The interplay between salt concentration and thermodynamic properties of electrolyte solutions is of vital importance for many physiological and technological applications, e.g. biochemistry, catalysis, energy storage, and materials science1,2. For example, the performance of energy storage devices, such as batteries and electrical double-layer capacitors depends entirely on the ionic conductivity, transference number, and electrochemical and thermal stability window of electrolytes. These electrolyte properties are determined by a complex interplay between the types of solvents, salts, and additives used, and are furthermore strongly dependent on the concentration and operating temperature of the environment3,4,5,6,7. Understanding the concentration and temperature dependence of the thermodynamic properties of an electrolyte for any given solvent/salt combination is therefore essential for selecting the proper electrolyte and further optimization of its properties.

In an electrolytic medium, the mean activity coefficient γ± relates to the change in chemical potential Δμ when going from an ideal system, in which all interactions (i.e. solvent-solvent, ion-solvent, ion-ion) are considered to be the same, to a real, nonideal system with component-specific intermolecular interactions8,9:

$$\Delta \mu =RT\ln {\gamma }_{\pm }$$

where R is the gas constant and T is the temperature of the solution. Knowledge of γ± allows for the derivation of the thermodynamic factor \(1+d\ln \left({\gamma }_{\pm }\right)/d\ln \left(c\right)\) (where c is the molar concentration), which is crucial for rationalizing and predicting the battery cell performance10,11,12,13. Additionally, γ± may be used to compute the freezing point depression when going from a pure solvent to the concentrated electrolyte14. Recently, the activity coefficient has been related to the upshift of the lithium metal anode potential15. Clearly, knowledge of activity coefficients of technologically relevant electrolytes is of primary importance, and many models have been developed to compute γ± as a function of electrolyte concentration and temperature for both weak and strong electrolytes2,14,16,17,18,19.

One of the most widely used theories to calculate γ± is the Debye-Hückel (DH) model, which provides a mathematical framework to quantify ion-ion interactions in dilute electrolytes2,20. The DH model predicts an asymptotic decay of γ± as the salt concentration c increases8,16,20. However, for many electrolyte systems, the measured γ± values show inflection points after which γ± increase with higher c, often exceeding values of 110,11,21. Therefore, the real behavior of electrolyte systems clearly deviates from the predicting capabilities of the DH model. This nonmonotonic behavior of real solutions has been associated with the ion-solvent interactions16, and may be restored by including the Born (B) solvation term22. Combining the DH and Born (B) equations (thereafter abbreviated as DH+B) has been shown to improve the predictive power of the model for activity coefficients of aqueous electrolytes17,23,24.

The main advantage of the DH+B model is that it is nearly parameter-free, except for the physically meaningful ionic radii and the static permittivity εr of the solution. This means that the DH+B model does not need to be fitted on experimental data, thereby circumventing the necessity for scarcely18,25 available experimental activity coefficients. Furthermore, the mathematical simplicity of the DH+B model (see later Eqs. (2) and (4)) makes it straightforward to understand and implement its governing equations in the treatment of electrolyte solutions.

Within the DH+B framework, previous studies made use of experimental values of εr and the Born solvation radii RB derived from experimental Gibbs solvation energies16,24,26. However, as εr and the Gibbs solvation energies (and thus RB) are often not available, in this work we set a rigorous protocol for predicting εr and RB from simulations, in particular, using molecular dynamics (MD) and density functional theory (DFT) calculations. This framework enables us to apply the DH+B model on any solvent/salt combination at any concentration and temperature of interest.

The aim of this work is twofold. Using our protocol, (i) we predict γ± for aqueous electrolytes and probe the performance of the DH+B model with computed values for εr and RB. (ii) We then apply the DH+B framework to nonaqueous organic electrolytes, which are highly relevant for commercial energy storage devices, including rechargeable batteries and supercapacitors. A useful guideline is provided to estimate the predictive accuracy of the DH+B model for technologically relevant electrolytes, for which the DH+B model may aid in understanding and predicting the thermodynamic properties.


