## Effects of periodic boundary conditions on the static properties of charged fluids

##### Abstract

I investigate the system-size dependence of the thermodynamic properties of classical Coulomb systems in three dimensions with periodic boundary conditions. The system of interest---at given density, composition, and temperature---is assumed to behave as a conductor in the thermodynamic limit V [right pointing arrow] [infinity symbol], where V is the volume of the unit cell. The leading-order finite-size correction to the Coulomb energy per unit volume is shown to be 1/2k[subscript B]T/V (with k[subscript B]T the thermal energy). This correction is the classical analog of the correction to the potential energy of the electron gas at zero temperature derived by Chiesa et al. [Phys.\ Rev.\ Lett.\ \textbf{97}, 076404 (2006)]. Also considered is the system-size dependence of the change in free energy due to the insertion of a localized charge distribution of net charge Q [does not equal] 0, which serves as a prototype for the size dependence of the chemical potential of an ion. It is found that the finite-size correction to the charging free energy is of order1/V. If the response of the system is linear, the leading-order correction is given by 1/2aQ²x⁽⁴⁾/V, where x⁽⁴⁾ is the fourth moment of the static charge response function of the bulk fluid, and $a$ is a constant that depends on the choice of units. This formula may provide a useful estimate of the system-size dependence of charging free energies (or ionic chemical potentials) when paired with an approximate theory that gives a closed-form expression for x⁽⁴⁾. Finite-size corrections to excess thermodynamic properties obtained from the linearized Debye--Hückel theory are reviewed; these analytic expressions are valid for ionic systems in the low-density limit. A subset of the theoretical results are compared with computer simulation data for two simple models: the classical one-component plasma and a symmetric primitive-model electrolyte solution. This work may have methodological implications for simulation studies of various models of charged fluids.