[Paper Review] Individual ion species chemical potentials in the Mean Spherical Approximation

Johan S. Høye, Dirk Gillespie|arXiv (Cornell University)|May 27, 2022

thermodynamics and calorimetric analyses13 references7 citations

TL;DR

This paper derives missing chemical potential terms in the Mean Spherical Approximation (MSA) for individual ion species, correcting a long-standing omission in the theory. By re-deriving the MSA from foundational equations without assuming global charge neutrality, the authors show that terms proportional to ion valence $z_i$—previously ignored as canceling in the mean—must be included for accurate individual species thermodynamics. The key result is a complete, physically consistent MSA formulation for $\mu_i$, significantly improving agreement with Monte Carlo simulations at high concentrations.

ABSTRACT

The Mean Spherical Approximation (MSA) is a commonly-used thermodynamic theory for computing the energetics of ions in the primitive model (i.e., charged hard-sphere ions in a background dielectric). For the excess chemical potential, however, the early MSA formulations (which were widely adopted) only included the terms needed to compute the mean excess chemical potential (or the mean activity coefficient). Other terms for the chemical potential $\mu_i$ of individual species $i$ were not included because they sum to $0$ in the mean chemical potential. Here, we derive these terms to give a complete MSA formulation of the chemical potential. The result is a simple additive term for $\mu_i$ that we show is a qualitative improvement over the previous MSA version. In addition, our derivation shows that the MSA's assumption of global charge neutrality is not strictly necessary, so that the MSA is also valid for systems close to neutrality.

Motivation & Objective

• . The paper addresses the long-standing omission of valence-dependent terms ($z_i$) in MSA formulations for individual ion species chemical potentials.
• It aims to correct the thermodynamic inconsistency in the standard MSA, where terms proportional to $z_i$ are discarded because they cancel in the mean chemical potential.
• The objective is to provide a complete, physically consistent MSA formulation for $\mu_i$ that enables accurate modeling of inhomogeneous systems like electrical double layers.
• The work demonstrates that the MSA remains valid even for systems near, but not strictly at, global charge neutrality, broadening its applicability.

Proposed method

• . The authors re-derive the MSA from Blum's original formulation, carefully isolating the terms that were previously lumped into a $+z_i \cdot \text{const}$ term.
• They show that the MSA's assumption of local charge neutrality (around a central ion) holds even without global charge neutrality, validating its use in inhomogeneous systems.
• The derivation uses the Ornstein-Zernike equation, Fourier transforms, and the Baxter factorization to solve for the direct correlation functions $c_{ij}(r)$ and total correlation functions $h_{ij}(r)$.
• A key step is the re-evaluation of integrals involving $J_{ij}(r)$, where charge neutrality was previously invoked prematurely; the authors show that the $r^3$ term from this step is asymptotically small but physically meaningful.
• They introduce an effective MSA system by embedding background charge into ion cores, preserving charge neutrality in the effective system and allowing derivation of the internal energy and chemical potential.
• The final expression for the individual species electrostatic excess chemical potential $\mu_i$ is derived as $\beta\mu_i = \beta u_{0i} + \beta\delta\mu_{0i} - 2z_i u^*$, with $u^*$ defined via the system's structure factors and densities.

Experimental results

Research questions

• RQ1 . Why do standard MSA formulations for individual ion chemical potentials fail at high electrolyte concentrations, and what is missing in the standard formulation?
• RQ2 Can the MSA be applied to systems that are not globally charge-neutral, such as those with local charge imbalances?
• RQ3 What is the physical origin and significance of the $z_i$-dependent terms that were previously omitted from $\mu_i$?
• RQ4 How can a consistent, complete MSA formulation for $\mu_i$ be derived from first principles without relying on global charge neutrality?
• RQ5 What is the quantitative impact of including these missing $z_i$-dependent terms on the accuracy of chemical potential predictions?

Key findings

• . The missing terms in the standard MSA formulation are proportional to the ion valence $z_i$, and their inclusion leads to a qualitatively improved description of individual ion chemical potentials.
• The derivation shows that the MSA remains valid for systems close to, but not strictly at, global charge neutrality, as long as local charge neutrality around each ion is preserved.
• . The new MSA formulation for $\mu_i$ is given by $\beta\mu_i = \beta u_{0i} + \beta\delta\mu_{0i} - 2z_i u^*$, where $u^*$ is a structure-dependent term derived from the system's correlation functions.
• The inclusion of the $-2z_i u^*$ term significantly improves agreement with Monte Carlo simulations, especially at high electrolyte concentrations where the standard MSA fails.
• The $z_i$-dependent terms were previously neglected because they cancel in the mean chemical potential, but they are essential for accurate individual species thermodynamics in inhomogeneous systems.
• The derivation confirms that the standard MSA's internal energy and free energy expressions remain valid when the effective valence $z_i^\text{eff}$ is used, and the correction term $\Delta E$ is properly accounted for.