Aqueous electrolytes

We begin by benchmarking the predictive capabilities of the DH+B model in reproducing reported experimental values of γ± of aqueous solutions with alkali metal chloride binary salts, with formula XCl (with X = Li, Na, K, Rb, and Cs) at 1 mol kg−1 (Fig. 1a). All DH+B data was generated with computed Born radii and with the solution static permittivity obtained from either available experimental data or MD simulations. In Fig. 1b the same approach is used for aqueous solutions of sodium halide salts, i.e. NaX (with X = F, Cl, Br, and I) at 1 mol kg−1. This concentration was chosen because it is close to the salt concentration commonly used in battery electrolytes and supercapacitors27. Supplementary Figure 1 explores a wider concentration range from the infinitely dilute situation to a highly concentrated regimes (4 mol kg−1).

Fig. 1: Mean activity coefficients γ±, Gibbs solvation energies and static permittivities εr of aqueous electrolytes.
figure 1

Panels (a, b) show the mean activity coefficient lnγ± and ΔG in kJ mol−1 at 1 mol kg−1 for XCl (X = Li, Na, K, Rb, Cs) and NaX (X = F, Cl, Br, I) in aqueous solution at 298 K. Experimental data taken from ref. 21. DH+B was computed by using εr from refs. 28,29, MD simulations or via Eq. (6). The \({\varepsilon }_{r}^{\exp }\) (solid line), \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) (dashed line) and \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) (dots) as a function of the concentration in mol kg−1 are shown in (c) for XCl and (d) for NaX. Dashed lines in (a, b) are shown for visual guidance.

The experimental γ± values become smaller for increasing cation sizes, following the order Li+ > Na+ > K+ > Rb+ > Cs+ (Fig. 1a). A smaller value for γ± corresponds to a more negative Gibbs free energy ΔG (right y-axes of Fig. 1a, b, see also Eq. (1)). A decrease in γ± with increasing cation size has been associated with the smaller cation binding more tightly to the solvent ions, resulting in a larger decrease in the static permittivity (Supplementary Fig. 2), and hence a decrease in ion solubility.

The dataset obtained with the DH+B model using the available experimental28,29 εr (i.e. \({\varepsilon }_{r}^{\exp }\)) results in all NaX electrolytes having a very similar values of γ±. However, at ~1.5 mol kg−1, the data points separate from each other, reproducing the experimental trend over the remaining concentration range up to 4 mol kg−1 (Supplementary Fig. 1). Replacing the experimental for the computational static permittivity εr (i.e. \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\)) in the DH+B model results in a poorer agreement between the predicted and experimental γ±. Indeed, \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) is almost identical for NaCl, KCl, RbCl and CsCl, but LiCl has the smallest, instead of the largest γ± in the XCl series.

These results can be understood by looking at the computed static permittivity (Fig. 1c). The MD simulations tend to overestimate εr for LiCl (i.e. \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) decreases too slowly), whereas the static permittivity of other alkali metal chlorides is underestimated (i.e. \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) decreases too rapidly). As a decrease in static permittivity results in an increase in \({\gamma }_{\pm }^{{{{\rm{B}}}}}\), underestimating the decrease in εr translates into \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) values that are too small for LiCl (see also Supplementary Fig. 3). Values of activity coefficients below 1 (i.e. a negative value for lnγ±) describe situations in which the ions prefer to be in the concentrated instead of infinitely dilute electrolyte, as is evident from the corresponding negative Gibbs free energy change. Hence, underestimating the activity coefficient for LiCl means that its preference to be in the concentrated electrolyte is overestimated.

