Van der Waals Theory of a Smooth Interface

Van der Waals was first to realize that the density varies continuously across a fluid-fluid interface. The fact that interfaces vary smoothly suggests that interfacial properties can be calculated with the Landau-Ginzburg functional. The following approach is originally due to van der Waals, but was subsequently reformulated by Landau and Lifshitz, and later was rediscovered and extended by Cahn and Hilliard.  

For a specific equation of state the differential equation (7.20) can be solved to yield an expression for the interfacial profile. The relationship between the interfacial profile and the surface tension takes the form (t = f m dp/dzf dz. Let /be the truncated Landau expansion given by eq 7.16. The equilibrium values of the liquid and vapour densities set the boundary conditions corresponding to the bulk phases. Taking the bulk liquid phase to be located at positive infinity, and solving the differential equation subject to this boundary condition, we find 

However, as we shall discuss in Section 7.4.3, the actual temperature dependence of the fluid interfacial tension is modified by the critical fiuctuations. 

Heretofore we have only addressed the properties of planar interfaces. For a curved surface, the radius of curvature affects the interfacial properties, in particular the interfacial profile and the surface tension. Consider a spherical bubble of vapour surrounded by liquid. In this case, the Laplace equation relates the pressure difference between inside the droplet and outside the droplet to the surface tension and the radius of curvature R as Ap = 2a/R. The curvature-dependent surface tension can be expanded in powers of the curvature as 

Fisher and Wortis have shown that Tohnan s length is zero for symmetric fluid coexistence and non-zero for asymmetric fluid coexistence. Symmetric fluids are represented by the lattice-gas (Ising) model in which the shape of the coexistence curve is perfectly symmetric with respect to the critical isochore. Real fluids always possess some degree of asymmetryAsymmetry in the vapour-liquid coexistence in helium, especially in He, is very small, but not zero. In the mean-field approximation, the asymmetry in the vapour-liquid coexistence is represented by the rectilinear diameter  

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