A modified steady-state Sine-Gordon equation model for the density profile of a supported thin film

In our research, we model the density profile of a thin glycerol film on silica, initially simulated by Cheng et al. through classical Molecular Dynamics. We represent this profile using a nonlinear partial differential equation (PDE). The film's density profile at the air-silica interface and the central region is effectively described by a kink solution of the sine-Gordon equation (SGE). To address density fluctuations near the silica surface, we introduce a term that includes the fourth-order spatial derivative to the SGE, accounting for the influence of the silica substrate's crystalline structure. This enables the development of a density functional theory (DFT) where the density profile emerges as the minimum free energy state. Further research will focus on finalizing the DFT formulation, assessing the stability of its one-dimensional solutions under different perturbations, and exploring dynamic, time-dependent solutions.