Microscopic derivation of the Hubbard–Onsager–Zwanzig expression of limiting ionic conductivity

An outstanding problem in the theory of ionic conductivity is a derivation of the well-known Hubbard–Onsager–Zwanzig expression for the dielectric friction on an ion from first principles. In fact, microscopic theories have repeatedly failed to reproduce the Hubbard–Onsager–Zwanzig expression under any limiting conditions. We show in this article that the existing molecular theories and the continuum model treatments calculate two entirely different contributions to the friction on the ion. While the former calculates the contribution from the solvent density mode alone, the latter includes only the contribution from the current mode. Thus the existing molecular theories can never be reduced to the Hubbard–Onsager (H–O) theory. In addition, we show that the existing molecular theories become inconsistent for larger ions where the H–O theory is accurate. The reverse is true for smaller ions. An expression is derived here which is valid at both the limits and for all ion sizes and its consequences discussed