# Comparisons of equation of state models for electrolytes: e-CPA and e-PPC-SAFT

Martin Due Olsen, Georgios M Kontog Georgis, Jean-Charles de Hemptinne,
Xiaodong Liang, Nicolas von Solms

► To cite this version:

Martin Due Olsen, Georgios M Kontog Georgis, Jean-Charles de Hemptinne, Xiaodong Liang, Nicolas von Solms. Comparisons of equation of state models for electrolytes: e-CPA and e-PPC-SAFT. Fluid Phase Equilibria, 2023, 571, pp.113804. 10.1016/j.fluid.2023.113804. hal-04082688

HAL Id: hal-04082688

https://hal.science/hal-04082688

Submitted on 26 Apr 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

# Comparisons of equation of state models for electrolytes: e-CPA and e-PPC-SAFT

Martin Due Olsen<sup>a</sup>, Georgios M. Kontogeorgis<sup>a</sup>, Jean-Charles de Hemptinne<sup>b</sup>, Xiaodong Liang<sup>a</sup>, Nicolas von Solms<sup>a,*</sup>

## ARTICLE INFO

**Keywords:**

Electrolytes

Equation of state

Permittivity

Parameterization

Thermodynamics

## ABSTRACT

Many different models have been published with the aim of describing electrolyte thermodynamics. In this study two successful Equations of State (EoS) are compared. The models investigated in this work are the electrolyte Cubic Plus Association (e-CPA) EoS and the electrolyte Polar Perturbed Chain Statistical Associating Fluid Theory (e-PPC-SAFT). The model terms are not changed from previous works, but they are reparameterized with various models for the relative static permittivity. The models are parameterized as similarly and fairly as possible by identifying key parameters in the models that are intended to describe the same physics. Several different sets of comparable parameters are investigated to test the models with the inclusion of various physics, and to avoid favoring one model due to the choice of adjustable parameters. The parameters of the models are optimized with an objective function containing the properties mean ionic activity coefficients (MIAC), osmotic coefficients and density in a wide temperature range of 273.15–473.15 K and they are tested for their ability to quantitatively describe these properties. The difference between e-CPA and e-PPC-SAFT is generally found to be relatively minor and the permittivity model and considered adjustable parameters are generally more important for the overall model performance. It is found that the permittivity models that only predict part of experimental decrease tend to perform the best for the properties MIAC, osmotic coefficients and density. It is also found that including both association parameters and dispersion/attractive parameters for ions is better than neglecting one type of parameters, even when the same number of adjustable parameters is included. To be able to capture the temperature trend of MIAC it is found to be important to include an additional temperature dependency, which in this work is a temperature dependent interaction parameter for the physical term. It is also found that it is important to include sufficiently strong ion association to be able to obtain the correct qualitative trend for individual ion activity coefficients.

## 1. Introduction

Having reliable models to describe electrolyte systems is important to a wide variety of industries [1] as ions often will be present in many processes. Having accurate models for predicting thermodynamic properties of electrolyte solutions was deemed to be important to industry in an review of the industrial needs from 2010 [2]. Ten years later, a follow-up review in 2020 [3] showed that electrolyte thermodynamics still was an important topic, and it might have become even more relevant due to the importance of electrolytes in biotechnology.

A lot of different models have been proposed to describe thermodynamic properties of electrolyte solutions [4], including both activity coefficient models [5–9] and equations of state (EoS) [4,10–13]. Comparisons of different types of models on a fair basis is important to

evaluate the capabilities and limitations of different models. However, it is not usually possible to directly compare the reported results of the performance of a model, because the experimental data used to optimize parameters of the model, and the data used to evaluate the performance are usually different between groups. This makes fair comparisons difficult.

For activity coefficient models there are quite a few published papers that compared various activity coefficient models [14–21]. For non-electrolyte systems there are also quite a few comparison of different EoS for some properties [22–25]. For electrolyte EoS (e-EoS) it is however very limited in terms of comparisons of different models, where they are tested on the same set experimental data. Some studies have been published where the same e-EoS is tested with

<footnote><sup>a</sup> Center for Energy Resources Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark (DTU), Lyngby, Denmark</footnote>

<footnote><sup>b</sup> IFP Energies Nouvelles, 1 et 4 Avenue de Bois-Préau, 92852, Rueil-Malmaison Cedex, France</footnote>

<footnote>* Corresponding author.
E-mail address: nvs@kt.dtu.dk (N. von Solms).</footnote>
different permittivity models for e-CPA [26], e-PPC-SAFT [27] and SAFT-VR-Mie [28].

In this work two EoS's are compared, electrolyte Cubic plus Association (e-CPA) and electrolyte Polar Perturbed Chain Statistical Associating Fluid Theory (e-PPC-SAFT). Both models have previously shown good results for various properties. The e-CPA model is based on the implementation used in previous works from the group of DTU [10, 26, 29–32]. The e-PPC-SAFT model is done with the implementation from the group of IFPEN [27,33,34].

The two models that have been chosen for this comparison features several differences. A main difference is in the description of the close range forces where e-CPA uses a SRK term and e-PPC-SAFT uses the SAFT framework. Another difference between the two models is in the description of the electrostatic interactions where the e-CPA version used in this work includes the full Debye–Hückel equation, whereas e-PPC-SAFT uses an unrestricted Mean Spherical Approximation (MSA) term. It has been shown that, in isolation, the two electrostatic models (DH and MSA) will perform similarly when the ion diameter is adjustable [28,35], so the impact of the choice of electrostatic model may be small. The models also have a similarity as both models describe association based on the theory of Wertheim [36]. The implementation of the Wertheim theory [36] is slightly different between the two models and the differences of the association terms of the two models are highlighted but it is not expected that these differences will be very important after parameterization of the models.

The objective of this study is two fold, one is to benchmark the models against the same data set for various choices of adjustable parameters and permittivity models, and the other aspect is to try to gain insight into how these two models function and what makes them different. This will be done by analysis of what the impact of different adjustable parameters has on some important properties as MIAC, osmotic coefficients and density. The analysis of the models will also include the impact of the terms of the models on the activity coefficients, the prediction of individual ion activity coefficients and extrapolation to high pressures for the density. This will contribute to the understanding of the importance of the various parameters and terms of the models.

The manuscript is organized as follows: In Section 2 the methods and theory of all of the models, choices of adjustable parameters and the parameter estimation will be covered. The permittivity models are described in detail in Appendix A. Section 3 is a results section, and it includes the overall results of the various parameterizations in terms of the optimized parameters and average deviations for MIAC, osmotic coefficients and density. The results section also includes the description of the relative static permittivity for all models. Section 4 is a discussion section that considers several aspects of this work, including comparisons of parameter trends, comparisons across different parameterizations evaluation of the contributions of terms to MIAC and extrapolation to other properties/conditions. The broader implications of this work are discussed by considering the qualitative and quantitative performance of various model choices; the EoS, the permittivity model and the adjustable parameters. Finally, it is discussed how to parameterize a new salt based on the investigation of this work. Section 5 sums up the main conclusions from this work.

## 2. Methods and theory

In the subsequent sections the key equations of the models that are used in this work are shown, highlighting the potential adjustable parameters. First e-CPA is described, then e-PPC-SAFT and finally the similar terms are presented together. The various models of the relative static permittivity are described in Appendix A. Finally, details of the optimization are covered including what sets of parameters that are considered adjustable for the comparisons of this work.

### 2.1. The e-CPA model

The electrolyte Cubic Plus Association (e-CPA) version used in this work has also been used in various other studies from our group at DTU [10,29,30,32]. The difference with the version used in this work is that the model for the relative static permittivity is varied. The permittivity models considered in this work will be described in Section 2.4. The version of e-CPA that is used in this work consists of 4 contributions, expressed as the Helmholtz energy:

$$A^{\text{res}} = A^{\text{SRK}} + A^{\text{Assoc}} + A^{\text{DH}} + A^{\text{Born}} \quad (1)$$

where SRK is short for Soave–Redlich–Kwong [37], which is a cubic EoS that describes the short range interactions, Assoc is association calculated using the theory of Wertheim [36], which describes hydrogen bonding, DH is the full Debye–Hückel equation [38] that describes the long-range electrostatic interactions between ions and the Born term [39] describes the solvation energy of ions. Full details of e-CPA are provided in previous works [10,29,32] and only key parts will be repeated here.

When no ions are present, e-CPA reduces to the classical CPA EoS, because the DH and Born terms are zero for uncharged species. Previously published water parameters of CPA can therefore be used, and none of the pure water parameters are re-estimated as a part of this work. The water parameters for e-CPA are shown in Table 1.

#### 2.1.1. Huron-Vidal mixing rule

The Huron-Vidal infinite pressure mixing rule is used for the SRK parameters and it is given by:

$$\frac{a}{b} = \sum_{i} x_{i} \frac{a_{i}}{b_{i}} - \frac{g^{E,\infty}}{\ln 2} \quad (2)$$

$$b = \sum_{i} x_{i} b_{i} \quad (3)$$

where a is an energy parameter, b is the co-volume with subscript i referring to pure component parameters and $g^{E,\infty}$ is the excess Gibbs energy of the Huron-Vidal/NRTL mixing rule which is given by:

$$\frac{g^{E,\infty}}{RT} = \sum_{i} x_{i} \frac{\sum_{j} x_{j} b_{j} \exp\left(-\alpha_{ji} \frac{\Delta U_{ji}}{RT}\right) \frac{\Delta U_{ji}}{RT}}{\sum_{j} x_{j} b_{j} \exp\left(-\alpha_{ji} \frac{\Delta U_{ji}}{RT}\right)} \quad (4)$$

This expression is simplified by setting the non-randomness parameter of NRTL $\alpha_{ji}$ to zero in all cases, and the expression becomes:

$$\frac{g_{\text{ions}}^{E,\infty}}{RT} = \frac{1}{b} \sum_{i} \sum_{j} x_{i} x_{j} b_{j} \frac{\Delta U_{ji}}{RT} \quad (5)$$

The interaction parameter, $\Delta U_{ij}$, is only considered non-zero for water–ion interactions, and it is made salt dependent by using the same value for water–cation and water–anion. It is furthermore made temperature dependent using the expression given in Eq. (6), which has the shape of a quadratic polynomial.

$$\frac{\Delta U_{ij}}{R} = \frac{\Delta U_{ij}^{ref}}{R} + \alpha \left[ \left(1 - \frac{T}{T_{a}}\right)^{2} - \left(1 - \frac{T_{ref}}{T_{a}}\right)^{2} \right] \quad (6)$$

The three parameters that (when included) are adjustable are $\Delta U_{ij}^{ref}$, $T_a$ and $\alpha$. $T_{ref}$ refers to the reference temperature of $\Delta U_{ij}^{ref}$. In this work it is set to $T_{ref} = 298.15$ K.

#### 2.1.2. DH term

The Debye–Hückel equation has been given more attention regarding its theoretical foundation recently [41–43] and several of the assumptions are found to have a small impact for a salt like NaCl. The
<table><caption>**Table 1**
The water parameters of e-CPA used in this work.</caption><thead><tr><th>Component</th><th>T<sub>L</sub><br>(K)</th><th>b<br>(cm<sup>3</sup>/mol)</th><th>c<sub>1</sub></th><th>Γ = a/(Rb)<br>(K)</th><th>β<sub>AB</sub><br>(10<sup>3</sup>)</th><th>ε<sub>AB</sub><br>(K)</th><th>Association<br>scheme</th><th>Reference</th></tr></thead><tbody><tr><td>Water</td><td>657.13</td><td>14.52</td><td>0.6736</td><td>1017.3</td><td>69.2</td><td>2003.25</td><td>4C</td><td>[40]</td></tr></tbody></table>

<table><caption>**Table 2**
The water parameters of e-PPC-SAFT used in this work.</caption><thead><tr><th>Compound</th><th>σ<sub>H<sub>S</sub>,w</sub><br>(Å)</th><th>m<sub>w</sub></th><th>ε/k<sub>b</sub><br>(K)</th><th>λ</th><th>μ<sub>w</sub><br>(D)</th><th>ε<sub>AB</sub>/k<sub>b</sub><br>(K)</th><th>κ<sub>w</sub></th><th>Association<br>scheme</th><th>x<sub>p</sub></th><th>Reference</th></tr></thead><tbody><tr><td>Water</td><td>T-dep<sup>1</sup></td><td>1.02122</td><td>201.747</td><td>0.203</td><td>1.85</td><td>1813</td><td>0.044394</td><td>4C</td><td>0.276</td><td>[47]</td></tr></tbody></table>

Helmholtz energy of the Debye–Hückel term is given by Eq. (7) when the full DH equation is used.

$$A^{\text{DH}} = \frac{k_B T V}{4\pi N_A \sum_i n_i z_i^2} \sum_i n_i z_i^2 \chi_i \quad (7)$$

$\chi_i$ is a function given by:

$$\chi_i = \frac{1}{\sigma_i^3} \left[ \ln \left( 1 + \kappa \sigma_i \right) - \kappa \sigma_i + \frac{1}{2} \left( \kappa \sigma_i \right)^2 \right] \quad (8)$$

Here $\sigma_i$ is the distance of closest approach for the ions, which can be considered comparable to the diameter of the ion. In this work scaled values of the crystal lattice values of Pauling [44] are used for the ions. $\kappa$ is the inverse Debye screening length, that can be calculated from:

$$\kappa^2 = \frac{e^2}{k_B T} \frac{1}{\epsilon_r \epsilon_0} \sum_i \rho_i Z_i^2 \quad (9)$$

## 2.2. The e-PPC-SAFT model

The electrolyte Polar Perturbed Chain Statistical Associating Fluid Theory (e-PPC-SAFT) that is used in this work is the e-EoS published by the group of IFPEN [27,33,34,45], where the model for the relative static permittivity will be varied. This e-EoS consists of the terms shown in Eq. (10) given in terms of the Helmholtz energy.

$$A^{\text{res}} = (m A^{\text{HS}} + A^{\text{chain}}) + A^{\text{disp}} + A^{\text{assoc}} + A^{\text{polar}} + A^{\text{MSA}} + A^{\text{Born}} \quad (10)$$

The first four terms are the same as in PC-SAFT [46] and they are the hard-sphere term (HS), chain term, dispersion term (disp) and the association term (assoc). The hard-sphere and chain terms are sometimes combined into a hard-chain term (hc). The polar term takes the polar interactions into account and is, in this work, only used for water. The last two terms are the electrolyte terms, where MSA is the Mean Spherical Approximation, which describes the long range interactions of the ions and the Born term [39] describes the energy of solvation associated with the ions. For full details of the model equations, readers are referred to the work by Roa Pinto et al. [27]. The hard sphere diameter is important for several of the terms of the model and it is temperature dependent based on the methodology of the PC-SAFT version of Gross and Sadowski [46]:

$$d(T) = \sigma_{HS} \left(1 - \lambda \exp\left(-3 \frac{\epsilon}{kT}\right)\right) \quad (11)$$

where $\epsilon$ is the dispersion energy and $\lambda$ is the softness factor, which is usually set to 0.12 and that value is used the ions in this work and value for water is given in Table 2 together with all the water parameters of e-PPC-SAFT. The water parameters are not optimized as part of this work. The water parameters were regressed by Ahmed et al. [47] with the goal to better describe the temperature dependency of the density and vapor pressure together with good agreement for LLE for water-hydrocarbon and oxygenates. The hard sphere diameter of water was given an additional temperature dependence (on top of Eq. (11)) to obtain better results in a wide temperature range including close to the freezing point of water where the density reaches a maximum.

### 2.2.1. MSA term

The Mean Spherical Approximation (MSA) term that is used in e-PPC-SAFT in this work is the unrestricted simplified MSA version that is given by:

$$\frac{A^{\text{MSA}}}{RT} = -\frac{N_A e^2}{4\pi\epsilon_0\epsilon_r RT} \sum_{i}^{\text{ions}} \frac{n_i Z_i^2 \Gamma}{1 + \Gamma \sigma_i^{\text{MSA}}} + \frac{V \Gamma^3}{3\pi} kT \quad (12)$$

Where $\sigma^{MSA}$ is the ion diameter for the MSA term, and $\Gamma$ is the shielding parameter calculated from:

$$4\Gamma^2 = \frac{N_A e^2}{\pi \epsilon_0 \epsilon_r RT} \sum_{i}^{\text{ions}} \frac{n_i}{V} \left( \frac{Z_i^2}{1 + \Gamma \sigma_i^{\text{MSA}}} \right) \quad (13)$$

## 2.3. For both models

### 2.3.1. Association terms

The association terms of both models are similar, because the Wertheim theory [36] is used to account for association with both models. This section will cover the association terms of both models where the small differences in the implementation will be highlighted. The Helmholtz energy from association is, for both models, given by:

$$\frac{A^{\text{Assoc}}}{RT} = \sum_{i} n_i \sum_{A \in i} \left( \ln X_{A_i} - \frac{1}{2} X_{A_i} + \frac{1}{2} \right) \quad (14)$$

where $X_{A_i}$ is the fraction of molecules of type i that is not bonded at site A, and $\Delta_{A_i B_j}$ is the association strength between site A on molecule i and site B on molecule j. The expressions for the association strength are slightly different between CPA and SAFT. In e-PPC-SAFT the association strength is given by:

$$\Delta_{A_i B_j}^{\text{SAFT}} = D_{ij}^3 \kappa_{A_i B_j} g_{ij}^{hs} \left[ \exp \left( \frac{\epsilon_{A_i B_j}}{kT} \right) - 1 \right] \quad (15)$$

where $D_{ij} = \frac{d_i + d_j}{2}$, where $d_i$ is the temperature dependent diameter (Eq. (11)) of component i, $\kappa_{A_i B_j}$ is the association volume, $g_{ij}^{hs}$ is the radial distribution function and $\epsilon_{A_i B_j}$ is the association energy. The corresponding expression for e-CPA is given by:

$$\Delta_{A_i B_j}^{\text{CPA}} = b_{ij} \beta_{A_i B_j} g(\rho) \left[ \exp\left(\frac{\epsilon_{A_i B_j}}{RT}\right) - 1 \right] \quad (16)$$

where $b_{ij} = \frac{b_i + b_j}{2}$ is the average co-volume of the two molecules, $\beta_{A_i B_j}$ is the association volume, $g(\rho)$ is the radial distribution function and $\epsilon_{A_i B_j}$ is the cross association energy.

Regardless of the EoS, Na+ has 7 positive sites and Cl- has 6 negative sites. It has been discussed by Rozmus et al. [45] that this is a reasonable estimate for the amount of sites for the ions. For both models there are two association parameters, the association energy ($\epsilon_{A_i B_j}$) and the association volume ($\kappa_{A_i B_j}$ or $\beta_{A_i B_j}$). The association parameters are treated in the same way for both models, when association is included. The association volume of the ions is set equal to the value of water for

<footnote n="1">¹The hard sphere diameter of water is temperature dependent with the expression ($\sigma_{HS}$ in Å, T in K): $\sigma_{HS}(T) = 2.2423 + 0.51212 \cdot \exp(0.001126 \cdot T) + \frac{9904.13}{T^2}$.</footnote>
each of the models given in Tables 1 and 2 for e-CPA and e-PPC-SAFT respectively, and the cross association volume will therefore also be equal to the value of water in all cases. When the association energy is considered to be an adjustable parameter, it is defined for the ions and the CR1 combining rule is used to calculate the cross association energy:

$$ \epsilon_{A,B_j} = \frac{\epsilon_i^{AB} + \epsilon_j^{AB}}{2} (1 - w_{ij}) \qquad (17) $$

where $w_{ij}$ is an interaction parameter and $\epsilon_i^{AB}$ is the association energy of component i. The difference in notation between cross-association ($\epsilon_{A,B_j}$) and component specific association ($\epsilon_i^{AB}$) energies is to clearly distinguish between them. In this work $w_{ij}$ for ion-water interactions is always set to zero, and when $w_{ij}$ is included as an adjustable parameter it is between cation and anion.

Both models include some form of average volume (diameter cubed in SAFT or the co-volume in CPA). There are two differences, that could have an impact when parameterization is included. The first difference is that the $D_{ij}$ parameter of e-PPC-SAFT is temperature dependent, while the corresponding parameter of e-CPA, the co-volume (b), is temperature independent. The difference between the models due to this will likely be small, because the change in temperature dependent diameter between 273 K and 473 K is only a few percent, so the impact of the temperature dependency will be small.

The second difference is that the expression for the radial distribution function is different in the two implementations. In the e-PPC-SAFT implementation the full expression of the hard sphere radial distribution function is used, which is given by:

$$ g_{ij}^{hs} = \frac{1}{1 - \zeta_3} + 3 \frac{d_{ij} \zeta_2}{(1 - \zeta_3)^2} + 2 \frac{(d_{ij} \zeta_2)^2}{(1 - \zeta_3)^3} \qquad (18) $$

$$ \zeta_k = \frac{\pi N_A}{6\nu} \left[ \sum_{i=1}^{n_{comb}} x_i m_i (d_{ii})^k \right] \qquad (19) $$

Whereas the expression used in e-CPA is a simplified expression given by:

$$ g(\rho) = \frac{1}{1 - 1.9\eta}, \qquad \eta = \frac{b}{4\nu} = \zeta_3 \qquad (20) $$

This simplified expression, first proposed by Elliot et al. [48], has been shown by Kontogeorgis et al. [40] to provide similar results to the full expression. Overall, the differences between the association terms of the two models are not likely to have a great effect on the overall performance of the models as the differences are relatively minor. This should especially be the case when parameters are optimized.

### 2.3.2. Born term

The Born term is the same for both models and it is based on the derivation of the energy from moving a charged ion from vacuum to a solvent with a given relative static permittivity by Born [39]. The Helmholtz energy of the Born term is given by:

$$ A^{\text{Born}} = \frac{N_A e^2}{4\pi\epsilon_0} \sum_i \frac{n_i z_i^2}{\sigma_{\text{Born},i}} \left(1 - \frac{1}{\epsilon_r}\right) \qquad (21) $$

In this equation, $\sigma_{\text{Born},i}$ is the diameter of the Born cavity, which will be called the Born diameter. When the Born diameter is included as an adjustable parameter the same value is used for both ions. When the Born diameters are not considered adjustable they are set to values that have been used in previous works [10,26,27,29,30,32]. These values are optimized to the energy of hydration at infinite dilution and are given in Table 3 together with the average value, which shows that the average Born diameter of e-PPC-SAFT is slightly bigger. The Born diameters for e-PPC-SAFT were fitted to Eq. (21), while the Born diameters for e-CPA were fitted to the full e-EoS where Eq. (21) is the most dominant contribution. The work of Schlaikjer et al. [29] shows that refitting of the e-CPA parameters (without adjusting the Born diameter) only results in a minor change to the calculated hydration energy.

