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Articles Depletion Flocculation of Nonaqueous Dispersions Containing Binary Mixtures of Nonadsorbing Polymers. Evidence for Nonequilibrium Effects Paul Jenkins and Brian Vincent* School of Chemistry, University of Bristol, Bristol BS8 1TS, U.K. Received November 14, 1994. In Final Form: April 3, 1996X Depletion flocculation, induced in dispersions of hydrophobic silica particles in nonpolar media by the presence of binary mixtures of nonadsorbing polymers, has been investigated both experimentally and theoretically. The binary mixtures used were either similar polymers having different molecular weights or mixtures of chemically different polymers. To this end various monodisperse samples of polystyrene and polydimethylsiloxane homopolymers were used. The order of mixing of the components was found to affect crucially the minimum value of the (total) volume fraction of polymer required for flocculation to be observed. If the particles are added to a solution of the premixed polymers, then the experimental results indicate that the larger polymer dominates the interaction. On the other hand, intuition would suggest that the smaller molecules should dominate, in the sense that they ought to partially “fill up” any depletion layer formed by the larger molecules. Indeed, theoretical modeling studies, presented here, do in fact predict that the smaller polymer should dominate the depletion interaction, under equilibrium conditions. Nonequilibrium effects (namely, an inhomogeneous distribution of the large polymer) have, therefore, been used to explain the discordant nature of the experimental and theoretical findings.
Introduction Much attention has been directed to the stability of colloidal dispersions in the presence of nonadsorbing polymers, with regard to the depletion interaction. Most academic studies have concentrated on dispersions of particles in solutions of reasonably monodisperse homopolymers. However, in practice, the continuous phase may well contain a complex mixture of dissolved polymers, i.e. of different molecular weights (polydisperse samples), or even chemically different polymers. A systematic study of depletion flocculation in such mixtures is, therefore, both desirable and necessary to understand the behavior of many industrial processes. In this paper we describe experiments in which the minimum (total) polymer concentration required to induce depletion flocculation of hydrophobed silica particles, dispersed in nonpolar solutions of binary homopolymer mixtures, was determined. The binary mixtures were different molecular weight samples of polystyrene (PS) and/or polydimethylsiloxane (PDMS). For binary mixtures of different molecular weight homopolymers of the same type, one might expect, intuitively, that the smaller molecular weight fraction ought to control the net depletion interaction, since the depletion layer thickness should effectively be set by this smaller molecular weight fraction (smaller molecules should “fill” the space left by the larger molecules in a depletion layer). We have performed calculations using the Schuetjens-Fleer self-consistent-field model1 for polymers at interfaces, and the results confirm this supposition. However, in the first set of experiments reported here, using such binary mixtures, we found the opposite trend, namely, that the larger molecular weight fraction polymer * Author to whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, June 1, 1996. (1) Fleer, G. J.; Cohen-Stuart, M. A.; Scheutjens, J. H. M. H.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993.
S0743-7463(94)00901-7 CCC: $12.00
dominates the depletion flocculation behavior. This puzzled us until we subsequently performed further experiments in which the order of mixing of the various components (silica particles, the higher molecular weight polymer, and the lower molecular weight polymer) was investigated. We then found that the predictions of the equilibrium theory could indeed be obtained experimentally but only if the higher molecular weight polymer component is added prior to the smaller molecular weight one. Experimental Section Preparation and Characterization of Silica Particles with Grafted n-Octadecyl Chains (SiO2-g-nC18). Monodisperse silica particles were prepared, either by the method of Sto¨ber et al.2 or by the procedure of Skuse et al.,3 using Ludox AS-40 as the seed particles. Particle diameters were determined by transmission electron microscopy. The esterification reaction of colloidal silica particles with n-octadecyl alcohol has been described by van Helden et al.4 The particle coverage by the n-octadecyl [nC18] chains was determined by elemental analysis. Data for the SiO2-g-nC18 particles used in this work are shown in Table 1. As may be seen from Table 1, the area per nC18 chain appears to decrease (or, conversely, the coverage increases) with increasing particle size. Such a trend has been observed before4 and has been ascribed to an increasing porosity of the particles with increasing size. For a monolayer of close-packed, n-alkyl chains, one would expect a value for the area per chain of around 0.22 nm2, from studies of surfactants at the planar air/water interface.5 Preparation and Characterization of Monodisperse Polystyrene (PS) and Polydimethylsiloxane (PDMS). PS samples were prepared in this laboratory by the anionic (2) Sto¨ber, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (3) Skuse, D. R.; Tadros, Th. F.; Vincent, B. Colloids Surf. 1986, 17, 343. (4) van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81, 354. (5) Rabonovitch, W.; Robertson, R. F.; Mason, S. G. Can. J. Chem. 1960, 38, 1881.
