Many materials containing colloids or polymers are polydisperse: they comprise particles with properties (such as particle diameter, charge, or polymer chain length) that depend continuously on one or several parameters. This review focuses on the theoretical prediction of phase equilibria in polydisperse systems; the presence of an effectively infinite number of distinguishable particle species makes this a highly non-trivial task. I first describe qualitatively some of the novel features of polydisperse phase behaviour, and outline a theoretical framework within which they can be explored. Current techniques for predicting polydisperse phase equilibria are then reviewed. I also discuss applications to some simple model systems including homopolymers and random copolymers, spherical colloids and colloid-polymer mixtures, and liquid crystals formed from rod- and plate-like colloidal particles; the results surveyed give an idea of the rich phenomenology of polydisperse phase behaviour. Extensions to the study of polydispersity effects on interfacial behaviour and phase separation kinetics are outlined briefly.

https://nms.kcl.ac.uk/peter.sollich/papers_pdf/PolydispReviewMMI.pdf

Predicting phase equilibria in polydisperse systems

We use self-consistent mean field methods and analytical theory to determine the behavior of AB copolymers at the interface between two incompatible homopolymers, A and B. We calculate the reduction in interfacial tension, γ, resulting from the copolymers localizing at the A/B interface. We examine the effects of chain length, composition, and molecular architecture on the efficiency of the copolymers. In particular, we compare the interfacial behavior of different linear copolymers (random, alternating, and diblock) and various branched copolymers (stars and combs). At fixed molecular weight, the diblock copolymers are the most efficient at reducing γ. However, when we compare random and comb copolymers with diblocks at different molecular weights, we observe that the longer random or comb copolymers are more efficient than short diblocks. These studies allow us to predict the reduction in interfacial tension produced by a wide variety of copolymers and, thereby, permit a rational design of cost-effectiv...

Designing Compatibilizers To Reduce Interfacial Tension in Polymer Blends

A polydisperse system contains particles with at least one attribute $\sigma$ (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 {\em distribution} $\rho(\sigma)$. The free energy depends on all details of $\rho(\sigma)$, making the analysis of phase equilibria in such systems intractable. However, in many (especially mean-field) models the {\em excess} free energy only depends on a finite number of (generalized) moments of $\rho(\sigma)$; we call these models truncatable. We show, for these models, how to derive approximate expressions for the {\em 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.

Moment Free energies for polydisperse systems

http://www.cchem.berkeley.edu/akcgrp/arups%20papers/69.pdf

Disordered heteropolymers: models for biomimetic polymers and polymers with frustrating quenched disorder

Motivated by practical issues that pertain to polymer adhesion, we consider the equilibrium behavior of a dilute solution of ideal A–B random block copolymers confined between two solid surfaces. We develop a general theory for the situation wherein the A–A, B–B, and A–B intersegment interactions are different, and furthermore, the A and B segments interact differently with the solid surfaces. Random block copolymers constitute a class of materials wherein a quenched disorder (the sequence distribution) is carried by the fluid whose statistical properties are of interest. In our theory, we perform quenched disorder averages using the replica trick. The nonlocal terms in our action functional are decoupled by introducing a set of random fields. The resulting equations for the propagator are analyzed within the framework of eigenfunction expansions. Since we consider long chains in confined geometries, we invoke the ground state approximation. We also carry out the functional integrals over the random field...

Surface‐induced ordering for confined random block copolymers

Ideal random AB copolymers with degree of polymerization N are mixtures of N+1 types of chains with different compositions (fractions fi of A monomers). Conditions for single phase and multiphase equilibria are studied using Flory–Huggins free energy of mixing with χ representing the A–B interaction parameter. The spinodal for one phase instability is given by χs=[2f(1−f)]−1 for all N, where f is the average A fraction in the system. The transition from one to two phases is continuous at χ=χs when f=0.5 and discontinuous at χ≤χs when f≠0.5. Three, four, and more phases become stable at larger values of χ. Our numerical solution suggests that the stability range for multiple phases approaches Δχ≊0.15 at large (but finite) N. Macroscopically and microscopically phase separated states are investigated with the Landau approach of Fredrickson, Milner and Leibler. The Landau method gives reasonable but inexact results for two macroscopic phases when the random copolymer has compositional symmetry (f=0.5). A dis...

A. Nesarikar

Papers

Phase transitions in random copolymers