

Depletion Flocculation of Latex Dispersions

Langmuir, Vol. 10, No. 2, 1994 461

the following expression for the case of constant surface charge density with the assumptions of low surface potentials, spherical particles of equal size, and electrical double layers which are thin compared to the particle size (KU > 1)

where (11)

and (12)

The permittivity of the aqueous medium is E, @O is the effective surface potential, K is the inverse Debye length, I is the ionic strength, C is the concentration of the electrolyte, and Za and zc are the valences of the anion and cation, respectively. The effective surface potential can be determined from the surface charge density using the Grahame equations u=

[

[

-

2JV,kBzci , exp( - ~ ~ ~ ) - l ] ] ' "(13)

Steric interactions arise when particles with adsorbed polymeric layers approach each other at distances less than twice the adlayer thickness (26). The close approach of the particles can be described in terms of the interpenetrational domain (6 < h < 26) and the interpenetrational-plus-compressional domain (h < 6).1° As the polymer adlayers overlap (h C 26), the chemical potential of the solvent in the overlap zone becomes less than that in the bulk. This difference in chemical potential sets up a driving force which causes solvent to flow into the overlap zone, thereby separating the particles. When 6 < h < 26, the steric potential energy due to the mixing of the overlapping adlayers for a pseudotails segment volume profile may be calculated using the expression given by Vincent et al.19 a 2

v, = 32=u(62

(i- x ) [

15~~6'

1062(6 -

3'-

126(6 -

4,"+

where 42a is the effective volume fraction of segments in the steric layer and u1 is the molecular volume of the solvent. Upon closer approach of the particles (h < 61, the presence of the opposing particle surface causes the adsorbed polymer chains to undergo elastic compression, thereby reducing the configurational entropy. This loss in configurational entropy generates a repulsive interaction; consequently, there is an elastic component as well as a mixing component of the steric potential energy in this domain. The following expressione describe the steric potential energy in the interpenetrational-plus-compessional domain1

v, = v, + Vel where p2 and Mads are the density and molecular weight, respectively, of the adsorbed polymer. A value of 0.031 was used for the effective volume fraction of polymer segments in the stericlayer. This was the best-fit modeling value determined previously for the case of electrosteric stabilization." The adlayer thickness of the L43 stabilizer was estimated from the data of Baker and B e r p to be approximately 20 A. The adlayer thickness of the L35 stabilizer was determined to be approximately45 A, based on PCS measurements for a steric stabilizer with a PEO molar mass of lo00 g/m01.~l (It should be noted that it is extremely difficult to measure thin adlayers using PCS because the width of the adlayer is almost within the experimental error of the method.) The Flory-Huggins x parameter was determined from the second virial coefficient

where VZis the molar volume of the polymer and VI is the molar volume of the solvent. The dependence of x on the salt concentration was incorporated into the model. The depletion potential energy can be described using the expression given by Fleer et al.3

where A is the thickness of the depletion layer for nonadsorbing polymer adjacent to hard spheres. Vincent et replaced the hard sphere depletion layer thickness with an effective depletion layer thickness, &E, for the case of particles with steric layers. The effective depletion layer thickness accounts for the thickness of the steric layer (ti), the distance that the free polymer can interpenetrate the steric layer @), and the amount by which the steric layer is compressed due to the presence of the free polymer (9) d 1 p 2

Aea = A

+ 6- p - 9

(20)

The hard sphere depletion layer thickness (A) was calculated using a modified version of the expression given by Vincent5for nonadsorbing polymer adjacent to a hard, flat surfece. In this derivation, Vincent equated the bulk osmotic pressure as given by the Flory-Huggins (FH) expression with the force of elastic compression per unit area on a coil next to the surface. FH theory for the thermodynamics of polymer solutions is based on the so the assumption of relatively concentrated FH osmotic pressure expression will not be valid over the range of free polymer fractions which were used experimentally. The virial expansion for the osmotic pressure, given below, is a better choice because it is valid for dilute solutions.

no= RTP2d4& + Bfl2@,b+ ...)

(21)

Incorporation of eq 21 into Vincent's derivation yields the following expression for the depletion layer thickness (64) Israelachvili, J. N. Zntermolecuhr and Surfaces Forces;Academic

Prese: New York, 1986.

(65) Baker, J. A.; Berg, J. C. hngmuir 1988,4, 1055. (66)Hiemenz, P. C. Polymer Chemistry: The Basic Concepts;Marcel Dekker, Inc.: New York, 1984.