When obtaining the static permittivity via Eq. (6), i.e. \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) (Fig. 1a), the NaX electrolytes recover a very similar value for γ± at 1 mol kg−1. Around ~ 1.5 mol kg−1, the data points separate from each other and correctly reproduce the experimental values for γ± over the remaining concentration range for each electrolyte (see Supplementary Fig. 1). As shown in Fig. 1c, the static permittivities obtained with Eq. 6 overlap almost perfectly with \({\varepsilon }_{r}^{\exp }\), which explains why the activity coefficients obtained with \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) are as good as those obtained with \({\varepsilon }_{r}^{\exp }\).

The DH+B model performs significantly better for the Na-halide series (Fig. 1b) compared to their Li-analogues. With NaF having the smallest and NaI having the largest values of γ±, the experimental trend is correctly reproduced by the DH+B model with \({\varepsilon }_{r}^{\exp }\). Furthermore, the values of \({\gamma }_{\pm }^{\exp }\) and \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) are in good agreement for NaCl, NaBr, and NaI. For NaF, the DH+B model gives a higher activity coefficient than \({\gamma }_{\pm }^{\exp }\), meaning that NaF preference to be in the concentrated electrolyte is underestimated by the DH+B model. Ion pairs of similar sizes, such as NaF, are less soluble than ion pairs of different sizes30. Furthermore, less soluble salts will show a less pronounced decrease in the static permittivity because fewer solvent molecules will participate in the solvation of the ions31. Should this be the case, we would expect lower predicted activity coefficients with the DH+B model. A possible explanation for the difference between \({\gamma }_{\pm }^{\exp }\) and \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) for NaF may be that the decrease in static permittivity of NaF is overestimated due to the highly localized charge on the bare fluoride ion, F.

When using static permittivities estimated from MD simulations \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\), the difference between computed values of γ± becomes smaller, but the relative order of the γ± values is still in agreement with the experimental trend. This improved performance (in comparison with the binary alkali metal chlorides) is mostly due to the computed values for \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\), which are much closer to their experimental values (Fig. 1d). When using \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) in the DH+B equations, the difference between the values for γ± increases, approaching the values for γ± as obtained with \({\varepsilon }_{r}^{\exp }\). Given the nearly perfect overlap between \({\varepsilon }_{r}^{\exp }\) and \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) (see Fig. 1d), these results are in line with expectations.

Nonaqueous electrolytes

It is instructive to analyze the contribution of εr to the activity coefficient associated with the DH term (\({\gamma }_{\pm }^{{{{\rm{DH}}}}}\)) and the Born term (\({\gamma }_{\pm }^{{{{\rm{B}}}}}\)), as nonaqueous electrolytes often display a lower static permittivity than their aqueous counterparts. Figure 2 displays the values of \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) as a function of the salt concentration and change in the solution static permittivity \({\varepsilon }_{r}^{{{{\rm{soln}}}}}\), respectively. In Fig. 2, each color corresponds to a different static permittivity of the solvent \({\varepsilon }_{r}^{{{{\rm{solv}}}}}\). On one hand, from Fig. 2a, \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) becomes smaller with a lower static permittivity, and this decrease becomes more pronounced for smaller values of εr. On the other hand, for the Born term, (Fig. 2b), \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) increases exponentially with a decrease in \({\varepsilon }_{r}^{{{{\rm{soln}}}}}\), and this increase in \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) becomes more pronounced for electrolytes with low solvent’s static permittivity \({\varepsilon }_{r}^{{{{\rm{solv}}}}}\). As a consequence, both \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) become increasingly sensitive to the quality of the static permittivity of the solution with lower permittivity electrolytes.

Fig. 2: The effect of the static permittivity on the Debye-Hückel and Born activity coefficients.
figure 2

Panel (a) shows \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) as a function of the concentration for LiCl at 298 K. The data were generated by using a constant εr (instead of the concentration-dependent εr) in the Debye-Hückel equation with a step size of 2 in the εr range of 81 to 3. Each color corresponds to a different value for \({\varepsilon }_{r}^{{{{\rm{solv}}}}}\), and the black line corresponds to \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) when using the concentration-dependent εr for aqueous LiCl. In panel (b) the \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) is depicted as a function of the change in the solution static permittivity \({\varepsilon }_{r}^{{{{\rm{soln}}}}}\). Different colors correspond to the static permittivity of the solvent \({\varepsilon }_{r}^{{{{\rm{solv}}}}}\) (i.e. at infinite dilution). The data were generated with a step size of 2 in the εr range of 81 to 3. Plots extending to larger values of γ± are given in Supplementary Fig. 4.

