Generalized hopping model for frequency-dependent transport in a dynamically disordered medium, with applications to polymer solid electrolytes

Protonic diffusion in hydrogen-bonded networks, ionic conduction in polymeric solid electrolytes, and other processes in which the carrier transport mechanism involves motion of the host medium on a time scale comparable to that of the carrier motion itself require generalization of the usual models based on carrier hopping in a static medium. Under the assumption that this concurrent motion of the host can be modeled by a random reassignment (or ``renewal'') of hopping probabilities, with a constant probability {λ} per unit time for renewal to occur, the effects of host motion on the frequency-dependent diffusion coefficient D({ω}) are now considered. We consider both the dynamic bond-percolation model (in which the site-to-site hopping probability is randomly assigned either the value w or the value 0) and the more general model based on a possibly continuous distribution of hopping rates randomly assigned between different pairs of sites. Under these assumptions, the diffusion coefficient D({ω}) with renewal is shown to be obtainable from D({ω}) without renewal through the formal substitution i{ω}{→}{λ}+i{ω}. For the {ω}=0 limit, an expression is obtained for the time-dependent mean-square displacement with renewal in terms of the mean-square displacement without renewal. These general formal results are applied to the one-dimensional dynamic percolation model, for which specific exact analytic results are thereby obtained, and D({ω}) is calculated and studied for this case.