Phase diagram of highly asymmetric binary hard-sphere mixtures

We study the phase behavior and structure of highly asymmetric binary hard-sphere mixtures. By first integrating out the degrees of freedom of the small spheres in the partition function we derive a formal expression for the effective Hamiltonian of the large spheres. Then using an explicit pairwise (depletion) potential approximation to this effective Hamiltonian in computer simulations, we determine fluid-solid coexistence for size ratios $q=0.033,0.05,0.1,0.2$, and 1.0. The resulting two-phase region becomes very broad in packing fractions of the large spheres as q becomes very small. We find a stable, isostructural solid-solid transition for $q<~0.05$ and a fluid-fluid transition for $q<~0.10$. However, the latter remains metastable with respect to the fluid-solid transition for all size ratios we investigate. In the limit $\overrightarrow{q}0$ the phase diagram mimics that of the sticky-sphere system. As expected, the radial distribution function $g\left(r\right)\mathrm{}$ and the structure factor $S\left(k\right)\mathrm{}$ of the effective one-component system show no sharp signature of the onset of the freezing transition and we find that at most points on the fluid-solid boundary the value of $S\left(k\right)\mathrm{}$ at its first peak is much lower than the value given by the Hansen-Verlet freezing criterion. Direct simulations of the true binary mixture of hard spheres were performed for $q>~0.05$ in order to test the predictions from the effective Hamiltonian. For those packing fractions of the small spheres where direct simulations are possible, we find remarkably good agreement between the phase boundaries calculated from the two approaches—even up to the symmetric limit $q=1\mathrm{}$ and for very high packings of the large spheres, where the solid-solid transition occurs. In both limits one might expect that an approximation which neglects higher-body terms should fail, but our results support the notion that the main features of the phase equilibria of asymmetric binary hard-sphere mixtures are accounted for by the effective pairwise depletion potential description. We also compare our results with those of other theoretical treatments and experiments on colloidal hard-sphere mixtures.