Statistical–mechanical derivation of transport equations for glass-forming ionic liquids under a weak electric field based on time-convolutionless mode-coupling theory

The transport equations for ionic liquids near the glass transition are derived under a weak electric field from a statistical–mechanical point of view based on the time-convolutionless mode-coupling theory recently proposed. The analytic form of ionic conductivity σ(T) is thus found as σ(T)=ρ_eff(T)e²D_e ^L(T)∕k_BT, where ρ_eff is an effective ion density, e an elementary charge, and D_e ^L a long-time ion-diffusion coefficient. This result is quite different from the well-known Nernst–Einstein relation because ρ_eff(T) depends on temperature and also because D_e ^L(T) is not just the summation of the cationic and anionic self-diffusion coefficients. The analytic function of ρ_eff(T) suggests that it increases drastically near the glass transition as temperature decreases. This behavior is checked by experiments. The physical origin of such a behavior is also discussed.