We present a numerical scheme for solving a sixth-order Cahn-Hilliard type equation that captures the dynamics of phase transitions in a ternary mixture consisting of two immiscible fluids and a surface active molecule that is amphiphilic. We show that by considering a suitable midpoint approximation for the nonlinear terms in the differential equation, we obtain an unconditionally energy-stable numerical scheme that is second-order in time. We demonstrate that our proposed numerical scheme satisfies these key properties for a wide range of physical parameters in two and three dimensions. Moreover, we present the results of a numerical study to report on the impact of each physical parameter on the behavior of the dynamics of the phase transitions, which are in agreement with the experimental observations.

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