^{1,a)}, Gerhard Nägele

^{1}, Johan Buitenhuis

^{1}, Remco Tuinier

^{2}and Jan K. G. Dhont

^{1}

### Abstract

We study polymer depletion-driven cluster aggregation and initial phase separation in aqueous dispersions of charge-stabilized silica spheres, where the ionic strength and polymer (dextran) concentration are systematically varied, using dynamic light scattering and visual observation. Without polymers and for increasing salt and colloid content, the dispersions become increasingly unstable against irreversible cluster formation. By adding nonadsorbing polymers, a depletion-driven attraction is induced, which lowers the stabilizing Coulomb barrier and enhances the cluster growth rate. The initial growth rate increases with increasing polymer concentration and decreases with increasing polymer molar mass. These observations can be quantitatively understood by an irreversible dimer formation theory based on the classical Derjaguin, Landau, Verwey, and Overbeek pair potential, with the depletion attraction modeled by the Asakura–Oosawa–Vrij potential. At low colloid concentration, we observe an exponential cluster growth rate for all polymer concentrations considered, indicating a reaction-limited aggregation mechanism. At sufficiently high polymer and colloid concentrations, and lower salt content, a gas-liquidlike demixing is observed initially. Later on, the system separates into a gel and fluidlike phase. The experimental time-dependent state diagram is compared to the theoretical equilibrium phase diagram obtained from a generalized free-volume theory and is discussed in terms of an initial reversible phase separation process in combination with irreversible aggregation at later times.

We thank Sylvia de Waal and Hui Ning for the TEM picture, and George Petekidis and Benoit Loppinet for their encouraging interest and valuable suggestions.

I. INTRODUCTION

II. THEORY

A. DLVO-type description of aggregation kinetics

B. Depletion-induced cluster aggregation

C. Generalized free-volume theory of equilibrium phase diagram

III. EXPERIMENTAL DETAILS

A. Sample materials

B. Experimental techniques and sample characterization

1. Colloidal particle size and polymer radius of gyration

2. Viscosity of the polymer solution

3. Evidence that dextran does not adsorb at the silica-water interface

IV. RESULTS AND DISCUSSION

A. Cluster-aggregation in pure silica dispersions

B. Effect of the salt concentration on the aggregation rate

C. Theoretical description of the aggregation rate

D. Influence of colloid volume fraction on aggregation rate

E. Effect of added nonadsorbing polymer chains

F. Effect of polymer-to-colloid size ratio on the aggregation rate

G. Nonequilibrium state diagrams

V. CONCLUSIONS

### Key Topics

- Colloidal systems
- 92.0
- Polymers
- 83.0
- Aggregation
- 55.0
- Silica
- 26.0
- Hydrodynamics
- 17.0

## Figures

Measured collective diffusion coefficient, , obtained by DLS as a function of the silica particle volume fraction, , for a salt concentration of .

Measured collective diffusion coefficient, , obtained by DLS as a function of the silica particle volume fraction, , for a salt concentration of .

(a) TEM picture of a dried Ludox particles dispersion. (b) Size distribution of Ludox particles observed from image analysis. The mean particle radius is (see text for details).

(a) TEM picture of a dried Ludox particles dispersion. (b) Size distribution of Ludox particles observed from image analysis. The mean particle radius is (see text for details).

Shear viscosity of a pure dextran solution as a function of the reduced concentration of polymer coils, , for three different molecular weights (see legend) at . The solid curve is a fit to the experimental data (see text).

Shear viscosity of a pure dextran solution as a function of the reduced concentration of polymer coils, , for three different molecular weights (see legend) at . The solid curve is a fit to the experimental data (see text).

Left: Collective diffusion coefficient, , of a Ludox silica dispersion at as a function of added dextran concentration, , , for , 0.1, and 0.2 mol/l, respectively. Right: Effective hydrodynamic radius, , obtained from the single-particle Stokes–Einstein relation using . The lower scale of the abscissa gives the polymer surface concentration, , that would result if all polymers did adsorb on the surface of the silica particles. As a rule of thumb, the surface is completely covered when (Ref. 48).

