Three-phase equilibria in density-functional theory: Interfacial tensions.

A mean-field density-functional model for three-phase equilibria in fluids (or other soft condensed matter) with two spatially varying densities is analyzed analytically and numerically. The interfacial tension between any two out of three thermodynamically coexisting phases is found to be captured by a surprisingly simple analytic expression that has a geometric interpretation in the space of the two densities. The analytic expression is based on arguments involving symmetries and invariances. It is supported by numerical computations of high precision, and it agrees with earlier conjectures obtained for special cases in the same model. An application is presented to three-phase equilibria in the vicinity of a tricritical point. Using the interfacial tension expression and employing the field variables compatible with tricritical point scaling, the expected mean-field critical exponent is derived for the vanishing of the critical interfacial tension as a function of the deviation of the noncritical interfacial tension from its limiting value, upon approach to a critical endpoint in the phase diagram. The analytic results are again confirmed by numerical computations of high precision.