(418f) Disjoining Pressure For Non-Uniform Thin Films

The effect of the attractive forces originating from van der Waals interactions on the dynamics of thin films (

~100 nm) is often approximated as the disjoining pressure between two unbounded parallel interfaces. However, it is known that this concept of the disjoining pressure, as a force per unit area between parallel interfaces cannot generally be extended to films of nonuniform thickness. Based on the analysis of Yeh and coworkers, we derive a formula for the disjoining pressure for a film of non-uniform thickness by minimizing the total Helmholtz free energy for a thin film residing on a solid substrate. Comparing to the augmented Young-Laplace equation, the disjoining pressure for a thin film of small slope on a flat substrate of small slope is shown to take the form:
,  where
 is the Hamaker constant for a phase 1 and 2 interacting through a phase 3;
,
 and 
 are the local film thickness, slope and second order derivative, respectively. Unlike the previous studies, we take into account the excess energy outside the thin film region. This affects the boundary conditions at the common contact lines but not the formula for the disjoining pressure. For the limiting case of parallel interfaces (e.g.,
), the disjoining pressure reduces to
 in agreement with the classical Lifshitz expression for the van der Waals force in this case. The derivation can be readily extended to more general non-uniform films by constructing tangential planes at both interfaces of the films. Because of the steric effects that prevent molecules from overlapping each other, the molecular size cannot be neglected when applying the mesoscopic concept of the disjoining pressure to films of thickness comparable to molecular scales.