© 1996 American Chemical Society
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Jenkins and Vincent
∫0∞Gfp(h) dh
Table 1. Details of SiO2-g-nC18 Particles code
diameter/nm
Si6 Si1 Si3 Si2 Si4 Si8
37 ( 4 46 ( 4 81 ( 8 116 ( 15 199 ( 17 404 ( 14
area per nC18
0.49 0.39 0.19 0.15 0.09 0.04
Table 2. Details of Polymer Samples
a
Gs(h) ) πa
chain/nm2
code
Mn
Mw/Mn
PS 1 PS 2 PS 3 PS 5 PS 6a PDMS 3 PDMS 7b PDMS 8 PDMS 9
6 630 47 200 15 800 195 000 523 000 24 970 102 000 53 00 580 000
1.20 1.14 1.12 1.14 1.06 1.20 1.22 1.08 1.21
Supplied by Polymer Laboratories. b Supplied by Dow Corning.
polymerization of styrene using a standard vacuum frame technique, as reported previously.6 Toluene was used as the solvent, and n-butyl lithium, as the initiator. PDMS was prepared in a similar manner via the ring opening of hexamethylcyclotrisiloxane by n-butyl lithium. Dimethyl sulfoxide was used as the promoting agent, as described by Lee and co-workers.7 The number-average (Mn) and weight-average (Mw) molecular weights of the various prepared polymer samples were kindly determined by Dr. S. Holding of RAPRA, using gel-permeation chromatography (GPC) (Table 2). Detection of Flocculation. The stability behavior of the various silica dispersions was studied in the presence of free polymer mixtures. Samples were prepared gravimetrically in small glass phials, and additions of solvent, polymer solution, and silica dispersion were made, from existing stock solutions, using Gilson micropipets. Using appropriate density values, the gravimetric data were converted to volume fractions for both the free polymer and the SiO2-g-nC18 particles (φ2 and φ, respectively). Note that values for φ2 were corrected to take account of the exclusion of polymer chains from the space occupied by the particles. Samples were allowed to stand at ambient temperature (22 ( 1 °C) for 24 h. Flocculation was readily detected, visually, by an increase in turbidity, leading to colloidal phase separation.8 In general, this increase in turbidity occurred within 1 h of sample preparation. The minimum volume fraction of free polymer, required to induce flocculation (φ2†), in each case was determined by successive subdivisions of the upper and lower limits found in earlier attempts.
Theoretical Section The Scheutjens-Fleer (SF) self-consistent-field lattice theory1 was used to calculate segment density profiles and the interaction free energy between two flat plates, in the presence of nonadsorbing polymer mixtures. To this end their Goliad program (available from the Department of Physical and Colloid Chemistry, Agricultural University of Wageningen, The Netherlands) was used to effect the numerical integrations involved. In order to convert the interaction free energy calculated for two flat plates, Gfp(h), into that between two spheres, each of radius a, Gs(h), the Derjaguin approximation9 was used, (6) Jones, A.; Vincent, B. Colloids Surf. 1989, 42, 113. (7) Lee, C. L.; Frye, C. L.; Johannson, O. K. Polym. Prep. Am. Chem. Soc. 1969, 10, 1361. (8) Vincent, B.; Emmett, S. N.; Edwards, J.; Croot, R., Colloids Surf. 1988, 31, 267. (9) Derjaguin, B. Kolloid Z. 1934, 69, 155; Trans. Faraday Soc. 1940, 36, 203.
