Phase transitions in random copolymers

Ideal random AB copolymers with degree of polymerization N are mixtures of N+1 types of chains with different compositions (fractions f_i 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 χ_s=[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=χ_s when f=0.5 and discontinuous at χ<χ _s 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 disordered mesophase is expected to be the most stable state at least over a range Δχ≈0.15 above the critical point when N≫1. The Landau approach with a single order parameter cannot be used for discontinuous transitions in random copolymers (f≠0.5).