Statistical characterization of an ensemble of functional neural networks

The Ginzburg-Landau free energy functional with two order parameters has been widely used to describe surfactant adsorption phenomena at the interface between two immiscible fluids such as oil and water. To model surfactant adsorption, additional surfactant related terms are added to the original free energy functional which models an immiscible binary mixture. In this paper, we present a detailed comparison of phase-field models for an immiscible binary mixture with surfactant. In particular, we investigate the effects of mathematical model parameters on equilibrium surfactant profile across the interface between the immiscible binary mixture. Most previous models have severe time-step constraints due to the nonlinear coupling of order parameters. To solve these stability problems, we propose a special case of these models which allows the use of a much larger time-step size. We also apply a type of unconditionally gradient stable scheme and a fast multigrid method to solve the proposed model efficiently and accurately.