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 ±{}q. The coexistence curve (CC) is acceptable with Tc*{≡}kB{T}c$a/${q}²$=1/16, which is roughly correct, but the critical density, ${{{ρ}}}c*≡{{{ρ}}}ca³=1/64{π}, is far too low. Allowing for association into ideal dipolar pairs, following Bjerrum, improves ρc and leaves Tc 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 (Tc*,ρc*) close to Monte Carlo based estimates: (0.057±{}1₅ , 0.030{∓}8).