462 Langmuir, Vol. 10, No. 2, 1994

Seebergh and Berg

where a0 is the range of depletion effect in the limit that the bulk polymer concentration goes to zero, u2 is the molecular volume of a free polymer chain, N Ais Avagadro 's number, and 42b is the bulk volume fraction of free polymer. In accordance with the calculations of Fleer et aL3 a0 was set equal to 1.4rg,where r, is the radius of gyration. The radius of gyration can be determined from the unperturbed radius of gyration, rgo,where

rg = arg0

lio-'

o

4

i

W

(23)

and @Ib

The intramolecular expansion factor, a,as a function of salt concentration was determined from viscosity measurements, as described above. The unperturbed root mean square end-to-end distance, (r2)01/2,for PEO as a function of molar mass was given by the Polymer Handbook (see value given above). It should be noted that although eq 22 was derived for the situation of free polymer adjacent to flat plates, calculations have shown that this expression is valid for spheres provided that the radius of gyration of the free polymer coils is much less than the radius of the spherical particles. The distance that the free polymer can interpenetrate the steric layer was given by Vincent' for the case of a pseudotails segment profile in the steric layer

Figure 5. Modeling results for 100 OOO molar mass PEO as a function of sodium chloride concentration. io4

3

i

W

@Ib

For all of the experimental cases which were modeled, p was found to be equal to the steric layer thickness, Le. the free polymer could completely penetrate the steric layer. In light of the low average value for the segment volume fraction in the steric layer and the near-theta solvency conditions, this outcome seems physically realistic. No attempt was made to account for the possible compression of the steric layer due to the presence of the free polymer. In order to compare experimental results with theory, the total potential energy must be converted to a theoretical stability ratio. Theoretical expressions exist which relate the potential energy to the stability ratio for either coagulation into the primary minimum,67flocculation into the secondary minimum,S7l or aggregation into both minima.72J3 Dispersions which are sterically stabilized are more likely to flocculate into the secondary minimum because of the large steric energy barrier which prevents coagulation into the primary minimum; however, primary minimum coagulation is possible under certain conditions. In previous modeling w0rk,~3!%the Marmur73-Wang72 model was used to calculate theoretical stability ratios for bare particles, L43-coated particles, and L35-coated particles in the presence of electrolyte. This model accounts for aggregation a t both the primary and secondary minima and requires knowledge of only the maximum, V,,, and the minimum, Vmin, in the particle pair potential function. The model did a reasonable job of predicting the salient qualitative features of the stability behavior. Based on (67)Fuchs, N.Z.Phys. 1934,89, 736. (68)Bagchi, P. Colloid Polym. Sci. 1976,254, 890. (69)Hogg, R.;Yang, K.C . J. Colloid Interface Sci. 1976,56, 573. (70)Ruckenstein, E.J. Colloid Interface Sci. 1978, 66, 531. (71)Cowell, C.;Vincent, B. J. Colloid Interface Sci. 1983, 95, 573. (72)Wmg, Q.J . Colloid Interface Sci. 1991, 145, 99. (73)Marmur, A. J. Colloid Interface Sci. 1979, 72, 41.

Figure 6. Modeling results for 18 500 molar mass PEO as a function of sodium chloride concentration.

M

--t 0.03

t

-

l0

lo6

-e-

0.1 M

-m-

0.4 M 1.0 M

los

10.~

@*b

Figure 7. Modeling results for 8000 molar mass PEO as a function of sodium chloride concentration. the previous results, the Marmur model will be used for this analy~is.~3 Results. Figures 5,6, and 7 show the modeling results for L35-coated particles with 100 000, 18 500, and 8000 molar mass free polymer, respectively, in the presence of NaC1. The trends in the stability ratio as a function of free polymer volume fraction and salt concentration for the different free polymers are qualitatively similar. For each salt concentration, there is a plateau in the stability ratio, followed by a decrease in stability above the critical depletion flocculation volume fraction. Depletion stabilization is predicted in some, but not all, cases. As the salt concentration is increased from 0.03 to 1.0M, the plateau value of the stability ratio decreases from approximately 2600 to approximately 350 in all cases. The critical

