Criticality in ionic fluids: Debye-Hückel theory, Bjerrum, and beyond

Debye-H\"uckel (DH) theory predicts phase separation in the primitive model electrolyte: hard spheres of diameter a with charges. The coexistence curve (CC) is acceptable with T₂^*k₁T₂a/q^2=1/16, which is roughly correct, but the critical density, ₂^*₂a^3=1/64, is far too low. Allowing for association into ideal dipolar pairs, following Bjerrum, improves ₂ and leaves T₂ unchanged but yields an unphysical CC. Extending DH theory to compute the dipole-ionic-fluid coupling yields a mean-field description with sensible CC and (T₂^*, ₂^*) close to Monte Carlo based estimates: (0. 0571₅, 0. 0308).