Exact solution of mean spherical approximation for dipolar discs

Abstract

The mean spherical approximation is solved exactly for a two dimensional fluid of dipolar hard discs. Using the basis functions 1, Δ = Ŝ ₁ · Ŝ ₂ and D = Ŝ ₁ · (2[rcirc][rcirc]-U) · Ŝ ₂, it is shown that certain linear combinations of the associated radial coefficients are the solutions of the Ornstein-Zernike equation for hard discs in the Percus-Yevick approximation at densities θ, Kθ and -Kθ with θ the number density and K a self-determined parameter of the solution. A numerical investigation of the thermodynamic properties of this exact result culminates in the finding of a liquid-gas critical point identified by θR ²=0·114 and K _B TR ²/m ²=0·200 with R the disc diameter, k _B Boltzmann's constant, m the dipole moment strength and T the absolute temperature. In addition, the dielectric constant of such a fluid within the mean spherical approximation is calculated and the resulting closed-form expression is compared with the values of the Debye and Onsager theories over a wide range of θm ²/(k _B T) values.