(322d) Diffuse-Interface Modeling and Simulation of Amphiphilic Fluids as Ternary Mixtures

Complex fluid formulations consisting of surfactant molecules added to immiscible fluid components are ubiquitous in many engineering and industrial applications. Recently, renewed interest in amphiphilic systems has emerged in view of the challenges posed by more efficient tertiary oil recovery, or design of phase-change colloid/emulsion thermal energy storage systems. In fact, interfacial processes in complex fluids are poorly understood on the mesoscale which is reflected in empirical design procedures for these systems and in predictions of numerical models which can differ widely from experimental measurements. Towards predictive modeling of amphiphilic fluids, we investigate binary liquid+surfactant systems via simulation and modeling of ternary liquid mixtures as regular solutions with a Flory-Huggins and Cahn-Hilliard representation of the excess and non-local components of the Gibbs free energy of mixing. In this diffuse-interface model, convection and diffusion are coupled via a non-equilibrium Korteweg force expressing the tendency of the ternary system to minimize its free energy. Using a hybrid compact/pseudo-spectral discretization, the governing equations are integrated in a two- or three-dimensional periodic box or in a channel in a sequence of problems for assessing the predictive ability of the model through subsequent comparisons with experimental results. For code verification, we perform simulations of isothermal composition-induced phase separation (Gupta {\it et al.}, {\it Ind.~Eng.~Chem.~Res.}~{\bf 35}, 2360 (1996)), showing consistency with the equilibrium behavior obtained from the ternary phase diagram. In application to binary liquid+surfactant phase separating systems, the model reproduces a characteristic surfactant-induced retardation of the phase segregation process. For validation purposes, we perform simulations of symmetric ternary phase separation, showing that, in agreement with previous simulations, the segregation process can be characterized as a primary spinodal decomposition giving rise to an almost binary mixture, followed by a secondary phase separation which produces, at large times, three phases at equilibrium. These numerical results show that the behavior of the ternary system as it phase segregates depends on the value of the Peclet number. In fact, when $Pe<100$, during secondary separation small inclusions of the third phase appear in regions where there was an interface between the primary phases of the pseudo-binary mixture. In the end, both phases of the pseudo-binary are still present. However, for $Pe \ge 1000$, enhanced coalescence of the single-phase microdomains of the pseudo-binary leads to an effective addtional delay before nuclei of the third phase start to form. Finally, diffuse-interface simulations have been conducted to investigate droplet interactions as well as droplet/homophase coalescence during sedimentation of an initially phase-separated mixture at equilibrium. As expected, the addition of surfactant slows down coarsening in the vertical direction with a similar (monotonic) dependence of the sedimentation rate. Work in progress addresses these detailed dependences as a function of the Bond, Peclet, and Schmidt numbers and surfactant concentration at the spherical caps.