A molecular theory of interfacial phenomena in multicomponent systems

The van der Waals theory of surface tensions is generalized to multicomponent systems. The local free energy density consists of a ’’local equilibrium’’ free energy (i.e., equilibrium free energy of a uniform mixture having species densities equal to the local species densities) plus a quadratic form in the gradients of the species densities. The coefficients in this quadratic form depend on the local species densities through the density dependence of the second moment of the local multicomponent direct correlation function. The requirement that the free energy be a minimum yields a system of partial differential equations (one for each component). A particular linear combination of the differential equations is the condition for mechanical equilibrium. It can be interpreted as a microscopic statement of the multicomponent Young–Laplace equation for the pressure variation across a curved interface. For two component systems the theory is a generalization of the treatment of Cahn and Hilliard in that it allows for pressure variations. If the local pressure fluctuations are suppressed, the differential equation for the concentration is very similar to theirs, except that the total density may vary across the interface. Similarly, when the theory is applied to liquid–vapor equilibrium in a binary system, the differential equation for the total number density reduces to that of a single component system when the local chemical potential difference (μ=μ1−μ2) is held constant.