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

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Abstract
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 iω→λ+iω. 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.