  Salt-in-ionic liquid electrolytes have attracted significant attention as
potential electrolytes for next generation batteries largely due to their
safety enhancements over typical organic electrolytes. However, recent
experimental and computational studies have shown that under certain conditions
alkali cations can migrate in electric fields as if they carried a net negative
effective charge. In particular, alkali cations were observed to have negative
transference numbers at small mole fractions of alkali metal salt that revert
to the expected net positive transference numbers at large mole fractions.
Simulations have provided some insights into these observations, where the
formation of asymmetric ionic clusters, as well as a percolating ion network
could largely explain the anomalous transport of alkali cations. However, a
thermodynamic theory that captures such phenomena has not been developed, as
ionic associations were typically treated via the formation of ion pairs. The
theory presented herein, based on the classical polymer theories, describes
thermoreversible associations between alkali cations and anions, where the
formation of large, asymmetric ionic clusters and a percolating ionic network
are a natural result of the theory. Furthermore, we present several general
methods to calculate the effective charge of alkali cations in ionic liquids.
We note that the negative effective charge is a robust prediction with respect
to the parameters of the theory, and that the formation of a percolating ionic
network leads to the restoration of net positive charges of the cations at
large mole fractions of alkali metal salt. Overall, we find excellent
qualitative agreement between our theory and molecular simulations in terms of
ionic cluster statistics and the effective charges of the alkali cations.
