HR: 13:45h
AN: H22G-01 INVITED     [PDF]
TI: The Poisson-Boltzmann Equation as a Dynamical System
AU: * Sposito, G
EM: sposito@ce.berkeley.edu
AF: Department of Civil and Environmental Engineering, Environmental Engineering Group, MC 1710, University of California, Berkeley, CA 94720-1710 United States
AU: * Sposito, G
EM: sposito@ce.berkeley.edu
AF: Department of Geophysics and Geomechanics, Division of Earth Sciences, MC 90/1116, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 United States
AB: The Poisson-Boltzmann equation is a nonlinear ordinary differential equation that provides a rather accurate model for the behavior of the diffuse ion swarm near charged colloidal particles suspended in dilute aqueous media. Analytical solutions of this equation are known only for planar charged surfaces (e.g. layer type clay minerals), but an iterative method of approximate solution suggested by J.-Y. Parlange in 1972 has enjoyed widespread application to both cylindrical (e.g. polyanions) and spherical (e.g. metal oxides) charged surfaces. Since the Poisson-Boltzmann equation is a special case of a generic nonlinear ordinary differential equation whose other realizations include well known mathematical models in astrophysics, nuclear engineering, and theoretical ecology, progress in the analytical solution of the Poisson-Boltzmann equation also contributes to the elucidation of its mathematical siblings. In this paper we show that the Poisson-Boltzmann equation can be derived from a Lagrangian variational principle and that it can be put into one-to-one correspondence with a Hamiltonian dynamical system. This correspondence opens the door to utilizing the vast array of mathematical results available in Lagrangian and Hamiltonian dynamics. For example, Noether's theorem can be applied to find first integrals and an exact Hamilton-Jacoby equation can be constructed, the solution of which is tantamount to solving the Poisson-Boltzmann equation. This latter avenue is explored for planar, cylindrical, and spherical geometries emphasizing the utility of the Parlange approximation scheme.
DE: 1045 Low-temperature geochemistry
DE: 1806 Chemistry of fresh water
DE: 1831 Groundwater quality
DE: 3210 Modeling
DE: 3220 Nonlinear dynamics
SC: Hydrology [H]
MN: 2003 Fall Meeting