Temperature-particle concentration phase diagrams for dispersions of weakly interacting particles

Temperature-Particle Concentration Phase Diagrams for Dispersions of Weakly Interacting Particles CHRISTOPHER

COWELL

AND BRIAN V I N C E N T ~

Department o f Physical Chemistry, University o f Bristol, Bristol BS8 1 TS, United Kingdom Received June 16, 1981; accepted October 19, 1981

INTRODUCTION

The objectives of this paper are to discuss the analogy between the types of phases obtained with weakly interacting (sterically stabilized) colloidal particles and those normally observed with molecular systems, and to suggest possible forms for the phase diagram. The equilibrium physical properties of both molecular and particulate systems are controlled by two main parameters: (i) the number density (p) or volume fraction (4~) of the species concerned; and (ii) the interaction free energy (G;) 2 which at low particle volume fraction may be approximated as pairwise additive. At high p, multibody interactions may have to be considered, rather than the single pairwise interaction. For charged colloidal particles, in aqueous media, GI is essentially given by the DLVO theory and modern extensions thereof (1). At low electrolyte concentrations the interactions are long range and repulsive; aggregation is, therefore, normally precluded. Various authors (1-15) have discussed the order-disorder transitions that occur for this type of dispersion. A coexistence region (i.e., ordered phase plus disordered phase) is shown to exist over the 4~range 0.50 to 0.55 1Author to whom correspondence should be addressed. 2 Free energy is preferred to potential energy (at least in the case of colloidal particles) since the pairwise interaction may contain entropic contributions.

(16, 17), where the particle radius is considered to be that of the core plus the "effective" thickness of the electrical double layer. (The value to be taken for the "effective" thickness is not clearly resolved (14). The origin of this disorder ~ order phase change has been ascribed to the Kirkwood-Alder transition (3-9), known to occur for hardsphere systems (16, 17). Ottewill (15) has suggested, from light diffraction and neutron scattering studies on polymer latices, that the "ordered" phase shows only relatively short-range positional correlations in the volume fraction range corresponding to the coexistence region and up to some higher critical value; he has, therefore, suggested this type of ordered phase is more "liquid like." Beyond this critical value of ~ much longer-range positional correlations are observed and the ordered phase in that region is truly pseudocrystalline; here iridescence, due to Bragg diffraction, may be observed. The reason for the existence of the "liquid-like" ordered state with this type of dispersion is a direct result of the "softness" of the long-range electrical double layers. For dispersions where there is a net attraction between the particles, somewhat different behavior may be expected: in particular, aggregation (coagulation or flocculation) may occur. Flocculation results when there is a weak, net attraction between the particles. A situation of this type may arise J

518 0021-9797/82/060518-09502.00/0 Copyright © 1982 by AcademicPress, Inc. All rights of reproductionin any form reserved.

Journal of Colloidand Interface Science, Vol. 87, No. 2, June 1982

TEMPERATURE-PARTICLE PHASE DIAGRAMS in several ways: e.g., a "secondary minimum" in charge-stabilized systems (1); as a result of bridging by higher-molecularweight polymers or polyelectrolytes (18); or in sterically stabilized dispersions where the van der Waals forces between the particles is of longer range than the short-range ("hard") steric interactions (19). Long et al. (20) suggested that with systems of the last kind a strong analogy exists with molecular phase changes. Indeed, the analogy here will be stronger than with the charge-stabilized systems referred to above, because, in the case of sterically stabilized dispersions, the form of the Gi-h curve (h is the particle separation) bears a much closer resemblance to that for molecules (e.g., the Lennard-Jones pairwise potential energy). In both cases a shallow minimum ( a m i n ) o c c u r s in the interaction curve, which is typically in the range 0 to several kT/ "particle" pair. Long et aL showed that a "two-phase" region could be established beyond some critical value of ~b, i.e., a "dispersed" phase (consisting essentially of singlet particles) and a "floc" phase (consisting of sedimented flocs); the molecular analogy here is the vapor-condensed phase equilibrium. The authors suggested that the equilibrium could be described by the Boltzmann relationship, in the form 4~d = (9r e x p ( - E / k T ) ,

