Conductivity in polymer ionics. Dynamic disorder and correlation

Theoretical constructs are developed for discussing diffusivity and conductivity in polymer ionic materials. Such materials are characterized by extensive disorder, either static (lack of long-range order) or static and dynamic (lack of long-range order with short-range order evolving with time). Beginning with a dynamic percolation model, we show that, in general, so long as the mean-square displacement of the charged particle obeys a certain growth law, the observed charged-particle motion will be diffusive, both in the ballistic regime, corresponding to electronic motion with strong scattering, and in the ionic-hopping regime, corresponding to dynamic disorder renewal of the hopping situation. Some general behaviour for transport under these conditions is predicted, including definite statements about the frequency dependence of the conduction, the relationship between the growth law in a single interval and the growth law for observation times long compared to scattering or renewal times, and the behaviour in the neighbourhood of the percolation threshold for the static problem. Interpretations are suggested both for ion and electron-hopping situations.A statistical thermodynamic model is developed for analysis of contact ion pair formation and its effect on conductivity in ion-conducting polymer systems. In this model, the energy (due to solvation and polarization) favouring formation of a homogeneous complex in which the cations are solvated by the polymer host, competes with an entropic term favouring the separated structures (free polymer and contact ion pairs). We derive general conditions for this phase separation, and an expression for the number of polymer-bound, homogeneously solvated ions. We show that this number will, in general, decrease monotonically with increase in temperature, due to entropic favouring of the phase-separated material, this is reminiscent of the lower consolute temperature phenomenon in liquid mixtures.