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

We discuss the 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 that have attempted to by-pass this difficulty. In the first, truncating the density expansion of the direct correlation functions, we have considered approximations simpler than the PY one. These Cn approximations can be systematically improved. We have been able to provide a complete analytical description of polydisperse SHS fluids using the simplest two orders C 0 and C 1. 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 behaviour, at the cost of considering only weak polydispersity. Although a final determination of the soundness of the latter method would require numerical simulations for the polydisperse Baxter model, we argue that this approach is expected to correctly take into account the effects of polydispersity, at least qualitatively