Finite size effects on the phase diagram of a binary mixture confined between competing walls

Abstract

A symmetrical binary mixture AB that exhibits a critical temperature T_cb of phase separation into an A- and a B-rich phase in the bulk is considered in a geometry confined between two parallel plates a distance D apart. It is assumed that one wall preferentially attracts A while the other wall preferentially attracts B with the same strength (“competing walls”). In the limit D → ∞, one then may have a wetting transition of first-order at a temperature T_w, from which prewetting lines extend into the one phase region both of the A- and the B-rich phase. It is discussed how this phase diagram gets distorted due to the finiteness of D: the phase transition at T_cb immediately disappears for D < ∞ due to finite size rounding, and the phase diagram instead exhibit two two-phase coexistence regions in a temperature range T_trip < T < T_c₁ = T_c₂. In the limit D → ∞ T_c₁,T_c₂ become the prewetting critical points and T_trip →T_w. For small enough D it may occur that at a tricritical value D_t the temperatures T_c₁ = T_c₂ and T_trip merge, and then for D < D_t there is a single unmixing critical point as in the bulk but with T_c(D) near T_w. As an example, for the experimentally relevant case of a polymer mixture a phase diagram with two unmixing critical points is calculated explicitly from self-consistent field methods.