### Abstract

The authors study the phase behavior of mixtures of monodisperse colloidal spheres with a depletion agent which can have arbitrary shape and can possess a polydisperse size or shape distribution. In the low concentration limit considered here, the authors can employ the free-volume theory and take the geometry of particles of the depletion agent into account within the framework of fundamental measure theory. The authors apply their approach to study the phase diagram of a mixture of (monodisperse) colloidal spheres and two polydisperse polymer components. By fine tuning the distribution of the polymer, it is possible to construct a complex phase diagram which exhibits two stable critical points.

I. INTRODUCTION

II. THEORY

A. Free-volume theory

B. Fundamental measure theory

C. Generalization to polydisperse depletion agent

III. DISTRIBUTION FUNCTIONS

IV. RESULTS

A. Influence of size polydispersity

B. Influence of shape polydispersity

C. Ternary mixtures with a bimodal polydisperse distribution

V. DISCUSSION

### Key Topics

- Colloidal systems
- 57.0
- Polymers
- 34.0
- Critical point phenomena
- 31.0
- Phase diagrams
- 27.0
- Phase separation
- 14.0

## Figures

(a) Phase diagrams of mixtures of colloids and polydisperse (spherical) polymer with an average size ratio and (b) the corresponding distributions. We employ a free-volume theory based on the BMCSL equation of state. For low degrees of polydispersity, corresponding to (full lines), the fluid-fluid phase separation is metastable with respect to fluid-solid coexistence. Upon increasing the degree of polydispersity, corresponding to (dash-dotted lines) and (dotted lines), the fluid-fluid coexistence is stabilized.

(a) Phase diagrams of mixtures of colloids and polydisperse (spherical) polymer with an average size ratio and (b) the corresponding distributions. We employ a free-volume theory based on the BMCSL equation of state. For low degrees of polydispersity, corresponding to (full lines), the fluid-fluid phase separation is metastable with respect to fluid-solid coexistence. Upon increasing the degree of polydispersity, corresponding to (dash-dotted lines) and (dotted lines), the fluid-fluid coexistence is stabilized.

Effect of shape polydispersity of the depletion agent on the phase behavior of a mixture of colloids and depletion agent. The depletion agent consists of ellipsoids with half axes and . The mean value of the distribution is fixed at while the size ratio between the depletion agent and the colloid is fixed at . For (full lines) the fluid-fluid coexistence is metastable and similar to the monodisperse case (not shown here). Upon increasing the degree of polydispersity we find two stable fluid phases. The phase diagram for (dashed lines) exhibits a stable critical point and a triple point.

Effect of shape polydispersity of the depletion agent on the phase behavior of a mixture of colloids and depletion agent. The depletion agent consists of ellipsoids with half axes and . The mean value of the distribution is fixed at while the size ratio between the depletion agent and the colloid is fixed at . For (full lines) the fluid-fluid coexistence is metastable and similar to the monodisperse case (not shown here). Upon increasing the degree of polydispersity we find two stable fluid phases. The phase diagram for (dashed lines) exhibits a stable critical point and a triple point.

The path of the two critical points in a ternary mixture of colloidal spheres and two polydisperse polymer components. If the mixing parameter , only the smaller polymer is present and we find a stable critical point at and . As we decrease the value of this critical point moves towards smaller values of and vanishes at (upper line). For a mixing parameter , when only the bigger polymer are present, we find a second critical point at and , which moves towards larger values of (lower line). For (indicated by the dotted lines) we find two critical points in the system.

The path of the two critical points in a ternary mixture of colloidal spheres and two polydisperse polymer components. If the mixing parameter , only the smaller polymer is present and we find a stable critical point at and . As we decrease the value of this critical point moves towards smaller values of and vanishes at (upper line). For a mixing parameter , when only the bigger polymer are present, we find a second critical point at and , which moves towards larger values of (lower line). For (indicated by the dotted lines) we find two critical points in the system.

Phase diagrams for a mixture of colloids and a bimodal distribution of polydisperse (spherical) polymer. For very close to 1 we recover the phase diagram for the monomodal distribution, see plot (a) for . The fluid-fluid coexistence (full line) is stabilized by polydispersity. Upon decreasing , i.e., by adding larger polymer, the spinodal (dotted line) exhibits a second minimum for in (b) and we observe a second critical point, which is metastable (diamond). For we observe two stable critical points. Besides a low density (in colloids) gas (G) there exist a low density (LDL) and a high density (HDL) liquid phase—see (c). For a value of , shown in (d), the critical point at high values of (triangle) is metastable.

Phase diagrams for a mixture of colloids and a bimodal distribution of polydisperse (spherical) polymer. For very close to 1 we recover the phase diagram for the monomodal distribution, see plot (a) for . The fluid-fluid coexistence (full line) is stabilized by polydispersity. Upon decreasing , i.e., by adding larger polymer, the spinodal (dotted line) exhibits a second minimum for in (b) and we observe a second critical point, which is metastable (diamond). For we observe two stable critical points. Besides a low density (in colloids) gas (G) there exist a low density (LDL) and a high density (HDL) liquid phase—see (c). For a value of , shown in (d), the critical point at high values of (triangle) is metastable.

## Tables

Geometrical measures for one- and two-dimensional depletion agents. All lengths are measured in units of the diameter of the spherical colloids. We consider needles of length and radius and platelets with radius and thickness . The value of plays the role of a size ratio and compares the size of the depletion agent to that of a colloid.

Geometrical measures for one- and two-dimensional depletion agents. All lengths are measured in units of the diameter of the spherical colloids. We consider needles of length and radius and platelets with radius and thickness . The value of plays the role of a size ratio and compares the size of the depletion agent to that of a colloid.

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