Moment Free Energies for Polydisperse Systems

Abstract

A polydisperse system contains particles with at least one attribute σ (such as particle size in colloids or chain length in polymers) which takes values in a continuous range. It therefore has an infinite number of conserved densities, described by a density distribution ρ(σ). The free energy depends on all details of ρ(σ), making the analysis of phase equilibria in such systems intractable. However, in many (especially mean-field) models the excess free energy only depends on a finite number of (generalized) moments of ρ(σ); we call these models truncatable. We show, for these models, how to derive approximate expressions for the total free energy which only depend on such moment densities. Our treatment unifies and explores in detail two recent separate proposals by the authors for the construction of such moment free energies. We show that even though the moment free energy only depends on a finite number of density variables, it gives the same spinodals and critical points as the original free energy and also correctly locates the onset of phase coexistence. Results from the moment free energy for the coexistence of two or more phases occupying comparable volumes are only approximate, but can be refined arbitrarily by retaining additional moment densities. Applications to Flory-Huggins theory for length-polydisperse homopolymers, and for chemically polydisperse copolymers, show that the moment free energy approach is computationally robust and gives new geometrical insights into the thermodynamics of polydispersity.

Phase behavior and particle size cutoff effects in polydisperse fluids

“…, ρ 4 , i.e. it is truncatable [24]. This allows us to exploit the MFE method to obtain accurate numerical predictions for the phase behaviour.…”

“…[4,23]. The reason for this is that such models are normally "truncatable" [24] so that the phase equilibrium conditions can be reduced to nonlinear equations for a finite number of variables. This approach has been applied to the study of phase separation in fluids exhibiting separate size and interaction strength polydispersity, yielding predictions for the cloud and shadow curves and critical parameters as a function of polydispersity [25].…”

“…The MFE method delivers (for the given model free energy) exact results for the location of critical points and the cloud and shadow curves. With appropriate refinements, it can also be used to obtain exact numerical solutions to the phase coexistence conditions when two or more phases coexist in finite amounts [24,26,27,28]. The MFE method has been applied to the study of phase behaviour in systems ranging from polydisperse hard rods to the freezing of hard spheres [5,29,30,31,32], and we shall deploy it again here to study the phase behaviour of the present model, based on a refined van der Waals-type approximation to the excess free energy.…”

“…This approach has been applied to the study of phase separation in fluids exhibiting separate size and interaction strength polydispersity, yielding predictions for the cloud and shadow curves and critical parameters as a function of polydispersity [25]. An alternative approach which more systematically exploits the advantages of truncatable models is the moment free energy (MFE) method [5,24]. This approximates the full free energy appropriately in terms of a "moment free energy" which depends on a small number of density variables, thereby permitting the efficient use of standard tangent plane construction to locate phase boundaries.…”

Microstructure and macroscopic properties of polydisperse systems of hard spheres

“…Theoretical descriptions of polydisperse systems are mainly based on a small set of moments of the particle size distribution [31,32,[49][50][51][52][53][54]. In order to reduce the strictly infinite number of equations for polydisperse systems to a finite set, Sollich, Cates and Warren [50,51] have assumed that the excess free energy in a polydisperse hard-sphere mixture depends only on a limited set of moment densities of the diameter distribution.…”