Generating true minima in constrained variational formulations via modified Lagrange multipliers

Conduction in heterogeneous systems in the low-frequency regime: variational principles and boundary integral equations

The response of a homogeneous material to the presence of an external low-frequency oscillating electric field can be described by means of an effective complex conductivity. Low frequencies are characterized by negligible magnetic and radiative effects. The properties of heterogeneous systems, composed of multiple homogeneous regions, can be determined from those of the individual components and their geometric arrangement. Examples of such heterogeneous systems include soft materials such as colloidal suspensions, electrolyte systems, and biological tissues. The difference in the intrinsic conductivities between the homogeneous regions leads to the creation of an oscillating charge density localized at the interfaces between these regions. We show how to express key properties of these systems using this dynamic charge as a fundamental variable. We derive a boundary integral equation for the charges and reconstruct potentials and fields from its solution. We present a variational principle that recovers the fundamental equations for the system in terms of the oscillating charge and show that, in some formulations, the associated functional can be interpreted in terms of the power dissipated in the system. The boundary integral equations are numerically solved using a finite element method for a few illustrative cases.

Surface Polarization Effects on Ion-Containing Emulsions

Surface polarization in ion-containing heterogeneous dielectric media such as cell media and emulsions is determined by and determines the positions of the ions. We compute the surface polarization self-consistently as the ions move and analyze their effects on the interactions between electro-neutral, ion-containing droplets using coarse-grained molecular dynamics simulations based on the true energy functional. For water droplets immersed in oil, the interdroplet interaction is attractive, and the surface polarization makes the major contribution. By contrast, for oil droplets in water, the ion-surface induced charge interaction is repulsive and counteracts the attraction between the ions, leading to a small attractive interaction between the droplets. This research improves our understanding of self-assembly in mixed phases such as metal extraction for recovering rare earth elements and nuclear waste as well as water purification.

Convexity and Stiffness in Energy Functions for Electrostatic Simulations

We study the properties of convex functionals which have been proposed for the simulation of charged molecular systems within the Poisson-Boltzmann approximation. We consider the extent to which the functionals reproduce the true fluctuations of electrolytes and thus the one-loop correction to mean field theory-including the Debye-Hückel correction to the free energy of ionic solutions. We also compare the functionals for use in numerical optimization of a mean field model of a charged polymer and show that different functionals have very different stiffnesses leading to substantial differences in accuracy and speed.