### 2.4. Permittivity models

In this work several models for the relative static permittivity are tested for both e-CPA and e-PPC-SAFT. In Table 4 an overview of all models investigated as part of this work is shown. Details of most of the permittivity models have been presented in previous works [26,27] and the key equations and models parameters of each of the models are presented in Appendix A. A comparison of how the permittivity models fit the data is shown in Section 3.7, and Fig. 3 of that section provides a general overview. The choice of models allow for detailed tests and comparisons. The Zuo and Furst model [49] is only dependent on the temperature so the relative static permittivity will have the value of the pure solvent regardless of the salt concentration. The MMB (Mollerup-Michelsen-Breil) solvent model [50] and the Schreckenberg model [51] are both developed to describe the relative static permittivity of the solvent based on the temperature and volume dependency. The volume dependency also makes the models express an apparent salt concentration dependency. This is because the volume change from the addition of salt will lead to a change in the calculated relative static permittivity. The MMB solvent model has previously been used for e-CPA [26] and the Schreckenberg model has been used with e-PPC-SAFT [27,33,34].

The Pottel model [52] predicts a decrease of the permittivity that is slightly larger than the solvent models (MMB solvent and Schreckenberg) due to a dependence that is related to the ion size. The Simonin [53] and MMB (Mollerup-Michelsen-Breil) ion [50] models are both developed to accurately describe the experimental relative static permittivity.

The Maribo-Mogensen models [54,55] are dependent on the parameterization strategy. This is because they rely on the association network to describe the permittivity of the mixture. When the ions are not considered associating species the model predicts the permittivity of the solution to be almost independent of the salt concentration (with accurate density/volume as input) [32]. In this case the model can best be compared to the model of Zuo and Furst. However, when the ions are associating the water-water association network will be affected, which leads to a decrease of the relative static permittivity with increasing ion concentration due to a reduced amount of water-water hydrogen bonds. The decrease of the permittivity will be smaller when the solvent version of the model is used, but both the solvent and the ion versions of the model are heavily dependent on the association parameters of the ions.

### 2.5. Permittivity models for pure water

Several of the permittivity models, presented in Table 4, are developed to describe the relative static permittivity of non-electrolyte solvents. The solvent permittivity models in this work are Zuo and Furst, Maribo-Mogensen, Schreckenberg and MMB (Mollerup-Michelsen-Breil). In Fig. 1 the temperature and pressure dependencies of the relative static permittivity of pure water are shown for these permittivity models. The results with the Maribo-Mogensen model and the Mollerup-Michelsen-Breil model were shown in a previous work [26], and these results are repeated here for clarity. Looking at the temperature dependency of the models (Fig. 1(a)) it can be seen that the overall agreement with the experimental data is quite similar, especially at lower temperatures. The largest deviations are observed at higher temperatures, where the Schreckenberg model slightly overestimates the relative static permittivity and the Zuo and Furst model is under predicting it. Regarding the Zuo and Furst model it is noted that it is very accurate in the temperature range that it was fitted to (288.15–403.15 K) and larger deviations are found in the extrapolated range. Overall, it is considered that the differences of the temperature dependency of the water models are small enough that they are not expected to significantly affect the results after fitting parameters of the rest of the model.
<caption>**Table 3**</caption>

The Born diameter that are used for e-CPA and e-PPC-SAFT when the Born diameter is not adjustable. These Born diameters have been optimized to hydration energy for both models in the given reference. Avg. is short for average, and is the arithmetic average value of the two ions.

<table><caption>**Table 4**</caption><thead><tr><th rowspan="2">Model</th><th rowspan="2">Fitted property</th><th colspan="3">Born diameter</th><th rowspan="2">Reference</th></tr><tr><th>Na+(Å)</th><th>Cl−(Å)</th><th>Avg. (Å)</th></tr></thead><tbody><tr><td>e-CPA</td><td>Δ<sub>hyd</sub> H</td><td>3.331</td><td>3.657</td><td>3.494</td><td>[10]</td></tr><tr><td>e-PPC-SAFT</td><td>Δ<sub>hyd</sub> G</td><td>3.236</td><td>4.513</td><td>3.875</td><td>[27]</td></tr></tbody></table>

Overview of the various models of the relative static permittivity, that are tested in this work for the models e-CPA and e-PPC-SAFT.

<table><thead><tr><th>Model</th><th>Dependencies</th><th>e-CPA</th><th>e-PPC-SAFT</th><th>Reference</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>T</td><td>x</td><td>x</td><td>[49]</td></tr><tr><td>Maribo-Mogensen solvent</td><td>T, V</td><td>x</td><td></td><td>[54]</td></tr><tr><td>MMB¹ solvent</td><td>T, V</td><td>x</td><td></td><td>[50]</td></tr><tr><td>Schreckenberg</td><td>T, V</td><td></td><td>x</td><td>[51]</td></tr><tr><td>Pottel</td><td>ε<sub>r,sw</sub>(T, V)<sup>2</sup>, n<sub>ion</sub>, V</td><td>x</td><td>x</td><td>[52]</td></tr><tr><td>Simonin</td><td>ε<sub>r,sw</sub>(T, V)<sup>2</sup>, n</td><td>x</td><td>x</td><td>[53]</td></tr><tr><td>MMB¹ ion</td><td>ε<sub>r,sw</sub>(T, V)<sup>2</sup>, n<sub>ion</sub>, V</td><td>x</td><td></td><td>[50]</td></tr><tr><td>Maribo-Mogensen ion</td><td>T, V, n</td><td>x</td><td></td><td>[55]</td></tr></tbody></table>

¹ MMB is short for Mollerup–Michelsen–Breil and refers to a solvent (Appendix A.3) or electrolyte (Appendix A.8) permittivity model.

²ε<sub>r,sw</sub> refers to the fact that these models require an input of the relative static permittivity of the solvent. For e-CPA the solvent model used is the Michelsen–Mollerup–Breil solvent model [50] Appendix A.3 and for e-PPC-SAFT the solvent model is from Schreckenberg et al. [51] Appendix A.2. Both solvent models are dependent on T and V.

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

**Fig. 1.** The temperature (left) and pressure (right) dependency of the relative static permittivity of water with the permittivity models (defined in Table 4) used in this work. For the T-dependency (left) the pressure is atmospheric below 373 K and the saturation pressure above 373 K. For the P-dependency (right) the temperature is 293.15 K. With Maribo-Mogensen and MMB models the volume is calculated with e-CPA water parameters, and for Schreckenberg’s model the volume is calculated with the e-PPC-SAFT water parameters. All experimental data are from [56].

Regarding the pressure dependency of the water models (Fig. 1(b)), it can be seen that there is a larger difference between the models. The pressure dependency is, for all models, due to the volume dependency of the models since none of the models include an explicit pressure dependency. Because the Zuo and Furst model is only dependent on the temperature it is constant regardless of the pressure. It can also be seen that the most accurate model for the pressure dependency is the Maribo-Mogensen model. The MMB model is less accurate at lower pressures but becomes more accurate as the pressure increases. The Schreckenberg model is accurate at lower pressures, but it does not follow the pressure trend and under predicts the relative static permittivity more as the pressure increases. It should also be noted that the dependency of the relative static permittivity with regards to the pressure is much smaller than the dependency on the temperature, which also makes the differences appear larger.

## 2.6. Contribution of terms to MIAC

Analysis of the contributions from the terms of a model to a property (mostly to MIAC) has been performed by various groups for different

models [27,31,33,57,58] and it is a tool to understand how the models function regarding what terms are important. The contributions to the various terms to MIAC will also be considered in this work.

Due to the additive nature of the various terms for the Helmholtz energy it is possible to calculate the size of the individual terms. This is shown in Eq. (22) for e-CPA.

$$ \ln \gamma_i = \left( \frac{\partial \Delta F^{SRK}}{\partial n_i} \right)_{V,T,n_j} + \left( \frac{\partial \Delta F^{Assoc}}{\partial n_i} \right)_{V,T,n_j} + \left( \frac{\partial \Delta F^{DH}}{\partial n_i} \right)_{V,T,n_j,\epsilon_r} + \left( \frac{\partial \Delta F^{Born}}{\partial n_i} \right)_{V,T,n_j,\epsilon_r} + \left( \frac{\partial \Delta F^{DH}}{\partial \epsilon_r} \right)_{V,T,n_j,n_i} \left( \frac{\partial \epsilon_r}{\partial n_i} \right)_{V,T,n_j} + \left( \frac{\partial \Delta F^{Born}}{\partial \epsilon_r} \right)_{V,T,n_j,n_i} \left( \frac{\partial \epsilon_r}{\partial n_i} \right)_{V,T,n_j} - \ln \left( \frac{Z}{Z_0} \right) \quad (22) $$

Where ΔF is the difference of the reduced Helmholtz energy at the composition compared to a reference state which is the pure solvent. This is expressed by:

$$ \Delta F = \left( \frac{A}{RT} \right)_{V,T,n} - \left( \frac{A}{RT} \right)_{V_0,T,n_0} \quad (23) $$
<table><caption>**Table 5**
A list of all the comparable parameters for the ions of the e-CPA and e-PPC-SAFT models.</caption><thead><tr><th>e-CPA</th><th>e-PPC-SAFT</th><th>Description</th></tr></thead><tbody><tr><td>b</td><td>σ<sup>HS</sup></td><td>Physical size</td></tr><tr><td>a</td><td>ε<sub>i</sub></td><td>Attractive/Dispersion energy</td></tr><tr><td>ΔU<sub>ij</sub></td><td>k<sub>ij</sub></td><td>Water-ion interaction parameter for physical term</td></tr><tr><td>ε<sub>i</sub><sup>AB</sup></td><td>ε<sup>AB</sup></td><td>Association energy of ions</td></tr><tr><td>β<sub>i</sub><sup>AB</sup></td><td>κ<sub>i</sub><sup>AB</sup></td><td>Association volume of ions</td></tr><tr><td>w<sub>ij</sub></td><td>w<sub>ij</sub></td><td>Cation–Anion interaction parameter for association energy</td></tr><tr><td>σ<sup>DH</sup></td><td>σ<sup>MSA</sup></td><td>Electrostatic diameter</td></tr><tr><td>σ<sup>Born</sup></td><td>σ<sup>Born</sup></td><td>Born/solvated diameter</td></tr></tbody></table>

The contributions of the individual terms to the activity coefficients are here exemplified with the SRK term:

$$ \ln \gamma_i^{SRK} = \left( \frac{\partial \Delta F^{SRK}}{\partial n_i} \right)_{V,T,n_j} \quad (24) $$

The procedure is the same for all terms. An equivalent analysis can be made for e-PPC-SAFT. The names for e-PPC-SAFT match the names given in Eq. (10). For e-PPC-SAFT the hard-sphere, chain and dispersion terms are combined to a single term that considers all the short range physical interactions. This “physical” term as it will be called, should be comparable to the SRK term of e-CPA. For e-CPA the impact of the derivatives of the concentration dependency of the permittivity is isolated by including a permittivity term (called Perm in later figures). It is the terms $\left(\frac{\partial \Delta F^{DH}}{\partial \epsilon_r}\right)_{V,T,n_j,n_i} \left(\frac{\partial \epsilon_r}{\partial n_i}\right)_{V,T,n_j} + \left(\frac{\partial \Delta F^{Born}}{\partial \epsilon_r}\right)_{V,T,n_j,n_i}$ of Eq. (22). This is done to follow the methodology of previous works with e-CPA [26,31,32].

## 2.7. Parameter comparison of e-CPA and e-PPC-SAFT

To have a fair comparison between two different EoS models it is necessary to parameterize both models in a similar fashion. To do this some key parameters of the two models are assumed to be “equal”, when they are used as adjustable parameters. Comparisons are performed between parameter sets where both models have the same set of “equal” adjustable parameters. What is meant by equal is not that the parameters should have the same value but instead that the parameters should describe the same physical phenomena. Because of this, when the parameters are optimized it is the assumption that they will have the same effect. It is of course two different models so while the parameters might correspond to the same physical phenomena, it is likely that the impact will be different, and this will likely be more apparent when more parameters are adjusted at the same time. This is why it is important to test several different parameterizations to make sure that one model is not favored because the selection of parameters are better for that model.

In Table 5 it is shown which parameters of the two models that are considered comparable. In the next sections additional details of the parameterization of the two models are given.

## 2.8. e-PPC-SAFT

In this section some additional details of the parameterization of the e-PPC-SAFT EoS are given. The hard sphere diameter is adjusted to maintain a realistic relative difference between the ion size of the two ions by scaling the Pauling values as follows:

$$ \sigma^{HS} = \lambda^{HS} \sigma^{Pauling} \quad (25) $$

where $\lambda^{HS}$ is the scaling factor, which is adjusted and is the same for both ions. The MSA diameter is scaled from the hard-sphere diameter by:

$$ \sigma^{MSA} = \lambda^{MSA} \sigma^{HS} \quad (26) $$

where $\lambda^{MSA}$ is an adjustable parameter. In this way the HS and the MSA diameters are estimated independently from each other, but the relative size between the ions is the same and at a reasonable value. When the MSA diameter is not adjusted it is set to the hard sphere diameter ($\lambda^{MSA} = 1$).

When the dispersion energy is adjusted the same value is used for both ions. When the interaction parameter ($k_{ij}$) is optimized without optimizing the dispersion energy, the value of the dispersion energy is set equal to the value for water. The interaction parameter that is optimized is the water-ion interaction parameter, and the same value is used for both ions. Like ion dispersion is neglected in all cases ($k_{aa} = k_{cc} = 1$). When the interaction parameter is temperature dependent it is calculated by a quadratic expression like for e-CPA (Eq. (6)):

$$ k_{ij} = k_{ij}^{(0)} + k_{ij}^{(1)} \cdot (T - T_{ref}) + k_{ij}^{(2)} \cdot (T - T_{ref})^2 \quad (27) $$

where $T_{ref} = 298.15$ K, like for e-CPA and $k_{ij}^{(0)}$ correspond to the value of the interaction parameter at 298.15 K.

## 2.9. e-CPA

The e-CPA EoS is parameterized in a similar way as e-PPC-SAFT. The ion diameter in the DH term is estimated as a scaled value of the Pauling diameter similarly to e-PPC-SAFT:

$$ \sigma^{DH} = \lambda^{DH} \sigma^{Pauling} \quad (28) $$

The co-volume has, in many works with e-Cubics/CPA EoS's [10,29, 30,32,58,59], been calculated from the ion diameter in the electrostatic term (DH/MSA) or the other way around. In this work the co-volume is allowed to be independent from the ion diameter with the following expression:

$$ b_i = \lambda^b N_A \pi \sigma_i^3 \quad (29) $$

where $\lambda^b$ is a scaling factor for both ions. When it is not adjusted $\lambda^b = 1/6$, which makes the co-volume equal to the volume of the sphere. In the parameterization either one or both of the scaling factors are included. All the parameterizations of this work include the $\lambda^{DH}$ for e-CPA and $\lambda^{HS}$ for e-PPC-SAFT as adjustable parameters. The two models are treated equally, as constant $\lambda^b$ and $\lambda^{MSA}$ will lead to both models estimating the physical and electrostatic sizes with one parameter. When $\lambda^b$ and $\lambda^{MSA}$ are adjustable, the physical and electrostatic sizes are independent of each other for both models.

## 2.10. Investigated scenarios

Several different sets of adjustable parameters are compared in this work and a given set of adjustable parameters is referred to as a scenario. An overview of the scenarios that are considered in this work is shown in Table 6. Scenarios 1 and 2 consider dispersion forces for the ions and no association. Scenarios 3 and 4 consider the ions as associating but disregards the dispersion forces. Finally, scenarios 5 and 6 consider both dispersion and association for the ions.

The choices of adjustable parameters allow for a thorough comparison of the two models, as it is possible to investigate the effect of including various forces for the ions in the model. Having two sets of parameters for each type of “force” (only dispersion, only association or both), can show the impact of adding an additional parameter. Another important aspect of this choice of parameters is what type of forces should be considered by the models can be evaluated. This is because scenarios 2, 4 and 5 all have 5 adjustable parameters, so the performance of the models will not be decided due to a differing
<table><caption>**Table 6**
List of the adjustable parameters of the e-CPA and e-PPC-SAFT models for each of the scenarios of this work. Disp is short for dispersion and Assoc is short for association.</caption><thead><tr><th>Scenario</th><th>e-CPA</th><th>e-PPC-SAFT</th><th>Adjustable parameters</th></tr></thead><tbody><tr><td>1 (Disp)</td><td>λ<sup>DH</sup>, U<sub>wi</sub>(T)</td><td>λ<sup>HS</sup>, k<sub>wi</sub>(T)</td><td>4</td></tr><tr><td>2 (Disp)</td><td>λ<sup>DH</sup>, U<sub>wi</sub>(T), σ<sub>Born</sub></td><td>λ<sup>HS</sup>, k<sub>wi</sub>(T), σ<sub>Born</sub></td><td>5</td></tr><tr><td>3 (Assoc)</td><td>λ<sup>DH</sup>, λ<sup>b</sup>, ε<sup>AB</sup>, w<sub>ca</sub></td><td>λ<sup>HS</sup>, λ<sup>MSA</sup>, ε<sup>AB</sup>, w<sub>ca</sub></td><td>4</td></tr><tr><td>4 (Assoc)</td><td>λ<sup>DH</sup>, λ<sup>b</sup>, ε<sup>AB</sup>, w<sub>ca</sub>, σ<sub>Born</sub></td><td>λ<sup>HS</sup>, λ<sup>MSA</sup>, ε<sup>AB</sup>, w<sub>ca</sub>, σ<sub>Born</sub></td><td>5</td></tr><tr><td>5 (Disp+Assoc)</td><td>λ<sup>DH</sup>, ε<sup>AB</sup>, ΔU<sub>wi</sub>(T)</td><td>λ<sup>HS</sup>, ε<sup>AB</sup>, k<sub>wi</sub>(T)</td><td>5</td></tr><tr><td>6 (Disp+Assoc)</td><td>λ<sup>DH</sup>, λ<sup>b</sup>, ε<sup>AB</sup>, ΔU<sub>wi</sub>(T)</td><td>λ<sup>HS</sup>, λ<sup>MSA</sup>, ε<sup>AB</sup>, k<sub>wi</sub>(T)</td><td>6</td></tr></tbody></table>

number of parameters, which is often the case when additional physics is added to models, as new terms often require additional parameters.

An additional reason for thoroughly investigating the effect of various forces that ions should have in the model, is that the choices in recent publications with e-CPA and e-PPC-SAFT have been diverging. Good results have been obtained with e-CPA [10,29,30,32] without considering the ions as associating species. On the other hand with e-PPC-SAFT good results have been shown while completely neglecting the dispersion of ions, and only considering the association of ions [33, 34,45]. This discrepancy in how the models have been parameterized calls for a comprehensive study on which method that is actually the best.

Some work on evaluating which forces that should be considered has been done for e-PPC-SAFT in a recent study by Roa Pinto et al. [27]. Here results are shown with various parameter sets that only consider either dispersion or association and not both, and the approaches are compared for the e-PPC-SAFT model. The choice of parameters and properties in the objective function is different from the current work, so different information can be extracted.

## 2.11. Parameter estimation

The data set, used for optimization of the parameters of both EoS's, have previously been used for e-CPA [26,32]. A complete overview of the experimental data is provided in a previous work [32]. The data include the properties MIAC, osmotic coefficients and density. For all properties the temperature range is 273.15–473.15 K. The pressure is either atmospheric or equal to the water vapor pressure for temperatures above 373.15 K to make sure that a liquid phase will be present. The concentration range is 0–6 mol/kg for MIAC and density and 0–8 mol/kg for the osmotic coefficients. 6 mol/kg is close to the solubility limit of NaCl in water at 298.15 K and 8 mol/kg is roughly the limit at 473.15 K. The extended solubility range of the osmotic coefficients is due to the availability of data. The objective function (OF), that is minimized to optimize the parameters of e-CPA and e-PPC-SAFT for the various scenarios, is given by:

$$OF = \frac{1}{N_{\gamma_{\pm}}} \sum_{i} \left( \frac{\gamma_{\pm,i}^{\text{exp}} - \gamma_{\pm,i}^{\text{calc}}}{\gamma_{\pm,i}^{\text{exp}}} \right)^2 + \frac{1}{N_{\phi}} \sum_{i} \left( \frac{\Phi_{i}^{\text{exp}} - \Phi_{i}^{\text{calc}}}{\Phi_{i}^{\text{exp}}} \right)^2 + \frac{1}{N_{\rho}} \sum_{i} \left( \frac{\rho_{i}^{\text{exp}} - \rho_{i}^{\text{calc}}}{\rho_{i}^{\text{exp}}} \right)^2 \quad (30)$$

where N is the number of data points for the property that is indicated by its subscript, γ± is MIAC, Φ is osmotic coefficients and ρ is density. The superscripts exp and calc refers to experimental and calculated values respectively.

## 3. Results

In the next sections the results of each of the investigated scenarios are presented individually (Sections 3.1 to 3.6). The models are evaluated for their quantitative agreement with experimental data, which is quantified with relative absolute deviation (RAD) given by:

$$RAD = \frac{1}{N} \sum_{i} \frac{|X_{i}^{\text{exp}} - X_{i}^{\text{calc}}|}{X_{i}^{\text{exp}}} \cdot 100\% \quad (31)$$

where X is a property. In the deviation tables of the following sections the permittivity models are listed in order from the least sensitive to salt content (i.e. Zuo and Furst) to the most sensitive (i.e. Simonin and MMB ion). The Maribo-Mogensen ion model is listed at the end even though it varies significantly between the scenarios because it is not included in all scenarios due to it being the same as the Maribo-Mogensen solvent model when the ions are not associating (scenarios 1 and 2). Section 3.7 shows the results of the predicted relative static permittivity of the various permittivity models. For the permittivity models that directly depend on one of the potentially adjustable parameters (Pottel and Maribo-Mogensen models) results are shown for all scenarios.

### 3.1. Scenario 1 (Dispersion)

In this scenario, 4 parameters are adjusted and only dispersion forces are considered for the ions so the ions are considered as non-associating. The parameters for e-CPA are λDH and 3 parameters for the temperature dependent ΔU. The corresponding parameters for e-PPC-SAFT are λHS and 3 parameters for the temperature dependent kij. There is not a difference between adjusting the hard sphere and the MSA diameters when they are connected (in this case the same) like the co-volume of e-CPA is calculated from the ionic diameter in the DH term. This choice of adjustable parameters is based on a previous work with e-CPA (set 4 from [32]) and since the results looked promising the methodology is tested for all the RSP models with both EoS's.

In Table 7 the deviations of the various models are shown for each of the models. The results of e-CPA are different from the previous work [32], because in that work the ion size was scaled from Marcus values [60] instead of Pauling values [44] that are used in this work.

It can be seen that the best performing models in this case are found with e-CPA with the most accurate model being the Maribo-Mogensen solvent model with e-CPA. The best performing model for e-PPC-SAFT is the Pottel model, but its performance is worse than the best e-CPA models. Especially the density is described worse with e-PPC-SAFT for all similar description of the RSP. For the results with Zuo and Furst it can be seen that e-CPA performs better for all properties. The same is observed for the results for the MMB solvent model with e-CPA and Schreckenberg with e-PPC-SAFT where e-CPA performs better for all properties. With Pottel's model, the results are overall similar with e-CPA performing better for density and e-PPC-SAFT performing better for MIAC and osmotic coefficients. Regardless if the physical model is e-CPA or e-PPC-SAFT the performance for the three properties considered in Table 7 gets significantly worse when the permittivity model captures the experimental RSP (Simonin and MMB ion models).

