Analytical solution of the Poisson–Boltzmann problem for two-layer spherical cell model

• A two-layer cell model for solution of the Poisson–Boltzmann equation is proposed.
• The general solution can be obtained by matching of solutions in each of the layers.
• Correct choice of matching conditions provides high accuracy of a general solution.
• The model can be used for highly charged particles and arbitrary volume fraction.
• The type of particle packing strongly affects the potential distribution.

The two-layer spherical cell model to describe the equilibrium electrical potential distribution between strongly charged particles in a monodisperse system is proposed. It is shown that the general solution of the Poisson–Boltzmann equation can be obtained by a combination of its solutions in the approximation of the excluded co-ions for the inner layer and in the Debye approximation for the outer one. The equation for the coordinate of the matching of solutions is proposed and analytically solved. It is shown that the correct choice of conditions for the matching provides a high precision of the solution in each layer. The article also analyzes the dependence of the characteristics of dispersed systems on the type of particle arrangement.

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