[l ]

where od and q~f are the volume fraction of particles in the dispersed (vapor-like) phase, and the floc (condensed) phase, respectively. E is the energy required to remove a particle from (the surface of) the floc phase to the dispersed phase, and is therefore given by E = (z/Z)Gmin,

[2]

where z is the coordination number of a particle in the center of the floc phase. The original experiments of Long et al. were carried out using charge-stabilized polystyrene latex particles, each carrying an adsorbed monolayer of an alkyl ethoxylate

519

nonionic surfactant (C12E6). Sufficient electrolyte was added to remove the electrical double-layer interactions and, hence to "expose" the minimum in the Gi-h curve. From the observed critical flocculation volume fraction (c.f.~), Groin could be calculated from Eqs. [ 1] and [2], by making some assumption concerning the floc structure (e.g., random close packing). Values of Gmin determined in this way were in reasonable agreement with those obtained by direct calculation of the interparticle interaction energies (20). More recently the existence of a c.f.~b has been established for polystyrene (PS) latices carrying layers of terminally anchored (grafted) poly(ethylene oxide) (PEO) chains (21), and also p h y s i c a l l y adsorbed PEO (22). A similar situation pertains with nonaqueous, sterically stabilized dispersions, e.g., silica particles carrying terminally anchored polystyrene chains in ethyl benzene (23). The vapor pressure above a condensed molecular phase is temperature dependent and one may, therefore, expect the c.f.~b to be temperature dependent also. An alternative, but essentially equivalent, statement is to say that the critical flocculation temperature (c.f.T.) of a sterically stabilized dispersion ought to be ~ dependent. Cowell et al. (24) showed that for neutral, aqueous PS-g-PEO latices (PEO molecular weight 750 or 2000), the c.f.T, decreased (albeit slightly) with increasing ~ over the range 10 -5 to 10 -3. A similar finding has been reported by Everett and Stageman (25) for nonaqueous dispersions, i.e., poly(methyl methacrylate) particles carrying polydimethylsiloxane chains in aliphatic hydrocarbons. In this paper we discuss the extension of the results obtained with the aqueous system referred to above to higher ~ values and so, in effect, construct the T-~b boundary line, by analogy with the T - p line for vapor/liquid equilibria. It is, of course, more convenient to control ~b in these systems, rather Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982

520

COWELLAND VINCENT

than the pressure. Pressure here, in fact, refers to the osmotic pressure, 7r.3 The phase rule requires that if T and q~ (or p) are the chosen variables, then 7r is determined. In fact, one may write an equation of state for each phase, e.g., a virial equation of the form, Ir= R T p + Bp 2 + . . . . [3] The other objective of this paper is to postulate a possible form for the complete Tphase diagram, taking into account evidence for the existence of ordered condensed phases that are known to occur in systems of weakly attracting particles. EXPERIMENTAL Materials

All the water used had been doubly distilled from an all-Pyrex apparatus. Salts were B.D.H. "AnalaR" grade. Styrene and divinylbenzene were B.D.H. reagent grade; both were freshly distilled under reduced pressure immediately prior to use. Azodiisobutyronitrile (ADIB) was obtained from the Aldrich Chemical Company and recrystallized from acetone. Homopolymer PEO (having nominal/~r 1500) was B.D.H. material. "PEO 1600 Ma" (i.e., a PEO chain, h~rn 1600, having a CH3 group at one end and a methacrylate monomer group at the other) was kindly supplied by ICI Paints Division Ltd (Slough). " H M T 6000" (HMT = hexamethylene triamine) having the structure (1) EO = ~CH2CH20~ n EO i

Eo-Po\

Pol

EO_Po/N(CH2)6

N {CH2) 6 N

/ Po-Eo PO_EO

Po= ~cH-c"2°÷~i CH 3 = 21,

n = 18

was kindly supplied by ICI Petrochemicals Division Ltd. (Billingham). 3The total pressure may also be varied, but is convenientlyheld at 1 atm. Journal of Colloid and Interface Science. Vol. 87, No. 2, June 1982

