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

A symmetrical binary mixture AB that exhibits a critical temperature Tcb 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 Tw, 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 Tcb immediately disappears for D<∞ due to finite size rounding, and the phase diagram instead exhibit two two-phase coexistence regions in a temperature range Ttrip<T<Tc1=Tc2. In the limit D→∞Tc1,Tc2 become the prewetting critical points and Ttrip→Tw. For small enough D it may occur that at a tricritical value Dt the temperatures Tc1=Tc2 and Ttrip merge, and then for D<Dt there is a single unmixing critical point as in the bulk but with Tc(D) near Tw. 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.