Depletion Flocculation of Latex Dispersions

10' W

Langmuir, Vol. 10, No. 2, 1994 463

use of a pH of 5.5 for water, the ion concentration is 3.16 X 10-8 M for the no salt case. This correspondsto a double layer thickness of approximately 150 nm and a surface potential of -0.21 V. The enormous electrostatic potential energy barrier will dominate all of the other interaction energies and a robust stability will be maintained. This implies that under normal experimental conditions, no aggregation would be observed;however, as seen in Figures 2 and 4, finite stability ratios were measured for the nosalt cases. The stability ratios were very large, indicating robust stability, but they were not of the order predicted by the model. One possible explanation is that the P h o n i c polymers and/or the free polymer contained electrolytic contaminants and the so-called no-salt case was actually a low-saltcase. This hypothesis was checked by measuring the conductivity of dispersions with free polymer but no added salt. Ideally, the conductivity of these dispersions would be the same as that of the distilled water (1.3 pS/ cm);however, values up to 190&/cm were measured. This corresponds to concentrations of 1.5 mM NaC1,0.66 mM BaC12, and 0.45 mM AlC13. If the contaminants were divalent or trivalent salts, which have ccc's 1to 2 orders of magnitude less than NaC1, it would only take a small amount to significantly lower the stability. No attempt was made to model the depletion behavior of the L43-coated particles. In order to predict the behavior observed in Figure 4, the model must be altered to account for the mechanism by which the stability is increased. This mechanism is not well understood at this time.

i t -8- 0.1

M

-+- 0.4 M --c- 1.0M

$tb

Figure 8. Modeling results for lo00 molar mass PEO as a function of sodium chloride concentration.

depletion flocculation volume fraction decreases with increasing free polymer molar mass. This result is in good agreement with experimental observations reported in the literature. The 1.0 M modeling results can be compared with the experimental results in Figure 3. The plateau values are in good agreement (Wz 350), and the critical depletion flocculation concentrations are in relatively good agreement. The model predicts a much sharper decrease in stability ratio above the critical volume fraction than is experimentally observed. Also, the model predicts depletion stabilization for some cases, but depletion stabilization was never observed experimentally. Depletion stabilization may in fact occur a t higher volume fractions; however, experimental limitations precluded measurements above volume fractions of 0.1 for PEO 1000,0.02 for PEO 8000 and 18 500,and 0.004 for PEO 100 000. At these volume fractions, it was difficult to completely dissolve the PEO, as evidenced by the cloudy appearance of the free polymer solutions. Figure 8 shows the modeling results for L35-coated particles with lo00 molar mass free polymer in the presence of NaC1. Interestingly, no flocculation is predicted for any salt concentration. Examination of the individual pair potentials revealed that the effective depletion layer thickness of the 100 molar mass free polymer is so thin that the attraction due to the depletion potential is dominated by the repulsion due to the steric potential. In essence, the depletion layer is "buried" within the steric layer, such that steric interactions dominate upon close approach of the particles. One possible explanation for the experimental observation of flocculation in the 1000 molar mass case is polydispersity of the free polymer. The presence of longer chains would effectively increase the depletion layer thickness, which would in turn lead to flocculation at lower volume fractions. A more likely explanation is that the model used for the depletion layer thickness is imprecise. This hypothesis was tested by incorporating other expressionsfor A into the model. When alinear expression given by Vincent or an expression based on scaling theory was used, flocculation was predicted to occur at a volume fraction between 0.01 and 0.04, in very good agreement with experimental observations (seeFigure 3). Unfortunately, when these expressions for A are used to model systems with higher molar mass free polymer, the agreement between theory and experiments is poor. An attempt was made to model the no-salt systems; however, infinitely large stability ratios were predicted in all cases. In the absence of electrolyte, the electrostatic double layer is diffuse and extends far into solution. By

Conclusions Depletion flocculation was observed upon the addition of homopolymer PEO to electrosterically-stabilizedpolystyrene dispersions. The flocculation followed widelyobserved trends in that the critical depletion flocculation concentration decreased with increasing free polymer molar mass and the stability was greater for systems with higher molar mass steric adlayers. For a given system, the stability with respect to flocculation was reduced as the sodium chloride concentration was increased due to the progressive destruction of polymer solvency. The decrease in PEO solvency as a function of NaCl concentration was confirmed by viscosity measurements. The stabilityof L43-coated particles in the presence NaCl was observed to increase upon the addition of PEO 1000. This unusual behavior has not been reported elsewhere, and the mechanism by which the stability is increased is not well understood. The pragmatic model did a reasonable job of predicting the salient features of the stability behavior, including the effects of salt due to solvency and the existence of stability plateaus at low free polymer concentrations. In some cases, there was good quantitative agreement between theory and experiment for values of W and the critical depletion flocculation concentration. Although the model is inadequate to describe observations under some conditions, the extent of agreement which was observed is promising in light of the complexity of the systems being modeled.

Acknowledgment. This work was supported in part by a gift from the Shell Development Corporation. The authors also thank Maryann Einarson and Anthony Swanda for assistance in the early stages of this work.