PEO-stabilized Latices

Polystyrene latices (PS-g-PEO) in which the particles carried terminally grafted polyethylene oxide (PEO) chains were prepared following a procedure similar to that described by Cowell et al. (24). A mixture of styrene and divinylbenzene (in the ratio 98.7:1.3 by weight) was dissolved in a methanol-water mixture, together with the required amount of PEO 1600 Ma stabilizer precursor. The system was heated to reflux (70°C) and the initiator (ADIB) added; the polymerization was allowed to proceed for 24 hr. After this time the latex was centrifuged and the supernatant was analyzed for free PEO using the tannic acid complex-formation method described by Attia and Rubio (26). The latex was redispersed in distilled water and subjected to two further centrifugation/redispersion cycles. Two latices (C3, C4) prepared in this way were shown to have mean particle diameters of 454 + 20 and 378 + 16 nm, respectively, from transmission electron micrographs, and to have effectively zero electrophoretic mobility (using a Rank Bros. Mark II microelectrophoresis apparatus). From a knowledge of the PEO concentration, prior to and subsequent to polymerization, the coverage of latex C4 particles by anchored PEO 1600 chains was estimated to be 1.5 mg m-2; this corresponds to an area per chain of 1.8 nm 2, implying that the PEO tails are in a somewhat extended conformation normal to the surface. (The projected area of a random PEO 1600 coil, based on its solution radius of gyration (21), is ~ 6 nm2.) A third latex (C6) was prepared in a similar way to C3 and C4, but without any divinylbenzene, and with " H M T 6000" replacing "PEO 1600 Ma" in the reaction mixture. This had a mean particle diameter of 233 _+ 20 nm. Critical Flocculation Temperatures

Two techniques were used to establish c.f.T.'s depending on the volume fraction

TEMPERATURE-PARTICLE PHASE DIAGRAMS

521

ature shows a sharp break (see Fig. 1 ); the t e m p e r a t u r e at the break point is taken to be the c.f.T, for that system. rn-I, $

Slow-Speed Centrifugation

x l O -3

2.5

T *C

FIG. 1. Low shear viscosity:m-1 (m = d log &h/dlog t) versus temperature; a typical plot. The c.f.T, is taken to be the temperature correspondingto the discontinuity (50.4°C).

This technique has been described previously (28, 29). It was used to study the onset of the ordered phase. The latex particles are placed in centrifuge tubes having a narrow bore capillary a t t a c h m e n t at the lower end. The tubes are then rotated at low speed in an M S E bench centrifuge place in an air thermostat box (37 + I ° C ) , The sediment in the capillary is observed as a function of time and rotor speed. In particular, in these experiments, the onset of iridescence was looked for. This indicates the formation of pseudocrystalline structure. RESULTS

range to be studied: (i) Turbidity (z)/wavelength (X) scans (10 -5 < @ < 10-2). This method has been described elsewhere (20, 24). A sharp b r e a k is observed in a plot of n ( = d log z/d log X) versus t e m p e r a t u r e at the c.f.T. A P y e - U n i c a m SP1800 with a thermostated cell-housing was used to monitor turbidity. (ii) Low-shear viscometry (10 -2 < < 0.2). A differential capillary viscometer (Polymer Consultants Ltd.) was used in these experiments. The difference in height (&h) of liquid in the two arms of the viscometer is monitored as a function of time (t), as Ah ~ 0. Under these conditions the shear forces operating are very small (corresponding to shear rates < 1 sec-~). Log &h-t plots were then constructed at different temperatures, the t e m p e r a t u r e of the thermostated (___0.1°C) water bath being raised in small increments. Since this is a perturbatory method the system was allowed to stand for at least 20 min to recover its initial state before each t e m p e r a t u r e adjustment. The slope ( m ) of the log &h-t plot is proportional to the (near-zero shear) viscosity of the system. A plot of m -~ versus temper-

Critical Flocculation Temperatures The c.f.T.'s of latices C3 ( P E O 1600 M a stabilizer) and C6 ( H M T 6000 stabilizer) as a function of the n u m b e r ( X ) of centrifugation redispersal in 0.26 mole dm -3 MgSO4 solution cycles are shown in Fig. 2a. It can 60 T,

'C 55--

c,

o

o C3

50

45

1

2

3

4.