(1)
This equation is valid provided the particle radius is much greater than the range of the interactions. Note, in eq 1, h is the separation distance; to obtain this as a length, rather than a numbers of lattice layers, an estimate has to be made for the effective lattice size, l. Gs(h) may also be calculated analytically, at least for solutions of single, nonadsorbing polymers, at small separations (h), using the following equation, derived by Scheutjens, Fleer, and Vincent,10
(
Gs(h) ) 2πa
)(
)
µ1 - µ°1 2 2∆ ∆ 1+ ν°1 3a
(2)
where µ1 and µ°1 are the chemical potential of the solvent in the bulk polymer solution and the pure solvent, respectively, ν°1 is the molar volume of the pure solvent, and ∆ is the depletion layer thickness. The first term in parentheses on the right-hand side of eq 2 is equal to the osmotic pressure of the bulk polymer solution; it is given by the standard Flory-Huggins expression,
µ1 - µ°1 1 ) φb2 1 - + ln(1 - φb2) + χ12(φb2)2 kT r
(
)
(3)
where r is the number of segments per polymer chain and χ12 is the Flory polymer/solvent interaction parameter. Alternatively, the osmotic pressure of the bulk polymer solution may be determined experimentally. This was not the case here. Various approaches1 have been used to estimate ∆. For example, Vincent11 has derived the following implicit equation for ∆ as a function of φ2.
()[
∆ rg rg r - ) rg ∆ l φb 2
2/3
(
ln(1 - φb2) + φb2 1 -
]
1 + χ(φb2)2 (4) r
)
Equations 2-4 were used to calculate values of Gs(h) for silica particles in single polymer solutions. Calculation of Second Virial Coefficients. The second virial coefficient B2 of a colloidal dispersion is defined by eq 5,
∫2a2(a+∆)
B2 ) 4vp + 2π
{
(
1 - exp -
)}
Gs(R) R2 dR kT
(5)
where vp is the particle volume and R is the particle center-center separation (R ) h + 2a). For hard-sphere particles with no long-range interactions, B2 ) 4vp. Any attractive interactions will reduce B2 and eventually make it negative. Values of B2 may be used as relative estimates of the stability of colloidal dispersions under different conditions. For the case of colloidal phase separation, occurring under spinodal conditions, de Hek and Vrij12 used the following expression to relate B2 to the particle volume fraction (φ),
B2 1 )vp 2φ
(6)
However, colloidal phase separation, closer to binodal conditions, is much more likely to be the case for (10) Fleer, G. J.; Scheutjens, J. H. M. H.; Vincent, B. ACS Symp. Ser. 1983, 240, 245. (11) Vincent, B. Colloids Surf. 1990, 50, 241. (12) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409.
Depletion Flocculation of Nonaqueous Dispersions
Langmuir, Vol. 12, No. 13, 1996 3109
Figure 1. φ2† as a function of polymer mixture composition (WL) for Si1 particles dispersed in toluene, in the presence of various mixtures of free PS (different Mn), as indicated. φ ) 0.01.
Figure 2. φ2† as a function of polymer mixture composition (WL) for Si2 particles dispered in (1) a theta solvent and (2) a good solvent, in the presence of mixtures of PS1 and PS2. φ ) 0.01.
dispersions undergoing depletion flocculation. In that case, the situation is more complex to analyze, and full statistical mechanical treatments (e.g. hard-sphere perturbation theory13,14) are required to relate the conditions for phase separation to the pair potential. However, for qualitative purposes, estimates of B2, calculated from the SF model using eqs 1 and 5, will be used here to compare experimental values of φ2†, obtained as a function of the relative concentration of the higher molecular weight fraction in binary polymer mixtures, for silica dispersions at the same value of φ. Results and Discussion Experiments Using Premixed Polymer Solutions. The composition of a bimodal polymer solution is given by
WL ) 100(wL/wT)
(7)
where WL is the weight of the larger molecular weight fraction present in the polymer mixture and WT is the total weight of polymer. The effects of different polymer molecular weight mixtures, giving rise to different net Mn values (Figure 1), polymer solvency conditions (Figure 2), particle radius (Figure 3), and particle volume fractions (Figure 4), were all investigated for SiO2-g-nC18 particles dispersed in nonpolar media containing bimodal polystyrene mixtures. In general, the molecular weight of the larger fraction in the mixture was an order of magnitude greater than that of the smaller one. The trends shown in Figures 1-4 are all in agreement with those seen previously with single polymer solutions. For bimodal polymer mixtures, φ2† (at any given value of the polymer mixture composition WL) was found to decrease with (i) increasing polymer mixture, Mn, (ii) decreasing free polymer solvency parameter, χ, (iii) increasing particle radius, a, and (iv) increasing particle volume fraction, φ. (13) Gast, A. P.; Han, C. K.; Russel, W. B. J. Colloid Interface Sci. 1983, 96, 251. (14) Van Megen, W.; Snook, I. Adv. Colloid Interface Sci. 1984, 21, 119.
