Profiles in Liquid Films on Solids

The density profile at a vapor-liquid interface was considered briefly in Chapter 1 (Section 5). Prominent in that discussion was the positive contribution to the local free energy density produced by sharp gradients in density (cf. Equation 1.41). The buildup of a thin film on a solid surface is also influenced by this gradient energy (Cahn, 1977 Teletzke et al., 1982). As we shall see, gradient effects sometimes prevent the continuous film buildup predicted by conventional adsorption isotherms as the bulk density of the vapor approaches its samration value. Moreover, when both liquid and vapor are present, the liquid spreads spontaneously on the solid for temperatures sirfficiendy near the eritical temperature, but has a finite equilibrium contact angle at lower temperatures. 

Suppose that a solid surface is placed in a bulk vapor phase having a pressure Pb and molecular density below the saturation values at the existing temperature. Attraction between the solid and molecules of the vapor favors buildup of an adsorbed film. However, the additional energy associated with the density gradient between the film and bulk vapor phase opposes film formation, an effect not considered in developing the BET adsorption isotherm of the preceding section. 

The local Helmholtz free energy density corresponding to Equation 1.41 in such a film is 

Since the term in dn/dz will lead to a flux, it has been set to zero at the wall as appropriate in a closed system and 

Recognizing that [(/o( b) + / b)/ b] i the chemical potential we see that Equation 2.50 is of the same form as the expression given following Equation 1.43. Invoking the calculus of variations, we find that the density profile that minimizes G must satisfy the following condition  

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