Theory of Ionic Conductivity with Morphological Control in Polymers
M. Muthukumar
Abstract
We present a general theory of ionic conductivity in polymeric materials consisting of percolated ionic pathways. Identifying two key length scales corresponding to interpath permeation distance ξ and one-dimensional hopping conduction path length mλ, we have derived closed-form formulas in terms of the energy U required to unbind a conductive ion from its bound state and the partition ratio ξ/mλ between the three-dimensional permeation and one-dimensional hopping pathways. The results provide design strategies to significantly enhance ionic conductivity in single-ion conductors. For large barriers to dissociate an ion, corrections to the Arrhenius law are presented. The predicted dependence of ionic conductivity on the unbinding time is in agreement with results in the literature based on simulations and experiments. This theory is generally applicable to conductive systems where the two mechanisms of permeation and hopping occur concurrently.