Energy variational approach to study charge inversion (layering) near charged walls
YunKyong Hyon  Department of Mechanical Engineering, University of Nevada, Reno, Reno, NV 89557, United States (email) Abstract: We introduce a mathematical model, which describes the charge inversion phenomena in systems with a charged wall or boundary. This model may prove helpful in understanding semiconductor devices, ion channels, and electrochemical systems like batteries that depend on complex distributions of charge for their function. The mathematical model is derived using the energy variational approach that takes into account ion diffusion, electrostatics, finite size effects, and specific boundary behavior. In ion dynamic theory, a wellknown system of equations is the PoissonNernstPlanck (PNP) equation that includes entropic and electrostatic energy. The PNP type of equation can also be derived by the energy variational approach. However, the PNP equations have not produced the charge inversion/layering in charged wall situations presumably because the conventional PNP does not include the finite size of ions and other physical features needed to create the charge inversion. In this paper, we investigate the key features needed to produce the charge inversion phenomena using a mathematical model, the energy variational approach. One of the key features is a finite size (finite volume) effect, which is an unavoidable property of ions important for their dynamics on small scales. The other is an interfacial constraint to capture the spatial variation of electroneutrality in systems with charged walls. The interfacial constraint is established by the diffusive interface approach that approximately describes the boundary effect produced by the charged wall. The energy variational approach gives us a mathematically selfconsistent way to introduce the interfacial constraint. We mainly discuss those two key features in this paper. Employing the energy variational approach, we derive a nonlocal partial differential equation with a total energy consisting of the entropic energy, electrostatic energy, repulsion energy representing the excluded volume effect, and the contribution of an interfacial constraint related to overall electroneutrality between bulk/bath and wall. The resulting mathematical model produces the charge inversion phenomena near charged walls. We compare the computational results of the mathematical model to those of MonteCarlo computations.
Keywords: Finite size effects, energetic variational approach, PoissonNernstPlanck equations,
MonteCarlo computations, hard sphere,
LennardJones repulsive potential, charge inversion, layering, numerical computations.
Received: April 2011; Revised: September 2011; Available Online: July 2012. 
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