The optimized parameters of e-CPA and e-PPC-SAFT are given in Tables 8 and 9 respectively. For e-CPA (Table 8) it can be seen that the interaction parameter at 298.15 K (U<sub>ref</sub>/R) becomes more positive as the permittivity model gets closer to capturing the experimental concentration dependency of the permittivity. This illustrates that the permittivity model has an significant impact on the model, and that the parameters needs to be adapted according to the different permittivity models. Regarding the parameters of e-PPC-SAFT (Table 9) it can be seen that the hard sphere diameter decreases as permittivity model gets closer to the experimental data (Pottel<Schreckenberg<Zuo and Furst) when disregarding the Simonin results since the performance with Simonin is a lot worse (see Table 7). For the interaction parameter
<caption>**Table 7**</caption>

The deviations (RAD as defined in Eq. (31)) for three properties for the water-NaCl system in the temperature range of 273.15-473.15 K with the different permittivity models (defined in Table 4) of e-CPA and e-PPC-SAFT and the optimized parameters for scenario 1 given in Tables 8 and 9.

<table><caption>**Table 8**</caption><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="3">e-CPA</th><th colspan="3">e-PPC-SAFT</th></tr><tr><th>MIAC</th><th>Osmotic coef.</th><th>Density</th><th>MIAC</th><th>Osmotic coef.</th><th>Density</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>3.23</td><td>1.88</td><td>0.53</td><td>3.51</td><td>2.07</td><td>1.97</td></tr><tr><td>Maribo-Mogensen</td><td><b>2.30</b></td><td><b>1.80</b></td><td><b>0.33</b></td><td></td><td></td><td></td></tr><tr><td>MMB solvent</td><td>2.40</td><td>1.73</td><td>0.46</td><td></td><td></td><td></td></tr><tr><td>Schreckenberg</td><td></td><td></td><td></td><td>3.22</td><td>1.76</td><td>1.03</td></tr><tr><td>Pottel</td><td>3.12</td><td>1.77</td><td>0.59</td><td><b>2.98</b></td><td><b>1.62</b></td><td><b>0.98</b></td></tr><tr><td>Simonin</td><td>7.39</td><td>6.05</td><td>1.14</td><td>6.40</td><td>3.87</td><td>7.63</td></tr><tr><td>MMB ion</td><td>8.20</td><td>7.02</td><td>0.90</td><td></td><td></td><td></td></tr></tbody></table>

The optimized parameters for NaCl for e-CPA for scenario 1 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 9**</caption><thead><tr><th rowspan="2">Model</th><th colspan="3">Interaction parameter</th><th colspan="2">Scaling parameters</th><th colspan="2">Ion diameter, σ<sub>DH</sub></th><th colspan="2">Co-volume, b</th></tr><tr><th>U<sub>ref</sub>/R<br>(K)</th><th>T<sub>e</sub><br>(K)</th><th>α<br>(K)</th><th>λ<sup>DH</sup></th><th>λ<sup>b</sup></th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>Na+<br>(cm<sup>3</sup>/mol)</th><th>Cl−<br>(cm<sup>3</sup>/mol)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>-1245.9</td><td>2668.7</td><td>11523.2</td><td>0.979</td><td>1/6</td><td>1.909</td><td>3.523</td><td>2.19</td><td>13.79</td></tr><tr><td>Maribo-Mogensen solvent</td><td>-1022.7</td><td>429.8</td><td>5335.5</td><td>1.031</td><td>1/6</td><td>2.010</td><td>3.710</td><td>2.56</td><td>16.10</td></tr><tr><td>MMB solvent</td><td>-814.6</td><td>396.0</td><td>5052.8</td><td>0.996</td><td>1/6</td><td>1.941</td><td>3.584</td><td>2.31</td><td>14.52</td></tr><tr><td>Pottel</td><td>-588.0</td><td>391.2</td><td>4308.7</td><td>0.963</td><td>1/6</td><td>1.878</td><td>3.468</td><td>2.09</td><td>13.15</td></tr><tr><td>Simonin</td><td>972.2</td><td>344.0</td><td>3840.7</td><td>0.742</td><td>1/6</td><td>1.447</td><td>2.672</td><td>0.96</td><td>6.02</td></tr><tr><td>MMB ion</td><td>1195.3</td><td>436.3</td><td>3967.3</td><td>0.658</td><td>1/6</td><td>1.283</td><td>2.368</td><td>0.67</td><td>4.19</td></tr></tbody></table>

The optimized parameters for NaCl for e-PPC-SAFT for scenario 1 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="2">Scaling parameters</th><th colspan="2">HS diameter, σ<sub>HS</sub></th><th colspan="3">Interaction parameter</th></tr><tr><th>λ<sup>HS</sup><br>(Å)</th><th>λ<sup>MSA</sup><br>(Å)</th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>k<sup>(0)</sup><sub>ij</sub><br>(K<sup>-1</sup>)</th><th>k<sup>(1)</sup><sub>ij</sub><br>(K<sup>-1</sup>)</th><th>k<sup>(2)</sup><sub>ij</sub><br>(K<sup>-2</sup>)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>0.897</td><td>1.00</td><td>1.705</td><td>3.248</td><td>-0.5200</td><td>-3.578 · 10<sup>-3</sup></td><td>2.299 · 10<sup>-6</sup></td></tr><tr><td>Schreckenberg</td><td>0.865</td><td>1.00</td><td>1.644</td><td>3.133</td><td>-0.4783</td><td>-3.063 · 10<sup>-3</sup></td><td>6.954 · 10<sup>-7</sup></td></tr><tr><td>Pottel</td><td>0.856</td><td>1.00</td><td>1.626</td><td>3.097</td><td>-0.4338</td><td>-2.911 · 10<sup>-3</sup></td><td>1.957 · 10<sup>-6</sup></td></tr><tr><td>Simonin</td><td>1.050</td><td>1.00</td><td>1.996</td><td>3.802</td><td>0.2337</td><td>2.703 · 10<sup>-4</sup></td><td>4.359 · 10<sup>-6</sup></td></tr></tbody></table>

at 298.15 K ($k_{ij}^{(0)}$) an increasing trend towards more positive values are seen for a more significant concentration dependency of the permittivity model. Both e-PPC-SAFT and e-CPA therefore show an increasing trend with the reference interaction parameter with regards to the permittivity model.

## 3.2. Scenario 2 (Dispersion)

Scenario 2 considers the same four adjustable parameters as scenario 1, with the addition of one additional adjustable parameter, the Born diameter. This means that there are five adjustable parameters in this case and the ions are still considered as non-associating. This is equivalent to a method previously studied with e-CPA [26].

In Table 10 the deviations are shown for the various permittivity models of scenario 2. The results are quite comparable to the results of scenario 1, with slight improvement for all models. It is still the Maribo-Mogensen model with e-CPA that performs the best, while the best performing permittivity model for e-PPC-SAFT is still Pottel, since it has the smallest deviations for all three properties. Especially the density description makes it better than e-PPC-SAFT/Schreckenberg.

The Zuo and Furst parameterization results are the same as in scenario 1, because the additional adjustable parameter of the Born diameter does not impact any of the properties in the objective function. For the Maribo-Mogensen permittivity model with e-CPA the results are almost the same between scenarios 1 and 2, because the relative static permittivity is very close to the solvent value when the ions are not associating and the Born term will be very small in both cases, so the adjustment of the Born diameter has a small effect for this permittivity model. For e-CPA/Pottel the agreement with MIAC and density has improved but osmotic coefficients are described worse. It can also be seen that the parameters for e-CPA/Pottel are significantly different

(Table 11) for this model, because the Born diameter is very large, and it is also much larger than with Simonin and MMB ion models, which have a larger concentration dependency of the relative static permittivity and therefore a larger impact of the Born term. This is even though the Born diameters of Simonin and MMB ion are much larger than the average values given in Table 3, where the Born diameters are optimized to hydration energy. The Maribo-Mogensen and MMB solvent models both have Born diameters that are a bit smaller than the average value when optimized to hydration energy (Table 3). A large Born diameter makes the Born term very small for the property MIAC and the other terms of the model will therefore have to capture this property.

The parameters of the various permittivity models for e-PPC-SAFT are given in Table 12. It can be seen that the trend of the hard sphere diameter is still present and even stronger in this case, compared to the results of scenario 1 (Pottel<Schreckenberg<Zuo and Furst). It can also be seen that the Born diameters are much larger than the average value when the Born diameter is optimized to the hydration energy (Table 3). This means that the inclusion of the Born diameter leads to a smaller contribution from the Born term, which was also seen in a previous work for e-CPA [26]. For the interaction parameter at 298.15 K ($k_{ij}^{(0)}$) a decreasing trend towards more negative values are seen for a more significant concentration dependency of the permittivity model. This is the opposite trend compared to what was observed for scenario 1.

## 3.3. Scenario 3 (Association)

In this scenario the ions are considered as associating and the parameters are the two scaling factors, which means that the electrostatic and physical ion sizes are optimized independently of each other, together with the association energy of the ions and an interaction
<caption>**Table 10**</caption>

The deviations (RAD as defined in Eq. (31)) for three properties for the water-NaCl system in the temperature range of 273.15-473.15 K with the different permittivity models (defined in Table 4) of e-CPA and e-PPC-SAFT and the optimized parameters for scenario 2 given in Tables 11 and 12.

<table><caption>**Table 11**</caption><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="3">e-CPA</th><th colspan="3">e-PPC-SAFT</th></tr><tr><th>MIAC</th><th>Osmotic coef.</th><th>Density</th><th>MIAC</th><th>Osmotic coef.</th><th>Density</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>3.23</td><td>1.88</td><td>0.53</td><td>3.51</td><td>2.07</td><td>1.97</td></tr><tr><td>Maribo-Mogensen</td><td><strong>2.28</strong></td><td><strong>1.79</strong></td><td><strong>0.32</strong></td><td></td><td></td><td></td></tr><tr><td>MMB solvent</td><td>2.37</td><td>1.74</td><td>0.47</td><td></td><td></td><td></td></tr><tr><td>Schreckenberg</td><td></td><td></td><td></td><td>2.95</td><td>1.70</td><td>1.14</td></tr><tr><td>Pottel</td><td>2.52</td><td>2.09</td><td>0.36</td><td><strong>2.84</strong></td><td><strong>1.66</strong></td><td><strong>0.78</strong></td></tr><tr><td>Simonin</td><td>6.23</td><td>4.77</td><td>0.33</td><td>9.02</td><td>6.16</td><td>3.67</td></tr><tr><td>MMB ion</td><td>7.64</td><td>5.83</td><td>0.42</td><td></td><td></td><td></td></tr></tbody></table>

The optimized parameters for NaCl for e-CPA for scenario 2 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 12**</caption><thead><tr><th rowspan="2">Model</th><th colspan="3">Interaction parameter</th><th colspan="2">Scaling parameters</th><th colspan="2">Ion diameter, σ<sub>DH</sub></th><th colspan="2">Co-volume, b</th><th>Born diameter</th></tr><tr><th>U<sub>ref</sub>/R<br>(K)</th><th>T<sub>a</sub><br>(K)</th><th>α<br>(K)</th><th>λ<sup>DH</sup></th><th>λ<sup>b</sup></th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>Na+<br>(cm<sup>3</sup>/mol)</th><th>Cl−<br>(cm<sup>3</sup>/mol)</th><th>σ<sub>Born</sub><br>(Å)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>-1245.9</td><td>2668.7</td><td>11523.2</td><td>0.979</td><td>1/6</td><td>1.909</td><td>3.523</td><td>2.19</td><td>13.79</td><td>–</td></tr><tr><td>Maribo-Mogensen solvent</td><td>-1007.7</td><td>423.0</td><td>5429.9</td><td>1.034</td><td>1/6</td><td>2.016</td><td>3.721</td><td>2.58</td><td>16.25</td><td>3.36</td></tr><tr><td>MMB solvent</td><td>-707.8</td><td>372.9</td><td>5353.2</td><td>1.000</td><td>1/6</td><td>1.949</td><td>3.599</td><td>2.34</td><td>14.70</td><td>3.06</td></tr><tr><td>Pottel</td><td>-1443.6</td><td>3756.5</td><td>19883.6</td><td>0.986</td><td>1/6</td><td>1.922</td><td>3.549</td><td>2.24</td><td>14.09</td><td>11.44</td></tr><tr><td>Simonin</td><td>-1197.9</td><td>779.0</td><td>5643.9</td><td>0.983</td><td>1/6</td><td>1.917</td><td>3.539</td><td>2.22</td><td>13.97</td><td>6.87</td></tr><tr><td>MMB ion</td><td>-1058.1</td><td>793.5</td><td>5240.3</td><td>0.970</td><td>1/6</td><td>1.892</td><td>3.492</td><td>2.13</td><td>13.43</td><td>6.39</td></tr></tbody></table>

The optimized parameters for NaCl for e-PPC-SAFT for scenario 2 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 13**</caption><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="2">Scaling parameters</th><th colspan="2">HS diameter, σ<sub>H,S</sub></th><th colspan="3">Interaction parameter</th><th>Born diameter</th></tr><tr><th>λ<sup>HS</sup></th><th>λ<sup>MSA</sup></th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>k<sup>(0)</sup><sub>ij</sub></th><th>k<sup>(1)</sup><sub>ij</sub><br>(K<sup>-1</sup>)</th><th>k<sup>(2)</sup><sub>ij</sub><br>(K<sup>-2</sup>)</th><th>σ<sub>Born</sub><br>(Å)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>0.897</td><td>1.00</td><td>1.705</td><td>3.248</td><td>-0.5200</td><td>-3.578 · 10<sup>-3</sup></td><td>2.299 · 10<sup>-6</sup></td><td>–</td></tr><tr><td>Schreckenberg</td><td>0.867</td><td>1.00</td><td>1.647</td><td>3.137</td><td>-0.6141</td><td>-3.648 · 10<sup>-3</sup></td><td>7.081 · 10<sup>-7</sup></td><td>6.649</td></tr><tr><td>Pottel</td><td>0.850</td><td>1.00</td><td>1.615</td><td>3.077</td><td>-0.6220</td><td>-3.701 · 10<sup>-3</sup></td><td>-7.178 · 10<sup>-8</sup></td><td>5.487</td></tr><tr><td>Simonin</td><td>0.742</td><td>1.00</td><td>1.410</td><td>2.686</td><td>-1.4575</td><td>-9.779 · 10<sup>-3</sup></td><td>-6.106 · 10<sup>-6</sup></td><td>5.324</td></tr></tbody></table>

The deviations (RAD as defined in Eq. (31)) for three properties for the water-NaCl system in the temperature range of 273.15-473.15 K with the different permittivity models (defined in Table 4) of e-CPA and e-PPC-SAFT and the optimized parameters for scenario 3 given in Tables 14 and 15.

<table><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="3">e-CPA</th><th colspan="3">e-PPC-SAFT</th></tr><tr><th>MIAC</th><th>Osmotic coef.</th><th>Density</th><th>MIAC</th><th>Osmotic coef.</th><th>Density</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>7.22</td><td>3.64</td><td>1.36</td><td>3.10</td><td>1.38</td><td>3.15</td></tr><tr><td>Maribo-Mogensen</td><td>5.01</td><td>2.19</td><td>0.60</td><td></td><td></td><td></td></tr><tr><td>MMB solvent</td><td>4.45</td><td>1.61</td><td>0.61</td><td></td><td></td><td></td></tr><tr><td>Schreckenberg</td><td></td><td></td><td></td><td><strong>3.02</strong></td><td><strong>1.14</strong></td><td><strong>2.55</strong></td></tr><tr><td>Pottel</td><td>4.94</td><td>1.58</td><td>0.62</td><td>3.33</td><td>1.23</td><td>2.36</td></tr><tr><td>Simonin</td><td>3.52</td><td>2.50</td><td>0.89</td><td>6.79</td><td>3.51</td><td>2.86</td></tr><tr><td>MMB ion</td><td>3.23</td><td>2.17</td><td>0.81</td><td></td><td></td><td></td></tr><tr><td>Maribo-Mogensen ion</td><td><strong>3.65</strong></td><td><strong>1.17</strong></td><td><strong>0.68</strong></td><td></td><td></td><td></td></tr></tbody></table>

parameter for the ion-ion association energy and therefore 4 adjustable parameters are considered in this scenario. The adjustable parameters are listed in Table 6.

In Table 13 the deviations of the different permittivity models are listed for scenario 3. It can be seen that e-PPC-SAFT is significantly worse at describing the density for all permittivity models, but in most cases perform better for MIAC and osmotic coefficients. The exception to this is with models that describe the experimental RSP (Simonin and MMB ion), which perform better with e-CPA for all properties.

In Tables 14 and 15 the parameters of e-CPA and e-PPC-SAFT are listed. Looking at the parameters for e-CPA it can be seen that the DH-diameter is larger than the Pauling values [44] for the models that have no/a small concentration dependency of the relative static permittivity. On the other hand the models, that describe the experimental concentration dependency of the permittivity (Simonin and MMB ion), have much smaller values. A smaller DH-diameter leads to a larger magnitude of a negative contribution from the DH-term to MIAC, which is likely necessary to counter balance a large positive contribution from

the Born term. While the scaling factor for the co-volume ($\lambda^b$) varies significantly, it should also be noted that the variation of the co-volume, $b$, is relatively small for most of the permittivity models, which is likely linked to the fact that the co-volume is very important for density description [32] and the large variation of $\lambda^b$ is mostly to counterbalance the variation of the DH-diameter. Finally, the association energy of the ions will be discussed for e-CPA. Here it is observed that the permittivity models that are in between the solvent and experimental values for the relative static permittivity, all obtain significantly smaller association energies than both Zuo and Furst, which is salt concentration independent, and Simonin and MMB ion, which describe the measured experimental data for the electrolyte solution. Additionally, the Maribo-Mogensen solvent, MMB solvent and Pottel models all have an ion-ion interaction parameter for the association energy ($\omega_{ca}$) that is 1, which means that the effective association energy/strength between the ions is zero, and cation-anion association is therefore neglected.

For the parameters of e-PPC-SAFT (Table 15) it can be seen that the hard sphere diameter is close to the Pauling values [44] for permittivity
<caption>**Table 14**</caption>

The optimized parameters for NaCl for e-CPA for scenario 3 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 15**</caption><thead><tr><th rowspan="2">Model</th><th colspan="2">Scaling parameters</th><th colspan="2">Ion diameter, σ<sub>DH</sub></th><th colspan="2">Co-volume, b</th><th colspan="2">Assoc. energy</th></tr><tr><th>λ<sup>DH</sup></th><th>λ<sup>b</sup></th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>Na+<br>(cm<sup>3</sup>/mol)</th><th>Cl−<br>(cm<sup>3</sup>/mol)</th><th>ε<sub>i</sub><sup>AB</sup>/R<br>(K)</th><th>ω<sub>ca</sub></th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>1.678</td><td>0.030</td><td>3.272</td><td>6.040</td><td>1.98</td><td>12.43</td><td>1798.2</td><td>0.179</td></tr><tr><td>Maribo-Mogensen solvent</td><td>1.513</td><td>0.047</td><td>2.950</td><td>5.446</td><td>2.28</td><td>14.35</td><td>627.3</td><td>1.000</td></tr><tr><td>MMB solvent</td><td>1.243</td><td>0.081</td><td>2.425</td><td>4.476</td><td>2.19</td><td>13.76</td><td>667.8</td><td>1.000</td></tr><tr><td>Pottel</td><td>0.975</td><td>0.158</td><td>1.902</td><td>3.511</td><td>2.06</td><td>12.98</td><td>540.5</td><td>1.000</td></tr><tr><td>Simonin</td><td>0.711</td><td>0.612</td><td>1.386</td><td>2.559</td><td>3.08</td><td>19.41</td><td>2912.8</td><td>-0.023</td></tr><tr><td>MMB ion</td><td>0.701</td><td>0.610</td><td>1.367</td><td>2.524</td><td>2.95</td><td>18.56</td><td>3277.6</td><td>-0.036</td></tr><tr><td>Maribo-Mogensen ion</td><td>1.303</td><td>0.068</td><td>2.540</td><td>4.690</td><td>2.11</td><td>13.30</td><td>314.3</td><td>-0.297</td></tr></tbody></table>

The optimized parameters for NaCl for e-PPC-SAFT for scenario 3 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="2">Scaling parameters</th><th colspan="2">HS diameter, σ<sub>HS</sub></th><th colspan="2">Assoc. energy</th></tr><tr><th>λ<sup>HS</sup></th><th>λ<sup>MSA</sup><br>(Å)</th><th>Na+<br>(Å)</th><th>Cl−<br>(K)</th><th>ε<sub>i</sub><sup>AB</sup>/k</th><th>ω<sub>ca</sub></th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>1.035</td><td>2.039</td><td>1.967</td><td>3.748</td><td>2146.2</td><td>1.000</td></tr><tr><td>Schreckenberg</td><td>1.076</td><td>1.562</td><td>2.044</td><td>3.894</td><td>4892.5</td><td>0.105</td></tr><tr><td>Pottel</td><td>0.988</td><td>1.506</td><td>1.878</td><td>3.578</td><td>1297.7</td><td>1.000</td></tr><tr><td>Simonin</td><td>0.849</td><td>2.050</td><td>1.614</td><td>3.075</td><td>4999.8</td><td>-0.024</td></tr></tbody></table>

models other than Simonin, because the scaling factor is close to 1. The MSA scaling factor is very different between the models. This is also what was observed for e-CPA, where the physical sizes (co-volume) were relatively similar, but the electrostatic diameters were very different for the various permittivity models. The association energies for e-PPC-SAFT also vary a lot with the permittivity models, but there do not appear to be the same trend as for e-CPA, but for e-PPC-SAFT they seem more “random”. There are also permittivity models that have no cation–anion association due to an interaction parameter (ω<sub>ca</sub>) that is 1. It is also quite interesting that overall very similar results are obtained with Schreckenberg and Pottel models (Table 13) given that the association parameters are so different (Table 15) with otherwise similar parameter sets. This indicates that there is some form of interrelation between the association energy and the cation–anion interaction parameter.

### 3.4. Scenario 4 (Association)

The adjustable parameters of this scenario are listed in Table 6, and they include the same four parameters as scenario 3, with the addition of the Born diameter, and there are therefore five adjustable parameters in total in this scenario. In Table 16 the deviations of the different models are listed for scenario 4. In this case the best performing model for e-CPA is the Maribo-Mogensen ion model and for e-PPC-SAFT it is the Schreckenberg model. The difference in performance between these two models is relatively small with a slightly better performance for MIAC and density with e-PPC-SAFT/Schreckenberg and slightly worse (but for both models very good) description of osmotic coefficients. When comparing the results with the same permittivity models for both e-CPA and e-PPC-SAFT (Pottel and Simonin) it can be seen that e-PPC-SAFT performs better for all properties and for MIAC the performance is significantly better.