X

10

20 30 C~, ppm

FIG. 2. (a) c.f.T, as a function of the number (X) of centrifugation/redispersal cycles in 0.26 mole dm-3 MgSO4 solution; (b) c.f.T, of latex C6 (after five"cleanup" cycles) as a function of added HMT 6000 stabilizer concentration (Cp). q~= 10-4. Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982

522

COWELL AND VINCENT

be seen that the c.f.T, of latex C3 remains invariant, whereas that of C6 falls to a constant, limiting value. The implication is that the PEO chains in latex C3 are chemically anchored to the polystyrene matrix, presumably because of the cross-linking role of the divinylbenzene, but in the case of latex C6 some (but only a fraction) of stabilizer is removed by the clean-up procedure. Figure 2b shows that the c.f.T, can be restored to its original value by equilibrating the latex with increasing (ppm) concentrations of the stabilizer. These results serve to illustrate the strong dependence of the c.f.T, on the packing density of the stabilizing moieties for particles stabilized by relatively low molecular weight chains. The c.f.T, plots for latices C3 and C6 in 0.26 mole dm -3 MgSO4, as a function of ~b, are shown in Fig. 3. The closed-circle points were obtained from turbidimetric analysis, and the open circles from low-shear viscometry. The decrease in c.f.T, with increasing q~ confirms the earlier trends observed by Cowell et al. (24) and Everett and Stageman (25), where a much lower range of @values was investigated. The theta temperature (O) for PEO in 0.26 mol dm -3 MgSO4 solution is 59°C (27). The limiting value of the c.f.T, for latex C3 as ~b~ 0 is 55°C, slightly lower than 0 (outside GC

T, *C

4(

o.1

o12

FIG. 3. c.f.T~ as a function of particle volume fraction (q~) in 0.26 mole dm -3 MgSO4 solution for latex C3 and latex C6 (after five "clean-up" cycles). Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982

0

~02

0~4

096

098

FIG. 4. c.f.T, as a function of particle volume fraction (@) in 0.26 mole dm -3 MgSO4 solution for latex C4, at different bulk concentrations of PEO 1500, as indicated by the figures (wt% PEO 1500).

experimental error). Cowell et al. (24), for a similar latex (PS-PEO 2000), reported that the c.f.T, fell from 43 ° to 40°C over the @ range 10-5 to 10-3. Clearly, although the decrease in c.f.T, with @ is confirmed, the absolute values were significantly lower in the earlier work, compared to the present results. The PS-PEO latices used in that earlier work, however, did not contain divinylbenzene as a cross-linking agent, and therefore it is likely that the stabilizing PEO chains were not present at maximum coverage, due to partial desorption during the clean-up procedure. However, it should be stated that the c.f.T, of those latices was reproducible over the time period of the experiments, and therefore the trends observed are real. The c.f.T, data for latex C4 (similar to C3), as a function of ~b and various concentrations of free homopolymer PEO 1500, are shown in Fig. 4. Again, the closed-circle points were obtained by turbidimetric analysis, and the open circles by low-shear viscometry. The results indicate that, at a given value of q~, the c.f.T, decreases with increasing PEO 1500 concentration, i.e., a different trend from that observed in Fig. 2b. Clearly,

TEMPERATURE-PARTICLE

tl

ii! D. F .f.T.

iF.o: !

o :b I D

¢

.

FIG. 5. Schematic T-~b phase diagram for PS-g-PEO particles in aqueous electrolyte solution. Roman numerals indicate the number of coexisting "phases": D = dispersed phase; F = ftoc phase; O = ordered phase.

in the case of the data shown in Fig. 4 we are not observing an effect of coverage, but one due to the presence of large concentrations of free polymer. This effect on the c.f.T. has been reported and discussed previously (21, 22, 24) for volume fractions up to 10-3; these results serve to indicate that similar effects occur in concentrated dispersions.

