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

Debye-Hückel (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 ${\mathit{T}}_{\mathit{c}}^{\mathrm{*}}$≡${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}$a/${\mathit{q}}^2$=1/16, which is roughly correct, but the critical density, ${\mathrm{\rho }}_{\mathit{c}}^{\mathrm{*}}$≡${\mathrm{\rho }}_{\mathit{c}}$${\mathit{a}}^3$=1/64π, is far too low. Allowing for association into ideal dipolar pairs, following Bjerrum, improves ${\mathrm{\rho }}_{\mathit{c}}$ and leaves ${\mathit{T}}_{\mathit{c}}$ 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 (${\mathit{T}}_{\mathit{c}}^{\mathrm{*}}$,${\mathrm{\rho }}_{\mathit{c}}^{\mathrm{*}}$) close to Monte Carlo based estimates: (0.057±$1_5$ , 0.030∓8).