Left: Collective diffusion coefficient, , of a Ludox silica dispersion at as a function of added dextran concentration, , , for , 0.1, and 0.2 mol/l, respectively. Right: Effective hydrodynamic radius, , obtained from the single-particle Stokes–Einstein relation using . The lower scale of the abscissa gives the polymer surface concentration, , that would result if all polymers did adsorb on the surface of the silica particles. As a rule of thumb, the surface is completely covered when (Ref. 48).

Normalized measured time-dependent (effective) collective diffusion coefficient of Ludox spheres as a function of the elapsed time after sample preparation, for varying (see legend) and .

Normalized measured time-dependent (effective) collective diffusion coefficient of Ludox spheres as a function of the elapsed time after sample preparation, for varying (see legend) and .

Semilogarithmic plot of the reduced time-dependent effective hydrodynamic radius of colloid clusters deduced for several salt concentrations as a function of time for . The straight line segments are fits to the form , with the aggregation time . (▽): (a) , (b) ; (○): (c) , (d) ; (◇): (e) ; (◻): (f) , respectively.

Semilogarithmic plot of the reduced time-dependent effective hydrodynamic radius of colloid clusters deduced for several salt concentrations as a function of time for . The straight line segments are fits to the form , with the aggregation time . (▽): (a) , (b) ; (○): (c) , (d) ; (◇): (e) ; (◻): (f) , respectively.

Initial aggregation time, , as a function of salt concentration , for .

Initial aggregation time, , as a function of salt concentration , for .

Effective colloid charge number as a function of added salt concentration, obtained from matching Eq. (5) to the experimentally determined initial aggregation time given in Fig. 7. The solid curve is a fit to the form , with , , and .

Effective colloid charge number as a function of added salt concentration, obtained from matching Eq. (5) to the experimentally determined initial aggregation time given in Fig. 7. The solid curve is a fit to the form , with , , and .

DLVO pair potential for parameters obtained from adjusting the effective colloid charge number , entering the calculation of , to the experimentally found initial aggregation time, for a system with at (see Table I).

DLVO pair potential for parameters obtained from adjusting the effective colloid charge number , entering the calculation of , to the experimentally found initial aggregation time, for a system with at (see Table I).

Semilogarithmic plot of the effective hydrodynamic radius of aggregated colloidal clusters as a function of the elapsed time. The straight lines (a)–(f) are fits to the form , with the parameters and determined right after the sample preparation. The data for , and are replotted for comparison from Fig. 6. : (⌂, f) , ; (▲, e) , ; (▽, c) , ; : (◼, d) , ; (◇, b) , ; (◻, a) , .

Semilogarithmic plot of the effective hydrodynamic radius of aggregated colloidal clusters as a function of the elapsed time. The straight lines (a)–(f) are fits to the form , with the parameters and determined right after the sample preparation. The data for , and are replotted for comparison from Fig. 6. : (⌂, f) , ; (▲, e) , ; (▽, c) , ; : (◼, d) , ; (◇, b) , ; (◻, a) , .

Aggregation time, , as a function of inverse colloid volume fraction, for (○) and (◻). The dashed and solid curves are the theoretically predicted aggregation times for and 0.3 mol/l, respectively, calculated from Eq. (5), using in Eq. (4) (see also Fig. 9).

Aggregation time, , as a function of inverse colloid volume fraction, for (○) and (◻). The dashed and solid curves are the theoretically predicted aggregation times for and 0.3 mol/l, respectively, calculated from Eq. (5), using in Eq. (4) (see also Fig. 9).

Semilogarithmic plot of the reduced effective hydrodynamic radius of colloid clusters as a function of elapsed time, for a mixture of Ludox silica spheres at and dextran at varying concentrations, with , and . The straight lines are fits to the form , with the characteristic time and the effective hydrodynamic radius measured right after sample preparation. For comparison, the data points for are replotted from Fig. 6. : (◻); : (△) ; : (○) .

Semilogarithmic plot of the reduced effective hydrodynamic radius of colloid clusters as a function of elapsed time, for a mixture of Ludox silica spheres at and dextran at varying concentrations, with , and . The straight lines are fits to the form , with the characteristic time and the effective hydrodynamic radius measured right after sample preparation. For comparison, the data points for are replotted from Fig. 6. : (◻); : (△) ; : (○) .

The data points (symbols) give the experimentally determined aggregation time as a function of the reduced polymer concentration, in a mixture of dextran with varying range of attractions, , , and , respectively, and silica particles with and . The curves are the theoretically predicted based on the AOV potential and values of as explained in Sec. II B.