Figure 3. φ2† as a function of polymer mixture composition (WL) for different size SiO2-g-nC18 particles dispersed in toluene, in the presence of mixtures of free PS3 and PS5. φ ) 0.01.
Figures 1-4 are for chemically-equivalent polymer pairs; Figure 5 is for two chemically different polymers. Chemically-different polymer pairs are, in general, immiscible and will only form a stable, single-phase solution at low (net) polymer concentrations. Hence, for any study involving depletion flocculation, the establishment of this one-phase region is important, in order to ensure that any phase separation observed is caused by the depletion mechanism and not polymer-polymer immiscibility. Figure 6 shows the phase diagram for the ternary polymer-polymer-solvent system PDMD3/PS5/toluene. It may be seen that a stable single-phase region is only observed for mixtures containing total polymer volume fractions of less than about 5%. The “curvatures” [i.e. dφ2/dWL] of the φ2† versus WL plots shown in Figures 1-5 are similar, i.e. dφ2/dWL decreases as WL increases. In dilute solution, the depletion layer thickness (∆) is of the order of the radius of gyration of
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Figure 6. Ternary phase diagram of PDMS3/PS5/toluene, 25 °C.
Figure 4. φ2† as a function of polymer mixture composition (WL) for different volume fractions of Si3 particles dispersed in toluene, in the presence of mixtures of free PS3 and PS5. φ ) 0.01.
Figure 7. φ2† as a function of polymer mixture composition (WL) for Si8 particles dispersed in cyclohexane, in the presence of mixtures of PDMS8 and PDMS9. φ ) 0.01. The dotted upper line illustrates the expected (equilibrium) stability boundary.
Figure 5. φ2† as a function of polymer mixture composition (WL) for Si6 particles dispersed in toluene, in the presence of the following polymer mixtures: (1) PDMS3/PS5 and (2) PS3/ PS5. φ ) 0.01.
the polymer.10 Hence, for the binary mixtures, the depletion layer associated with the larger polymer (∆L) will be greater than that of the smaller polymer (∆S). However, for osmotic reasons, the depletion zone of the large polymer may be expected to “draw in” molecules of the smaller polymer. Hence, a binary mixture would possess an effective depletion zone having a thickness comparable to ∆S. Thus, a binary polymer solution, containing low- and high-molecular weight polymers would be predicted to behave similarly to a solution containing only the smaller polymer but at an equivalent osmotic pressure. A schematic plot of how φ2† would be expected to behave as a function of WL, based on this intuitive argument, is shown by the dotted line in Figure 7. However, the actual experimental data, shown in Figures 1-5, together with those obtained for a system
based on PDMS mixtures [Si8/PDMS8/PDMS9/cyclohexane], given by the solid line in Figure 7, imply that it is the larger polymer which dominates the interaction. Clearly, an explanation for this behavior is required. To this end it was decided, firstly, to see if the SF theory predicts similar behavior to that implied by the intuitive reasoning presented above. Comparison of Experiment and Theory. As indicated earlier, the SF analysis requires use of the Derjaguin approximation9 in order to calculate Gs(h) curves and hence B2 values. As mentioned earlier, this approximation is only valid when the radius of the particles is large in comparison to the range of the depletion interaction (i.e. a . ∆). It was decided, therefore, to use the experimental data shown in Figure 7, i.e. for the phase separation of Si8 particles (a ) 202 nm), dispersed in cyclohexane, induced by bimodal PDMS mixtures, as a basis for a theoretical comparison. Figure 8 shows the relative segment volume fraction profile, computed for this mixture of PDMS polymers in a good solvent (χ ) 0.4), in the presence of a nonadsorbing, planar wall (χs ) 0). The bulk polymer volume fraction used for each profile was that found to induce flocculation for that particular polymer composition, i.e. φ2†. It can be
Depletion Flocculation of Nonaqueous Dispersions
Langmuir, Vol. 12, No. 13, 1996 3111 Table 3. Calculation of Gs(0) Values Using the Analtyical Approach10 polymer φ2† r χ ∆/nm Gs(0)/kT WL/%
PDMS 8 0.0022 163 0.40 8.9 -8.1 0
PDMS 9 0.0008 1781 0.40 29.5 -3.5 100
Figure 8. Relative segment volume fraction profiles computed for bimodal mixtures of PDMS8 and PDMS9. χ ) 0.40, χs ) 0, rs ) 163, and rL ) 1781. z is the number of lattice layers.
