Statistical Mechanics of a Multipolar Gas:  A Lattice Field Theory Approach

A recently proposed lattice field theory approach to the statistical mechanics of a classical Coulomb gas ( J . Chem . Phys . 1992, 97, 5653) is generalized to treat particles with arbitrary electric multipole moments. Explicit development of the mean-field approximation is given for the case of mobile dipoles and monopoles (ions) surrounding an arbitrary collection of fixed charges embedded in macroions. In particular, a modified Poisson−Boltzmann (PB) equation is derived in which the mobile dipoles provide an effective spatially dependent dielectric constant. This equation implies a self-consistent iteration procedure by which the monopole and dipole densities are simultaneously determined in terms of single scalar (PB) field. Moreover, this equation can be derived from a minimum principle; an annealing strategy for computing the PB field is thereby suggested. In addition, explicit mean-field expressions for thermodynamic free energies are obtained as simple functionals of the PB field. Connection to the well-known Langevin dipole model is made. A numerical application to a system consisting of two parallel plates is presented in order to illustrate the utility of the formulation.