A Numerical Study Of Self-Assembling Amphiphilic Systems

Mixing processes in ternary mixtures involve immiscible fluids such as oil and water, and a surface-active molecule called surfactant. These physical processes find applications in various fields, including enhanced oil recovery, drug delivery design systems, and the formulation of cleaning products. Even though this process has several applications, the mathematical models describing it and the numerical methods solving it are not well understood. The underlying mathematical model is a nonlinear initial-boundary value problem involving sixth-order derivatives and belongs to the class of sixth-order Cahn-Hilliard equations. Authors Sharma and Tierra recently proposed a numerical method to approximate its solutions in two and three dimensions. The proposed numerical method satisfies the key properties of being solvable, satisfying the discrete energy dissipation, being mass conservative, and being second-order accurate in time. Furthermore, numerical results were presented to demonstrate the scheme's effectiveness in capturing the dynamics of phase transitions for a broad parameter range and the self-assembly of the molecules into bilayers. In this work, with the help of a ternary phase diagram, we provide additional numerical studies and demonstrate the numerical scheme's further capability in capturing other self-assembly morphologies, such as micelles. We also demonstrate the effectiveness of the scheme across a broad parameter range in two-dimensional simulations. Our numerical investigations reveal how these parameters govern system morphology across three composition regimes, including: (1) surfactant-dominated (60% surfactant), (2) balanced oil-water (40% each), and (3) oil-dominated (60% oil) mixtures. The results quantitatively correlate parameter variations with interfacial pattern formation, matching established experimental behavior in ternary amphiphilic systems. This work establishes a robust computational framework for studying composition-dependent phase transitions, with natural extensions to three-dimensional systems and more complex interfacial phenomena.