Disjoining pressure for nonuniform thin films

The effect of the attractive forces originating from van der Waals interactions on the dynamics of thin films (<= similar to 100 nm) is often approximated in fluid dynamics 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. In this paper, we derive a formula for the disjoining pressure for a film of nonuniform 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 is shown to take the form Pi=-A(123)(4-3h(x)(2)+3hh(xx))/24 pi h(3), where A(123) is the Hamaker constant for phases 1 and 3 interacting through phase 2; h, h(x), and h(xx) are the local film thickness, slope and second order derivative, respectively. For the limiting case of parallel interfaces (e.g., h(x) = h(xx) equivalent to 0), the disjoining pressure reduces to Pi=-A(123)/6 pi h(3) in agreement with the classical Lifshitz expression for the van der Waals force. The derivation can be readily extended to more general nonuniform films by constructing tangential planes at both interfaces of the films. Because of 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.