A Continuum Model with Differential Evolution on Binary Phase Diagram of Ionic Surfactant Aqueous Solution

We develop a continuum thermodynamic model basing on the cell model at three perspectives. First, incorporate the Helfrich free energy as amphiphilic molecules aggregate surface free energy; Second, modify the Poisson-Boltzmann equation by introducing the ion-specic dispersion interaction energy of the counter-ion in aqueous region with the aggregate surface to obtain the concentration distributions of both the surfactant monomer and the counter-ions; Third, include the temperature dependence of chemical potential for the standard state transition, allowing for calculations on the binary phase diagram of a series of potassium carboxylate as well as of sodium carboxylate soaps. The differential evolution algorithm is applied to obtain the global minimum of the required criteria, including the boundary conditions of the electrostatic potential, the optimization of aggregate size with respect to the total free energy and the equilibrium of monomers transferring between the aggregate and aqueous region. The specific-ion effect are presented in the aggregate surface tensions and in the counter-ions distribution within the aqueous regions. The continuum model gives good agreement with the dimension sizes and phase boundaries (lamellar-cylindrical and cylindrical-micellar) which are determined with thermodynamic measurements.