Response to “Comment on ‘The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion–ion correlations’” [J. Chem. Phys. 134, 157101 (2011)

Abstract

Fraenkel commented1 on our paper2 in which we offered a different view of the activity coefficient (or, equivalently, the excess chemical potential) of electrolytes. Fraenkel summarized the main points of our theory nicely. Also, he offered his smaller-ion-shell (SiS) model3, 4 as an alternative to our approach. In this response, we categorize our thoughts along a few main issues. (1) The most important issue for us is the question whether the (concentration-dependent) dielectric constant, (c), of the solution should be used in the model, or the dielectric constant of the pure solvent (that is equal to the dielectric constant of the solution at infinite dilution). We argue that is a macroscopic physical parameter whose value is well-defined and measurable for a given thermodynamic state point (different concentration corresponds to a different state point). In the implicit solvent model of electrolytes the ions are modeled on the molecular level, therefore, they are characterized by microscopic parameters such as the ionsize parameters (ISP). The solvent, on the other hand, is modeled on the basis of its average dielectric response, which is characterized by a macroscopic physical parameter: the dielectric constant. In our view, physical variables that enter the calculations as external parameters, such as the dielectric constant, should have the experimental values for the given state point. The structure of the solvent is influenced by the presence of ions and decreases with increasing concentration (dielectric saturation). Unfortunately, measurements of the static dielectric constant of an electrolyte have several technical difficulties. The extrapolation to zero frequency based on dielectric relaxation data over a frequency range requires special instrumentation to determine the dielectric loss. For example, newer experiments by Buchner et al.5 for NaCl gave (c) values that are larger than those reported by Barthel et al. previously.6 The absence of experimental data for (c) makes the applicability of our approach limited and, at the same time, from a practical point of view, necessarily requires the use of theories applying the dielectric constant of the solvent. The absence or availability of experimental data, however, does not influence the correctness of the physical picture behind our model. Our opinion is that solvation should be taken into account by a concentration-dependent dielectric constant rather than by an increased, solvated ionic radius. (2) Our paper might give the (false) impression that the question whether adjustable parameters are used or not is a central issue for us. It is not. It depends on the intention of the investigator: if one wants to adjust molecular parameters to fit with experiments, it is legitimate to do so. Actually, that is what we do in our ion channel studies, for example.7 There are parameters, however, whose values are well-defined so they should not be adjusted. Even if we adjust a parameter, the resulting value should have a sensible physical meaning. In general, we can shed a light on our standpoint regarding modeling if we cite Occam’s razor that “admonishes us to choose from a set of otherwise equivalent models of a given phenomenon the simplest one. In any given model, Occam’s razor helps us to ‘shave off’ those concepts, variables, or constructs that are not really needed to explain the phenomenon. By doing that, developing the model will become much easier, and there is less chance of introducing inconsistencies, ambiguities, and redundancies.”8 Paraphrasing this idea for the case of adjustable parameters, we can offer a less strong statement: “In a model, we should minimize the number of adjustable parameters.” (3) The third issue is the question of the statistical– mechanical method with which the results for the activity coefficient [and the ion–ion (II) term in our approach] are calculated. Our opinion is that the best method should be used even if it is computationally expensive (especially in the age of fast computers). Technically simpler methods might be justified by computational convenience, but when scientific knowledge is sought, the most accurate method is favorable. According to the complexity of the studied many-body systems, more complex methods usually provide more accurate results. (Alas, Occam’s razor does not help here.) For a well-defined molecular model of a many-particle system, computer simulations, if performed properly, provide exact results apart from statistical uncertainties and system-size effects. Theories necessarily use approximations. Their results, therefore, should be tested against simulation data. The simulation results contain only the error due to the imperfectness of the model, while the theoretical results also contain the error due to the imperfectness of the approximation to derive the theory. Separation of these two errors is important for the activity coefficient of electrolytes as demonstrated in our paper using grand canonical Monte Carlo (MC) simulations9 and the mean spherical