^{1}and Peter Sollich

^{1}

### Abstract

We study theoretically the equilibrium phase behavior of a mixture of *polydisperse* hard-sphere colloids and monodisperse polymers, modeled using the Asakura–Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys.22, 1255 (1954)] within the free volume approximation of H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren [Europhys. Lett.20, 559 (1992)]. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagramtopologies as the polymer-colloid size ratio and the colloid polydispersity are varied. Quantitatively, we find that polydispersity disfavors fluid-solid against gas-liquid separation, causing a substantial *lowering* of the threshold value above which stable two-phase gas-liquid coexistence appears. Phase splits involving two or more solids can occur already at low colloid concentration, where they may be kinetically accessible. We also analyze the strength of colloidal size fractionation. When a solid phase separates from a fluid, its polydispersity is reduced most strongly if the phase separation takes place at low colloid concentration and high polymer concentration, in agreement with experimental observations. For fractionation in gas-liquid coexistence we likewise find good agreement with experiment, as well as with perturbative theories for near-monodisperse systems.

I. INTRODUCTION

II. FREE VOLUME THEORY

III. MOMENT FREE ENERGY METHOD

IV. PHASE DIAGRAMTOPOLOGIES

V. QUANTITATIVE ANALYSIS OF PHASE DIAGRAMS

A. Cloud and shadow curves

B. Fractionation effects

C. Inner phase boundaries

VI. CONCLUSION AND OUTLOOK

### Key Topics

- Colloidal systems
- 183.0
- Polymers
- 145.0
- Phase diagrams
- 41.0
- Free energy
- 33.0
- Phase separation
- 22.0

## Figures

Phase diagram sketch for colloid-polymer mixtures with three different values of the size ratio . For the depletion interaction is weak and the only effect is the widening of the fluid-solid coexistence region; at (right) the longer polymers create a sufficiently long-range interaction that induces the formation of gas and liquid phases and hence of a critical point, marked with a circle. The crossover value (middle) is the value where a region of gas-liquid coexistence first appears.

Phase diagram sketch for colloid-polymer mixtures with three different values of the size ratio . For the depletion interaction is weak and the only effect is the widening of the fluid-solid coexistence region; at (right) the longer polymers create a sufficiently long-range interaction that induces the formation of gas and liquid phases and hence of a critical point, marked with a circle. The crossover value (middle) is the value where a region of gas-liquid coexistence first appears.

Phase diagram on a grid of values, . In each region the nature of the phase(s) coexisting at equilibrium is indicated (, fluid; , solid; , gas; , liquid). The dashed lines indicate the best guess of the phase boundary in regions where our numerical data become unreliable.

Phase diagram on a grid of values, . In each region the nature of the phase(s) coexisting at equilibrium is indicated (, fluid; , solid; , gas; , liquid). The dashed lines indicate the best guess of the phase boundary in regions where our numerical data become unreliable.

Phase diagram for polydisperse hard spheres without added polymer. Note the appearance of regions with multiple solid coexistence as polydispersity increases. Horizontal cuts at and 0.08 give the behavior along the base lines of the graphs in the previous figure. From Ref. 1.

Phase diagram for polydisperse hard spheres without added polymer. Note the appearance of regions with multiple solid coexistence as polydispersity increases. Horizontal cuts at and 0.08 give the behavior along the base lines of the graphs in the previous figure. From Ref. 1.

Phase diagram at and . Note the topology at high polymer concentrations where the region of gas-liquid phase splits terminates, giving way to phase separation involving only a single fluid phase and one or more solids. The circles labelled (a–c) indicate at which points in the phase diagram the diameter distributions in Fig. 13 below are calculated.

Phase diagram at and . Note the topology at high polymer concentrations where the region of gas-liquid phase splits terminates, giving way to phase separation involving only a single fluid phase and one or more solids. The circles labelled (a–c) indicate at which points in the phase diagram the diameter distributions in Fig. 13 below are calculated.

Sketch of expected phase diagram topologies as a function of polymer size and colloid polydispersity . The sequence of phases along the base line in each row corresponds to , 0.07, and 0.08, respectively. Middle row, right: Dashed lines indicate a possible alternative topology which can still be connected smoothly to the one below but is physically less plausible. Bottom row, middle: The dotted line indicates the remnant of the three-phase region, whose area shrinks to zero at the transition between the two topologies on the left and right.

Sketch of expected phase diagram topologies as a function of polymer size and colloid polydispersity . The sequence of phases along the base line in each row corresponds to , 0.07, and 0.08, respectively. Middle row, right: Dashed lines indicate a possible alternative topology which can still be connected smoothly to the one below but is physically less plausible. Bottom row, middle: The dotted line indicates the remnant of the three-phase region, whose area shrinks to zero at the transition between the two topologies on the left and right.

Plots of cloud and shadow curves at and 0.08. Dotted lines connect example cloud-shadow pairs. Left: small polymers, ; the cloud curve marks the onset of fluid-solid phase separation. Right: large polymers, ; the cloud curve now has two branches, which meet in a triple point (solid squares). Fluid-solid phase separation occurs at low , with the solid shadow curve just visible in the bottom right corner; at higher , the initial phase separation is into gas and liquid. Polydispersity effects are weak there: the critical point (marked by the circle and triangle for and 0.08, respectively) is independent of on the scale of the plot, as are the gas-liquid cloud and shadow curves. These curves in fact nearly coincide, as they would in a monodisperse system; therefore only the part of the cloud curve below the critical point is shown, with the corresponding shadow curve above.

