Statistical mechanics of a Coulomb gas with finite size particles: A lattice field theory approach

A recently proposed lattice field theory approach to the statistical mechanics of a classical Coulomb gas [J. Chem. Phys. 97, 5653 (1992)] is generalized to treat gas particles of finite size. To do this, a repulsive Yukawa interaction between all pairs of gas particles is added to the usual pairwise Coulombic interactions of the gas particles with each other and also with an arbitrary collection of immobile charges. Such a model is directly relevant for understanding the energetics of systems composed of macroions in electrolytic solutions when the simple ions that comprise the electrolyte are sufficiently large. A field theoretic representation of the grand partition function for the modified Coulomb gas is derived. Two coupled three-dimensional scalar fields are involved. Physically, one is related to the electrostatic potential and the other to the Yukawa potential. The field theory expression, once discretized onto an appropriate lattice, can be evaluated via saddle point expansion. The zeroth order or mean field approximation is found to be analogous to the Poisson-Boltzmann equation in the simple (infinitesimal particle) Coulomb gas problem. Higher order corrections can be obtained via a loop expansion procedure. Successful numerical application is reported for systems consisting of two spherical, equally charged macroions immersed in an electrolytic solution. Imbuing the simple ions in the solution with finite size prevents the degree of polarization of the ion cloud which is found in the infinitesimal ion limit.