Dynamic Percolation Theory Research Articles

Applications of dynamic bond percolation theory to the dielectric response of polymer electrolytes

Our recently-developed dynamic bond percolation model is extended and applied to polymer electrolytes by assuming an approximate form for the relaxation of the carrier mean-square displacement to its asymptotic value below the percolation threshold, with similar assumptions for the short-range motion of the ionic charges bound to the polymer host. The parameters characterizing the long-range-motion part of the carrier response are shown to be fully determined in terms of the bond percolation parameters, thereby allowing comparison with the exact analytic solution presently available only in one dimension. The behavior of the dielectric response based on the assumed functional form of 〈r 2> 0(t) (the mean-square carrier displacement from its initial position in a frozen lattice) is shown to be given by a sum of Debye dielectric loss peaks, and is compared with frequency-dependent data for PEO·NaSCN.

Dynamic bond percolation theory: A microscopic model for diffusion in dynamically disordered systems. I. Definition and one-dimensional case

A dynamic bond percolation model is defined and studied. The model is intended to describe diffusion of small particles (ions, electrons) in a medium which is statistically disordered (as in ordinary bond percolation), but which is also undergoing dynamic rearrangement processes on a timescale short compared to the observation time. The model should be applicable to polymeric solid electrolytes, where the orientational motions of the polymer (which are responsible for configurational entropy) cause the dynamic motion of the medium (polymer) in which the small particles (alkali ions) diffuse. The model is characterized by three parameters: an average hopping rate w which appears in the master equation for hopping, a percentage of available bonds f, and a mean renewal time τ̄ren for dynamic motion of the medium to rearrange the assignments of closed and open bonds. We show that the behavior is always diffusive for observation times long compared to τ̃ren, in agreement with experiment on polymeric solid electrolytes. We also derive a closed-form expression for the diffusion coefficient. For observation times smaller than the renewal time there is no diffusion, again in accord with the behavior of polymeric solid electrolytes below the glass transition temperature. The diffusion coefficient is a monotonically increasing function of the inverse renewal time and hence of the free volume, the configurational entropy, and the temperature.