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.
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