Wertheim association theory for ion pairing in electrolytes: effect of neutral clusters

TL;DR

This work develops a Wertheim-based framework to describe ion pairing and neutral-cluster formation in the Restricted Primitive Model (RPM) of electrolytes. By splitting the Coulomb attraction into a long-range reference part solved via hypernetted-chain (HNC) theory and a short-range association part treated with Wertheim thermodynamic perturbation theory, the authors obtain a tractable, parameter-free route to quantify ion pairing through the association integral Δ_1 and the unpaired-ion fraction x, while fixing the splitting parameter κ via a stationarity condition on the total free energy. Extending the theory to include neutral clusters through isodesmic assembly (with Δ_2 = α^2 Δ_1) and an Arrhenius-like temperature dependence for α, the approach improves agreement with Monte Carlo phase boundaries and provides a framework to rationalize anomalous underscreening by accounting for the density of free ions contributing to screening. The results offer a physically transparent, systematically improvable route to capture low-temperature vapor-liquid condensation in strong electrolytes and illuminate the interplay between ion pairing, cluster formation, and electrostatic screening in the RPM.

Abstract

We address the problem of the vapor-liquid phase transition in the restricted primitive model (RPM) using Wertheim's statistical associating fluid theory to capture the effects of ion pairing which dominate the low-temperature vapor phase. For this we employ a reference system in which ion-pairing is suppressed by a judicious modification of the interaction between unlike charges from 1/r to erf(kappa r)/r, where kappa is a state-dependent parameter chosen so that the Helmholtz free energy A is at a null point (dA/d(kappa) = 0). Unlike the original RPM, this reference fluid admits real solutions to the hypernetted-chain (HNC) closure of the Ornstein-Zernike equations over a wide range of densities and temperatures. In the present study, we go beyond previous work [M. Li, Ph.D. thesis, University of Manchester (2011)] to allow for isodesmic assembly of ion pairs into neutral clusters. We find this has the potential to improve significantly the agreement with the Monte-Carlo results for the RPM vapor phase boundary. We can also match recent results on anomalous underscreening in the RPM [Härtel et al., Phys. Rev. Lett. 130, 108202 (2023)] assuming that only the free ions contribute to the screening length.

Figures (12)

• Figure 1: Vapor-liquid phase coexistence in the restricted primitive model (RPM) showing the coexistence region from Monte-Carlo simulations, redrawn from Luijten et al.*[] [. The line in Fig. \ref{fig:lims} is a fit to the Monte-Carlo data in this paperassuming Ising universality and the law of rectilinear diameters in $\surd\rho^*$ (following up a suggestion by the authors). The fit expression is $\surd\rho^*=A+Bt\pm Ct^\beta$ where $t\equiv1-T^*/T^*_c\ge0$the critical temperature $T^*_c=0.05069$the exponent $\beta=0.326$and the fit parameters are $A=0.2779$$B=0.1925$$C=0.4872$.] luijten_2002. The solid marker is the critical point at $T^*\simeq0.0507$ and $\rho^*\simeq 0.079$. The vapor (left) contains mostly ion pairs or quasi-neutral clusters caillol_1995haertel_2023, and coexists with a disordered liquid (right). The region where the hyper-netted chain (HNC) closure to the Ornstein-Zernike (OZ) equations for the RPM has real solutions lies to the right of the dashed line lims-note.
• Figure 2: Schematic of our strategy for solving the RPM: (1) the attraction between unlike charges is split into a weakened reference part plus a short-range correction; (2) the reference fluid is solved by HNC; (3) the short-range part is taken into account using Wertheim association theory; (4) the results are combined to make a thermodynamic theory for the RPM; (5) vapor-liquid phase coexistence boundaries are calculated; (6) the results are compared to the known phase diagram.
• Figure 3: Reference fluid: (a) potential between unlike ions for several values of $\kappa$ (the limit $\kappa\to\infty$ recovers the Coulomb law), and (b) the corresponding pair distribution functions at $T^*=0.05$ and $\rho^*=10^{-3}$ from HNC; for comparison the inset shows the HNC pair functions in the RPM liquid at the same reduced temperature and $\rho^*=0.3$.
• Figure 4: Splitting parameter: dependence of $\kappa$ on $\rho^*$ according to Eq. \ref{['eq:stat']} for two values of $T^*$ (main plot), and dependence of $\beta a_N$ on $\kappa$ for three values of $\rho^*$ at $T^*=0.05$ (inset; curves arbitrarily offset) with the stationary points (all minima) indicated by open circles. Results are shown for the pure ion pairing model; the case where neutral clusters are included is similar.
• Figure 5: Dependence of $\Delta_1$ on $T^*$ at two values of $\rho^*$ and two values of $\alpha$, compared to the association constant $K(T^*)$ from the DHBj Bjerrum pairing model in Eqs. \ref{['eq:Kdef']} and \ref{['eq:Qex']}.