Free Energy of a Fluctuating Interface

In order to study the fluctuations of interfaces and surfaces, one first needs an expression for the free energy of an interface with a shape that is not necessarily flat. Phenomenologically, one can write that the free energy is the product of the interfacial tension, y, and the area, which in the Monge representation for the surface, z = h Xy y), reads  

To solve for the profile that minimizes F, one must solve the Euler-Lagrange equation taking into account the variations in x (x, y,z — h x, y)) in three dimensions  

In our approximate analysis, we assume that the spatial variations in the z direction are much larger than those in the Jc or y directions. We thus keep only those terms proportional to the second derivative in z. This is equivalent to assuming that the spatial variations of the interface position occur over length scales much longer than the interface thickness. One then has the approximate equation for x, 

The physical meaning of this expression is that to a first approximation, valid when the interface varies slowly in the x and y directions compared to its width in the z direction, is that the composition is just M(z) calculated with an argument that represents the normal component of the distance to the interface  

Using this equation, one can find the surface free energy, F, by subtracting from the free energy of the system with a single interface the free energy of the bulk system with no interface — ie., Fg = F — Ft where Fb is the total bulk free energy Fb = f dr fb). We find an expression identical to Eq. (3.1), where now the surface tension y is determined from the one-dimensional profile  

---