Figure 10. Reduced second virial coefficient B2/vp as a function of polymer mixture composition WL for the Si8/PDMS8/PDMS9/ cyclohexane system. Plots 1-5 illustrate the effect of increasing the segment length l for PDMS.
Figure 9. Depletion interaction energy Gsph dep as a function of particle surface-surface separation h, for the Si8/PDMS8/ PDMS9/cyclohexane system. Same parameters as for Figure 8.
inferred that, as WL increases, so the distance from the wall (z lattice layers) at which φi/φb attains a value of unity becomes progressively greater; i.e., ∆ for the mixtures is increasing. Unfortunately, given the limited plate separation (i.e. maximum number of lattice layers, M ) 100) for which the Goliad program could be run, the segment volume fraction did not quite reach the bulk polymer concentration value (i.e. φi/φb ) 1) by z ) 50 in all cases. Gs(h) plots, each corresponding to the equivalent φi/φb plot for the various mixtures shown in Figure 8, are presented in Figure 9. One problem which needed to be addressed in these calculations was the choice of the
effective segment length l for PDMS (i.e. the size of a lattice element in the Scheutjens-Fleer model). For the Gs(h) data shown in Figure 9 a value of l ) 3.02 nm was used. The basis of this choice was a comparison of the Gs(0) (i.e. “contact energy”, h ) 0) values obtained using this SF approach and those calculated, for the two singlepolymer solution cases (PDMS 8 and 9 alone), using the analytical approach of Scheutjens, Fleer, and Vincent,10 outlined in the Theoretical Section. Some of the parameters used or derived in these latter calculations are given in Table 3. The values shown in the penultimate row of Table 3 agree well with the corresponding ones obtained from Figure 9 (SF theory) for WL ) 0 and 100, i.e. ∼ -8kT and ∼ -4kT, respectively. The value of l used (3.02 nm) may seem somewhat high for PDMS. Indeed, Vincent et al.8 had previously estimated a value of 1.09 nm for PDMS (based on comparing experimental values of the radius of gyration and the equivalent contour length of the chains from molecular models). B2 values have been calculated using eq 5, for a range of values of l between these two extremes. The results are shown in Figure 10. As may be seen, the choice of l is critical. A choice of l ) 1.09 nm would yield B values which are positive at all WL values (curve 1, Figure 10). Phase separation could simply not occur in that case. For the dispersions corresponding to the experimental data shown in Figure 7, φ ) 0.01. According to eq 6 spinodal phase separation would occur at B/vp values ∼ -50, at this particle volume fraction. Although, as indicated earlier, colloidal phase separation is more likely to occur closer to binodal conditions (and presumably, therefore, lower B/vp values), it is of interest to note from
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Figure 11. φ2† as a function of polymer mixture composition (WL) for Si3 particles dispersed in toluene, in the presence of mixtures of PS3 and PS5. An investigation into the effect of adding polymers in different orders. φ ) 0.01.
Figure 12. φ2† as a function of polymer mixture composition (WL) for Si6 particles dispersed in toluene, in the presence of mixtures of PS3 and PS5. An investigation into the effect of adding polymers in different orders. φ ) 0.01.
Figure 10 that a choice of l ) 3.02 nm leads to B/vp values in the range -70 to -180, over the whole WL range. Although not too great an emphasis should be placed on the actual magnitudes of these B/vp values, the trend in these values with varying WL, shown in curve 5 of Figure 10, is significant. One might expect B/vp to be roughly independent of WL, or at most vary monotonically with WL, if flocculation were to occur under equilibrium conditions. However, it should be recalled that the φ2 values chosen for the computations shown in Figure 10 were the experimental φ2† values. What the trend in curve 5 is suggesting, therefore, is that, for equilibrium conditions to be achieved with mixed polymer solutions, a higher φ2† value (giving a higher B2/vp value) would be required than is actually observed. Thus, it would seem that depletion flocculation, induced using premixed binary polymer solutions, proceeds under nonequilibrium conditions, at least in the experiments reported here. Order of Mixing Experiments. In order to establish whether the order of polymer addition had any effect on the observed stability of the dispersions, three sets of complimentary experiments were performed, as outlined below: (i) φ2† values were determined, as described previously, using premixed polymer solutions. (ii) The required amounts of solvent, low molecular weight polymer solution, and dispersion were added to a vial, which was then sealed and allowed to stand for 8 h. The necessary quantity of high molecular weight polymer solution to induce flocculation was then added. (iii) The experiment was repeated as in (ii) above, but this time the high molecular weight polymer was added before and low molecular weight polymer. In all three experiments the samples were left to stand for 24 h before detection of flocculation was attempted. Figures 11 and 12 illustrate the stability boundaries that were established for the Si3/PS3/PS5/toluene and Si6/PS3/PS5/toluene systems, respectively. The striking aspect of these results is the difference between the stability boundaries observed for the three distinct experiments described. The boundaries established for experiments (i) and (ii) were coincident, whilst that for experiment (iii) occurred at higher φ2† values for inter-
