How wrong is the Debye–Hückel approximation for dilute primitive model electrolytes with moderate bjerrum parameter?

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Abstract
Monte Carlo (MC) simulations have been performed for primitive model electrolytes for moderate Bjerrum parameters (B= 1, B= 1.546 and a few with B= 1.681) and values of κa in the range ca. 0.015–0.45. Emphasis has been laid on very dilute systems. Several millions of configurations have been used, and the number of ions in each simulation (N) was varied between N= 32 and N= 1000 (N= 1728 in a single case). It is shown, by means of ‘universal scaling’ comparing the Debye length 1/κ with the periodic length of the boundary conditions, that excess energies (Eex/NkT) should be extrapolated using a polynomial commencing with the power 2/3 in 1/N rather than the usual plots against 1/N. The same seems to be the case for the electrostatic, excess Helmholtz free energy and the excess heat capacity. However, the logarithm of the activity coefficients (In y±, In y+ and In y–) as calculated by the test particle method of Widom should be extrapolated with a leading term, which is the cube root of 1/N. The extrapolated values of the thermodynamic parameters compare well with high-precision HNC calculations and with DHX calculations. In summary, the conclusion is drawn, that the usual Debye–Hückel laws (including the κa correction) and not the Debye–Hückel limiting laws are the true low-correlation limits (limit of low plasma parameter Bκa) for the electrostatic contributions to the thermodynamic properties. However, even for 1 : 1 electrolytes there are small, but significant deviations from the DH laws. These deviations increase with B. For the excess heat capacity, the deviations are quite large. They are predicted accurately by the DHX approximation. A consistency check has been made of the DHX theory, calculating In y± in various ways. This model is quite consistent at low values of κa, when we depart from Eex, but less consistent when using the Kirkwood–Buff formalism, dependent on the second moment with respect to the radial distance of the radial distribution functions. The Monte Carlo results for the radial distribution functions are statistically indistinguishable from the DHX radial distribution functions. Also, the DHX and the HNC radial distribution functions cannot be distinguished up to at least κa= 0.45. For unequal radii of the ions, a straightforward generalisation of DHX (GDHX) is found to hold true comparing with MC simulations. The thermodynamic properties in dilute systems are dominated by the contact distance between the cation and the anion. Fixing this (B= 1.546), and increasing the ratio between the radii of the two ions to 3 leads to quite small, though systematic changes in the thermodynamic properties at κa=ca. 0.10 and 0.14. In particular, one observes that In y+= In y–= In y± to a very good approximation, at least for this dilute system.

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