Phase behavior of polydisperse sticky hard spheres: analytical solutions and perturbation theory

TL;DR

This paper explores the phase behavior of polydisperse sticky hard spheres using analytical solutions and perturbation theory, addressing the challenges of solving coupled equations in polydisperse colloidal suspensions with adhesion forces.

Contribution

It introduces simplified analytical approximations and a perturbative approach to better understand phase coexistence in polydisperse sticky hard sphere fluids.

Findings

Analytical description of phase behavior using $C_{0}$ and $C_{1}$ approximations.

Perturbative expansion around monodisperse solutions captures effects of weak polydispersity.

Comparison of methods highlights trade-offs between accuracy and analytical tractability.

Abstract

We discuss phase coexistence of polydisperse colloidal suspensions in the presence of adhesion forces. The combined effect of polydispersity and Baxter's sticky-hard-sphere (SHS) potential, describing hard spheres interacting via strong and very short-ranged attractive forces, give rise, within the Percus-Yevick (PY) approximation, to a system of coupled quadratic equations which, in general, cannot be solved either analytically or numerically. We review and compare two recent alternative proposals, which we have attempted to by-pass this difficulty. In the first one, truncating the density expansion of the direct correlation functions, we have considered approximations simpler than the PY one. These $C_{n}$ approximations can be systematically improved. We have been able to provide a complete analytical description of polydisperse SHS fluids by using the simplest two orders $C_{0}$ and $C_{1}$, respectively. Such a simplification comes at the price of a lower accuracy in the phase diagram, but has the advantage of providing an analytical description of various new phenomena associated with the onset of polydispersity in phase equilibria (e.g. fractionation). The second approach is based on a perturbative expansion of the polydisperse PY solution around its monodisperse counterpart. This approach provides a sound approximation to the real phase behavior, at the cost of considering only weak polydispersity. Although a final seattlement on the soundness of the latter method would require numerical simulations for the polydisperse Baxter model, we argue that this approach is expected to keep correctly into account the effects of polydispersity, at least qualitatively.