mediate values of WL. In fact, the form of the stability boundary observed for experiment (iii), in both Figures 11 and 12, is very similar to the schematic one shown in Figure 7, which is the result to be expected if depletion flocculation were to occur under equilibrium conditions! Explanation of Nonequilibrium Behavior. One plausible explanation for the observed nonequilibrium behavior when either a premixed polymer solution is added to the particulate dispersion or the lower molecular weight polymer is added first (experiments (i) and (ii) above) is as follows. Consider a stable colloidal dispersion being added to a homogeneous polymer solution, containing polymer molecules of a single molecular weight (but at a polymer volume fraction below φ2†). Mixing of such a system would be expected to produce a homogeneous solution, with respect to the distribution of particles and polymer molecules. However, this situation may not pertain; the attraction between the particles (induced by the depletion interaction), whilst insufficient to cause flocculation per se, may be sufficient that small, fluctuating regions, containing higher particle concentrations and relatively few polymer molecules, could form. These nonmixed regions will be termed “pseudoflocs” in the subsequent discussion. If a second polymer, smaller in size than the original one, is added to this dispersion of “pseudoflocs”, then, after mixing, these smaller chains can penetrate inside the solution space occupied by the “pseudoflocs”, as well as the open, bulk volume space between them. However, if the second polymer added were significantly larger than the first one, it can be envisaged that these large polymer molecules may not be able to enter the solution space inside the “pseudoflocs”, because of entropic restrictions. Only the bulk solution space, between the “pseudoflocs”, would be available to the large polymer molecules, thus creating a solution which is very inhomogeneous with respect to the large polymer. Hence, the effective bulk solution volume fraction of the large polymer would be significantly increased and flocculation might be expected to occur at correspondingly lower polymer volume fractions than predicted by equilibrium theories. In the former situation [i.e. addition of the large polymer followed by the small polymersexperiment (iii)], floc-
Depletion Flocculation of Nonaqueous Dispersions
culation of “primary” colloidal particles occurs as expected, with the depletion layer thickness around the particles being mainly determined by the small polymer. However, in the latter case [i.e. addition of the small polymer followed by the large polymersexperiment (ii)], only the small polymer may adopt the expected concentration profile around the “primary” particles. Because of the exclusion of the large polymer from the solution space within the “pseudoflocs”, it may be envisaged that the large polymer sees these “pseudoflocs” as the main particle species. Hence, the decrease in φ2†, compared to equilibrium expectations, might be ascribed not only to the increase in the effective polymer volume fraction in the bulk solution but also to the increased size of these “pseudoflocs” compared to the “primary” particles. The coincidence of the stability boundaries for experiments (i) and (ii) can be readily explained using this hypothesis. In the premixed solutions, experiment (i), the smaller polymer molecules, having a higher diffusion coefficient, reach the particles first and form the “pseudofloc” system already described. By the time the larger polymer molecules arrive at the particles, the “pseudoflocs” are already present and the large polymers are unable to penetrate them. This situation is analogous to that proposed for the addition of small followed by large polymers, and thus it is not surprising that similar stability boundaries are observed in both cases. The mechanism just outlined is supported by the work of MacMillan, Garvey, and Vincent.15 These authors studied the stability of aqueous dispersions of hydrophilic silica particles in the presence of sodium poly(acrylic acid) (NaPAA), at a pH where adsorption of NaPAA onto the silica particles did not occur. The order of mixing the water, NaPAA, and silica was found to greatly affect the type of flocculation observed. To investigate this further, MacMillan et al. took an initially stable dispersion, at a NaPAA concentration lower than φ2†, and concentrated the solution by slow evaporation (15) MacMillan, R. A. Ph.D. Thesis, Bristol, 1989. MacMillan, R. A.; Garvey, M. J.; Vincent, B. To be published.
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of water. Microscopy revealed that, on the verge of instability, the discrete silica particles aggregated to form flocs (
