Fluid-fluid phase separation in hard spheres with a bimodal size distribution

– The effect of polydispersity on the phase behaviour of hard spheres is examined using a moment projection method. It is found that the Boublik-Mansoori-Carnahan-StarlingLeland equation of state shows a spinodal instability for a bimodal distribution if the large spheres are sufficiently polydisperse, and if there is sufficient disparity in mean size between the small and large spheres. The spinodal instability direction points to the appearance of a very dense phase of large spheres. The phase behaviour of hard sphere mixtures has received much attention recently following the observation by Biben and Hansen [1] that, with an improved closure approximation, integral equation theory indicates the presence of a fluid-fluid spinodal instability in a bidisperse mixture if the ratio of the large to small sphere diameters is ∼ 5. Several experimental groups report confirmatory evidence [2, 3, 4]. The dense phase of the large spheres is sometimes observed to be an ordered phase and at other times to be an amorphous colloidal glass, although some discrepancies in detail remain [3, 4]. Numerical solutions to integral equations frequently only suggest the existence of a spinodal instability, from the apparent divergence of the structure factor for instance. Thus for binary hard spheres, Caccamo and Pellicane [5] can argue that the integral equation evidence supports fluid-solid segregation rather than fluid-fluid separation. Recent computer simulations support this scenario, by establishing that a region of fluid-fluid coexistence exists, but is metastable with respect to fluid-solid coexistence [6]. Analytic solutions to integral equations, although rare and generally perceived to be less accurate than numerical solutions, do not suffer from the same problems. Most notable of these for hard spheres is the Percus-Yevick closure [7]. As reported by Lebowitz and Rowlinson [8] though, this closure predicts that a binary hard sphere fluid is stable for all size ratios and physically accessible compositions. The appeal of the Percus-Yevick approximation is tempered by an internal inconsistency between the compressibility equation of state (EOS) and the virial EOS. It is precisely this Typeset using EURO-LTEX (does not imply endorsement by Europhysics Letters) 2 EUROPHYSICS LETTERS inconsistency that the new closure approximations mentioned above are designed to cure. A phenomenological way around this problem though is to interpolate between the two EOS’s. For monodisperse hard spheres, a suitable interpolation leads to the highly succesful Carnahan and Starling EOS [9]. For an arbitrary mixture, the same interpolation leads to an EOS first considered by Boublik and Mansoori et al (BMCSL EOS) [10]. This improved EOS also predicts phase stability for a binary hard sphere fluid for any size ratio and composition. It is well known though that polydispersity enhances phase instability. An interesting and experimentally relevant question therefore concerns the effects of size polydispersity on the phase behaviour of hard sphere mixtures. Fortunately the above EOS’s have been extended to polydisperse systems, for instance the Helmholtz free energy and related thermodynamic quantities corresponding to the BMCSL EOS are given by Salacuse and Stell [11] in terms of the number density and the first three moments of the size distribution. For the Percus-Yevick compressibility EOS, Vrij [12] established that an arbitrarily polydisperse hard sphere fluid is stable, thus confirming a conjecture of Lebowitz and Rowlinson [8]. However the question of whether, under the BMCSL EOS, a binary mixture becomes unstable if one or both of the species is allowed to become polydisperse has not to my knowledge been previously investigated. Perhaps surprisingly in the light of Vrij’s result the answer to this question is “yes”. Moreover the instability reflects the experimental observations, albeit at a much larger degree of polydispersity and size ratio than are seen in reality. Very recently Cuesta has found a similar spinodal instability in hard spheres with a unimodal, log-normal size distribution [13]. I address the spinodal stability problem using a novel moment projection method developed recently by Sollich and Cates [14] and myself [15]. The approach reduces the free energy to one which depends only on moment densities (defined below). It gives exact results for cloud and shadow curves (the polydisperse analogues of the binodal) and for spinodal curves, provided the excess free energy is a function only of the corresponding moments of the size distribution. Since the BMCSL excess free energy is precisely of this form, the new tool may be aplied to this situation. By way of contrast, Cuesta approaches the spinodal problem by casting it as an integral equation [13]. He also finds that if the excess free energy depends only on a few moments, the integral equation is reducible to a matrix problem, which can in fact be proved to be identical to the results obtained below. Such a reduction in dimensionality for the spinodal problem has been noted several times in the past [16]. I outline the approach in general first to indicate its application to spinodal curves, which was not previously covered in detail. Suppose that the excess free energy depends only on a few moment densities, defined to be φn = ∑N i=1 fn(σi)/V , where V is the system volume and the functions fn(σ) are arbitrary. The sum is over all particles in a homogeneous phase, and the σi are individual particle properties, such as size. If we set f0(σ) = 1 then we include the number density ρ = φ0 amongst the moment densities (in doing this though the n = 0 term should be omitted from certain sums below). As previously reported, moment densities can be treated as independent thermodynamic density variables provided the ideal part of the free energy density is suitably generalised: f (id) = ρkBT (ln ρ− 1)− ρTs(mn) (1) where s(mn) is a generalised entropy of mixing per particle, and mn = φn/ρ = ∑N i=1 fn(σi)/N (n > 0) are generalised moments. An expression for s(mn) can be given only if the parent or feedstock distribution, p(σ), is specified. This is because one must know the parent distribution from which particles are drawn, to conclude anything about the distributions in daughter phases. The generalised entropy of mixing is then given by a Legendre transform (I will set P. B. WARREN: FLUID-FLUID PHASE SEPARATION ETC. 3 kB = T = 1 in the following for simplicity):