Electrostatic Modeling of Surfactant Liquid-Crystalline Aggregates:  The Modified Poisson−Boltzmann Equation

A model of liquid-crystalline aggregates is developed to calculate areas per surfactant head group in high-concentration cylindrical and planar geometries. Two free energy contributions, electrostatic and interfacial, are summed to generate the total cell free energy for a range of head-group areas. The modified Poisson−Boltzmann equation (MPBE) including dielectric saturation and hydration energies is used to calculated the electrostatic free energy. A Stern layer is used to prevent counterions at the charged interface, the thickness of which is decreased for large head-group areas to simulate the intercalation of counterions between head groups. The hydration constant was decreased at high surfactant concentration to account for reduced ion hydration as water is depleted. The calculation is performed for a one-dimensional variation of physical properties. Solutions to the MPBE were achieved for higher surface charge densities than previously reported, using a finite difference to estimate the derivatives and solving the resulting nonlinear algebraic equations. The optimized head-group areas for lamellar phase samples are compared to published X-ray data to estimate the extent of ion binding.