Systematic ionic screening theory of macroions

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Abstract
Techniques of lattice field theory are utilized to compute the free energy of a system of fixed charged macroions surrounded by small (atomic size) mobile ions. The grand partition function for the simple ions is written down as a functional integral over a three‐dimensional auxiliary field. This functional integral is discretized on a lattice, and then subjected to saddle point analysis. The lowest order or ‘‘mean field’’ result of the analysis isolates a field which satisfies the Poisson–Boltzmann equation, and from which the Helmholtz free energy can be extracted. The formalism also provides a minimum principle for the Poisson–Boltzmann field that can be realized numerically by elementary annealing techniques. Most importantly, the mean field approximation can be systematically corrected by evaluating fluctuations around the saddle point to successive orders in an appropriate interaction strength. It is shown by numerical tests on a two‐macroion system that the hierarchy of corrections converges rapidly in experimentally interesting regions of parameter space.