The trends of some of the adjustable parameters are also significantly different for scenario 4 compared to scenario 3 which can be seen in Tables 17 and 18 for e-CPA and ePPC-SAFT respectively. For e-CPA (Table 17) it can be seen that the smallest optimized value for the Born diameter (with Maribo-Mogensen solvent) is around the average value of Born diameters estimated from hydration energy (Table 3), and the rest of the models have larger values, which reduces the impact of the Born term. For e-PPC-SAFT (Table 18) both Schreckenberg and Pottel have significantly smaller values for the Born diameter, which increases the impact of the Born term. It is also noted that for e-PPC-SAFT

there are either no cation–anion association or it is very small for all of the permittivity models, because ω<sub>ca</sub> is close to 1. The trend that was observed for e-CPA/scenario 3 (Table 14) that the association energy is smallest for permittivity models that are in between the solvent value and the value of the experimental values for the electrolyte solution. This trend is not completely clear for e-CPA/scenario 4 (Table 17).

### 3.5. Scenario 5 (Disp+Assoc)

This scenario includes parameters for dispersion and association forces for the ions, and the adjustable parameters of this scenario are listed in Table 6. The adjustable parameters include the temperature dependent interaction parameter for the SRK term/dispersion energy (3 parameters), the association energy of the ions and the size of the ions, where only one parameter is adjusted to scale both the physical and the electrostatic size of the ions together. In total five adjustable parameters are included in this scenario.

In Table 19 the deviations of the different models are listed for scenario 5. It can be seen that e-CPA performs better in this case, almost regardless of the permittivity model, as the Zuo and Furst, Maribo-Mogensen solvent, MMB solvent and Pottel models all perform better than the best performing model of e-PPC-SAFT/Schreckenberg. It is interesting that Zuo and Furst now performs similarly to the other models as Zuo and Furst was generally performing worse for scenarios 1–4. This suggests that having contributions from both the physical term and the association term is more beneficial when there is no contribution from the Born term due to the lack of a concentration dependency of the relative static permittivity. It is also notable that the models that describe the experimental RSP (Simonin and MMB ion) are performing significantly worse for both models, which was also found to be the case in many of the previous scenarios for both EoS's. Regarding the performance of e-PPC-SAFT, it can be seen that the best performing permittivity model is Schreckenberg, followed by Pottel. While Zuo and Furst is significantly better than Simonin, it is also clearly worse than Schreckenberg and Pottel especially due to the density description.

In Table 20 the parameters of e-CPA are listed for each of the permittivity models. It can be seen that both the MMB ion model and the Maribo-Mogensen ion model have essentially no association energy for the ions. In this case the combining rule (Eq. (17)) leads to a cation–anion cross-association that is zero and there will not be any cation–anion association. The water–ion cross-association will
<caption>**Table 16**</caption>

The deviations (RAD as defined in Eq. (31)) for three properties for the water-NaCl system in the temperature range of 273.15-473.15 K with the different permittivity models (defined in Table 4) of e-CPA and e-PPC-SAFT and the optimized parameters for scenario 4 given in Tables 17 and 18.

<table><caption>**Table 17**</caption><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="3">e-CPA</th><th colspan="3">e-PPC-SAFT</th></tr><tr><th>MIAC</th><th>Osmotic coef.</th><th>Density</th><th>MIAC</th><th>Osmotic coef.</th><th>Density</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>7.22</td><td>3.64</td><td>1.36</td><td>3.10</td><td>1.38</td><td>3.15</td></tr><tr><td>Maribo-Mogensen</td><td>4.98</td><td>2.18</td><td>0.58</td><td></td><td></td><td></td></tr><tr><td>MMB solvent</td><td>4.71</td><td>1.91</td><td>0.67</td><td></td><td></td><td></td></tr><tr><td>Schreckenberg</td><td></td><td></td><td></td><td><strong>3.19</strong></td><td><strong>1.28</strong></td><td><strong>0.46</strong></td></tr><tr><td>Pottel</td><td>4.49</td><td>1.66</td><td>0.79</td><td>3.29</td><td>1.39</td><td>0.59</td></tr><tr><td>Simonin</td><td>5.88</td><td>4.10</td><td>1.01</td><td>4.55</td><td>2.34</td><td>0.55</td></tr><tr><td>MMB ion</td><td>5.61</td><td>4.07</td><td>0.55</td><td></td><td></td><td></td></tr><tr><td>Maribo-Mogensen ion</td><td><strong>3.51</strong></td><td><strong>1.17</strong></td><td><strong>0.81</strong></td><td></td><td></td><td></td></tr></tbody></table>

The optimized parameters for NaCl for e-CPA for scenario 4 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 18**</caption><thead><tr><th rowspan="2">Model</th><th colspan="2">Scaling parameters</th><th colspan="2">Ion diameter, σ<sub>DH</sub></th><th colspan="2">Co-volume, b</th><th colspan="2">Assoc. energy</th><th rowspan="2">σ<sub>Born</sub><br>(Å)</th></tr><tr><th>λ<sup>DH</sup></th><th>λ<sup>b</sup></th><th>Na+<br>(Å)</th><th>Cl-<br>(Å)</th><th>Na+<br>(cm<sup>3</sup>/mol)</th><th>Cl-<br>(cm<sup>3</sup>/mol)</th><th>ε<sub>i</sub><sup>AB</sup>/R<br>(K)</th><th>w<sub>ca</sub></th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>1.678</td><td>0.030</td><td>3.272</td><td>6.040</td><td>1.98</td><td>12.43</td><td>1798.5</td><td>0.179</td><td>-</td></tr><tr><td>Maribo-Mogensen solvent</td><td>1.510</td><td>0.047</td><td>2.944</td><td>5.435</td><td>2.29</td><td>14.43</td><td>618.9</td><td>0.996</td><td>3.44</td></tr><tr><td>MMB solvent</td><td>1.348</td><td>0.061</td><td>2.629</td><td>4.853</td><td>2.08</td><td>13.09</td><td>755.6</td><td>0.999</td><td>4.11</td></tr><tr><td>Pottel</td><td>1.196</td><td>0.087</td><td>2.331</td><td>4.304</td><td>2.10</td><td>13.19</td><td>1039.0</td><td>0.174</td><td>4.76</td></tr><tr><td>Simonin</td><td>0.911</td><td>0.167</td><td>1.776</td><td>3.279</td><td>1.77</td><td>11.14</td><td>476.3</td><td>0.933</td><td>4.94</td></tr><tr><td>MMB ion</td><td>1.086</td><td>0.138</td><td>2.117</td><td>3.908</td><td>2.48</td><td>15.62</td><td>2007.1</td><td>0.006</td><td>5.26</td></tr><tr><td>Maribo-Mogensen ion</td><td>1.326</td><td>0.065</td><td>2.585</td><td>4.772</td><td>2.14</td><td>13.44</td><td>626.3</td><td>-0.154</td><td>4.06</td></tr></tbody></table>

The optimized parameters for NaCl for e-PPC-SAFT for scenario 4 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 19**</caption><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="2">Scaling parameters</th><th colspan="2">HS diameter, σ<sub>HS</sub></th><th colspan="2">Assoc. energy</th><th rowspan="2">Born<br>σ<sub>Born</sub><br>(Å)</th></tr><tr><th>λ<sup>HS</sup></th><th>λ<sup>MSA</sup></th><th>Na+<br>(Å)</th><th>Cl-<br>(Å)</th><th>ε<sub>i</sub><sup>AB</sup>/k<br>(K)</th><th>w<sub>ca</sub></th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>1.035</td><td>2.039</td><td>1.967</td><td>3.748</td><td>2146.2</td><td>1.000</td><td>-</td></tr><tr><td>Schreckenberg</td><td>0.916</td><td>1.505</td><td>1.740</td><td>3.316</td><td>901.2</td><td>0.881</td><td>1.717</td></tr><tr><td>Pottel</td><td>0.858</td><td>1.491</td><td>1.631</td><td>3.107</td><td>635.3</td><td>0.980</td><td>2.182</td></tr><tr><td>Simonin</td><td>0.939</td><td>1.429</td><td>1.784</td><td>3.398</td><td>5000.0</td><td>0.890</td><td>9.745</td></tr></tbody></table>

The deviations (RAD as defined in Eq. (31)) for three properties for the water-NaCl system in the temperature range of 273.15-473.15 K with the different permittivity models (defined in Table 4) of e-CPA and e-PPC-SAFT and the optimized parameters for scenario 5 given in Tables 20 and 21.

<table><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="3">e-CPA</th><th colspan="3">e-PPC-SAFT</th></tr><tr><th>MIAC</th><th>Osmotic coef.</th><th>Density</th><th>MIAC</th><th>Osmotic coef.</th><th>Density</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>2.16</td><td>1.63</td><td>0.36</td><td>2.86</td><td>1.80</td><td>1.30</td></tr><tr><td>Maribo-Mogensen</td><td><strong>1.92</strong></td><td><strong>1.53</strong></td><td><strong>0.34</strong></td><td></td><td></td><td></td></tr><tr><td>MMB solvent</td><td>1.97</td><td>1.56</td><td>0.37</td><td></td><td></td><td></td></tr><tr><td>Schreckenberg</td><td></td><td></td><td></td><td><strong>2.35</strong></td><td><strong>1.51</strong></td><td><strong>0.82</strong></td></tr><tr><td>Pottel</td><td>2.01</td><td>1.47</td><td>0.41</td><td>2.77</td><td>1.55</td><td>0.75</td></tr><tr><td>Simonin</td><td>4.61</td><td>3.44</td><td>1.12</td><td>5.46</td><td>3.16</td><td>4.57</td></tr><tr><td>MMB ion</td><td>8.23</td><td>7.05</td><td>0.89</td><td></td><td></td><td></td></tr><tr><td>Maribo-Mogensen ion</td><td>2.76</td><td>1.67</td><td>0.45</td><td></td><td></td><td></td></tr></tbody></table>

be half the value of the association energy of water's self-association and this is the smallest water-ion association that is allowed in this parameterization. Additionally, when the association energy of the ions is zero with the Maribo-Mogensen ion model, there will only be a small reduction of the relative static permittivity from the addition of the ions (see Fig. 5).

Since four of the permittivity models (Zuo and Furst, Maribo-Mogensen, MMB solvent and Pottel, see Table 19) for e-CPA quantitatively describe the experimental data equally well, the difference in their optimized parameters (Table 20) should be directly related to the choice of permittivity model. It can be seen that the ion diameters are almost the same with all four models, with a value that is 6-7.5% larger than the Pauling values [44]. The interaction parameter at 298.15 K ($U_{ref}/R$) seems to follow a trend where it becomes more negative the closer the relative static permittivity is to the solvent value. This apparent trend was also observed for scenario 1 (Table 8) and it also

extends to the models that describe the experimental relative static permittivity as they have the most positive $U_{ref}$-values. Interestingly, between the four best performing models the association energy is lowest for the Maribo-Mogensen model and for the MMB solvent model. This is interesting because these models predict the relative static permittivity to be in between what is predicted with the models of Zuo and Furst (solvent value) and that of Pottel (smaller value), so there is not a purely increasing/decreasing trend for the association energy with the behavior of the concentration dependency of the relative static permittivity.

The parameters of e-PPC-SAFT are given in Table 21. It can be seen that the parameters with the Simonin model are significantly different from the other three permittivity models but this model also performs significantly worse (Table 19). For the other three models it can be seen that the hard sphere diameters and the association energy values are fairly close between the models. For the hard-sphere diameter a trend is
<caption>**Table 20**</caption>

The optimized parameters for NaCl for e-CPA for scenario 5 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 21**</caption><thead><tr><th rowspan="2">Model</th><th colspan="3">Interaction parameter</th><th colspan="2">Scaling parameters</th><th colspan="2">Ion diameter, σ<sub>DH</sub></th><th colspan="2">Co-volume, b</th><th rowspan="2">Assoc. energy<br>ε<sub>i</sub><sup>AB</sup>/R<br>(K)</th></tr><tr><th>U<sub>ref</sub>/R<br>(K)</th><th>T<sub>a</sub><br>(K)</th><th>α<br>(K)</th><th>λ<sup>DH</sup></th><th>λ<sup>b</sup></th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>Na+<br>(cm<sup>3</sup>/mol)</th><th>Cl−<br>(cm<sup>3</sup>/mol)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>-659.8</td><td>499.7</td><td>3885.5</td><td>1.069</td><td>1/6</td><td>2.08</td><td>3.85</td><td>2.85</td><td>17.96</td><td>1756.8</td></tr><tr><td>Maribo-Mogensen</td><td>-282.0</td><td>387.9</td><td>4431.7</td><td>1.075</td><td>1/6</td><td>2.10</td><td>3.87</td><td>2.90</td><td>18.26</td><td>1209.8</td></tr><tr><td>MMB solvent</td><td>-307.8</td><td>360.8</td><td>4898.9</td><td>1.069</td><td>1/6</td><td>2.08</td><td>3.85</td><td>2.85</td><td>17.96</td><td>1321.2</td></tr><tr><td>Pottel</td><td>76.1</td><td>342.6</td><td>5129.4</td><td>1.061</td><td>1/6</td><td>2.07</td><td>3.82</td><td>2.79</td><td>17.56</td><td>2128.9</td></tr><tr><td>Simonin</td><td>1999.9</td><td>233.4</td><td>2201.5</td><td>0.914</td><td>1/6</td><td>1.78</td><td>3.29</td><td>1.78</td><td>11.22</td><td>4999.9</td></tr><tr><td>MMB ion</td><td>1578.1</td><td>417.3</td><td>3457.3</td><td>0.658</td><td>1/6</td><td>1.28</td><td>2.37</td><td>0.67</td><td>4.20</td><td>0.7</td></tr><tr><td>Maribo-Mogensen ion</td><td>-370.1</td><td>419.2</td><td>3651.1</td><td>1.003</td><td>1/6</td><td>1.96</td><td>3.61</td><td>2.36</td><td>14.85</td><td>0.0</td></tr></tbody></table>

The optimized parameters for NaCl for e-PPC-SAFT for scenario 5 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="2">Scaling parameters</th><th colspan="2">HS diameter, σ<sub>HS</sub></th><th colspan="2">Assoc. energy</th><th colspan="3">Interaction parameter</th></tr><tr><th>λ<sup>HS</sup></th><th>λ<sup>MSA</sup></th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>ε<sub>i</sub><sup>AB</sup>/k<br>(K)</th><th>k<sub>ij</sub><sup>(0)</sup><br>(K<sup>-1</sup>)</th><th>k<sub>ij</sub><sup>(1)</sup></th><th>k<sub>ij</sub><sup>(2)</sup></th><th>ε<sub>i</sub><sup>AB</sup>/R<br>(K)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>0.954</td><td>1.000</td><td>1.813</td><td>3.454</td><td>981.93</td><td>-0.2720</td><td>-2.776 · 10<sup>-3</sup></td><td>5.000 · 10<sup>-6</sup></td><td></td></tr><tr><td>Schreckenberg</td><td>0.939</td><td>1.000</td><td>1.783</td><td>3.398</td><td>1027.87</td><td>-0.2091</td><td>-2.180 · 10<sup>-3</sup></td><td>5.000 · 10<sup>-6</sup></td><td></td></tr><tr><td>Pottel</td><td>0.908</td><td>1.000</td><td>1.725</td><td>3.286</td><td>981.57</td><td>-0.1850</td><td>-1.651 · 10<sup>-3</sup></td><td>-1.833 · 10<sup>-6</sup></td><td></td></tr><tr><td>Simonin</td><td>1.091</td><td>1.000</td><td>2.072</td><td>3.949</td><td>4647.99</td><td>0.4626</td><td>1.936 · 10<sup>-4</sup></td><td>3.843 · 10<sup>-6</sup></td><td></td></tr></tbody></table>

observed where the values become smaller when the permittivity model becomes more dependent on the salt concentration. For the interaction parameter at 298.15 K ($k_{ij}^{(0)}$) the same trend is found as for scenario 1 with an increasing trend towards more positive values for permittivity models with more significant concentration dependencies.

### 3.6. Scenario 6 (Disp+Assoc)

The adjustable parameters of this scenario are listed in Table 6. The adjustable parameters are the same as scenario 5, with one additional parameter for scaling the ion size, leading to the physical and electrostatic ion sizes being independently optimized. This scenario therefore has six adjustable parameters and considers both dispersion and association for the ions.

In Table 22 the deviations of the different permittivity models are listed for scenario 6. Looking at the first five listed permittivity models (from Zuo and Furst until Pottel) it can be seen that the performance of e-CPA and e-PPC-SAFT is now much closer compared to scenario 5 (Table 19), with the osmotic coefficients being better described with e-PPC-SAFT, while MIAC and density are better described with e-CPA. Overall, this scenario leads to the best agreement with experimental data for both models. The performance with e-CPA is slightly better especially due to the better agreement with the density data and slightly better performance for MIAC. The osmotic coefficients are generally slightly better described with e-PPC-SAFT.

Since e-PPC-SAFT improved more than e-CPA from scenario 5 to scenario 6, it suggests that the set scaling ratios used in scenario 5 was worse for e-PPC-SAFT compared to e-CPA. For e-CPA the relative difference in the performance of the permittivity models is similar between scenarios 5 and 6, which suggests that none of the permittivity models are specifically favored in the new choice of adjustable parameters. For e-PPC-SAFT three of the permittivity models (Zuo and Furst, Schreckenberg and Pottel) now perform very similarly, whereas for scenario 5 (Table 19) there was a clear difference between the performance of these models.

In Tables 23 and 24 the parameters of e-CPA and e-PPC-SAFT are listed. As for scenario 5 the very similar and good performance for several of the models allows for good comparisons of the impact of the permittivity models on the parameters. The parameter trends that were observed for scenario 5 are also found for scenario 6. For e-CPA (Table 23) the interaction parameter still shows a trend to become more positive when there is a large decrease of the permittivity. The co-volumes are very similar between the permittivity models as has also

been noted for other scenarios but for three of the models (Zuo and Furst, Maribo-Mogensen solvent and MMB solvent) the DH-diameter is significantly increased. This will lead to a smaller impact of the DH-term on the activity coefficients, but because the co-volumes are similar the performance is still similar for the density. The order of the association energy is still the same as in scenario 5 meaning that the Maribo-Mogensen solvent and MMB solvent models are in between Zuo and Furst and Pottel even through the permittivity of the first two models are in between the second two.

For e-PPC-SAFT three of the permittivity models of this scenario performed very similarly (Zuo and Furst, Schreckenberg and Pottel, see Table 22), so the parameters of these can be compared. The parameters are shown in Table 24 and it can be seen that the hard sphere diameters are very similar between the models, which is similar to e-CPA, where the co-volumes are similar. For the MSA diameter a trend is found where the diameter gets smaller for larger reductions of the relative static permittivity with salt concentration. This is possibly related to a balancing of the Born term. The association energy is also very similar between the models, which was also the case for scenario 5, but it is also different compared to what is found for e-CPA (Table 23), where there is a significant difference between the models. Another difference between the parameters with the different permittivity models is the value of the interaction parameter ($k_{ij}^{(0)}$). At the reference temperature of 298.15 K, the interaction parameter ($k_{ij}^{(0)}$) follows a similar trend as observed for scenarios 1 and 5, where the value becomes more positive with stronger concentration dependency of the permittivity model.

In Fig. 2 the density is shown for a permittivity model for each of the EoS's. It can be seen that there are some differences in the performance of the two models. The description at 273.15 K and low concentrations are better for e-PPC-SAFT, which is due to the used water parameters being better at describing the density at lower temperatures close to the density maximum. The used water parameters for e-PPC-SAFT were optimized to liquid density and vapor pressure with a specific focus on describing the temperature dependency of liquid density close to the density maximum [47]. To achieve this, the hard sphere diameter of water was made temperature dependent. The water parameters for e-CPA [40] were optimized to liquid density and the vapor pressure in a wide temperature and pressure range without special considerations to the low temperature region where the density reaches a maximum and the density maximum is not accurately captured. The difference in the water parameterization is the reason that e-PPC-SAFT performs better at lower temperature. As the salt concentration increases the impact
<caption>**Table 22**</caption>

The deviations (RAD as defined in Eq. (31)) for three properties for the water-NaCl system in the temperature range of 273.15-473.15 K with the different permittivity models (defined in Table 4) of e-CPA and e-PPC-SAFT and the optimized parameters for scenario 6 given in Tables 23 and 24.

<table><caption>**Table 23**</caption><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="3">e-CPA</th><th colspan="3">e-PPC-SAFT</th></tr><tr><th>MIAC</th><th>Osmotic coef.</th><th>Density</th><th>MIAC</th><th>Osmotic coef.</th><th>Density</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>1.87</td><td>1.56</td><td>0.36</td><td>1.97</td><td>1.30</td><td>0.80</td></tr><tr><td>Maribo-Mogensen</td><td><strong>1.61</strong></td><td><strong>1.36</strong></td><td><strong>0.37</strong></td><td></td><td></td><td></td></tr><tr><td>MMB solvent</td><td>1.67</td><td>1.34</td><td>0.40</td><td></td><td></td><td></td></tr><tr><td>Schreckenberg</td><td></td><td></td><td></td><td><strong>2.01</strong></td><td><strong>1.23</strong></td><td><strong>0.70</strong></td></tr><tr><td>Pottel</td><td>1.94</td><td>1.38</td><td>0.42</td><td>2.09</td><td>1.27</td><td>0.69</td></tr><tr><td>Simonin</td><td>4.58</td><td>3.58</td><td>0.63</td><td>4.79</td><td>2.48</td><td>3.78</td></tr><tr><td>MMB ion</td><td>6.27</td><td>4.93</td><td>0.58</td><td></td><td></td><td></td></tr><tr><td>Maribo-Mogensen ion</td><td>2.69</td><td>1.61</td><td>0.46</td><td></td><td></td><td></td></tr></tbody></table>

The optimized parameters for NaCl for e-CPA for scenario 6 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><caption>**Table 24**</caption><thead><tr><th rowspan="2">Model</th><th colspan="3">Interaction parameter</th><th colspan="2">Scaling parameters</th><th colspan="2">Ion diameter, σ<sub>DH</sub></th><th colspan="2">Co-volume, b</th><th rowspan="2">Assoc. energy<br>ε<sub>i</sub><sup>AB</sup>/R<br>(K)</th></tr><tr><th>U<sub>ref</sub>/R<br>(K)</th><th>T<sub>a</sub><br>(K)</th><th>α<br>(K)</th><th>λ<sup>DH</sup><br>(Å)</th><th>λ<sup>b</sup><br>(Å)</th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>Na+<br>(cm<sup>3</sup>/mol)</th><th>Cl−<br>(cm<sup>3</sup>/mol)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>-557.6</td><td>467.5</td><td>4006.5</td><td>1.279</td><td>0.096</td><td>2.493</td><td>4.603</td><td>2.83</td><td>17.79</td><td>2038.8</td></tr><tr><td>Maribo-Mogensen solvent</td><td>-169.3</td><td>377.4</td><td>4159.8</td><td>1.241</td><td>0.108</td><td>2.419</td><td>4.467</td><td>2.88</td><td>18.13</td><td>1321.1</td></tr><tr><td>MMB solvent</td><td>-175.7</td><td>348.3</td><td>4984.8</td><td>1.294</td><td>0.094</td><td>2.523</td><td>4.657</td><td>2.86</td><td>18.00</td><td>1586.7</td></tr><tr><td>Pottel</td><td>166.2</td><td>339.0</td><td>5223.8</td><td>1.096</td><td>0.150</td><td>2.137</td><td>3.945</td><td>2.76</td><td>17.37</td><td>2313.7</td></tr><tr><td>Simonin</td><td>957.0</td><td>299.5</td><td>4451.7</td><td>0.747</td><td>0.427</td><td>1.456</td><td>2.688</td><td>2.49</td><td>15.68</td><td>9388.9</td></tr><tr><td>MMB ion</td><td>1235.4</td><td>295.9</td><td>3031.9</td><td>0.683</td><td>0.486</td><td>1.331</td><td>2.458</td><td>2.17</td><td>13.66</td><td>9056.1</td></tr><tr><td>Maribo-Mogensen ion</td><td>-352.8</td><td>416.9</td><td>3641.8</td><td>1.025</td><td>0.156</td><td>1.998</td><td>3.689</td><td>2.35</td><td>14.78</td><td>0.4</td></tr></tbody></table>

The optimized parameters for NaCl for e-PPC-SAFT for scenario 6 (adjustable parameters defined in Table 6) with the different relative static permittivity models (defined in Table 4). When one value is given it is used for both ions.