Slow-Speed Centrifugation Four latex samples were centrifuged (750 rpm and 37°C), namely, C3 and C6 (initial = 10-3), each with and without added electrolyte (0.26 mole dm -3 MgSO4). After several weeks the tubes were examined and iridescence was observed in each case, except for latex C6 in the presence of 0.26 mol dm 3 electrolyte. However, it can be seen from Fig. 3 that the c.f.T, for latex C6 is 37 ° at q~ ~ 0.02. An equivalent statement, therefore, is that the critical flocculation volume fraction (c.f.40 for this system is 0.02 at 37°C. Thus, the reason iridescence was not observed here was because flocculation occurred when the c.f.q~ was exceeded during the concentration of the latex by sedimentation. In the other three cases the system remains stable up to high q~ values. These results are similar to those reported previously from this laboratory (28, 29) for polystyrene latices having physically ad-

PHASE DIAGRAMS

523

sorbed monolayers of poly(vinyl alcohol) (PVA) in electrolyte solution. In that case, only samples containing PVA above a critical molecular weight (i.e., adsorbed layer thickness) showed iridescence colors. Scanning electron microscopy (29) confirmed the ordered structure of the latex particles in these cases. Those samples not showing iridescence appeared to be flocculated; again the c.f.q~ appears to have been exceeded during centrifugation. DISCUSSION

We have described in this work evidence for three types of phase structure: (i) a "dispersed" phase (D), consisting of singlet particles; (ii) a "floe" phase (F), consisting of randomly packed particles; and (iii) an ordered phase (O), consisting of particles in a pseudocrystalline array. Clearly, there are direct analogies here with the vapor, condensed (liquid or amorphous solid), and crystalline solid phases observed with molecular systems. Figure 5 represents a schematic T-~ phase diagram for the systems studied here. Roman numerals indicate the number of coexisting "phases." The line ab represents the flocculation boundary line and be the disorder ~ order (disperse phase ---ocrystalline phase) transition. We have previously discussed (21, 24) the flocculation boundary in terms of an equation of the form A G f = A G i -- T A S h s ,

[4]

where AShs is the configurational entropy change associated with aggregation in a hard-sphere system; AShs is --re but IAShsl becomes smaller as 4~ increases. AGi is the free energy change associated with the other interactions (steric and van der Waals) between the particles. AGi is essentially a function of Gmin; it is also negative and increases with increasing IGmin]. The line ab in Fig. 5 represents the boundary condition AGf = 0. The boundary line bc may also be deJournal of Colloid and Interface Science, Vol. 87, N o . 2, J u n e 1982

524

COWELL

AND VINCENT

scribed by a condition of the form AG = 0, where AG is the free energy change associated with the phase change concerned. It seems reasonable to suppose that in this case AG = - TASh, where AShs changes sign from negative to positive at a critical ¢, i.e., a Kirkwood-Alder type transition. The temperature T¢ in Fig. 5 corresponds to the limiting value of the c.f.T, at high ¢. It represents a critical temperature for the system, i.e., below which no flocculation (analogous to condensation) can occur. The apparent limiting value of the c.f.T. at low ck is more difficult to interpret. Both Cowell et al. (24) and Everett and Stageman (25, 30) find that the c.f.T, is still ¢ dependent at very low particle volume fractions (down to 10-5). Everett (30) has suggested that we can express the equilibrium between singlet particles (chemical potential ~1) in the disperse phase and those in the floc phase at the c.f.T. (Tf) by a relationship of the form, #~,D q_ R T f In cD = #~,F [5] or

RTf In ~b~ = gl~'F

--

~?,D

:

Af/~,

[6]

where ¢o is equivalent to the c.f4b and Aftz~ is the standard free energy change associated with flocculation. From Eq. [6], the volume fraction dependence of the c.f.T, is given by a In ¢~

R LOT\

T /J

[7]

