Search

In this work, we analyse a broad class of generalized Poisson–Boltzmann equations and reveal a common mathematical structure. In the limit of a wide electrode, we show that a broad class of generalized Poisson–Boltzmann equations admits a reduction that affords an explicit connection between the functional form of the corresponding free energy and the associated differential capacitance data. We exploit the relation to we show that differential capacitance curves generically undergo an inflection transition with increasing salt concentration, shifting from a local minimum near the point of zero charge for dilute solutions to a local maximum point near the point of zero charge for concentrated solutions. In addition, we develop a robust numerical method for solving generalized Poisson–Boltzmann equations which is easily applicable to the broad class of generalized Poisson–Boltzmann equations with very few code adjustments required for each model [ABSTRACT FROM AUTHOR]

The functionalized Cahn-Hilliard (FCH) free energy models phase separation in amphiphilic systems, which possess a component with both hydrophilic and hydrophobic tendencies. We establish the existence of a family of bilayer structures, described by homoclinic solutions of a fourth-order dynamical system arising from the FCH energy in the nongeneric case of an equal-depth double well, W ≥ 0. The free parameter accommodates bilayers of variable width, which correspond to the compression of the bilayer under external pressure. Modulo a nondegeneracy condition, which we conjecture is associated to an orbit-flip bifurcation within the fourth-order dynamical system, we show that its homoclinic orbits are well approximated by those of a corresponding second-order dynamical system obtained by perturbing the potential well W with the primitive of Modica and Mortola's √W. [ABSTRACT FROM AUTHOR]