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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.

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 [

The stability of colloids is indeed known to depend on the presence of charge at their surfaces [

Mathematically, the stability depends on the details of the pair-wise energy as a function of separation of colloidal particles [

The exact PB nonlinear equation in 3D is not amenable to analytical solutions even for a single colloidal particle in the electrolyte. 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 then used to minimize the PB functional with respect to the parameters.

The PB equation is typically obtained by combining Poisson’s equation [

with

where n is the ion bulk concentration of electrolyte, T is the absolute temperature, e the ion charge magnitude of anions and cations, and k_{B} is Boltzmann’s constant.

Equation (2) is converted into dimensionless form [

where

the absolute dielectric constant,

Equation (3) can be derived from a variational principle, by applying Euler-Lagrange to the action

where V is volume. The minimum of I occurs for the function φ that satisfies the Euler-Lagrange equation, which gives rise Equation (3).

Taking z as the axis that joins the centers of the two colloidal particles, and due to the axial symmetry of the problem, we rewrite the action in cylindrical coordinates as

where η is the radial polar coordinate in the xy plane, while the angular polar integration is readily performed and gives 2π. The additional factor of 2 comes from integrating z in half space and multiplying by 2 due to mirror symmetry. To make progress, we propose the following ansatz for the density and corresponding electrostatic potential which depends on the parameter k,

where

To emphasize, this choice of potential is dictated first by the fact that the potential must rapidly approach its bulk value away from the spheres, and second by the fact that the electrostatic potential must have a constant value at the surface of the colloidal particles (Dirichlet boundary conditions).

To find the sought solution to the original PB equation (Equation (3)), we minimize the PB action functional with respect to the parameter k. For fixed potential φ_{0} and fixed separations d, we find the constant k (i.e. minimum point k = k(φ_{0}, d)) for the proposed φ(x, y) that minimizes the action.

As _{best}, and the separation d for each

boundary condition φ_{0}. It leads us to further investigate the relationship between the linear relationship and the boundary condition φ_{0}.

We obtain polynomial approximations for the functions that relate the linear parameters (that are the slopes and η-intercepts) and the boundary conditions (as shown in

Moreover, for large separations between the colloidal particles, the best constant k converges to 0.1 for all of the boundary conditions φ_{0}. This should be a universal feature regardless of the model used since at large separations we should obtain a simple superposition the potential around single spheres.

The functional forms for the k_{best} allow us to write a simple function as follows (

where A(φ) is the polynomial approximation between the linear parameter-η-intercepts and φ_{0}, B(φ) is the polynomial approximation between the other linear parameter-slope and φ_{0}, and d is the center-to-center separation between the colloids.

Since the charge is distributed in the whole space that surrounds the colloidal particles, we have the energy as a function of separation d [

where to recall ρ is density and φ is voltage, which are now known from the previous section. For each boundary condition, the integral in (8) is performed for the corresponding optimal value of k.

Equation (8) then provides the sought sphere-sphere energy-separation curves. The computed results are shown in

_{0} the particles attract each other at small separations. This is consistent with all the published experimental literature [_{0} the energy decreases monotonically giving rise to repulsion at large separations. While, for small φ_{0} there are plateaus that suggest the existence of secondary minima. [_{0} there are local minima at distances larger than 30, but they cannot be expected to represent experimental behavior since they correspond to distance too large compared to the size of the particles. We also notice that the peak positions of the energy curves shift to larger distances as φ_{0} increases, as expected. Finally the behavior of the screening parameter as shown in

For the AFM community, these results are useful in comparing experimental forces with the derivative of the curves in

Funding for this project comes from the National Science Foundation grant #CHE- 1508085.

Lam, W.-T. and Zypman, F.R. (2016) Interaction Energy between an Atomic Force Microscope Tip and a Charged Particle in Electrolyte. Journal of Applied Mathematics and Physics, 4, 1989-1997. http://dx.doi.org/10.4236/jamp.2016.411199