<table><thead><tr><th rowspan="2">ε<sub>r</sub>-model</th><th colspan="2">Scaling parameters</th><th colspan="2">HS diameter, σ<sub>HS</sub></th><th rowspan="2">Assoc. energy<br>ε<sub>i</sub><sup>AB</sup>/k<br>(K)</th><th colspan="3">Interaction parameter</th></tr><tr><th>λ<sup>HS</sup></th><th>λ<sup>MSA</sup></th><th>Na+<br>(Å)</th><th>Cl−<br>(Å)</th><th>k<sup>(0)</sup><sub>ij</sub></th><th>k<sup>(1)</sup><sub>ij</sub><br>(K<sup>-1</sup>)</th><th>k<sup>(2)</sup><sub>ij</sub><br>(K<sup>-2</sup>)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>0.911</td><td>1.479</td><td>1.732</td><td>3.299</td><td>1256.5</td><td>-0.3276</td><td>-2.622 · 10<sup>-3</sup></td><td>2.173 · 10<sup>-6</sup></td></tr><tr><td>Schreckenberg</td><td>0.916</td><td>1.293</td><td>1.740</td><td>3.315</td><td>1131.1</td><td>-0.1945</td><td>-1.853 · 10<sup>-3</sup></td><td>4.978 · 10<sup>-6</sup></td></tr><tr><td>Pottel</td><td>0.913</td><td>1.173</td><td>1.734</td><td>3.304</td><td>1217.4</td><td>-0.1578</td><td>-1.772 · 10<sup>-3</sup></td><td>5.000 · 10<sup>-6</sup></td></tr><tr><td>Simonin</td><td>1.041</td><td>1.369</td><td>1.978</td><td>3.768</td><td>5000.0</td><td>0.6744</td><td>1.285 · 10<sup>-3</sup></td><td>5.000 · 10<sup>-6</sup></td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** The salt concentration dependency of the system water-NaCl at various temperatures with e-CPA/Maribo-Mogensen (left) and e-PPC-SAFT/Schreckenberg (right) with scenario 6 parameters (Tables 23 and 24). For e-PPC-SAFT/Schreckenberg the results at 273.15 K and 298.15 K are almost the same. The points are from a correlation of experimental data by Al-Ghafri et al. [61].</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 3.** The concentration dependency of the relative static permittivity for the system water-NaCl at 298.15 K with e-CPA scenario 2 (left) and e-PPC-SAFT scenario 4 (right) with the various permittivity models. The experimental data are from Shcherbakov et al. [62] and Buchner et al. [63]. MMB is short for Mollerup–Michelsen–Breil.</figcaption></figure>

of the ion parameters become greater and, as it can be seen in Fig. 2, the accuracy at higher salt concentrations is better with e-CPA in this case. This is the reason e-CPA has lower deviations overall in the entire temperature and concentration ranges.

### 3.7. The relative static permittivity

The results of the concentration dependency of the various models are shown in this section. The results of one scenario for each of e-CPA and e-PPC-SAFT will be presented. Additionally, the permittivity models that vary between scenarios due to a direct dependency from an adjustable (and not due to different volume/density description) will be shown and discussed as well. The permittivity models considered in this work, that have such a direct dependency on one of the adjustable parameters, are the Pottel model due to the dependence on the ion size, and the models of Maribo-Mogensen (solvent and ion) that relies on the association energy of the ions.

Fig. 3 shows the concentration dependency of the relative static permittivity for each of the EoS. For permittivity models that are not shown in subsequent figures, Fig. 3 can be considered an illustration of the results with the permittivity models. It can be seen that the Zuo and Furst model as well as the Maribo-Mogensen model are (essentially) salt composition independent. For the Maribo-Mogensen model this is only the case for scenarios 1 and 2, because the ions are not associating. It can also be seen that the Schreckenberg and the MMB solvent models have a very similar trend with a small reduction of the permittivity, but they are still fairly close to the solvent value. The MMB solvent model predicts slightly smaller values for the relative static permittivity compared to Schreckenberg. The Pottel model shows a larger reduction of the permittivity, but is still quite far from the experimental values. Finally, it can be seen that the Simonin and the MMB ion models are able to describe the experimental values well for both models.

In Fig. 4 the results with the Pottel model with the parameters from the different scenarios are shown. It can be seen that the results with e-CPA show lower values of the relative static permittivity compared to e-PPC-SAFT. This is due to the fact that the optimized $\sigma_{DH}$ of e-CPA tends to be larger, than the optimized $\sigma_{MSA}$ of e-PPC-SAFT. The spread of the different scenarios with the same model appears to be quite similar. It is also interesting to note that the trend of the different scenarios is completely different between the models. For example scenario 4 has the largest reduction of the permittivity with increasing salt concentration for e-CPA, whereas scenario 4 has the smallest reduction with e-PPC-SAFT. The opposite behavior is observed for scenario 3. This indicates that the choice of adjustable parameters does not lead to the same optimized ion size for both models.

The results of the concentration dependency of the relative static permittivity for the various parameters with the Maribo-Mogensen solvent and ion models are shown in Fig. 5. These models vary between the different scenarios because they are dependent on the association energy, which is optimized for scenarios 3–6. As mentioned, it can be seen that when the ions are not associating the model(s) predicts an almost salt concentration independent permittivity (scenarios 1 and 2 in Fig. 5(a)). It is also interesting to note that when the association energy of the ions is optimized, the experimental permittivity is not described well. This means that the experimental relative static permittivity is not captured when parameters are optimized to the properties in the objective function (Eq. (30)).

It can also be seen that the solvent version of the Maribo-Mogensen model generally predicts higher values of the permittivity compared to the ion version, which corresponds to the theory. Scenarios 5 and 6 both have an ion association energy that is (basically) zero for the ion model and it is therefore the closest that the model can get to the solvent value. The actual values of the relative static permittivity of the solvent model is actually quite similar to the ion model for scenarios 5 and 6 due to the optimized value of the association energy. This suggests that for the optimization of the association energy for the Maribo-Mogensen models, the impact that the association energy has on the relative static permittivity is more important than the impact on the association term. The performance for the properties in the objective function is better with the solvent model (in terms of deviations, see Tables 19 and 22), which suggests that the larger association energy for the solvent model yields better overall results in this case. This shows that the association term is important and it is likely a significant factor to why the solvent model performs better than the ion model for scenario 5 and 6.

## 4. Discussion

In the following sections various aspects of the results are discussed to highlight the capabilities and limitations of various approaches (permittivity models, incorporated physics, choice of adjustable parameters and physical model). First the performance of the models is investigated when they are extrapolated to other properties/conditions. In Section 4.1 the models are evaluated for their prediction of individual ion activity coefficients (IIAC) and in Section 4.2 the density is extrapolated to high pressures for some of the models. The MIAC description is discussed in Section 4.3 focusing on the temperature range where the data shows a maximum. Afterwards, the contributions of terms to MIAC are investigated in Section 4.4. In Section 4.5 the three scenarios with five adjustable parameters are compared and discussed. The trends of
<figure><img src="image_8.png" /><figcaption>Fig. 4. The concentration dependency of the relative static permittivity for the system water-NaCl at 298.15 K with e-CPA (left) and e-PPC-SAFT (right) with the various parameter sets for the Pottel model. The experimental data are from Shcherbakov et al. [62] and Buchner et al. [63].</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. The concentration dependency of the relative static permittivity for the system water-NaCl at 298.15 K with the Maribo-Mogensen solvent model (left) and the Maribo-Mogensen ion model (right) with the various parameter sets of e-CPA. The experimental data are from Shcherbakov et al. [62] and Buchner et al. [63].</figcaption></figure>

the interaction parameters (Section 4.6) and the electrostatic diameters (Section 4.7) are then discussed. Finally, the general performance, both qualitatively and quantitatively, is discussed.

## 4.1. IIAC

The concept of measuring Individual Ion Activity Coefficients (IIAC) has been considered controversial as evidenced by the discussions between Vera/Wilczek-Vera [64–66] versus Malatesta [67] and Zaru-bin [68–70]. Nevertheless, IIAC is a very interesting property, as it can distinguish between the ions, which thermodynamic bulk properties generally cannot, due to the principle of electroneutrality. In this work, we assume that the reported data for IIAC is valid at least in a qualitative sense of which ion has a larger (and smaller) activity coefficient compared to the mean value. The qualitative trend of several ions in various salts has also been investigated in a molecular simulation study [71] and the IIAC qualitative trends are in agreement with the trends of the experimental data, which increases the trust in the experimental data. The main result of this section is the evaluation of which models can capture the qualitative trend of the experimental data and these results are summarized in Tables 25 and 26 for e-CPA and e-PPC-SAFT respectively. These tables also include the deviations for each of the models to quantify the accuracy of the models. Some of the models are still quantitatively very inaccurate even though they predict the correct qualitative trend. Figures, showing the results of all

the models, can be found in the supplementary material and only plots that are discussed are presented in this section.

Tables 25 and 26 provide an overview of parameter sets of e-CPA and e-PPC-SAFT respectively, that are capable of getting the correct trend of IIAC in terms of which ion is higher (and lower) than the mean value. In the case of NaCl, the experimental data show that Na+ should have larger activity coefficients compared to Cl-. It should be noted that none of the properties that are included in the objective function are able to distinguish between the ions and several of the fitted parameters are the same for both ions. The parameter that mainly contributes to the difference between the ions is the ion diameter, because the relative difference between the ion diameter is kept the same, because the ion diameters are scaled from the Pauling values (Eqs. (25) and (26)). With this method there are some common trends for the different terms, which have been discussed for e-CPA in previous works [26,32]. The general trends for smaller ions are that the DH-term becomes more negative, the SRK term (typically) becomes less positive and the association term becomes more positive. Since the Born diameter in those works were disconnected from the physical and electrostatic ion size, it was not inter correlated with the other terms. However, a larger Born diameter also leads to a less positive contribution from the Born term but in most cases it is a smaller difference than what is observed for the other terms. For e-CPA it is observed that a larger value for ion association is required to be able to get the correct qualitative trend of IIAC. This why the tables also include the association energy for the
<caption>**Table 25**</caption>

An overview of which of the models/parameter sets with e-CPA that describe the correct trend of the IIAC data for NaCl. The numbers are the average deviations (in percent, average of both ions) and the numbers in the brackets are the association energy ($e^{AB}/R$ in K) of the ions. Scenarios 1 and 2 do not consider the ions as associating. Green means that the parameter set agrees with the qualitative trend of the experimental data and red means that the parameter set does not follow the qualitative trend of the experimental data.

<table><caption>**Table 26**</caption><thead><tr><th>Scenario</th><th>1 (disp)</th><th>2 (disp)</th><th>3 (assoc)</th><th>4 (assoc)</th><th>5 (disp+assoc)</th><th>6 (disp+assoc)</th></tr></thead><tbody><tr><td>ε<sub>r</sub>-model</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Zuo and Furst</td><td>12.0</td><td>12.2</td><td>3.2 (1798.2)</td><td>3.2 (1798.2)</td><td>6.5 (1756.8)</td><td>2.5 (2038.8)</td></tr><tr><td>Maribo-Mogensen</td><td>13.6</td><td>12.8</td><td>12.9 (627.3)</td><td>13.1 (618.9)</td><td>14.0 (1209.8)</td><td>11.9 (1321.1)</td></tr><tr><td>MMB solvent</td><td>11.8</td><td>12.9</td><td>12.4 (667.8)</td><td>12.1 (755.6)</td><td>12.6 (1321.2)</td><td>8.0 (1586.7)</td></tr><tr><td>Pottel</td><td>11.4</td><td>13.2</td><td>12.9 (540.5)</td><td>12.9 (1039.0)</td><td>2.7 (2128.9)</td><td>4.5 (2313.7)</td></tr><tr><td>Simonin</td><td>6.0</td><td>12.9</td><td>10.3 (2912.8)</td><td>12.6 (476.3)</td><td>26.3 (4999.9)</td><td>23.2 (4999.2)</td></tr><tr><td>MMB ion</td><td>7.2</td><td>12.7</td><td>11.6 (3277.6)</td><td>3.4 (2007.1)</td><td>7.0 (0.7)</td><td>24.0 (4998.4)</td></tr><tr><td>Maribo-Mogensen ion</td><td></td><td></td><td>11.2 (314.3)</td><td>12.4 (626.3)</td><td>12.1 (0.0)</td><td>12.0 (0.4)</td></tr></tbody></table>

An overview of which of the models/parameter sets with e-PPC-SAFT that describe the correct trend of the IIAC data. The numbers are the average deviations (in percent, average of both ions) and the numbers in the brackets are the association energy ($e^{AB}/k$ in K) of the ions. Scenarios 1 and 2 do not consider the ions as associating. Green means that the parameter set agrees with the qualitative trend of the experimental data and red means the parameter set does not follow the qualitative trend of the experimental data.

<table><thead><tr><th>Scenario</th><th>1 (disp)</th><th>2 (disp)</th><th>3 (assoc)</th><th>4 (assoc)</th><th>5 (disp+assoc)</th><th>6 (disp+assoc)</th></tr></thead><tbody><tr><td>ε<sub>r</sub>-model</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Zuo and Furst</td><td>15.7</td><td>15.7</td><td>12.3 (2146.2)</td><td>12.3 (2146.2)</td><td>32.0 (981.93)</td><td>27.3 (1256.5)</td></tr><tr><td>Schreckenberg</td><td>15.1</td><td>16.5</td><td>33.6 (4892.5)</td><td>19.0 (901.2)</td><td>31.2 (1027.87)</td><td>29.2 (1131.1)</td></tr><tr><td>Pottel</td><td>15.0</td><td>16.7</td><td>11.5 (1297.7)</td><td>19.3 (635.3)</td><td>31.3 (981.57)</td><td>27.6 (1217.4)</td></tr><tr><td>Simonin</td><td>4.3</td><td>22.0</td><td>27.5 (4999.8)</td><td>27.2 (5000.0)</td><td>27.7 (4647.99)</td><td>32.4 (5000.0)</td></tr></tbody></table>

ions, as it can clearly be seen that a larger association energy seems to be required for the model to be able to predict the correct trend.

For the results for e-CPA (Table 25) it can be seen that several of the models are not able to capture the qualitative IIAC trend. None of the permittivity models of scenarios 1 and 2 are able to do so. This is due to the choice of forces that is applied to the ions based on the selected adjustable parameters, because the ions are not considered as associating. In this case both the SRK term and the DH term make the bigger ion (Cl-) have larger activity coefficients, which is opposite of the data and the Born term is not impactful enough to counteract this. Another aspect of Table 25 is that none of the Maribo-Mogensen models, that describe the relative static permittivity from water-ion association, are able to get the correct trend. This is likely because the association energy of these models is lower than the models that are able to get the correct trend. With the Maribo-Mogensen models the association energy will impact both the association term, and the salt concentration dependency of the relative static permittivity. The permittivity heavily influences the DH and Born terms and the impact of the association energy is therefore different for the Maribo-Mogensen models compared to the other permittivity models. Given that none of the parameter sets with the Maribo-Mogensen models get the correct qualitative trend, this additional impact of the association energy seems to be leading to lower values of the association energy.

As discussed, the performance of the first four models (Zuo and Furst, Maribo-Mogensen, MMB solvent and Pottel) for scenario 6 with e-CPA is very similar and very good for the properties in the objective function (MIAC, osmotic coefficients and density, see Table 22). However, only Zuo and Furst and Pottel are able to capture the qualitative IIAC trend even though they are (very) slightly worse for the properties of the objective function compared to Maribo-Mogensen and MMB solvent. The results with Zuo and Furst and Pottel are shown in Fig. 6 for e-CPA/scenario 6. It can be seen that both of the models are fairly close to the experimental IIAC values as it would be expected based on the deviations in Table 25. Pottel shows a larger difference between the IIAC of the two ions compared to Zuo and Furst and this larger difference appears to make Pottel more accurate at lower salt concentrations and less accurate at higher concentrations compared to the results with Zuo and Furst.

In Table 25 it can also be seen that the MMB solvent model is not able to capture the qualitative trend of the IIAC data with any of the parameter sets. This is related to the association energy, as it is too low

a value to capture the trend, which can be seen because the association energy is usually lower than the obtained values for the parameter set that captures the trend. It can also be seen that the Zuo and Furst model as well as the Pottel model for some parameter sets are able to predict the correct trend for IIAC. As for the other models this is closely related to the value of the association energy, and for the Zuo and Furst model, this can be explained by the fact that the lack of a Born term makes the model need a larger association term, which can be accomplished by a larger association energy. The results for the Zuo and Furst model also showcase that while the association energy is very important for the qualitative trend of IIAC, it is also not the only parameter that impacts it since the value of the association energy with scenario 5 is quite close to scenarios 3/4. As previously discussed, the ion size also has an impact on the qualitative IIAC trend, and this is why scenario 5/Zuo and Furst does not capture the trend.

It can also be seen that the Simonin and MMB ion models are able to get the correct trend in a lot of cases, when ion association is included. It is interesting given that the accuracy of these permittivity models are lower for the mean properties in most cases. It is also related to the association energy, given that the value of the association energy of the Simonin and MMB ion models, that are able to capture the qualitative IIAC trend, are quite large. Due to these very large values for the association energy, the models tend to over predict the difference in IIAC between the ions and they are not quantitatively very accurate. This can be inferred from the listed deviations in Table 25 and it is illustrated for scenario 6/Simonin in Fig. 7(a).

To discuss the IIAC deviations in Table 25 in more details, it can be seen that there are significant differences in the quantitative accuracy of the models, also between models that capture the qualitative IIAC trend, where the lowest average deviation is 2.5% and the largest is 26.3%. As shown in Fig. 6 the data show a lot of scattering, and it seems like the results with scenario 6/Zuo and Furst is close to the lowest possible calculated deviation (2.5%). It can be seen that the Zuo and Furst and Pottel scenarios (3, 4 and 6 for Zuo and Furst and 5 and 6 for Pottel), that capture the qualitative IIAC trend, are all relatively accurate considering the scattering of the experimental data. All of the scenarios for Simonin, that are able to predict the qualitative IIAC trend (scenarios 3, 5 and 6), are somewhat inaccurate. Similarly for MMB ion, two of the three scenarios that predicts the correct qualitative IIAC trend are inaccurate (scenarios 3 and 6). This is likely a results of too large values for the association energy of the ions. It seems like the
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** The IIAC of the system water-NaCl at 298.15 K with e-CPA and the permittivity models of Zuo and Furst (left) and Pottel (right) with scenario 6 parameters. The experimental data are from [64,72–74].</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 7.** The IIAC of the system water-NaCl at 298.15 K with e-CPA (left) and e-PPC-SAFT (right) with Simonin scenario 6 parameters. The experimental data are from [64,72–74].</figcaption></figure>

optimal value of the association energy for the ions is in the range of 1800–2300 K for accurate quantitative description of the IIAC data with e-CPA. As previously discussed other parameters than the association energy also influence the IIAC to a degree, which explains the broad range.

In Table 26 an overview of which parameter sets can capture the qualitative trend of IIAC is shown for e-PPC-SAFT together with the association energy parameter of the ions and the average deviations. It can be seen that e-PPC-SAFT/scenario 1/Simonin is the only model able to describe the correct IIAC trend when ion association is not included (scenarios 1 and 2) for both e-PPC-SAFT and e-CPA. This must be due to the Born term and possible a slightly different impact of the physical part (HS+disp) of e-PPC-SAFT compared to the SRK term of e-CPA. It can also be seen that all of the e-PPC-SAFT parameterizations with Simonin that include ion association are able to get the right IIAC trend. As shown in Fig. 7(b) it over predicts the difference between the ions, and just like for e-CPA it is due to the size of the association energy of the ions which is very large. The overestimation is also reflected in the high average deviations. It can also be seen that Schreckenberg is only getting the correct trend with scenario 3 and the Zuo and Furst model gets the correct trend for scenarios 3 and 4. The size of the association energy of the ions is also the reason for this.

It is mostly the size of the association energy that seems to determine if e-PPC-SAFT is able to predict the qualitative trend of IIAC, when the relative ion size is kept constant and the water-ion interaction energy is the same for both ions. This is the same trend as for e-CPA. The size of the association term for each of the ions is determined by the size of the association strength (Eqs. (15) and (16) for e-PPC-SAFT and e-CPA respectively) where it can be seen why the association energy is an important parameter. From the equations it can also be seen that the ion size will impact the difference between the ions, because the ion diameters are scaled linearly, but the volume of the ions are used ($σ_{HS}^3$), so when the ion diameters are scaled the relative difference of the ion volume will change.

Looking at the deviations for e-PPC-SAFT (Table 26) it can be seen that there is a significant difference in the accuracy of the models that are capable of capturing the qualitative trend. Only one of the models is reasonably accurate, and it is scenario 1/Simonin with a deviation of 4.3%. While all Simonin parameterization except scenario 2 are capable of capturing the qualitative IIAC trend, scenarios 3–6 are very inaccurate. This is due to the very large values for the association energy of the ions, especially considering that it is possible to (accurately) capture the IIAC data without ion association (scenario 1).
<figure><img src="image_12.png" /><figcaption>**Fig. 8.** The pressure dependency of the density of water at 298.15 K with the water parameters for e-CPA and e-PPC-SAFT given in Tables 1 and 2 respectively. The water data are from NIST.</figcaption></figure>

## 4.2. Pressure dependency of the density

The models are investigated for their ability to extrapolate to high pressure conditions for the density. In Fig. 8 the performance for the pressure dependency of the density of pure water is shown with the previously published water parameters for both models. It can be seen that e-PPC-SAFT describes the density better at low pressures and e-CPA performs better at higher pressures. It is also noted that both models underpredict the increase in density as the pressure increases.

