Abstract
A variational principle to the nonlinear Poisson-Boltzmann equation (PB) in three dimensions is used to first obtain solutions to the electrostatic potential surrounding a pair of spherical colloidal particles, one of them modeling the tip of an Atomic Force Microscope. Specifically, we consider the PB action integral for the electrostatic potential produced by charged colloidal particles and propose an analytical ansatz solution. This solution introduces the density and its corresponding electrostatic potential parametrically. The PB action is then minimized with respect to the parameter. Polynomial-exponential approximations for the parameters as functions of tip-particle separation and boundary electrostatic potential are obtained. With that information, tip-particle energy-separation curves are computed as well. Finally, based on the shape of the energy-separation curves, we study the stability properties predicted by this theory.
Highlights
A variational principle to the nonlinear Poisson-Boltzmann equation (PB) in three dimensions is used to first obtain solutions to the electrostatic potential surrounding a pair of spherical colloidal particles, one of them modeling the tip of an Atomic Force Microscope
We consider the PB action integral for the electrostatic potential produced by charged colloidal particles and propose an analytical ansatz solution
An open problem of current scientific and technological interest is the theoretical prediction of the force between an Atomic Force Microscope (AFM) probe and a charged particle, in particular when both are immersed in an electrolytic environment [1]
Summary
An open problem of current scientific and technological interest is the theoretical prediction of the force between an Atomic Force Microscope (AFM) probe and a charged particle, in particular when both are immersed in an electrolytic environment [1]. One approach to obtain that energy is to first solve the Poisson-Boltzmann (PB) equation, whose solution gives the charge density and electrostatic potential in the liquid surrounding the colloidal particles [10]. PB equation provides the distribution of the electric potential in solution with charged ions present. This distribution, in turn, provides information to determine how the electrostatic interactions will affect colloidal forces. We here consider the problem of two interacting particles by introducing an ansatz for the charge density function and corresponding electrostatic potential parametrically; the variational method is used to minimize the PB functional with respect to the parameters
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