Terraced spreading of nanofilms under a nonmonotonic disjoining pressure

A thin (similar to nanometer) film of a viscous, essentially nonvolatile liquid spreads over a substrate controlled by the disjoining pressure Pi(h) exerted by the two interfaces on one another. Such films are commonly used as hard disk lubricants in the magnetic recording industry. Macroscopic nonuniformities in the film caused by a perturbation of the uniformly spread state flow away and the film is "healed" in a time frame governed by the appropriate hydrodynamic equations. Lubrication theory may be used to derive a diffusion equation for the local film height h(x,t) as a function of position and time which shows that an effective height-dependent diffusion coefficient D(h)=-[h(3)/3 mu(B)xi(h)][d Pi(h)/dh] controls the spreading dynamics, where mu(B) is the bulk liquid viscosity and xi(12) is a function accounting for any variation of local viscosity near the substrate due to molecularity of the liquid. Such an approach is possible due to the very small ratio of the film height to the in-plane length scale of the disturbance. Provided the disjoining pressure is positive and monotonically decreasing with film thickness, the motion of the film is unexceptional, exhibiting the usual smooth profiles associated with diffusive flow with time. However, for nonmonotonic disjoining pressures, the film is experimentally observed to exhibit vertical terraces. These abrupt jumps in height do not disappear with time and they move slowly or are stationary. This phenomenon is investigated here. We demonstrate how a physically consistent "weak" solution of the diffusion equation can be constructed, where only positive values of the diffusion coefficient are sampled. The film heights at the jump discontinuity are determined by an equal area rule for the disjoining pressure. Numerical simulations for a realistic nonmonotonic disjoining pressure exhibit finite termination on the low-side and vertical terraces, thereby matching the behavior observed in experimental systems. (C) 2011 American Institute of Physics. [doi:10.1063/1.3541968]