Free-energy-based lattice Boltzmann method for emulsions with soluble surfactant

Abstract

A diffuse interface model to simulate a system of two immiscible liquids with surfactants is proposed. This model is based on a modified Ginzburg-Landau free energy functional to account for non-ionic soluble surfactants and couples the Navier-Stokes equations with two Cahn-Hilliard equations representing the transport of immiscible liquids and surfactant concentration. The governing equations are solved using a free-energy-based lattice Boltzmann framework. The proposed model is validated for Langmuir and Frumkin adsorption isotherms when applied to simulate a planar interface and a spherical drop equilibration. The model ensures numerical stability and accurate results for low and high surfactant concentrations, and the diffuse interface thickness remains constant even for high surfactant loads. The methodology is suggested to correlate the strength of the nonlinear surfactant coupling with the packing of surfactant molecules in the interfacial region. The proposed model allows for a reduction in surface tension of 45-50% compared to the clean system at low surfactant concentrations for the Langmuir isotherm, which follows the experimental results observed for liquid-liquid systems. The appropriate surfactant mobility and solubility values are recommended for different values of the strength of nonlinear surfactant coupling.