Primitive and non-primitive model electrolytes: Comparing ion-related Helmholtz energies using molecular simulations

Two main frameworks are commonly used to describe electrolyte solutions: the non-primitive model, which rigorously accounts for all interactions between ions and solvent molecules; and the primitive model, which treats the solvent as a dielectric continuum, only describing ion-ion interactions explicitly. The primitive model offers simple Helmholtz energy expressions, including the Debye-Hückel (DH) equation, the primitive mean spherical approximation (MSA), and the Born theory of solvation. In this work, we evaluate the accuracy of primitive model approaches by comparing their Helmholtz energies with data from molecular simulations obtained for non-primitive model electrolyte solutions. We model electrolyte solutions as mixtures of equally sized, charged, and (non-polarizable) dipolar Lennard-Jones particles. Using thermodynamic integration, we isolate the Helmholtz energy contributions related to solvent-solvent, ion-solvent, and ion-ion interactions. Molecular simulations are performed across two temperatures and two densities, a range of charges, dipole moments, and ion mole fractions (0.005 ≤ xions ≤ 0.05). Our results show that while the primitive model expressions provide a qualitatively reasonable description of electrolyte solutions, they systematically underestimate the Helmholtz energy contributions associated with ion-solvent and ion-ion interactions. Achieving quantitative agreement requires empirical adjustments to the Born radius. Notably, the optimized Born radii are significantly larger than the actual ion sizes used in the molecular simulations, questioning the primitive model's applicability. This work presents rigorous benchmarks for the use of MSA, DH, and Born theories, along with molecular simulation data for non-primitive model electrolytes. These benchmarks provide insights for refining existing models and advancing the development of new equations of state for electrolyte solutions.