Error in the Debye–Hückel approximation for dilute primitive model electrolytes with Bjerrum parameters of 2 and ca. 6.8 investigated by Monte Carlo methods. Excess energy, Helmholtz free energy, heat capacity and Widom activity coefficients corrected for neutralising background
Monte Carlo (MC) simulations have been performed for primitive model electrolytes with a Bjerrum parameter B= 2 for five values of κa in the region ca. 0.022–0.11. Also, an extremely dilute 2 : 2 electrolyte (B= 6.8116 and κa ca. 0.0276) has been investigated. Between five and eight million configurations have been used in each simulation, and the number of ions in each simulation (N) was varied between 32 and 1728. The universal scaling of the results using the ratio of the Debye length to the half period of the periodic boundary conditions (minimum image cut-off distance), which was found for dilute systems with B= 1, 1.546 and 1.681 in an earlier paper, is found to hold also for the present simulations. In this way, precise extrapolations of excess energies (Eex/NkT), excess electrostatic Helmholtz free energies and excess heat capacities can be found.
By means of an analytic correction to the Widom test particle method for simulation of activity coefficients most of the variation, with the number of ions (N), of the excess chemical potentials may be removed. The trick is to introduce a homogeneous neutralising background. The N-dependence left scales exactly like Eex/NkT, so that precise extrapolated values for the excess chemical potentials can be found.
The deviations from the Debye–Hückel values increase with increasing B(at fixed κa). With B= 2, all the thermodynamic quantities except the excess heat capacity are situated between the Debye–Hückel law and the Debye–Hückel limiting law. With B= 6.8116 and κa≈ 0.0276, however, all quantities are on the ‘wrong’ side of the limiting law, and deviations from the Debye–Hückel values are very large, e.g. ca. 40% for Eex/NkT and 400–500% for the excess heat capacity. These deviations are well accounted for by the DHX theory, using the Debye–Hückel electric potential around a central ion as potential of mean force.