The data points (symbols) give the experimentally determined aggregation time as a function of the reduced polymer concentration, in a mixture of dextran with varying range of attractions, , , and , respectively, and silica particles with and . The curves are the theoretically predicted based on the AOV potential and values of as explained in Sec. II B.

Total pair potential for , , and , with , , , , and fixed polymer concentration, . The black dashed curve is the repulsive electrostatic part, and the dashed-dotted (turquoise) curve the short-ranged vdW part of . The inset displays the Coulomb barrier part of located at smaller particle separations.

Total pair potential for , , and , with , , , , and fixed polymer concentration, . The black dashed curve is the repulsive electrostatic part, and the dashed-dotted (turquoise) curve the short-ranged vdW part of . The inset displays the Coulomb barrier part of located at smaller particle separations.

Photographs of samples containing a colloid-polymer mixture with [see also Fig. 16(c)], and polymer molar mass . The picture has been taken two days after sample preparation. Sample (i) with and has become turbid and forms a gel at later times. In sample (ii), where and , and sample (iii), where and , two phases are observed. The total height, , of the dispersion and the height, , of the more turbid bottom phase are indicated by arrows.

Photographs of samples containing a colloid-polymer mixture with [see also Fig. 16(c)], and polymer molar mass . The picture has been taken two days after sample preparation. Sample (i) with and has become turbid and forms a gel at later times. In sample (ii), where and , and sample (iii), where and , two phases are observed. The total height, , of the dispersion and the height, , of the more turbid bottom phase are indicated by arrows.

Nonequilibrium state diagrams of aqueous mixtures of Ludox silica particles and dextran for varying salt concentrations: [chart (a)] , (b) , (c) , (d) , and (e) for at room temperature. The phase diagrams of the samples have been recorded by visual inspection two weeks after sample preparation. Open circles (○) indicate fluidlike homogeneous mixtures. The half-filled circles describe samples, where a turbid viscous phase is observed at the container bottom. The triangles (▲) mark samples that form a gel throughout the sample. The solid dividing curves are guides to the eye, separating the single-fluid phase region from the region where phase separation or a system-spanning gel is observed after 2 weeks. For all salt concentration considered, these dividing curves are summarized in (f).

Nonequilibrium state diagrams of aqueous mixtures of Ludox silica particles and dextran for varying salt concentrations: [chart (a)] , (b) , (c) , (d) , and (e) for at room temperature. The phase diagrams of the samples have been recorded by visual inspection two weeks after sample preparation. Open circles (○) indicate fluidlike homogeneous mixtures. The half-filled circles describe samples, where a turbid viscous phase is observed at the container bottom. The triangles (▲) mark samples that form a gel throughout the sample. The solid dividing curves are guides to the eye, separating the single-fluid phase region from the region where phase separation or a system-spanning gel is observed after 2 weeks. For all salt concentration considered, these dividing curves are summarized in (f).

Reduced total pair potential, , for , 0.15, and 0.2 mol/l with , , and . The inset shows the Coulomb barrier part.

Reduced total pair potential, , for , 0.15, and 0.2 mol/l with , , and . The inset shows the Coulomb barrier part.

Time evolution of the nonequilibrium state diagram for an aqueous mixture of silica particles and dextran, with and . The gray (blue) symbols give the state of the sample two days after sample preparation. The black symbols describe the state after 2 weeks. The theoretically predicted binodal [dashed (green) curve] and spinodal [solid (red) curve] are obtained form GFVT on assuming -solvent conditions. The asterisks denote the critical point.

Time evolution of the nonequilibrium state diagram for an aqueous mixture of silica particles and dextran, with and . The gray (blue) symbols give the state of the sample two days after sample preparation. The black symbols describe the state after 2 weeks. The theoretically predicted binodal [dashed (green) curve] and spinodal [solid (red) curve] are obtained form GFVT on assuming -solvent conditions. The asterisks denote the critical point.

## Tables

Parameters characterizing for several experimentally analyzed salt concentrations. The effective charge number, , has been adjusted to obtain the experimental values of without added polymers as described in the text.

Parameters characterizing for several experimentally analyzed salt concentrations. The effective charge number, , has been adjusted to obtain the experimental values of without added polymers as described in the text.

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