In Figs. 9 and 10 the results with scenario 4 parameters are shown for salt concentrations of 0.5 and 5 mol/kg respectively. In Figs. 11 and 12 the results with the scenario 6 parameters are shown at salt concentrations of 0.5 and 5 mol/kg respectively. Looking at the results at 0.5 mol/kg (Figs. 9 and 11) it can be seen that all permittivity models of both e-CPA and e-PPC-SAFT follow the trend that was observed for pure water for each of the models. Simonin with e-PPC-SAFT with scenario 6 parameters looks different than the others due to the much worse density agreement in general, but the impact of increasing pressure appears to be similar.

For the results at 5 mol/kg (Figs. 10 and 12) the impact of the ion parameters is much larger. In Fig. 10 it can be seen that the accuracy of the models varies quite a lot, and e-CPA is the most inaccurate in this case. However, the shape of the curves match the experimental data quite well, and is also better compared to the results for pure water (Fig. 8). This means that the extrapolation to high pressure is improved when ions are also included even through the high pressure density data were not included in the fitting of the parameters. The highest considered pressure in the optimization is around 16 bar, which is the water vapor pressure at 473.15 K.

For scenario 6 at 5 mol/kg (Fig. 12) it can be seen that aside from Simonin with e-PPC-SAFT, all of the permittivity models of both e-CPA and e-PPC-SAFT are able to accurately describe the density at low pressure at 5 mol/kg. It can also be seen that e-CPA is able to capture the trend of the pressure on the density better. It is also noted that the different permittivity models have an impact on the pressure trend for e-CPA, whereas there is almost no difference from the permittivity models for e-PPC-SAFT.

## 4.3. Maximum of MIAC

It has been pointed out [26,76] that MIAC of NaCl in the water-NaCl system shows a maximum with an increasing trend with temperature

until around 310 K, and at higher temperatures MIAC start to decrease with increasing temperature. The temperature isotherms of the data can be seen in Figs. 13–16 for some of the models. On the left plot of the figures the lower temperatures are shown and on the right plot the higher temperatures are shown. The data are increasing with temperature on the left plot and decreasing on the right.

In Figs. 13 and 14 results are shown for scenario 4 parameters with e-PPC-SAFT/Schreckenberg and e-CPA/Maribo-Mogensen ion respectively. It can be seen that those two models do not get the correct temperature trend, as MIAC is predicted to be decreasing at all temperatures. This limits the potential accuracy of the model. Scenario 4 does not include the dispersion parameters and the additional temperature dependency that is introduced for the interaction parameter (Eqs. (6) and (27)) is therefore not included.

In Figs. 15 and 16 results with scenario 6 parameters are shown for e-PPC-SAFT/Schreckenberg and e-CPA/Maribo-Mogensen respectively. These two parameter sets are the overall best performing ones for the two EoS's. In this case the temperature dependent interaction parameter is included in the parameter optimization together with the association parameter. For both models the correct qualitative temperature trend of MIAC is captured, but for e-PPC-SAFT (Fig. 15) the temperature isotherms are very close in the low temperature region. It seems that the temperature dependency is more prominent for e-CPA, and this might be the reason that the e-CPA is overall performing better for this case (see Table 22). This is likely related to the difference in the interaction parameter that has been made temperature dependent in the two models, which is the Huron-Vidal mixing rule for e-CPA compared to the quadratic temperature dependency of the interaction parameter of the dispersion energy for e-PPC-SAFT. However, since there are also other differences in the models it could also be due to those.

Figs. 13–16 also show another aspect of both e-CPA and e-PPC-SAFT. It can be seen that all models struggle to accurately capture the experimental data at molalities in the range of 1–3 mol/kg at lower temperatures (T < 290). This is even the case for the parameterizations that are able to capture the maximum of the temperature dependency. On the other hand, all models in Figs. 13–16 are able to reasonably capture the trends at higher temperatures.

It has been mentioned by Selam et al. [76] that it is necessary to include an additional temperature dependency in the model to be able to capture this maximum. In the work of Novak et al. [77] it is also reported that it was necessary to make the water-ion dispersion energy temperature dependent for SAFT-VR-Mie to be able to accurately capture the temperature trend of MIAC. However, it has been shown by Roa Pinto et al. [27] with e-PPC-SAFT that this is not always the case as one of the parameterizations is able to capture the temperature trend of MIAC without some additional temperature dependency. Even with this result it still seems beneficial to include an additional temperature dependency in the model for capturing the temperature trend of MIAC because it is very difficult to capture the trend without such a temperature dependency. This is because none of the many models of this work that do not include the additional temperature dependency are able to capture the MIAC temperature trend. On the other hand, many of the models that include such a trend are able to capture the MIAC temperature trend. Additionally, the work of Roa Pinto et al. [27] show six parameterizations that optimized 9–12 adjustable parameters for the water-NaCl system and only one of these parameterizations appear to accurately capture the temperature trend. While it is possible to capture the MIAC temperature trend without an additional temperature dependent parameter, it is much easier to do so when such a parameter is included. It might also be the case that it is necessary to include a temperature dependent parameter to be able to capture the trend with a reasonable amount of adjustable parameters.
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** The pressure dependency of the system water-NaCl at 0.5 mol/kg and 298.15 K with e-CPA (left) and e-PPC-SAFT (right) with the various permittivity models with scenario 4 parameters. The experimental data are from [75]. MMB is short for Mollerup-Michelsen-Breil.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 10.** The pressure dependency of the system water-NaCl at 5 mol/kg and 298.15 K with e-CPA (left) and e-PPC-SAFT (right) with the various permittivity models with scenario 4 parameters. The experimental data are from [75]. MMB is short for Mollerup-Michelsen-Breil.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** The pressure dependency of the system water-NaCl at 0.5 mol/kg and 298.15 K with e-CPA (left) and e-PPC-SAFT (right) with the various permittivity models with scenario 6 parameters. The experimental data are from [75]. MMB is short for Mollerup-Michelsen-Breil.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 12. The pressure dependency of the system water–NaCl at 5 mol/kg and 298.15 K with e-CPA (left) and e-PPC-SAFT (right) with the various permittivity models with scenario 6 parameters. The experimental data are from [75]. MMB is short for Mollerup–Michelsen–Breil.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 13. The concentration dependency of MIAC for NaCl in the water–NaCl system at various temperatures with scenario 4 e-PPC-SAFT/Schreckenberg parameters. The left plot shows results for lower temperatures ($T \le 303.15$ K) and the right plot shows results for higher temperatures ($T \ge 313.15$ K).</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 14. The concentration dependency of MIAC for NaCl in the water–NaCl system at various temperatures with scenario 4 e-CPA/Maribo-Mogensen ion parameters. The left plot shows results for lower temperatures ($T \le 303.15$ K) and the right plot shows results for higher temperatures ($T \ge 313.15$ K).</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Fig. 15. The concentration dependency of MIAC for NaCl in the water–NaCl system at various temperatures with scenario 6 e-PPC-SAFT/Schreckenberg parameters. The left plot shows results for lower temperatures (T ≤ 303.15 K) and the right plot shows results for higher temperatures (T ≥ 313.15 K).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 16. The concentration dependency of MIAC for NaCl in the water–NaCl system at various temperatures with scenario 6 e-CPA/Maribo-Mogensen parameters. The left plot shows results for lower temperatures (T ≤ 303.15 K) and the right plot shows results for higher temperatures (T ≥ 313.15 K).</figcaption></figure>

## 4.4. Balance of contributions for MIAC

In this section the models are analyzed by plotting the contributions of the various terms to MIAC as defined in Section 2.6. The impact of various choices in the model development can be evaluated from this as the influence of the various parameters can be made clear. First the general trends of the models are discussed by focusing on the dominant terms of the models. Afterwards, results for some of the models are presented to discuss the various details of the models/model development. Results of all the models can be found in the supplementary material and only the plots, that are discussed, are shown in the following sections.

### 4.4.1. Balance of contributions: General overview

In Table 27 an overview of the most dominant positive terms for the permittivity models/scenarios of e-CPA is presented. When two terms are listed both terms almost contribute with the same magnitude and terms given in parenthesis are significant positive contributions that only have slightly smaller magnitude than the largest contribution. In all cases, the dominant negative contribution is the DH term, which is the reason the overview focuses on the positive contributions. There

are some apparent trends of what terms are important depending on the permittivity model and parameterization strategy. The models, that predict the experimental relative static permittivity (Simonin and MMB ion), have the Born term as the most significant positive contribution. This is strongly related to the fact that the Born term becomes larger when the concentration dependency of the relative static permittivity is stronger, which is clearly shown in the work of Novak et al. [78] where the Born contribution is plotted as a function of the relative static permittivity. For the models that either have no/a small concentration dependency of the relative static permittivity (Zuo and Furst, Maribo-Mogensen and MMB solvent) the Born term is not significant enough to counterbalance the negative DH term and some other term becomes the most positive contribution. The term that compensates for the smaller Born term varies between the parameterization strategies and the largest positive contribution is either the SRK term (scenarios 1, 2, 5 and 6) or the association term (scenario 3 and 4). This means that, if the short-range water-ion interactions (dispersion) are included in the parameterization, the SRK-term will be the largest contribution. Only the parameterizations that disregards the dispersion (3 and 4) has the association term as the largest positive contribution.
<table><caption>**Table 27**
An overview of the terms of e-CPA with the largest positive contributions to MIAC for each of the permittivity models/parameterization. Terms in parenthesis are close to the largest term. The largest negative contribution is in all cases the DH term.</caption><thead><tr><th>ε<sub>z</sub>-model</th><th>1 (disp)</th><th>2 (disp)</th><th>3 (assoc)</th><th>4 (assoc)</th><th>5 (disp+assoc)</th><th>6 (disp+assoc)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>SRK</td><td>SRK</td><td>Assoc</td><td>Assoc</td><td>SRK</td><td>SRK</td></tr><tr><td>Maribo-Mogensen</td><td>SRK</td><td>SRK</td><td>Assoc (SRK+Born)</td><td>Assoc (SRK+Born)</td><td>SRK</td><td>SRK</td></tr><tr><td>MMB solvent</td><td>SRK</td><td>SRK</td><td>Assoc (Born)</td><td>Assoc</td><td>SRK</td><td>SRK</td></tr><tr><td>Pottel</td><td>Born (SRK)</td><td>SRK</td><td>Born</td><td>Born</td><td>Born (SRK)</td><td>SRK</td></tr><tr><td>Simonin</td><td>Born</td><td>Born</td><td>Born</td><td>Born</td><td>Born</td><td>Born</td></tr><tr><td>MMB ion</td><td>Born</td><td>Born</td><td>Born</td><td>Born</td><td>Born</td><td>Born</td></tr><tr><td>Maribo-Mogensen ion</td><td>Born</td><td>Born</td><td>Born</td><td>Born</td><td>SRK+Born</td><td>SRK+Born</td></tr></tbody></table>

It is also observed that it is not purely the choice of permittivity model that determines if the Born term is the most dominant positive contribution. This can clearly be seen for the Pottel model, where some of the parameterizations have the SRK term as the dominant positive contribution (2 and 6), some have the Born term (3 and 4) and some have the Born term with the SRK term being close (1 and 5). In Fig. 4(a) it was shown that permittivity predicted by the Pottel model varies between the scenarios because of the direct dependence on the ion size. While this will have an impact on the size of the Born term, other aspects of the parameterization are important as well. This is evident when comparing scenarios 5 and 6, where scenario 5 has the smallest reduction of the permittivity (see Fig. 4(a)) and therefore the smallest Born term of the two scenarios because the Born diameters are the same in both scenarios. However, scenario 5 also has the Born term as the largest contribution while scenario 6 has the SRK term as the largest contribution.

An overview of the dominant positive terms of e-PPC-SAFT is presented in Table 28. Some of the discussed trends for e-CPA are also apparent for e-PPC-SAFT. It can be seen that the Born term in all cases is dominant for Simonin's model. The reason that scenario 4/Simonin has both the association and Born terms as the most dominant positive contributions is due to the very large value of the Born diameter (9.745 Å, see Table 18), which significantly reduces the size of the Born term. It is also observed that the Pottel model has scenarios where the Born term is the dominant positive contribution and other scenarios where another term dominates. For Zuo and Furst and Schreckenberg the Born term is not the largest positive term and instead it is either the physical term or the association term that is the most significant. For the Schreckenberg model only scenario 4 has the Born term as being close to the largest term and it is due to all the terms having small magnitudes combined with small values of the Born diameter (1.717 Å, see Table 18) leading to a relatively large Born term for the permittivity model.

There are also differences regarding the trends of which terms dominate between e-CPA and e-PPC-SAFT. While the DH term is the dominating negative contribution in all cases for e-CPA, for e-PPC-SAFT there are some of the parameterizations that leads to terms that are comparable to MSA or even slightly larger depending on the salt concentration as noted in Table 28. This does not necessarily originate from the difference of the DH versus MSA terms. It can also be due to other parts of the models having a different impact, possibly due to the water parameters affecting the other terms of the model to have a larger impact. Another key difference between e-CPA and e-PPC-SAFT is that for e-PPC-SAFT the association term is very significant even for parameterizations where the ions are not associating (scenarios 1 and 2/Zuo and Furst and scenario 1/Schreckenberg). This is due to the water association, which is a lot more impactful for e-PPC-SAFT compared to for e-CPA. Since both models have very similar association terms because both models use Wertheim's theory, this difference in the impact of water association is likely linked to the difference in the water parameters.

### 4.4.2. Balance of contributions: Zuo and Furst

In Fig. 17 the contributions of terms to MIAC are shown for e-PPC-SAFT with the Zuo and Furst permittivity model with four different parameter sets. Because the relative static permittivity is independent of the salt concentration only three of the shown terms have a significant impact, the physical term, the association term and the MSA term. The physical term consists of three terms (hard-sphere, chain and dispersion) and typically both the hard sphere and dispersion terms have significant impacts on this combined term. As mentioned, the combined physical term is shown because it more closely resembles the SRK term of e-CPA than any of the individual terms. The polar term is also non zero, but it is in all cases very small, and it is there due to the reduction of the mole fraction of water when salt is added since the ions are not considered as polar. The only term that shows the same trend for all four parameter sets is the MSA term, which is always a significant negative contribution that at very low concentrations completely dominates, and it is the most negative contribution in almost the entire concentration range. The other two terms that significantly influence the results vary not only in terms of the size, but also in terms of the sign of the contribution being positive or negative.

For scenarios 1 and 2 (Fig. 17(a)) where ions only have dispersion forces it can be seen that the physical and association terms are positive and almost have the same magnitude. The reason that the association term contributes even when the ions are not associative, is due to the presence of water. In scenarios 3 and 4 (Fig. 17(b)) the ions do not have dispersion forces, but are associative. In this case it can be seen that it is the association term that is the only significant positive contribution. The physical term is now a very significant negative contribution that at high molality is more impactful than the MSA term. Even with dispersion forces being neglected the physical term can still be a significant contribution, because the impact from the ions taking up space is still considered with the hard sphere diameter/term.

Figs. 17 (c) and (d) shows the contribution for scenarios 5 and 6 with e-PPC-SAFT with the permittivity model of Zuo and Furst. For both of these sets the ions have dispersion and association forces. They have the same trends in the sense that for both parameter sets the physical term is a positive contribution and the association and MSA terms are negative contributions. We conclude that considering both dispersion and association forces for the ions significantly impacts the way the model functions as the trends are completely different compared to having only either dispersion or association. The trend of the association term for (c) and (d) is flipped compared to both (a) and (b) and instead the physical term is much larger.

In Fig. 18 the contributions to MIAC for e-CPA with the permittivity model of Zuo and Furst are shown for all scenarios. It can be seen that, similarly to the e-PPC-SAFT results (Fig. 17), the electrostatic term (DH) is a major negative contribution. In the case of e-CPA, the SRK term is positive in all cases whereas the physical term for scenarios 3 and 4 with e-PPC-SAFT (Fig. 17(b)) was negative. In general the trend of the association term is the same for both e-CPA and e-PPC-SAFT, but the size of the term is generally much larger for e-PPC-SAFT.

For scenarios 3 and 4 (Fig. 18(c)) the association term is the largest positive contribution, which makes sense given that scenarios 3 and 4 includes ion association. Scenarios 1, 2, 5, and 6 all behave very
<caption>Table 28</caption>

An overview of the terms of e-PPC-SAFT with the largest positive contributions to MIAC for each of the permittivity models/parameterization. Terms in parenthesis are close to the largest term. The largest negative contribution is the MSA term unless otherwise stated.

<table><thead><tr><th>ε<sub>r</sub>-model</th><th>1 (disp)</th><th>2 (disp)</th><th>3 (assoc)</th><th>4 (assoc)</th><th>5 (disp+assoc)</th><th>6 (disp+assoc)</th></tr></thead><tbody><tr><td>Zuo and Furst</td><td>Physical+Assoc</td><td>Physical+Assoc</td><td>Assoc<sup>1</sup></td><td>Assoc<sup>1</sup></td><td>Physical</td><td>Physical<sup>2</sup></td></tr><tr><td>Schreckenberg</td><td>Physical (Assoc)</td><td>Physical</td><td>Assoc</td><td>Assoc (Born)</td><td>Physical</td><td>Physical</td></tr><tr><td>Pottel</td><td>Born+physical (Assoc)</td><td>Physical</td><td>Assoc (Born)</td><td>Born</td><td>Physical</td><td>Physical</td></tr><tr><td>Simonin</td><td>Born</td><td>Born</td><td>Born<sup>2</sup></td><td>Assoc+Born</td><td>Born</td><td>Born</td></tr></tbody></table>

1: The physical term is a comparable/larger negative contribution than the MSA term at higher molalities.

2: The association term is a comparable/larger negative contribution than the MSA term at higher molalities.

<figure><img src="image_21.png" /><figcaption>Fig. 17. The results with e-PPC-SAFT with the Zuo and Furst permittivity model for the contribution of the different terms to the MIAC of NaCl at 298.15 K with the various scenarios. Scenarios 1 and 2 are the same for Zuo and Furst, and scenarios 3 and 4 likewise. The Physical term is the sum of the hard sphere term, the chain term and the dispersion term.</figcaption></figure>

similarly, showing a balance between the positive SRK term and the negative DH term, and the association term being a comparably small contribution. This shows that for e-CPA, even when both association and attractive energy of the ions are considered (but no solvation/Born term), MIAC is dominated by a balance of the physical short-range forces and the electrostatic long-range forces, and not by association. This is a significant difference from e-PPC-SAFT (Fig. 17) where the association term usually contributes a significant amount even when only water-water association is considered.

### 4.4.3. Balance of contributions: Scenario 6

In Fig. 19 the contributions with e-CPA with scenario 6 parameters are shown for various permittivity models that perform fairly similarly for MIAC. With MMB solvent the relative static permittivity is only slightly decreasing from the pure solvent value, and with the Pottel

model this decrease is slightly larger. The impact of this change can be investigated by comparing Figs. 19(a) and (b). It can be seen that the Born term is larger for Pottel, which is to be expected due to the additional decrease of the relative static permittivity. The SRK term is significantly reduced, which is due to the significantly changed interaction energy, but also to a smaller extend due to the slightly different co-volumes. The association term is essentially the same for both models. The Perm term, describing the impact of the composition derivative of the relative static permittivity, goes from being zero for the MMB solvent model to being a positive contribution with the Pottel model. The DH term is more negative for Pottel which is counterbalancing the overall increase of the positive terms.

When comparing the two versions of the Maribo-Mogensen model (Fig. 19(c) and (d)) it can be seen that the association term is negative when using the Maribo-Mogensen (solvent) model, but when it is
<figure><img src="image_22.png" /><figcaption>**Fig. 18.** The results with e-CPA with the Zuo and Furst permittivity model for the contribution of the different terms to the MIAC of NaCl at 298.15 K. Scenarios 1 and 2 are the same for Zuo and Furst, and scenarios 3 and 4 likewise.</figcaption></figure>

extended to the ion model the association term becomes a positive contribution. This is likely related to the much smaller association energy that is found for the Maribo-Mogensen ion model, where the association energy of the ions is almost zero, which means that the water-ion cross association strength is roughly half of the water self-association energy. It can also be seen that the permittivity term goes from being negative to positive, which means that the derivative from the Born term is now more significant than the derivative from the DH term.

In Fig. 20 the contributions to MIAC for e-PPC-SAFT are shown for scenario 6 (Zuo and Furst is shown in Fig. 17(d)). It can be seen that Schreckenberg and Pottel (and Zuo and Furst) have similar contributions of the association term, which is due to the very similar hard-sphere diameters and association energies. Interestingly, the difference for the physical term between the Schreckenberg and Pottel models is also quite small. It is also evident that there is a significant difference for the contribution from the Born term, which is due to the different concentration dependency of the permittivity models. Since the rest of the terms have similar contributions the term that is changed to counterbalance the change to the Born term is the MSA term, which becomes more negative as the Born term becomes more impactful. This was reflected in the parameters by having smaller MSA diameters while having almost the same hard-sphere diameters.

Finally, for the contributions for e-PPC-SAFT/Simonin (Fig. 20(c)) it can be seen that while the overall accuracy of the model is worse than the other permittivity models (Table 22) the description of MIAC

at 298.15 is reasonably accurate. It can also be seen that the trends are different from the other permittivity models, because the physical term is now almost zero, and the only important positive contribution is the Born term. For the negative contributions there is a contribution from the association term which is similar in size to the other permittivity models, but a much larger negative contribution from the MSA term makes the association term seem like it has a lesser relative importance.

### 4.5. Comparing scenarios with five adjustable parameters

Three of the scenarios shown in the earlier sections have the same amount of adjustable parameters (5), and it is therefore possible to directly compare the impact of the physics that the choice of parameters incorporate. One parameter set considers the dispersion of ions (scenario 2), another considers the ions as only associative (scenario 4) and finally a parameter set considers both dispersion and association forces for the ions (scenario 5). The deviations are shown as bar plots in Figs. 21 and 22 for e-CPA and e-PPC-SAFT respectively for the three sets of adjustable parameters. The figures are a visualization of the results shown in Tables 10, 16 and 19.

Looking at the results for e-CPA (Fig. 21) it can be seen that there is a clear trend in performance for all permittivity models except with Simonin and MMB ion, which in most cases perform worse. The trend is that the best results are obtained when both association and dispersion (scenario 5) is considered. Only considering dispersion (scenario 2)
<figure><img src="image_23.png" /><figcaption>(a) Scenario 6, MMB solvent</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(c) Scenario 6, Maribo-Mogensen</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(b) Scenario 6, Pottel</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(d) Scenario 6, Maribo-Mogensen ion</figcaption></figure>

<caption>**Fig. 19.** The results with e-CPA with scenario 6 parameters with some of the permittivity models for the contribution of the different terms to the MIAC of NaCl at 298.15 K.</caption>

performs the second best and finally the association parameterization (scenario 4) is the worst of the three approaches. Given that the trend is clear for multiple permittivity models that all perform well it can be concluded that better results are obtained when more physics is included, even with the same amount of adjustable parameters. It can also be seen that the property that is mostly affected by the choice of physics is MIAC, as this is the property that comparably to the other parameter sets is described the worst for the association parameterization.