The sign of d T f / d In Co depends (30) on the sign of d ( A f # ~ / T ) / d T : d T f / d In ~bD is negative, as observed here, if Afar1 ~ becomes more negative as T increases, i.e., IGmi,Eincreases with increasing T, which we have argued previously (21, 24) to be the case for aqueous systems of the type described here. If the above thermodynamic analysis holds (that is, provided we are observing true equilibrium between the particles in the dispersed and floc phases), then clearly the c.f.T, should rise rather steeply as ~b ~ 0. It is difficult, however, to carry out experiJournal of Colloid and Interface Science, Vol. 87, No. 2, June 1982

ments at ¢ much less than 10 -5, at least using light-scattering techniques. It is, therefore, only meaningful to quote c.f.T, values if ¢ is specified. Napper (31) has suggested that the c.f.T. is independent of ¢. However, most of his experimental work has been carried out using sterically stabilized dispersions, where the stabilizing moieties were, in general, terminally anchored chains of relatively high molecular weight (>5000). In that case the major contribution to the interparticle interaction (Gt) is the mixing term (Grnix), in the steric interaction (31); contributions from the van der Waals interaction (GA), and the elastic component (Gel) of the steric interaction are negligible (at least at low degrees of polymer sheath-polymer sheath interpenetration for two contacting particles). Gmin, therefore, deepens considerably over a very short temperature interval in the vicinity of the 0 temperature. Hence, changes in AG; dominate changes in AShs (Eq. [4]) at the c.f.T., which will correspond closely to the 0 temperature, where Gmix changes sign (positive to negative), at all ~b. With lower molecular weight, terminally anchored chains the role of Gel and GA become increasingly more significant (24). Thus, first, the change in Groin with temperature near 0 becomes smoother, because the effect of G~I is to "mask" the large change in Groincaused by the change in sign of Gmin, hence the correlation between the c.f.T, and the 0 temperature is weakened. Second, the increasing contribution from GA means that, at any given temperature, IGminlis greater the lower the molecular weight of the chains. Hence, at a given ~b, the c.f.T, can be expected to be lower. Moreover, the greater the contribution from GA, the greater the ~b dependence of the c.f.T. The effect of stabilizer chain length is depicted schematically in Fig. 6. The full lines indicate the T-¢ phase boundaries for three chain lengths: a > b > c. For high-molecularweight chains (case a) the c.f.T, is virtually independent of ~ (as observed by Napper),

TEMPERATURE-PARTICLE PHASE DIAGRAMS T

tl

FIG. 6. Effect of tail length on the form of the (schematic) T-~b phase diagram for sterically stabilized particles in aqueous solution: (a) > (b) > (c). (Note: see Fig. 5 for a description of the various phases concerned.)

but the ~ dependence becomes much stronger, the smaller the chain length. No cases of stability in worse-than-O solvents have as yet been reported for particles having terminally anchored tails as the stabilizing moieties. However, so-called "enhanced stabilization" (31, 32) has been reported, for example, by Dobbie et al. (32), for latex particles stabilized by a physically adsorbed PS-PEO block copolymer at low coverage. It was suggested that, at low coverages, the PEO chains (as well as the PS chains) are adsorbed at the surface (presumably in a loop and train type of configuration). As the coverage increased, so the PEO chains gradually became more extended, finally adopting a tail-like configuration. The low coverage situation is, in some respects, analagous to the short-tail case reported in this work. That is, one might expect Gel to contribute significantly to Gi, due to the interaction of the loops on opposing surfaces. Hence, the correlation between the c.f.T, and the 0 temperature is again masked. Whether GA would play a significant role would depend on the overall thickness of the adsorbed layer. By analogy with the terminally anchored tail systems, one would only expect a strong ~b dependence of the c.f.T. if GA does play a significant role. With increasing coverage of the adsorbed PS-PEO block copolymer the c.f.T. (fixed

