Depletion induced demixing in polydisperse mixtures of hard spheres

Polydisperse mixtures are those in which components with a whole range of sizes are present. It is shown that the fluid phase of polydisperse hard spheres is thermodynamically unstable unless the density of large spheres decreases at least exponentially as their size increases. The instability is with respect to the large spheres crystallising out into multiple solid phases. PACS: 64.10.+h, 64.75.+g, 82.70.Kj Mixtures of hard spheres in which spheres with a wide range of diameters are present are a good first model of emulsions. Emulsions are suspensions of droplets of oil or fat in water; milk is perhaps the most familiar example. The droplets of an emulsion interact via a short ranged repulsion, which is well represented by a hard-sphere interaction. They are typically present with a wide range of diameters: from 0.1 to a few micrometers [1–3]. Mixtures in which a continuous range of sizes are present are termed polydisperse [4]. They are much less well understood than systems which contain only one or two components. For example, the phase behaviour of single component hard spheres [5] has been understood for thirty years: the fluid phase is stable up the point where the spheres occupy a little less than half the volume of the suspension, there is then a first order transition to a solid. In contrast there are no phase diagrams known for polydisperse hard spheres. Below, we examine polydisperse spheres with particular emphasis on the largest spheres. We show that unless their density decreases at least exponentially with increasing size, they crystallise out of the mixture at all densities. The mixture is then never stable as a single fluid phase. The crystallisation is driven by a depletion attraction [6, 7] between the large spheres, due to the smaller spheres. Depletion-induced separation of the largest spheres has been observed in emulsions [3] but there the floating of the droplets to the surface due to gravity complicates the situation. Our demonstration applies to spheres at equilibrium. Specifying a polydisperse mixture requires specifying the number density of spheres of every size. This is done with a distribution function x(σ) [4]. The number density of spheres with diameter σ is then ρx(σ)dσ, where ρ is the total number density of spheres. Although our final result will apply to a whole class of distribution functions we choose a specific function for definiteness and because it is widely used to describe emulsions [1, 2] and powders [8]. The distribution is called the log-normal distribution, and it is defined by