Particle motion through a dynamically disordered medium: The effects of bond correlation and application to polymer solid electrolytes

To study the diffusion of small particles through a dynamically disordered medium, a dynamic bond percolation model has been developed. This differs from the standard percolation theory, in that the lattice is no longer static but undergoes rearrangements which reassign the open and closed bonds. Physically, these rearrangements correspond to orientational motions of the (polymer) host lattice. An interesting feature of this model is that, even below the percolation threshold, diffusive behavior can occur, as long as the renewal time τ_ren, the time characteristic of the rearrangement of bonds, is short compared to the observation time. Ionic conductivity in polymeric electrolytes is one of the systems for which this model is useful. In these materials, carrier ions diffuse through a medium (the polymer) which is undergoing dynamic motion caused by configurational motions of the polymer. Since polymer chain motions will affect several ion binding sites simultaneously or serially, the model is further elaborated to account for correlations in the segmental motions of the polymer host. The renewal of the bonds is changed from occurring randomly to include simple correlation effects. Of principal concern in this study is the effect that correlated renewals have on the transport behavior of the model. Simulations were done on a 1-D lattice and a diffusion coefficient calculated. The values of the diffusion coefficients from the two systems (with and without correlated renewal) are studied and their behavior as a function of the fraction of available bonds, f, and the renewal time is compared. For both correlated and uncorrelated renewals, the systems were diffusive. The diffusion coefficients, in both cases, increased with increasing f and decreasing τ_ren, corresponding to an increase in the free volume, the configurational entropy, and the temperature of the polymer systems. The diffusion coefficient from the correlated systems were always smaller than those from the uncorrelated systems, except for the limit f = 100% and τ_ren ≪ τ_hop. The ratio of the diffusion coefficients for the correlated and uncorrelated systems was studied as a function of τ_ren and f. This ratio falls off to a constant value as τ_ren is increased and reaches minimum value at f = 50%. This behavior of the ratio as a function of f can be explained by considering the diffusion of the bonds in the lattice for the correlated case.