For e-PPC-SAFT (Fig. 22) it can be seen that the trend of which scenario performs the best is not as clear as for e-CPA, because there is a difference in how well the relative description of the properties is for the various parameter sets. In all cases the association set (scenario 4) describes the density better, but MIAC is usually described the worst. Aside from Simonin's model, the osmotic coefficients are described similarly between the models. Comparing the dispersion sets (scenario 2) and the dispersion+association sets (scenario 5) it can be seen that having both dispersion and association is better almost regardless of the property and permittivity model. The only property where scenario 5 is not better than scenario 2 is the density with Simonin's model, but both parameter sets describe the density poorly compared to most other models.

Comparing the association results with the other two sets gives less clear trends. Disregarding the Simonin results, the association set generally has the worst description of MIAC, which is likely linked to the lack of a more impactful temperature dependency that is included with dispersion in the temperature dependent $k_{ij}$ as discussed in Section 4.3. The osmotic coefficients are generally close for Zuo and Furst, Schreckenberg and Pottel, with association performing better for both

Schreckenberg and Pottel which are the two best performing models for e-PPC-SAFT regardless of parameterization strategy in this work. As mentioned the density is better described with the association set for all permittivity models. Overall either the association set or the dispersion+association set can be considered the best, depending on which property is considered the most important.

### 4.6. *Trends of the interaction parameter*

In the results sections (Section 3.1–3.6) the interaction parameters have been presented and their trends have been discussed for each scenario. The general trends of the reference interaction parameter at 298.15 K ($\Delta U_{ref}$ and $k_{ij}^{(0)}$) are summarized in Table 29. It can be seen that scenario 2 of both EoS's behaves differently than the other scenarios. For the rest of the scenarios (scenarios 1, 5 and 6) it is found that the interaction parameter will increase (i.e. become more positive) when the chosen permittivity model has a stronger concentration dependency that will make the permittivity be closer to the experimentally measured permittivity. It is very interesting that it is the same scenario (scenario 2) that deviates from the trend for both models as this suggests that e-CPA and e-PPC-SAFT have a similar reliance on the adjustable parameters. It is possible that the reason for the differing trend of scenario 2 is linked to the inclusion of the Born diameter as an adjustable parameter. The optimized Born diameters vary significantly between the permittivity models (Tables 11 and 12). This variation is especially large for e-CPA (Table 11) and while e-PPC-SAFT (Table 12) shows less variation between permittivity models it will still impact the model performance. Such a difference in the Born diameter will have a
<figure><img src="image_27.png" /><figcaption>Fig. 20. The results with e-PPC-SAFT with scenario 6 parameters with some of the permittivity models for the contribution of the different terms to the MIAC of NaCl at 298.15 K.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 21. Results with e-CPA in terms of deviations from experimental data for MIAC, osmotic coefficients and density for the three scenarios that have 5 adjustable parameters. Disp refers to scenario 2, assoc refers to scenario 4 and disp+assoc refers to scenario 5, where the adjustable parameters of the scenarios are defined in Table 6. Numerical values of the deviations are given in Tables 10, 16 and 19.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>**Fig. 22.** Results with e-PPC-SAFT in terms of deviations from experimental data for MIAC, osmotic coefficients and density for the three scenarios that have 5 adjustable parameters. Disp refers to scenario 2, assoc refers to scenario 4 and disp+assoc refers to scenario 5, where the adjustable parameters of the scenarios are defined in Table 6. Numerical values of the deviations are given in Tables 10, 16 and 19.</figcaption></figure>

<caption>**Table 29**</caption>

An overview of the trends of the reference interaction parameter at 298.15 K, $\Delta U_{ref}$ and $k_H^{(0)}$ for e-CPA and e-PPC-SAFT respectively. The trend is in regards to how the interaction parameter changes with the choice of permittivity model, where incr. (increasing) means that it becomes more positive with permittivity models with a stronger concentration dependency on the permittivity (Zuo and Furst → Simonin/MMB ion). Decr. is short for decreasing and is opposite of increasing.

<table><thead><tr><th>Scenario</th><th>e-CPA</th><th>e-PPC-SAFT</th></tr></thead><tbody><tr><td>1 (disp)</td><td>Incr.</td><td>Incr.</td></tr><tr><td>2 (disp)</td><td>No clear trend</td><td>Decr.</td></tr><tr><td>5 (disp+assoc)</td><td>Incr.<sup>1</sup></td><td>Incr.</td></tr><tr><td>6 (disp+assoc)</td><td>Incr.<sup>1</sup></td><td>Incr.</td></tr></tbody></table>

great effect on size of the Born term that will impact the overall model. This will affect the other adjustable parameters as they will need to compensate for the changed Born contribution.

## 4.7. DH and MSA terms

It has been mentioned that the EoS's use a different term to describe the long range ion-ion interactions with e-CPA using a DH term and e-PPC-SAFT using a MSA term. The equations for each of the models are given in previous sections (Section 2.1.2 for DH and Section 2.2.1 for MSA). The implications of this difference between the models can be discussed. It has previously been shown that very similar results can be obtained for the Helmholtz energy and its first order derivatives when the diameter of closest approach from DH is slightly smaller (at a ratio of roughly 5/6) compared to the hard sphere diameter of the MSA theory [35]. As pointed out by Maribo-Mogensen et al. [35], this suggests that, when the ion diameter is considered to be a fitted parameter there should not be a significant difference between the two models. This has also been confirmed by Walker et al. [28], as similar results are obtained with DH and MSA terms in e-EoS. The DH and MSA theories should therefore perform similarly when the two models are compared in isolation and the ion diameters are fitted. In the context of this work this is the case when the physical and electrostatic size of the ions are fitted independently of each other.

However, considering that the electrostatic diameters used in the DH or MSA terms sometimes are related to the physical diameter, there

might be a “conflict” of what value to use because the optimal value for the physical and electrostatic size might not be the same. Correlating the physical and electrostatic ion size has been done in many e-EoS like e-CPA [10,29,30,32,59] where the co-volume is calculated from the ion diameter (or vice versa) and for e-SAFT type models [11,27,45,51,79] where the hard sphere diameter is set equal to (or with a constant scaling factor) the ion diameter of the electrostatic term.

When looking at the results of this work it can be seen that when the ratio that is used to correlate the physical and electrostatic diameters are held constant it is usually found that the ion size with e-PPC-SAFT/MSA is smaller than the ion size of e-CPA/DH (parameters for scenarios 1, 2 and 5). This is the opposite compared to what was found by Maribo-Mogensen et al. [35], where the MSA parameter needed to be slightly larger than the DH-diameter to get the same results for the Helmholtz energy and its derivatives. The reason for this is likely that the value of the physical diameter is more important than the electrostatic diameter for the properties, and therefore has a larger impact on what the optimized value of the ion diameter should be. When the physical and electrostatic sizes are independent of each other the trend is less clear, but it does often appear that the MSA diameter is larger than the DH diameter. A less clear trend indicates that the size of the other terms has an impact on the optimized ion diameter, which means that there is inter correlation between the optimized parameters.

## 4.8. Broader considerations

In this study a total of 40 e-CPA parameterizations and 24 e-PPC-SAFT parameterizations are presented and compared for a total of 64 model parameterizations. In the previous sections various results for these parameterizations have been shown and discussed on a “per-property” basis. The aim of the following sections is to provide an overview of the general results from both a quantitative and a qualitative standpoint to discuss the more general capabilities and limitations of the EoS’s, permittivity models and parameterization approaches.

### 4.8.1. Impact of the permittivity model

The results, shown in this work, clearly show that the choice of permittivity model greatly impacts the overall performance of the e-EoS even after parameter optimization because the deviations of the models are quite different for the different permittivity models (Tables 7, 10, 13, 16, 19 and 22). The general trend is that the best performance

<footnote n="1">¹The Maribo-Mogensen ion model does not strictly follow the reported trend, but it is close.</footnote>
is obtained with a slightly decreasing relative static permittivity with salt concentration, that is somewhere in between the solvent value and the experimentally measured value of the electrolyte solution. This is because the best performing models are typically the Schreckenberg, MMB solvent, Pottel or one of the Maribo-Mogensen models and they generally predicts such a concentration dependency (see Figs. 3–5). The Simonin and MMB ion models, which describe the experimental relative static permittivity well, generally, when incorporated in e-EoS's, perform worse than the models that predict the permittivity to be in between the solvent and the experimental value. The Zuo and Furst model, which is only dependent on the temperature and therefore describes the solvent value, is also often worse when implemented in e-EoS's than the models that have a small decrease of the permittivity with increasing salt concentration.

For the performance with the Zuo and Furst permittivity model, it has been noted that the performance is very close to the models with a small decrease of the permittivity with ion concentration for scenarios 5 and 6 where ions are both associative and has dispersion forces. This suggests that the impact of a salt concentration dependency of the relative static permittivity can almost be neglected when both dispersion and association are considered.

### 4.8.2. Quantitative performance

In Section 3.6 it is mentioned that the lowest deviations for the properties MIAC, osmotic coefficients and density are found when scenario 6 parameters are used for e-CPA. However, the results with scenario 5 parameters are only slightly worse than scenario 6, so if one has a different salt and needs to optimize parameters and optimization speed is important, scenario 5 parameters could be used as well.

The actual best performing model for e-CPA of this work can be debated. If only the mean properties included in the objective function (MIAC, osmotic coefficients and density) are considered, it is scenario 6 with the Maribo-Mogensen solvent model that has the overall lowest deviations. However, this model does not capture the qualitative trend of IIAC, and for certain applications this property may be important. If it is a requirement that IIAC is accurately described the best models are instead either Zuo and Furst or Pottel for e-CPA. The main difference between them is that Zuo and Furst describe MIAC and IIAC a bit better but osmotic coefficients slightly worse. Zuo and Furst is also slightly better for the density description, but both models are very accurate for this property.

For e-PPC-SAFT it depends more on the needs of the model, because the density is better described when only association parameters (scenario 4) are used, but scenario 6 shows much better agreement for MIAC. This is at least the case for the best performing permittivity models Schreckenberg and Pottel (and Zuo and Furst for scenario 6). This means that for e-PPC-SAFT it depends on the requirement of the model, which parameterization to use. For general use it is believed that scenario 6 is better overall, because of the ability to somewhat capture the temperature trend of MIAC while the density is fairly well described with an average deviation of around 0.7% in a wide temperature range. None of the models that perform the best overall for MIAC, osmotic coefficients and density are able to capture the IIAC data accurately, which is an issue if this is an important property for the application that the EoS is used to model.

### 4.8.3. Qualitative performance

In Table 30 a qualitative overview for the models performance in describing the properties in the objective function (MIAC, osmotic coefficients and density) reasonably well. It can be seen that some of the permittivity models perform well regardless of the parameterization approach. For e-PPC-SAFT, Schreckenberg and Pottel are the best in all cases, and Zuo and Furst often performs well but in many cases slightly worse. On the other hand, Simonin is never one of the better performing permittivity models for e-PPC-SAFT, and in many cases performs significantly worse. This is different compared to when the

ability to capture the experimental trend of the IIAC data qualitatively is considered. Here, e-PPC-SAFT/Simonin is able to capture the qualitative IIAC trend in 5 out of 6 scenarios (Table 26). On the other hand, none of the Pottel parameterizations of e-PPC-SAFT are able to capture the qualitative IIAC trend and for Schreckenberg only scenario 3 can capture the trend. Zuo and Furst is able to capture the qualitative IIAC trend with the association parameterizations (scenarios 3 and 4). This shows that there is a discrepancy in which models perform well for the mean properties in the objective function (MIAC, osmotic coefficients and density) and the IIAC data.

For e-CPA all of the permittivity models are among the best performing models for at least one of the parameterizations (Table 30). For the association scenarios (scenario 3 and 4) it is found that having a permittivity model that accurately describe the experimental permittivity is better for the description of other properties (MIAC, osmotic coefficients and density). For the scenarios that include dispersion (1, 2, 5 and 6) it is the permittivity models that are either concentration independent or have a small concentration dependency that perform the best. It should also be remembered that many of the permittivity models perform significantly worse for scenarios 3 and 4 (Fig. 21) and this is part of the reason why there is a discrepancy between the scenarios.

Another interesting observation for e-CPA is that at least one of (and for scenarios 5 and 6 both) of the Maribo-Mogensen models in all cases performs well for the properties in the objective function (MIAC, osmotic coefficients and density). It has also been discussed in Section 4.8.2 that the overall lowest deviations with the properties in the objective function (MIAC, osmotic coefficients and density) are found with e-CPA/scenario 6/Maribo-Mogensen solvent. However, as discussed in Section 4.1, none of the Maribo-Mogensen models of this work are able to capture the qualitative IIAC trend. This is linked to the choice of adjustable parameters as it has been shown that it is possible to accurately capture the IIAC data with the Maribo-Mogensen ion model [26]. Other choices of parameters could also make the models able to capture the qualitative IIAC trend, especially if the parameter fitting includes the IIAC data and parameters that are unique for each of the ions (for example different association energies for Na+ and Cl-).

Another aspect to consider when looking at Table 30 is that for a lot of the considered scenarios there are several permittivity models that are able to describe the properties in the objective function (MIAC, osmotic coefficients and density) almost equally well. This has already been discussed for scenarios 5 and 6 for e-CPA where all permittivity models aside from Simonin and MMB ion, that both describe the relative static permittivity accurately, perform equally well. For scenario 6 the difference between the best performing models for e-CPA and e-PPC-SAFT is also relatively small. To better differentiate the performance of the models it is necessary to investigate additional properties, including extrapolations. In this work, this is done by investigating the models ability to predict the behavior of IIAC and for some of the scenarios the ability to extrapolate the density calculation to high pressures is investigated. The density extrapolations, shown in Section 4.2, only revealed small differences between the permittivity models and the biggest factor in accurately describing the high pressure density data was the accuracy of the models at low pressures.

The prediction of IIAC, shown in Section 4.1, provided a different viewpoint on the performance of the models. In general, one of the major differences compared to the performance for the properties in the objective function (MIAC, osmotic coefficients and density), is that the permittivity models that accurately describe the experimental data (Simonin and MMB ion) are able to capture the qualitative IIAC trend in more cases than any of the other permittivity models across both EoS's. However, the quantitative agreement with the IIAC data is in many cases worse than the other permittivity models, even compared to the models that are incapable of predicting the correct qualitative trend. It is also likely that some of the well performing models would be able to capture the IIAC trend without completely losing the performance for
<caption>**Table 30**</caption>

Overview of the qualitative performance of the various parameterizations of e-CPA and e-PPC-SAFT. Columns 2 and 3 shows the best permittivity models for the properties in the objective function (equation (30)) for e-CPA and e-PPC-SAFT respectively for each of the scenarios. Permittivity models in parenthesis perform well but distinctly worse compared to the best permittivity models of a given scenario. MM is short for Maribo-Mogensen (solvent), MM ion is Maribo-Mogensen ion, MMBs is MMB (Mollerup-Michelsen-Breil) solvent, MMBi is MMB (Mollerup-Michelsen-Breil) ion and ZF is Zuo and Furst.

<table><thead><tr><th rowspan="2">Scenario</th><th colspan="3">Deviations of MIAC, osmotic coefficients and density</th></tr><tr><th>Best e-CPA</th><th>Best e-PPC-SAFT</th><th>Comment</th></tr></thead><tbody><tr><td>1 (Dispersion)</td><td>MM, MMBs, (ZF, Pottel)</td><td>Schreckenberg, Pottel, (ZF)</td><td>Simonin, MMBi significantly worse</td></tr><tr><td>2 (Dispersion)</td><td>MM, MMBs, Pottel, (ZF)</td><td>Schreckenberg, Pottel, (ZF)</td><td>Simonin, MMBi significantly worse</td></tr><tr><td>3 (Association)</td><td>MM ion, MMBi, Simonin</td><td>Schreckenberg, Pottel, ZF</td><td>Discrepancy in best permittivity models</td></tr><tr><td>4 (Association)</td><td>MM ion</td><td>Schreckenberg, Pottel</td><td>e-CPA performs worse expect for MM ion</td></tr><tr><td>5 (Disp+Assoc)</td><td>MM solvent, MMBs, ZF, Pottel, (MM ion)</td><td>Schreckenberg, Pottel, (ZF)</td><td>Simonin, MMBi significantly worse</td></tr><tr><td>6 (Disp+Assoc)</td><td>MM solvent, MMBs, ZF, Pottel, (MM ion)</td><td>Schreckenberg, Pottel, ZF</td><td>Simonin, MMBi significantly worse</td></tr></tbody></table>

other properties if the IIAC data was included in the objective function. This was at least the case for e-CPA with the Maribo-Mogensen ion permittivity model in a previous work [26]. The loss of performance from including IIAC data with the adjustable parameters of this work is uncertain.

## 4.9. General recommendations

In this study, results for many different combinations of e-EoS, permittivity models and adjustable parameters are evaluated for their performance for the water-NaCl system. The trends of the results are highly useful for a discussion of what the best practices could be when parameterizing e-EoS's for other salts/ions. There are limits to how conclusive/general the results can be considered, since only NaCl is considered where both ions are univalent and relatively small. It is expected that the general phenomena for other ions will be similar enough that the best parameterization methods also should work reasonably well for a significant amount of other ions.

When comparing the performance for the mean properties considered in this work (MIAC, osmotic coefficients and density) of the two physical models (CPA and PPC-SAFT), there are small differences in the performances of the models for the individual parameterizations, but when looking at the overall results, the differences are relatively small. Especially, if the consideration is how an unknown/unparameterized salt/ions should be parameterized, the differences between e-CPA and e-PPC-SAFT are usually not significant enough that one of the models is guaranteed to perform better than the other for the mean properties.

Regarding the choice of permittivity model most of the considered models perform well in some cases. However, the models, that are fitted to describe the experimental permittivity (Simonin and MMB ion), are generally performing worse for the mean properties (see Table 30) and are unlikely to perform better than the other permittivity models when parameterizing a different salt. The only case where these permittivity models (Simonin and MMB ion) perform the best are for e-CPA/scenario 3 (Table 30), however this scenario is also one of the worst scenarios for e-CPA.

In general, it appears to be important to include an additional temperature dependency in the model, as both models struggle to capture the temperature trend of MIAC where a maximum is observed if such a temperature dependency is omitted. If the temperature trend of MIAC

is predicted to only be decreasing, then the model is guaranteed to be inaccurate in part of the important temperature range of 273–373 K. In this work, the interaction parameter in the SRK/dispersion terms was made temperature dependent and it was found that it would make the models able to capture the MIAC temperature trend better. This method of including an additional temperature dependency in the interaction parameter is expected to work well for other ions as well. However, it is possible that making another parameter temperature dependent would also make the models able capture the MIAC temperature trend.

It also seems to be important to include ion association to reasonably be able to capture the qualitative IIAC trend, based on the results in Tables 25 and 26. When considering the ability to predict the IIAC data, the best performing permittivity models are different compared to the models performance for mean properties. When the ions are associating, both Simonin and MMB ion are able to predict the qualitative trend in most cases regardless of the physical model. However, in most cases these models (Simonin and MMB ion) are quantitatively inaccurate, and only e-CPA/scenario 3/MMB ion and e-PPC-SAFT/scenario 1/Simonin are very accurate of the Simonin and MMB ion permittivity models.

In the context of the parameterizations considered in this study, it is therefore recommended to use the adjustable parameters of scenarios 5 or 6. These scenarios include both association parameters and a temperature dependent interaction parameter, which both seem to be important to be able capture the qualitative trend of IIAC and the maximum of the temperature dependency of MIAC. The permittivity models that perform the best are in most cases a permittivity model that decreases slightly with increasing salt concentration, while the models, that accurately describe the experimental permittivity, are not as accurate for other properties (MIAC, osmotic coefficients and density). Limiting this discussion to scenarios 5 and 6 due to those being the recommended choice of adjustable parameters, several of the permittivity can be used. This can be seen based on the qualitative overview (Table 30) where all but the Simonin and MMB ion models perform well.

## 5. Conclusion

This study compares two EoS's for electrolytes, e-CPA and e-PPC-SAFT, with the inclusion of various permittivity models where the parameters are optimized for each model. The comparisons between EoS's are done as “fair” as possible by selecting sets of parameters (called scenarios) that can be considered to be comparable in terms of the physics considered. Six scenarios have been considered; two that include the dispersion/attractive energy parameter for the ions, two that consider the ions as associative and finally two scenarios that consider both dispersion and association. This choice of parameter sets allows for broad comparisons that can highlight the impact of the: EoS, permittivity model, considered physics of the ions and the choice of adjustable parameters.

Overall, it is found that the difference between the used EoS is relatively small and other choices in the model development have a much greater impact on the model performance. The used permittivity model is shown to impact the overall performance much more than the choice of EoS. In most cases permittivity models that predict a small decrease of the permittivity tend to perform better than the models predicting the solvent permittivity value. However, when both dispersion and association are considered for the ions the performance of the permittivity models that do not describe the experimental permittivity, becomes similar. Generally, the worst performing permittivity models are the ones that are fitted to capture the experimental permittivity, which in this work are the Simonin and MMB ion models. These permittivity models are only the best for one thermodynamic model (e-CPA) for a single scenario (scenario 3). They yield however worse results for MIAC, osmotic coefficients and density compared to the best permittivity models of the other scenarios. Overall, it seems more
promising to prioritize permittivity models that do not fully capture the concentration dependency of the permittivity when developing e-EoS and the focus is on phase equilibria.

The models and various parameterizations have been examined for their ability to capture the temperature trend of MIAC which for NaCl shows a maximum at around 310 K. It has been shown that only models that include an additional temperature dependency in the form of a quadratic temperature dependency of an interaction parameter were able to predict this behavior. It is found that e-CPA obtains a stronger temperature dependency for MIAC, which leads to better description of this property at lower temperatures, and this may be due to a combination of the SRK-term and the Huron-Vidal mixing rule.

It has also been shown that inclusion of ion association is highly important to be able to describe the correct trend of individual ion activity coefficients (IIAC) where the trend is regarding which of the ions that have a larger IIAC compared to the mean value. The only model that is able to capture the IIAC trend without ion-water association and sufficiently large association energies, had much worse performance for other properties. The permittivity models that describe the experimental permittivity accurately is found to be more likely to be able to capture the qualitative IIAC trend. However, these permittivity models also tend to be quantitatively inaccurate, due to the optimized values of the association energy being too large.

