Derivation of PB Equations and Associated Thermodynamical Relations: A Lattice Field Theory Approach

Up to this point, the PB Equation has been justified by somewhat heuristic arguments, and the thermodynamic connections noted above have been asserted without proof. In fact the PB Eq. can be derived as a mean field approximation to the full statistical mechanics of the classical Coulomb gas problem in such a way that the attendant thermodynamic connection emerges naturally. One way to do this [47] is via a Lattice Field Theory formulation of the full problem [38]. Using the identity $\vert r-r'\vert^{-1} = -4\pi <r\vert \Delta^{-1} \vert r'\gt$ (essentially, ``the Coulomb potential is the Green's function of the Laplacian operator"), it is possible to derive the following exact expression for the Grand Partition Function for the Coulomb gas Hamiltonian (an arbitrary collection of mobile ions surrounding an arbitrary assembly of fixed charges) [38]:

$$Z_{gc} = \int \prod_{\vec n} d\chi_{\vec n} e^{S(i\chi)}$$ (43)

Here $\chi$ is an auxiliary scalar field, discretized onto a cubic lattice (with value $\chi_{\vec n}$ at lattice site $\vec n$), and S is precisely the functional defined above in Eq. 38, evaluated at $\psi =i\chi$ [48]. According to Eq. 43, it is necessary to integrate over all field configurations obtained by letting each $\chi_{\vec n}$ independently take on all real values. This is a daunting exercise in multidimensional integration, particularly since the limit of arbitrarily fine lattice mesh, hence arbitrarily large number of $\chi_{\vec n}$ variables, must be taken. Monte-Carlo sampling provides a possible way of carrying out the required integrations. An alternate route, which has proven useful in a variety of applications, is a saddle point or steepest descent expansion procedure. In this approach we find the $\chi$ field configuration which extremizes the argument of the exponent, then perform a Taylor's series expansion about this configuration, as described below. The extremization conditions are obtained by setting $\partial S / \partial \chi_{\vec n} =0$.It turns out that the extremizing $\chi$ field is completely imaginary, so it is useful to write $\chi_{\vec n} = -i \bar\psi_{\vec n} + \xi_{\vec n}$,where $\bar \psi_{\vec n}$ is the value of the (real) extremizing field, and $\xi$ is a real-valued field that characterizes fluctuations about the extremizing field. Expressed in terms of the $\psi$ field the extremizing conditions are $\partial S(\psi) / \partial \psi_{\vec n} = 0$. These generate the Lattice PB Equations noted above. Once this field configuration (denoted $\bar \psi$)has been obtained, we then expand about it to write:

$$Z_{gc} = e^{S(\bar \psi)} \int \prod_{\vec n} d \xi_{\vec n} e^{ \{ \cdots \} }$$ (44)

where "$\{ \cdots \}$" are the quadratic and higher order terms in the expansion of $S(i\chi)$ about the reference configuration $-i\bar \psi$.To this point no approximation has been made. However, this form suggests a systematic approximation procedure. Assuming first that the fluctuations $\xi$ are small [49], we neglect the fluctuation integrals and approximate:

$$Z_{gc} \cong e^{S(\bar \psi)}$$ (45)

hence in this approximation $- \beta \Phi = ln Z_{gc} \cong S$. This establishes the thermodynamic connection of the PB field asserted above.

Now, to see how important the neglected terms (fluctuation integrals) are, we can attempt to perform them. If cubic and higher fluctuations are neglected the integrals are Gaussian and can be reduced to the calculation of a single determinant (this is referred to as the one loop correction). Finally, to include cubic and higher order effects, we expand down the anharmonic terms and perform ``Gaussian moment" integrals (two loop corrections and higher).

To give an illustration of the LFT method, we show in Figure 4 computations of the pair-correlation function of a fluid of charged polyballs (spherical colloids). This figure is taken from Ref. [50], where a full specification of relevant parameters is given. Calculations were done via standard Metropolis sampling, with LFT used at each trial configuration of the many-polyball system to calculate the effective inter-polyball potential ab initio. By ab initio we mean that the potential of mean force for the many-polyball system was computed from statistical mechanics using LFT. No assumption of pairwise decomposition of the potential of mean force was made. The centers of the polyballs were restricted to lattice positions, and Metropolis moves were carried out ``on lattice". Most LFT evaluations were done at the mean field (PB) level, but occasionally 1-loop corrections were computed. These were found to be orders of magnitude smaller than the mean field contributions, and hence negligible for the system under study. In Fig. 4 the LFT results for the pair-correlation function are compared with pairwise additive approximations. The deviation in quantitative details of LFT and pair-potential results suggests the desirability of going beyond pair-wise additive theories in the study of dense, highly charged macroion/electrolyte systems.