Microscopic theory of activated ion hopping in polymerized ionic liquids and glasses.

We combine polymer integral equation theory of structure with microscopic dynamical theories of activated relaxation to formulate a theory of ion hopping in supercooled polymerized ionic liquids (PolyILs) and glasses. Activation barriers and the mean ion relaxation time are analyzed as a function of the ion-to-monomer size ratio, polymer persistence length, intrachain degree of dynamic cooperativity, anion-cation Coulomb attraction strength, and dielectric constant. A general finding is the dominance of Coulomb cage correlations and anion-cation attractions in determining the hopping rate of the small ions studied. A critical finding is that the activation barrier exists only above a threshold value of the system-specific dimensionless Coulomb attraction strength. As a consequence, the barrier grows in a highly nonlinear manner with anion-cation attraction energy. This suggests a route to super-ionic transport via a relatively modest reduction of the Coulombic association energy, an effect that becomes more dramatic the smaller the mobile ion. The temperature-dependent growth of the ion relaxation time is non-Arrhenius in the supercooled liquid, but may, or may not, crossover to an apparent Arrhenius form in the glass depending on how the dielectric constant on the relevant timescale changes with temperature. The magnitude of dynamic decoupling between the ion and polymer alpha relaxation times at the laboratory glass transition, the degree of trajectory level coupling of the ion and monomer motion, and ion jump lengths are also determined. A high level discussion of the connections between theory, experiments, and simulations, and a quantitative application to specific lithium, sodium, and potassium PolyILs, are presented.