Theory for the mixed alkali effect in glasses

The mixed alkali or mixed mobile ion effect in glasses manifests itself by strong nonlinear variations of ionic transport properties upon mixing of different types of mobile ions. We develop a theory for this effect based on thermally activated hopping transport in disordered site energy landscapes that consistently incorporates the statistical-mechanical and kinetic aspects of a mobile ion mixture. This includes a consideration of the joint probability density of site energy states, generalized Fermi distributions for mean site occupations, and cross terms in the current response described by nondiagonal Onsager coefficients. The theory shows that a mixed alkali effect can arise even when the two ion species share identical site energy distributions. It suffices that sites have distinct energies when occupied by ions of different type. Taking into account that a mismatch energy is needed for ions of one type to occupy sites adapted to the other type, the mixed alkali effect becomes stronger. Spatial correlations between site energies are needed for the mobility of the majority ion to decrease stronger than exponential upon replacement by the minority ion. The theory agrees well with kinetic Monte Carlo simulations. Application to mixed alkali phosphate glasses yields good agreement with measured conductivity activation energies.