Coulombic Criticality in General Dimensions

Abstract

Debye-Hückel theory for spheres of diameter $a$ with charges ${}_{}{}^{\pm }{}_{}{}^{}q$ is extended to general dimension $d$. Explicit results include Bjerrum association of +/- ions and the dipole-ionic solvation energy. For all $d>\mathrm{}2\mathrm{}$ a critical point terminates conducting-liquid-conducting-vapor coexistence: The critical density ${\rho }_c^{*}={\rho }_c{a}^d$ falls with $d\searrow 2$, but $T_c^{*}\equiv \frac{k_B{T}_c{a}^{d-2\mathrm{}}}{q^2}$ rises. When $d=2\mathrm{}$, an insulating vapor appears for $T^{*}\le \frac{1}{4}$ (the Kosterlitz-Thouless point), separated from conducting fluid by an infinite-order critical line ending at a tricritical point: ${\rho }_t^{*}={\rho }_c^{*}(d\to 2)\simeq 0.004$, $T_t^{*}=T_c^{*}(d\to 2)\lesssim \frac{1}{4}\mathrm{for}{\rho }^{*}>{\rho }_t^{*}$ the transition is first order.