The models have also been investigated for their ability to predict the density at pressures up to 2000 bar at 298.15 K. For pure water, e-PPC-SAFT is found to be more accurate at lower pressures, but e-CPA is more accurate at higher pressures. At low salt concentrations (0.5 mol/kg), the results follow the same trend with pressure as the pure water case for both models. At a higher salt concentration (5 mol/kg), it is shown that in several cases both e-CPA and e-PPC-SAFT are able to follow the trend for the pressure dependency better than the pure water case. This is even though the density data, considered in the parameter optimization, do not include this high pressure density data.

It is found that the generally best approach for both models is to consider both dispersion and association and to have a model for the relative static permittivity that does not describe the experimental permittivity. The overall best performing model for mean properties in this study is found with e-CPA, with a permittivity model that calculates the permittivity based on association. However, this model is incapable of capturing the qualitative IIAC trend. The best performing model, that also captures the IIAC data accurately, also include adjustable parameters for both dispersion and association for the ions, and either the Zuo and Furst (solvent permittivity) or the Pottel (mild decrease) permittivity models with e-CPA. Some of the parameterizations of e-PPC-SAFT perform comparably for the mean properties, but none of the best performing models of e-PPC-SAFT are able to capture the IIAC data.

### CRediT authorship contribution statement

**Martin Due Olsen:** Methodology, Software, Formal analysis, Writing - Original Draft, Visualization. **Georgios M. Kontogeorgis:** Conceptualization, Writing - review & editing, Funding acquisition. **Jean-Charles de Hemptinne:** Methodology, Software, Supervision, Writing - review & editing. **Xiaodong Liang:** Software, Writing - review & editing. **Nicolas von Solms:** Project administration, Supervision.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

All data used in this work is available in the literature.

### Acknowledgments

The authors wish to thank the European Research Council (ERC) for funding of this research under the European Union's Horizon 2020 research and innovation program (grant agreement No 832460), ERC Advanced Grant project "New Paradigm in Electrolyte Thermodynamics".

The authors also thank DTU Chemical Engineering for partial funding of this work.

### Appendix A. Permittivity models

#### A.1. Zuo and Furst model

The relative static permittivity model of Zuo and Furst [49] is a purely temperature dependent model that has been fitted to data in the temperature range of 288.15-403.15 K. The model is tested with both e-CPA and e-PPC-SAFT using the originally proposed parameters from [49]. Previously, this model has been used in e-SAFT-VR-Mie [28, 76] and e-SAFT-HR [80]. Given that most of the properties in this work are considered in the temperature range of 273.15-473.15 K the model of Zuo and Furst is slightly extrapolated. For water the relative static permittivity is given by:

$$ \epsilon_r = -19.2905 + \frac{29814.5}{T} - 0.019678 \cdot T + 1.3189 \cdot 10^{-4} \cdot T^2 - 3.1144 \cdot 10^{-7} \cdot T^3 \quad (A.1) $$

#### A.2. Schreckenberg model

The model of Schreckenberg et al. [51] is dependent on the temperature and the volume. It has a mild ion concentration dependency due to the volume change from the addition of salt. The model has been used in various EoS's [33,51]. In this work it is used to describe the value of the relative static permittivity of the solvent for e-PPC-SAFT. The relative static permittivity is calculated by:

$$ \epsilon_r = 1 + \rho_{solv} d_j \quad (A.2) $$

where $\rho_{solv}$ is the number density of the solvent ($\rho_{solv} = N_{solv}/V$) and $d_j$ is a solvent dependent parameter, which is calculated by:

$$ d_j = d_{v,j} \left( \frac{d_{T,j}}{T} - 1 \right) \quad (A.3) $$

where $d_{v,j}$ and $d_{T,j}$ are volume and temperature parameters of the solvent $j$. The parameter values for water are given in Table A.31.

#### A.3. Mollerup-Michelsen-Breil solvent model

The model of Mollerup, Michelsen and Breil (MMB) [50] for relative static permittivity of solvents is in this work used in e-CPA for the permittivity models that require an input of the solvent value of the relative static permittivity. The model is dependent on the temperature and volume. As it is the case with the model of Schreckenberg et al. [51] (Appendix A.2) the volume dependency of the solvent leads to a mild dependency on the ion concentration due to the change in volume of the solution, when ions are added. This model was also used in our previous work [26]. The value of the relative static permittivity of pure water is calculated by:

$$ \epsilon_{r,w}(T, V) = \epsilon_{r,w}(T_0) + \frac{\beta_1 N_A \mu_{w,0}^2}{2 k_B \epsilon_0} \left( \frac{\rho_w}{T} - \frac{\rho_{w,0}}{T_0} \right) \quad (A.4) $$

where $T_0$ is the reference temperature (273.15 K), $\epsilon_{r,w}(T_0)$ (87.82) is the static permittivity of water at the reference temperature, $\mu_{w,0}$ (2.5 D-value from Chen et al. [81]) is the dipole moment of liquid water, $\beta_1$ (3.1306) is a fitting constant, $\rho_w$ is the molar density of water and
<table><caption>**Table A.31**
The parameters of the relative static permittivity model by Schreckenberg et al. [51].</caption><thead><tr><th>d<sub>v</sub><br>(dm<sup>3</sup>/mol)</th><th>d<sub>T</sub><br>(K)</th></tr></thead><tbody><tr><td>0.3777</td><td>1403</td></tr></tbody></table>

$\rho_{w,0}$ is the molar density of water at $T_0$. The volume dependency is in the molar density of water ($\rho_w = n_w/V$) where the volume is calculated with e-CPA.

### A.4. Maribo-Mogensen solvent model

The solvent model of Maribo-Mogensen et al. [54] is a permittivity model that calculates the relative static permittivity based on the dipole moment of solvent molecules and the association network of the mixture. It is developed to accurately describe the relative static permittivity of pure and mixed solvents. This model has been used in various works with e-CPA [10,26,29,30,32]. The relative static permittivity is with the Maribo-Mogensen solvent model calculated from:

$$ \frac{(2\epsilon_r + \epsilon_\infty)(\epsilon_r - \epsilon_\infty)}{\epsilon_r} = \left(\frac{\epsilon_\infty + 2}{3}\right)^2 \frac{N_A}{\epsilon_0 k_B T \nu} \sum_i x_i \epsilon_i \mu_{0,i}^2 \quad (A.5) $$

where $\mu_{0,i}$ is the dipole moment of component i. The infinite frequency permittivity, $\epsilon_\infty$, is calculated from the Clausius–Mossotti [82] equation:

$$ \frac{\epsilon_{\infty} - 1}{\epsilon_{\infty} + 2} = \frac{1}{3\epsilon_0} \frac{N_A}{\nu} \sum_i x_i \alpha_{0,i} \quad (A.6) $$

where $\alpha_{0,i}$ is the polarizability of component i and the sum is of all species. The Kirkwood g-factor, $g_i$, in Eq. (A.5) is given by:

$$ g_i = 1 + \sum_j \frac{z_{ij} P_{ij} \cos \gamma_{ij}}{P_i \cos \theta_{ij} + 1} \frac{\mu_{0,j}}{\mu_{0,i}} \quad (A.7) $$

where $z_{ij}$ is the coordination number of component j around component i, $P_{ij}$ is the probability that component i is associating with component j, $P_i = \sum_j P_{ij}$ is the probability that component i is associating, $\cos \gamma_{ij}$ is the angle between the dipole moments of components i and j, and $\cos \theta_{ij}$ is the rotation angle between shells in the network of hydrogen bonding. $P_{ij}$ is calculated by:

$$ P_{ij} = \rho x_j \Delta_{A_i B_j} X_{A_i} X_{B_j} \quad (A.8) $$

where X is the site fraction and the subscripts, $A_i$ and $B_j$, refer to different sites. The site fraction is from the association term and it is determined by:

$$ \frac{1}{X_{A_i}} = 1 + \sum_j \rho_j \sum_{B \in j} X_{B_j} \Delta_{A_i B_j} \quad (A.9) $$

$\Delta_{A_i B_j}$ is the association strength between two sites. It is calculated from the Wertheim formalism [36] and the expressions were presented in Eqs. (15) and (16) for e-PPC-SAFT and e-CPA respectively.

The values of the additional parameters needed for this model are given in Table A.32. When ions are considered to be non-associating, the model will predict a relative static permittivity that is essentially independent of the ion concentration (assuming accurate volume prediction). This is because the ions in this case do not impact the association network, and the ions do not have a dipole moment. However, when ions are considered as associating, the relative static permittivity will be predicted to decrease with increasing amounts of ions in the solution.

### A.5. Maribo-Mogensen ion model

The solvent model by Maribo-Mogensen et al. [54] (described in the previous section) has also been extended to include the effects of

<table><caption>**Table A.32**
The parameters used in the calculation of the static permittivity with Maribo-Mogensen permittivity models.</caption><thead><tr><th>Compound</th><th>μ<sub>0</sub> (debye)</th><th>α<sub>0</sub> (10<sup>40</sup> C m<sup>2</sup>/J)</th><th>γ (deg)</th><th>θ (deg)</th></tr></thead><tbody><tr><td>water</td><td>1.8546</td><td>1.6133</td><td>63.4715</td><td>94.7939</td></tr><tr><td>Na+</td><td>0</td><td>2.221</td><td></td><td></td></tr><tr><td>Cl-</td><td>0</td><td>3.557</td><td></td><td></td></tr></tbody></table>

ions [55]. The ions are accounted for by assuming that any solvent molecule that is associated to an ion will have its dipole moment canceled. The relative static permittivity is therefore calculated by:

$$ \frac{(2\epsilon_r + \epsilon_\infty)(\epsilon_r - \epsilon_\infty)}{\epsilon_r} = \left(\frac{\epsilon_\infty + 2}{3}\right)^2 \frac{N_A}{\epsilon_0 k_B T \nu} \sum_i x_i \Theta_{i} \epsilon_i \mu_{0,i}^2 \quad (A.10) $$

The extension from the solvent model (Eq. (A.5)) is the addition of $\Theta_i$, which describes the fraction of component i that is not bound to an ion. This leads to the result that any solvent molecule that is associating with an ion will effectively be removed from the calculation of the permittivity. The expression for $\Theta_i$ is:

$$ \Theta_i = 1 - \sum_j^{\text{ions}} P_{ij} \quad (A.11) $$

where $P_{ij}$ is the probability that two sites are hydrogen bonded. If there is no solvent-ion association the model $\Theta_i$ will be 1 and the model will be the same as the solvent model.

### A.6. Pottel model

The model of Pottel [52] estimates the reduction of the relative static permittivity due to the addition of ions from the space that the ions take up. The theory behind the model is that ions will dilute the permittivity of the mixture because the ions will occupy a volume of low permittivity. The Pottel model is given by:

$$ \epsilon_r - 1 = (\epsilon_{r,w} - 1) \frac{1 - \xi_3''}{1 + \xi_3''/2} \quad (A.12) $$

where $\xi_3''$ is the volume of low permittivity, that the ions occupy in the solution. Ions are assumed to be spherical, so $\xi_3''$ can be approximated by:

$$ \xi_3'' = \frac{\pi N_A}{6} \sum_i^{\text{ions}} \frac{n_i d_i^3}{V} \quad (A.13) $$

where $d_i$ refers to the diameter of the ion. The choice of diameter was discussed in a previous work [26]. In this work the DH or hard sphere diameter is used for e-CPA and e-PPC-SAFT respectively. The model has previously been applied in e-EoS like e-CPA [26,58] and ePPC-SAFT [33].

### A.7. Simonin model

The Simonin model [53] describes the relative static permittivity with an adjustable parameter. The expression for the model is:

$$ \epsilon_r = \frac{\epsilon_{r,w}}{1 + \sum_i^{\text{ions}} \alpha_i x_i} \quad (A.14) $$

where $\alpha_i$ is an ion/salt specific adjustable parameter. In this work a constant value of $\alpha_i = 5.08$ for NaCl from the work of Inchekel et al. [58] is used for both models. It has been shown that this value lets the Simonin model accurately describe the experimental relative static permittivity of NaCl [26,58]. The relative static permittivity of the solvent, $\epsilon_{r,w}$, is calculated with the Mollerup–Michelsen–Breil solvent model (Eq. (A.4)) for e-CPA and the Schreckenberg model (see Appendix A.2) for e-PPC-SAFT.
<table><caption>**Table A.33**
The parameters of the relative static permittivity model by Mollerup, Michelsen and Breil [50] (Eq. (A.15)).</caption><thead><tr><td>Parameter</td><td>Value (L/mol)</td></tr></thead><tbody><tr><td>β<sub>2</sub></td><td>0.160</td></tr><tr><td>β<sub>3</sub></td><td>0.010</td></tr><tr><td>α<sub>Na</sub><sup>+</sup></td><td>0.1062</td></tr><tr><td>α<sub>Cl</sub><sup>-</sup></td><td>0.1172</td></tr></tbody></table>

## A.8. Mollerup-Michelsen-Breil ion model

The ion model by Mollerup, Michelsen and Breil (MMB) [50] was developed to accurately describe the impact of ions on the relative static permittivity of electrolyte mixtures. With the model the relative static permittivity is calculated by:

$$ \epsilon_r = \epsilon_{r,w} \cdot \left( 1 + \sum_{i \text{ions}} \left( \beta_3 c_i - \frac{\alpha_i c_i}{1 + \beta_2 c_i} \right) \right) \quad (A.15) $$

Where $\beta_2$ and $\beta_3$ are fitting parameters of the model and $\alpha_i$ is an ion-specific parameter. $c_i = n_i/V$ is the molar concentration of the ions, and is the reason that the model has both a volume and composition dependency. For the water-NaCl system the parameters values used in this work are given in Table A.33. These values are from [50].

## Appendix B. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.fluid.2023.113804.

## References

[1] Georgios M. Kontogeorgis, Georgios K. Folas, Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories, John Wiley & Sons Ltd, 2010.

[2] Eric Hendriks, Georgios M. Kontogeorgis, Ralf Dohrn, Jean-Charles de Hemptinne, Ioannis G. Economou, Ljudmila Fele Zilnik, Velisa Vesovic, Industrial requirements for thermodynamics and transport properties, Ind. Eng. Chem. Res. 49 (22) (2010) 11131-11141.

[3] Georgios M. Kontogeorgis, Ralf Dohrn, Ioannis G. Economou, Jean Charles de Hemptinne, Antoonten Kate, Susanna Kuitunen, Miranda Mooijer, Ljudmila Fele Zilnik, Velisa Vesovic, Industrial requirements for thermodynamic and transport properties: 2020, Ind. Eng. Chem. Res. 60 (13) (2021) 4987-5013.

[4] Georgios M. Kontogeorgis, Anders Schlaikjer, Martin Due Olsen, Bjørn Maribo-Mogensen, Kaj Thomsen, Nicolas von Solms, Xiaodong Liang, A review of electrolyte equations of state with emphasis on those based on cubic and cubic-plus-association (CPA) models, Int. J. Thermophys. 43 (4) (2022).

[5] Chau-Chyun Chen, L.B. Evans, A local composition model for the excess Gibbs energy of aqueous electrolyte systems, AIChE J. 32 (3) (1986) 444-454.

[6] Maria C. Iliuta, Kaj Thomsen, Peter Rasmussen, Extended UNIQUAC model for correlation and prediction of vapour-liquid-solid equilibria in aqueous salt systems containing non-electrolytes. Part A. methanol-water-salt systems, Chem. Eng. Sci. 55 (14) (2000) 2673-2686.

[7] Kaj Thomsen, Martin Due Olsen, Lucas F.F. Corrêa, Modeling vapor-liquid-liquid-solid equilibrium for acetone-water-salt system, Pure Appl. Chem. 92 (10) (2020) 1663-1672.

[8] Peiming Wang, Andrzej Anderko, Robert D. Young, A speciation-based model for mixed-solvent electrolyte systems, Fluid Phase Equilib. 203 (1-2) (2002) 141-176.

[9] Kenneth Sanborn Pitzer, Activity Coefficients in Electrolyte Solutions, CRC Press, 1991, p. 542.

[10] Bjørn Maribo-mogensen, Kaj Thomsen, Georgios M Kontogeorgis, An electrolyte CPA equation of state for mixed solvent electrolytes, AICHE J. 61 (9) (2015) 2933-2950.

[11] Daniel K. Eriksen, Georgia Lazarou, Amparo Galindo, George Jackson, Claire S. Adjiman, Andrew J. Haslam, Development of intermolecular potential models for electrolyte solutions using an electrolyte SAFT-VR Mie equation of state, Mol. Phys. 114 (18) (2016) 2724-2749.

[12] Mark Bülow, Moreno Ascani, Christoph Held, ePC-SAFT advanced - Part I: Physical meaning of including a concentration-dependent dielectric constant in the born term and in the Debye-Hückel theory, Fluid Phase Equilib. 535 (2021) 112967.

[13] Willigert Raatschen, Allan H. Harvey, John M. Prausnitz, Equation of state for solutions of electrolytes in mixed solvents, Fluid Phase Equilib. 38 (1-2) (1987) 19-38.

[14] S. Balasubramonian, N.K. Pandey, R.V. Subba Rao, Comparison of activity coefficient models for the estimation of uranyl nitrate and nitric acid distribution coefficients in phosphoric solvent, Prog. Nucl. Energy 128 (September) (2020) 103472.

[15] Yi Lin, Antoon ten Kate, Miranda Mooijer, Javier Delgado, Philip Loldrup Fosbøl, Kaj Thomsen, Comparison of activity coefficient models for electrolyte systems, AIChE J. 56 (5) (2010) 1334-1351.

[16] Kamalodin Momeni, Seyed Hesam Najibi, Mohammad Thaghi Sadeghi, An evaluation of four electrolyte models for the prediction of thermodynamic properties of aqueous electrolyte solutions, 6 (2), 2017, pp. 46-60.

[17] F. Pérez-Villaseñor, S. Carro-Sánchez, G.A. Iglesias-Silva, Comparison among Pitzer-type models for the osmotic and activity coefficients of strong electrolyte solutions at 298.15 K, Ind. Eng. Chem. Res. 50 (18) (2011) 10894-10901.

[18] Javier Temoltzi-Avila, Gustavo A. Iglesias-Silva, Mariana Ramos-Estrada, Comparison among pitzer model and solvation models. Calculation of osmotic and activity coefficients and dilution enthalpy for single-electrolyte aqueous solutions, Ind. Eng. Chem. Res. 57 (31) (2018) 10684-10700.

[19] Chinghang Tong, Simon L. Clegg, John H. Seinfeld, Comparison of activity coefficient models for atmospheric aerosols containing mixtures of electrolytes, organics, and water, Atmos. Environ. 42 (21) (2008) 5459-5482.

[20] Congyu Zhang, Yubo Xing, Dongping Tao, A two-parameter theoretical model for predicting the activity and osmotic coefficients of aqueous electrolyte solutions, J. Solut. Chem. 49 (5) (2020) 659-694.

[21] Jiding Li, Hans Martin Polka, Jürgen Gmehling, A gE model for single and mixed solvent electrolyte systems. 1. Model and results for strong electrolytes, Fluid Phase Equilib. 94 (C) (1994) 89-114.

[22] Sourabh Ahitan, John M. Shaw, Quantitative comparison between predicted and experimental binary n-alkane + benzene phase behaviors using cubic and PC-SAFT EOS, Fluid Phase Equilib. 428 (2016) 4-17.

[23] Sukit Leekumjorn, Kristian Krejbjør, Phase behavior of reservoir fluids: Comparisons of PC-SAFT and cubic EOS simulations, Fluid Phase Equilib. 359 (2013) 17-23.

[24] M.T.M. Martinez, Maaike C. Kroon, Cor J. Peters, Modeling the complex phase behavior of methane, ethane and propane in an ionic liquid up to 11 MPa - A comparison between the PR EoS and the GC EoS, J. Supercrit. Fluids 101 (2015) 63-71.

[25] Wei Yan, Farhad Varzandeh, Erling H. Stenby, PVT modeling of reservoir fluids using PC-SAFT EoS and Soave-BWR EoS, Fluid Phase Equilib. 386 (2015) 96-124.

[26] Martin Due Olsen, Georgios M. Kontogeorgis, Xiaodong Liang, Nicolas von Solms, Comparison of models for the relative static permittivity with the e-CPA equation of state, Fluid Phase Equilib. (2022).

[27] Juan Sebastián Roa Pinto, Nicolas Ferrando, Jean Charles de Hemptinne, Amparo Galindo, Temperature dependence and short-range electrolytic interactions within the e-PPC-SAFT framework, Fluid Phase Equilib. 560 (2022) 113486.

[28] Pierre J. Walker, Xiaodong Liang, Georgios M. Kontogeorgis, Importance of the relative static permittivity in electrolyte SAFT-VR Mie equations of state, Fluid Phase Equilib. 551 (2022) 113256.

[29] Anders Schlaikjer, Kaj Thomsen, Georgios M. Kontogeorgis, Simultaneous description of activity coefficients and solubility with eCPA, Ind. Eng. Chem. Res. 56 (4) (2017) 1074-1089.

[30] Anders Schlaikjer, Kaj Thomsen, Georgios M. Kontogeorgis, eCPA: An ion-specific approach to parametrization, Fluid Phase Equilib. 470 (2018) 176-187.

[31] Li Sun, Xiaodong Liang, Nicolas Von Solms, Georgios M. Kontogeorgis, Analysis of some electrolyte models including their ability to predict the activity coefficients of individual ions, Ind. Eng. Chem. Res. 59 (2020) 11790-11809.

[32] Martin Due Olsen, Georgios M. Kontogeorgis, Xiaodong Liang, Nicolas von Solms, Investigation of the performance of e-CPA for a wide range of properties for aqueous NaCl solutions, Fluid Phase Equilib. 548 (2021) 113167.

[33] Saifuddin Ahmed, Nicolas Ferrando, Jean Charles de Hemptinne, Jean Pierre Simonin, Olivier Bernard, Olivier Baudouin, Modeling of mixed-solvent electrolyte systems, Fluid Phase Equilib. 459 (2018) 138-157.

[34] Juan Sebastian Roa Pinto, Pierre Bachaud, Tiphaine Fargetton, Nicolas Ferrando, Laurent Jeannin, Floriane Louvet, Modeling phase equilibrium of hydrogen and natural gas in brines: Application to storage in salt caverns, Int. J. Hydrogen Energy 46 (5) (2021) 4229-4240.

[35] Bjørn Maribo-Mogensen, Georgios M. Kontogeorgis, Kaj Thomsen, Comparison of the Debye-Hückel and the mean spherical approximation theories for electrolyte solutions, Ind. Eng. Chem. Res. 51 (14) (2012) 5353-5363.

[36] M.S. Wertheim, Fluids with highly directional attractive forces. I. Statistical thermodynamics, J. Stat. Phys. 35 (1-2) (1984) 19-34.

[37] Giorgio Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chem. Eng. Sci. 27 (6) (1972) 1197-1203.

[38] Peter Debye, Erich Hückel, Zur Theorie der Elektrolyte, Physikalische Zeitschrift 9 (1923) 185-206.

[39] M. Born, Volumen und Hydratationswärme der Ionen, Phys. Ges. 21 (13) (1920) 45-48.