Poisson–Boltzmann equation for spherical cell model: approximate analytical solution and applications

Using the cell model approach, a distribution of the equilibrium electric potential is obtained within an electrolyte solution being the continuous phase of a concentrated disperse system. The distribution is a solution of the Poisson–Boltzmann problem for a spherical cell. A method, which was developed in earlier works of Dukhin, is employed to derive an approximate analytical expression for the potential field. The derived analytical expressions are compared with results of numerical simulation. A discussion is provided on how the derived distributions can be used to predict physically measurable parameters of concentrated disperse systems. Expressions for the osmotic pressure in disperse systems is obtained as a function of the dispersed phase volume fraction and surface potential. The surface charge is obtained as a function of surface potential changes of electrolyte activity coefficient due to the presence of dispersed particles are predicted using the approximate solutions of the Poisson–Boltzmann equation.