525

4) gradually decreased and eventually reached the 0 temperature, in line with the known behavior of high-molecular-weight tails. The effect of coverage on the c.f.T, in these systems, therefore, is in direct contrast to the results reported in this work. Hence the c.f.T, was observed to increase with increasing coverage (Fig. 2b), at fixed q~. In conclusion, therefore, one may say that, first, a strong dependence of the c.f.T, on q~ will only be observed when the van der Waals forces contribute significantly to the pairwise interparticle interactions, i.e., for relatively thin adsorbed layers. How "thin" depends on the nature and size of the particle cores. Second, the correlation between the c.f.T, and the 0 temperature depends strongly on the configuration and the coverage of the stabilizing moieties. Finally, we have demonstrated that three phases, analogous to the vapor, condensed, and crystalline phases observed with molecular systems, may also be observed with sterically stabilized particulate systems. The form of the T-~ phase diagram, however, and in particular the flocculation boundary line, depends strongly on the depth of the minimum (Groin) in the interaction free energy curve and its temperature dependence. Thus, for most nonaqueous systems, where IGminl increases with decreasing T, one can expect the T-~b flocculation boundary lines to curve in the opposite sense to that depicted in Figs. 5 and 6. REFERENCES 1. Ottewill, R. H., J. Colloid Interface Sci. 58, 357 (1977). 2. Hiltner, P. A., and Krieger, I. M., J. Phys. Chem. 73, 2386 (1969). 3. Hachisu, S., Kobayashi, Y., and Kose, A., J. Colloid Interface Sci. 42, 342 (1973). 4. Kose, A., Osaki, M., Takano, K., Kobayashi, Y., and Hachisu, S., J. Colloid Interface Sci. 44, 330 (1973). 5. Hachisu, S., and Kobayashi, Y., J. Colloid Interface Sci. 46, 470 (1977). 6. Hachisu, S., Kose, A., Kobayashi, Y., and Takano, K., J. Colloid Interface Sci. 55, 499 (1976). Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982

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7. Takano, K., and Hachisu, S., J. Phys. Soc. Japan 42, 1775 (1977). 8. Takano, K., and Hachisu, S., J. Chem. Phys. 67, 2604 (1977). 9. Takano, K., and Hachisu, S., J. Colloid Interface Sci. 66, 124, 130 (1978). 10. Snook, I., and van Megen, W., Chem. Phys. Lett. 33, 156 (1975). 11. van Megen, W., and Snook, I., J. Colloid Interface Sci. 53, 172 (1975). 12. Snook, I., and van Megen, W., J. Chem. Soc. Faraday 2 72, 216 (1976). 13. van Megen, W., and Snook, I., J. Colloid Interface Sci. 57, 40, 47 (1976). 14. van Megen, W., and Snook, I., J. Chem. Soc. Faraday Discuss. 65, 92 (1978). 15. Ottewill, R. H., Progr. Colloid Polym. Sci. 67, 71 (1980). 16. Alder, B. J., and Wainwright, T. C., Phys. Rev. 127, 352 (1962). 17. Alder, B. J., Hoover, H. G., and Young, D. A., J. Chem. Phys. 49, 3681 (1968). 18. Vincent, B., Advan. Colloid Interface Sci. 4, 193 (1974). 19. Vincent, B., and Whittington, S., in "Colloids and Surfaces" (E. Matijevi6, Ed.), Vol. 12. Plenum, New York, 1981.

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20. Long, J. A., Osmond, D. W. J., and Vincent, B., J. Colloid Interface ScL 42, 545 (1973). 21. Vincent, B., Luckham, P. F., and Waite, F. A., J. Colloid Interface ScL 73, 508 (1980). 22. Cowell, C., and Vincent, B., in "The Effect of Polymers on Dispersion Properties" (Th.F. Tadros, Ed.). S.C.I., London, 1982. 23. Bridger, K., Ph.D. thesis, Bristol, t979. 24. Cowell, C., Li-In-On, R., and Vincent, B., J. Chem. Soc. Faraday Trans. 2 74, 337 (1978). 25. Everett, D. H., and Stageman, J. F., ColloidPolym. Sci. 255, 293 (1977). 26. Attia, Y. A., and Rubio, J., Brit. Polym. J. 7, 135 (1975). 27. Boucher, E. A., and Hines, P. M., J. Polym. Sci. (Phys.) 14, 2241 (1976). 28. Garvey, M. J., Tadros, Th.F., and Vincent, B., J. Colloid Interface ScL 55, 440 (1976). 29. Garvey, M. J., J. Colloid Interface ScL 61, 194 (1977). 30. Everett, D. H., and Stageman, J. F., Faraday Discuss. Chem. Soc. 65, 230, 314 (1978). 31. Napper, D. H., J. Colloid Interface Sci. 58, 390 (1977). 32. Dobbie, J. W., Evans, R., Gibson, D. V., Smitham, J. B., and Napper, D. H., J. Colloid Interface Sci. 45, 557 (1973).