Incorporation of Mobile Dipoles via a PB-like Equation:

A strength of the LFT formalism is that it provides a rigorous starting point for treating problems involving interacting charge distributions and a route (via saddlepoint analysis) to systematic and practical approximation schemes. To illustrate the flexibility of the method, we consider a system in which in addition to mobile ions in the Coulomb gas, there are mobile (point) dipoles as well [51]. In the situation of direct interest here, these dipoles could represent polar solvent molecules in which the mobile ions ``swim" and which surround a collection of macroions. The mobile dipoles affect electrostatic interactions via the polarization density $P(\vec r) = \sum_j \vec p_j \delta (\vec r - \vec r_j)$, where $\vec r_j$ is the location of dipole j characterized by dipole moment $\vec p_j$. This polarization density corresponds to an effective charge density $-\vec \nabla \cdot P(\vec r)$ [5]. The overall electrostatic contribution of the mobile dipoles can be included simply by adding the dipole charge density to the contributions associated with charge monopoles (mobile ions and the charges embedded on the macroions). The LFT formalism can then be carried through analogous to the case without the mobile dipoles [52]. This results in the following expression for the grand canonical partition function of the Coulomb gas:

All effects of the mobile dipoles are contained in the penultimate term in the exponent, where $\bar p$is the magnitude of the microscopic dipole (in units of ea, e being the magnitude of the electron charge and a the lattice spacing), $\gamma_d$ is the dimensionless bulk dipole density (i.e., $\gamma_d = \bar n_d a^3$, where $\bar n_d$ is the dimensionful bulk dipole density and a the lattice spacing), and $v_{d \vec n}$ is the dimensionless excluded volume potential experienced by the dipoles at lattice point $\vec n$. Carrying through the saddle point analysis in the manner discussed in the previous section, we obtain a Poisson-Boltzmann type set of lattice field equations:

with

which corresponds to extremization of the following ``action" functional:

As in the dipole-free system, the extremum defined by Eq. 50 is a strict and unique minimum of S. Hence the extremizing field can be identified using a simple annealing algorithm, in spite of the fact that the equations are strongly nonlinear in general. Expanding the partition function integrals about the saddle point leads to an expression with exactly the same structure as Eq. (44). Thus S in Eq. (50), evaluated at the extremizing $\psi$ configuration, can be identified as the mean field approximation to $-\beta \Phi$, where $\Phi$ is the thermodynamic grand potential. Corrections to the mean field approximation can be computed via loop expansion, analogous to the case where there are no mobile dipoles. Finally it is interesting to note from the structure of Eq. 47, that $\epsilon_{\vec n}$ plays the role of an electric-field dependent polarizability. At low field strengths it tends to a field-independent constant which corresponds to the orientational polarization predicted by the well-known Langevin dipole model [45]. At higher field strengths $\epsilon_{\vec n}$ becomes field-strength dependent. It reflects the nonlinear dependence of the induced dipole moment on the electric field strength at high field strengths.