Distribution of electropotential in the electrode area of a solid state ion electrolyte

A solid state electrolyte is considered as a system consisting of cations moving through the volume of a solid body and anions whose mobility can be neglected due to their large size compared to the size of the cations. Accordingly, in the homogeneous case local charge compensation takes place. Under the action of an external electric field, cations create in the near electrode area inhomogeneous redistribution of mobile charges and electric field. The model is used for the statistical­mechanical description of high­temperature ionic conductors and current sources. To obtain the free energy functional of the mobile charge subsystem depending on the distribution of their density, the cluster expansion scheme for the renormalized Mayer functions is used. The Hamiltonian of a system consisting of electric charges moving in the field of single­particle cell potentials of average forces is used as the basis one. The binary function of the host system is expressed in terms of the screened potentials and the potentials of the average forces based on the results of the method of collective variables. The internal energy of the system is calculated taking into account the short­ and long­range effects. The Gibbs – Duhem relation was used for calculating the free energy functional of the system. The distribution of the number density of moving particles and the electric potential in the near electrode region were found from the condition of extremality of the free energy. The potentials of average forces are obtained as a result of solving a system of integral equations in the lattice approximation, with accounting of the short­ and long­range effects. The transition from the correlative function to the correlation function allowed us to identify the correlated and uncorrelated parts of the electric potential. The linear contributions of the deviation of the charge concentrations from a uniform distribution to the chemical potential are considered. The calculations take into account the contribution of the correlation between the particles in the first three coordination spheres that leads to attraction of the first, repulsion of the second and third neighbors. The description is carried out using a linear differential equation of the fourth­order with complex values of the roots of the characteristic equation. The paper analyzes the results of the analytical solution.