Plots of cloud and shadow curves at and 0.08. Dotted lines connect example cloud-shadow pairs. Left: small polymers, ; the cloud curve marks the onset of fluid-solid phase separation. Right: large polymers, ; the cloud curve now has two branches, which meet in a triple point (solid squares). Fluid-solid phase separation occurs at low , with the solid shadow curve just visible in the bottom right corner; at higher , the initial phase separation is into gas and liquid. Polydispersity effects are weak there: the critical point (marked by the circle and triangle for and 0.08, respectively) is independent of on the scale of the plot, as are the gas-liquid cloud and shadow curves. These curves in fact nearly coincide, as they would in a monodisperse system; therefore only the part of the cloud curve below the critical point is shown, with the corresponding shadow curve above.

(a) Spinodal curves obtained for different values of together with the cloud curve at and critical point. (b) Change in the position of the critical point with colloid polydispersity at fixed polymer size .

(a) Spinodal curves obtained for different values of together with the cloud curve at and critical point. (b) Change in the position of the critical point with colloid polydispersity at fixed polymer size .

Plot of the phase boundaries defining the onset of (solid line) and (dashed line) coexistence at colloid polydispersities (left) and (right). To make the curves at different polymer sizes visually distinguishable, those for the lowest and highest values have been shifted along the horizontal axis by and 0.1, respectively.

Plot of the phase boundaries defining the onset of (solid line) and (dashed line) coexistence at colloid polydispersities (left) and (right). To make the curves at different polymer sizes visually distinguishable, those for the lowest and highest values have been shifted along the horizontal axis by and 0.1, respectively.

Left: Polydispersity of the fluid cloud and solid shadow phases vs their colloid volume fraction at the onset of coexistence for and parent polydispersity 0.05 and 0.08. See Fig. 6 (left) for the corresponding polymer volume fractions . Solid lines refer to the fluid cloud phase, dot-dashed lines to the solid shadow; and dotted lines connect sample cloud-shadow pairs. Middle and right: Example plots of the normalized colloid size distributions of the phases indicated by the circles in the left plot.

Left: Polydispersity of the fluid cloud and solid shadow phases vs their colloid volume fraction at the onset of coexistence for and parent polydispersity 0.05 and 0.08. See Fig. 6 (left) for the corresponding polymer volume fractions . Solid lines refer to the fluid cloud phase, dot-dashed lines to the solid shadow; and dotted lines connect sample cloud-shadow pairs. Middle and right: Example plots of the normalized colloid size distributions of the phases indicated by the circles in the left plot.

Log-log plot of the difference in mean colloid diameter between coexisting gas and liquid phases, as a function of the parent polydispersity. The polymer-colloid size ratio is . Solid and dashed lines show our predictions for two different choices of colloid and polymer concentration, as indicated in the legend. The dotted line is a power law with exponent . The circles and error bars indicate the experimental data points from Ref. 9, which were obtained from a set of samples in the range .

Log-log plot of the difference in mean colloid diameter between coexisting gas and liquid phases, as a function of the parent polydispersity. The polymer-colloid size ratio is . Solid and dashed lines show our predictions for two different choices of colloid and polymer concentration, as indicated in the legend. The dotted line is a power law with exponent . The circles and error bars indicate the experimental data points from Ref. 9, which were obtained from a set of samples in the range .

Properties of the coexisting phases, for a system with and , along the phase boundary shown in (a) by the dark line. (b) Fractions of system volume occupied by the various phases; the newly forming solid has vanishing fractional volume. (c) Fractional volumes of the solids normalized by the total fractional volume occupied by solid phases. (d–f) Polymer and colloid volume fractions and colloid polydispersity of the coexisting phases.

Properties of the coexisting phases, for a system with and , along the phase boundary shown in (a) by the dark line. (b) Fractions of system volume occupied by the various phases; the newly forming solid has vanishing fractional volume. (c) Fractional volumes of the solids normalized by the total fractional volume occupied by solid phases. (d–f) Polymer and colloid volume fractions and colloid polydispersity of the coexisting phases.

Left: Pressure plot of a polymer-free system with polydispersity . The values of the pressure at the phase transitions are marked by the horizontal lines. Right: Phase diagram of a colloid-polymer mixture with the same colloid polydispersity and polymer size , plotted as vs . Two phase boundaries are extrapolated (dashed) to . The agreement between the extrapolated values of the intercepts, 13.7 and 18.7, and the equilibrium coexistence pressures in the polymer-free system, 13.0 and 18.4, is good.

Left: Pressure plot of a polymer-free system with polydispersity . The values of the pressure at the phase transitions are marked by the horizontal lines. Right: Phase diagram of a colloid-polymer mixture with the same colloid polydispersity and polymer size , plotted as vs . Two phase boundaries are extrapolated (dashed) to . The agreement between the extrapolated values of the intercepts, 13.7 and 18.7, and the equilibrium coexistence pressures in the polymer-free system, 13.0 and 18.4, is good.

Normalized colloidal size distributions in coexisting phases; the parent distribution is shown for comparison. The three graphs correspond to the values of indicated in Fig. 4.

Normalized colloidal size distributions in coexisting phases; the parent distribution is shown for comparison. The three graphs correspond to the values of indicated in Fig. 4.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content