Abstract:
We describe how Monte Carlo simulation within the grand canonical ensemble can be applied to the study of phase behaviour in polydisperse fluids. Attention is focused on the case of fixed polydispersity in which the form of the `parent' density distribution $\rho^\circ(\sigma)$ of the polydisperse attribute $\sigma$ is prescribed. Recently proposed computational methods facilitate determination of the chemical potential distribution conjugate to $\rho^\circ(\sigma)$. By additionally incorporating extended sampling techniques within this approach, the compositions of coexisting (`daughter') phases can be obtained and fractionation effects quantified. As a case study, we investigate the liquid-vapor phase equilibria of a size-disperse Lennard-Jones fluid exhibiting a large ($\delta=40%$) degree of polydispersity. Cloud and shadow curves are obtained, the latter of which exhibit a high degree of fractionation with respect to the parent. Additionally, we observe considerable broadening of the coexistence region relative to the monodisperse limit.

Abstract:
The conditions of multi-phase equilibrium are solved for generic polydisperse systems. The case of multiple polydispersity is treated, where several properties (e.g. size, charge, shape) simultaneously vary from one particle to another. By developing a perturbative expansion in the width of the distribution of constituent species, it is possible to calculate the effects of polydispersity alone, avoiding difficulties associated with the underlying many-body problem. Explicit formulae are derived in detail, for the partitioning of species at coexistence and for the shift of phase boundaries due to polydispersity. `Convective fractionation' is quantified, whereby one property (e.g. charge) is partitioned between phases due to a driving force on another. To demonstrate the ease of use and versatility of the formulae, they are applied to models of a chemically-polydisperse polymer blend, and of fluid-fluid coexistence in polydisperse colloid-polymer mixtures. In each case, the regime of coexistence is shown to be enlarged by polydispersity.

Abstract:
It is shown that the van der Waals free-energy of polydisperse fluids, as introduced previously (L. Bellier-Castella, H. Xu and M. Baus, {J. Chem. Phys.} {113}, 8337 (2000)), predicts that for certain thermodynamic states (e.g. low temperatures and large polydispersities) the ordinary two-phase coexistences become metastable relative to a fractionation of the system into three phases, reducing thereby the polydispersity of each of the coexisting phases.

Abstract:
A new type of phase separation in the polyelectrolyte solutions consisting of several types of charged macromolecules differing in their degree of ionization is predicted via a general thermodynamic consideration. We show that even a small difference in the degree of ionization of otherwise equivalent components results in their spatial separation occurring upon decreasing the temperature much earlier than precipitation of the components from the solution. Some implications of charge fractionation in biological processes are discussed.

Abstract:
The equilibrium phase behaviour of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to non-equilibrium effects. We find no evidence of re-entrant melting; instead, sufficiently compressed solids phase separate into two or more solid phases. Under appropriate conditions, coexistence of multiple solids with a fluid phase is also predicted. The solids have smaller polydispersities than the parent phase as expected, while the reverse is true for the fluid phase, which contains predominantly smaller particles but also residual amounts of the larger ones. The properties of the coexisting phases are studied in detail; mean diameter, polydispersity and volume fraction of the phases all reveal marked fractionation. We also propose a method for constructing quantities that optimally distinguish between the coexisting phases, using Principal Component Analysis in the space of density distributions. We conclude by comparing our predictions to perturbative theories for near-monodisperse systems and to Monte Carlo simulations at imposed chemical potential distribution, and find excellent agreement.

Abstract:
We dynamically simulate fractionation (partitioning of particle species) during spinodal gas-liquid separation of a size-polydisperse colloid, using polydispersity up to ~40% and a skewed parent size distribution. We introduce a novel coarse-grained Voronoi method to minimise size bias in measuring local volume fraction, along with a variety of spatial correlation functions which detect fractionation without requiring a clear distinction between the phases. These can be applied whether or not a system is phase separated, to determine structural correlations in particle size, and generalise easily to other kinds of polydispersity (charge, shape, etc.). We measure fractionation in both mean size and polydispersity between the phases, its direction differing between model interaction potentials which are identical in the monodisperse case. These qualitative features are predicted by a perturbative theory requiring only a monodisperse reference as input. The results show that intricate fractionation takes place almost from the start of phase separation, so can play a role even in nonequilibrium arrested states. The methods for characterisation of inhomogeneous polydisperse systems could in principle be applied to experiment as well as modelling.

Abstract:
The recently proposed universal relations between the moments of the polydispersity distributions of a phase-separated weakly polydisperse system are analyzed in detail using the numerical results obtained by solving a simple density functional theory of a polydisperse fluid. It is shown that universal properties are the exception rather than the rule.

Abstract:
The statistical mechanics of phase transitions in dense systems of polydisperse particles presents distinctive challenges to computer simulation and analytical theory alike. The core difficulty, namely dealing correctly with particle size fractionation between coexisting phases, is set out in the context of a critique of previous simulation work on such systems. Specialized Monte Carlo simulation techniques and moment free energy method calculations, capable of treating fractionation exactly, are then described and deployed to study the fluid-solid transition of an assembly of repulsive spherical particles described by a top-hat "parent" distribution of particle sizes. The cloud curve delineating the solid-fluid coexistence region is mapped as a function of the degree of polydispersity $\delta$, and the properties of the incipient "shadow" phases are presented. The coexistence region is found to shift to higher densities as $\delta$ increases, but does not exhibit the sharp narrowing predicted by many theories and some simulations.

Abstract:
We study the polydisperse Baxter model of sticky hard spheres (SHS) in the modified Mean Spherical Approximation (mMSA). This closure is known to be the zero-order approximation (C0) of the Percus-Yevick (PY) closure in a density expansion. The simplicity of the closure allows a full analytical study of the model. In particular we study stability boundaries, the percolation threshold, and the gas-liquid coexistence curves. Various possible sub-cases of the model are treated in details. Although the detailed behavior depends upon the particularly chosen case, we find that, in general, polydispersity inhibits instabilities, increases the extent of the non percolating phase, and diminishes the size of the gas-liquid coexistence region. We also consider the first-order improvement of the mMSA (C0) closure (C1) and compare the percolation and gas-liquid boundaries for the one-component system with recent Monte Carlo simulations. Our results provide a qualitative understanding of the effect of polydispersity on SHS models and are expected to shed new light on the applicability of SHS models for colloidal mixtures.

Abstract:
We study the phase behaviour of the Zwanzig model of suspensions of hard rods, allowing for polydispersity in the lengths of the rods. In spite of the simplified nature of the model (rods are restricted to lie along one of three orthogonal axes), the results agree qualitatively with experimental observations: the coexistence region broadens significantly as the polydispersity increases, and strong fractionation occurs, with long rods found preferentially in the nematic phase. These conclusions are obtained from an analysis of the exact phase equilibrium equations. In the second part of the paper, we consider the application of the recently developed ``moment free energy method'' to the polydisperse Zwanzig model. Even though the model contains non-conserved densities due to the orientational degrees of freedom, most of the exactness statements (regarding the onset of phase coexistence, spinodals, and critical points) derived previously for systems with conserved densities remain valid. The accuracy of the results from the moment free energy increases as more and more additional moments are retained in the description. We show how this increase in accuracy can be monitored without relying on knowledge of the exact results, and discuss an adaptive technique for choosing the extra moments optimally.