Depletion-Driven Solid–Solid Coexistence in Colloid–Polymer Mixtures

Abstract

Hard spheres mixed with penetrable hard spheres display an isostructural solid–solid transition. This phase transition is fully driven by the entropy gain of the depletants without invoking explicit pair potentials between the colloidal particles. The solid–solid phase coexistence exists for size ratios $q\equiv \delta /R \lesssim 0.09$, with $\delta$ the penetrable hard sphere radius and R the hard sphere radius. This coexistence is revealed using a modified free volume theory, where the free volume fraction for depletants in the solid phase is calculated on geometrical grounds. Due to a better account of the small depletant partitioning, the fluid branch of the fluid–solid coexistence also decreases with decreasing q. Colloid–polymer mixtures are an excellent candidate for the experimental realization of this intricate solid–solid transition, first predicted by Bolhuis and Frenkel for hard spheres with short range pair attractions [PRL 72, 2211–2214 (1994)].

3.A TPT for the AOV Potential

In this short section we describe the approach followed to justify the shift of the fluid–solid (F–S) coexistence towards higher packing fractions upon addition of small depletants. Following standard thermodynamic perturbation theories [22] (previously applied to highly-screened repulsive interactions [37]), we consider an effective sphere of interaction whose diameter $\sigma '$ is calculated via

$$\begin{aligned} {\sigma '}/{\sigma } = 1 +\int _{1}^{\infty }\text {d}\tilde{r} \left(1-\exp {[-\beta W_\text {AOV} (\tilde{r})]}\right), \end{aligned}$$

(3.15)

with the Asakura–Oosawa–Vrij (AOV) depletion pair potential given as in Eq. (1.2). The integral given in Eq. (3.15) can be solved numerically for all $\{q,\phi _\text {d}^\text {R}\}$, providing $\sigma '$. We then map the thermodynamic functions of a pure HS suspension with an effective packing fraction:

$$\begin{aligned} \phi _\text {c}' = (\sigma '/\sigma )^3\phi _\text {c}. \end{aligned}$$

By substituting $\phi _\text {c} \leftrightarrow \phi _\text {c}'$ on all canonical expressions, calculation of the F–S binodal is straightforward.