We then move on to the application of the DH+B model to nonaqueous electrolytes, which are prevalent in Li and Na-ion batteries and supercapacitors. In Fig. 3, the activity coefficients γ± (left y-axes) and solution’s static permittivity as computed with MD \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) (right y-axes) are plotted as a function of the salt concentration. Figure 3a charts the behavior of LiClO4 (a common salt used in Li-ion batteries) in three different solvents, namely acetonitrile (ACN, \({\varepsilon }_{r}^{{{{\rm{solv}}}}}=36.0\) at 298 K), acetone (AC, \({\varepsilon }_{r}^{{{{\rm{solv}}}}}=21.0\) at 298 K), and dimethyl carbonate (DMC, \({\varepsilon }_{r}^{{{{\rm{solv}}}}}=3.1\) at 298 K). As shown in Fig. 3a, ACN displays the highest value of \({\gamma }_{\pm }^{\exp }\), whereas DMC has the lowest \({\gamma }_{\pm }^{\exp }\) over the whole range of concentration (solid lines). This trend is correctly reproduced by the DH+B model (dashed lines). Furthermore, \({\gamma }_{\pm }^{\exp }\) and \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) are in good agreement for the ACN and AC-based electrolytes. However, for DMC, the agreement between \({\gamma }_{\pm }^{\exp }\) and \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) appears extremely poor – this trend appears more evident when plotting the data in logarithmic scale (see Supplementary Fig. 5). For example, at 1 mol kg−1 \({\gamma }_{\pm }^{\exp }=0.002\), whereas \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}=3.61\times 1{0}^{-16}\), which equates to a difference in ΔG of 72.7 kJ mol−1. This is a consequence of εr increasing (instead of decreasing) with higher salt concentrations, an effect that has been previously reported for low permittivity electrolytes14,32,33. As a result, both \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\ll 1\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\ll \,1\) (Supplementary Fig. 5), resulting in extremely small total values of \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\).

Fig. 3: Mean activity coefficients γ± and static permittivity \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) at 298 K as a function of the concentration in mol kg−1.
figure 3

Panel (a) shows LiClO4 in ACN, AC, and DMC. Panel (b) shows LiBr, LiCl, and LiNO3 in DMSO. Panel (c) shows LiBr, LiPF6, and LiTFSI in ACN. Experimental data obtained from ref. 43 are used in panel (a) and ref. 25 are used in (b, c). Solid lines: \({\gamma }_{\pm }^{\exp }\). Dashed lines: \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\). Dots: \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\).

We then analyze the effect of changing the type of salt while keeping the solvent constant, starting from LiBr, LiCl, and LiNO3 in dimethyl sulfoxide (DMSO, \({\varepsilon }_{r}^{{{{\rm{solv}}}}}=46.5\)). From Fig. 3b, the experimental γ± (solid lines) is largest for LiBr and smallest for LiNO3. The opposite trend is obtained with the DH+B model up to 1.5 mol kg−1 (dashed lines). These results can be understood from \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) used in the DH+B model. For LiNO3, the decrease in εr is most pronounced, presumably due to the larger anion size, whereas the decrease in εr is least pronounced for LiBr (see dots in Fig. 3b). While this larger decrease in εr results in both a smaller \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and larger \({\gamma }_{\pm }^{{{{\rm{B}}}}}\), the effect on \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) is more pronounced (Supplementary Fig. 5). Hence, the resulting trend in \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) is completely determined by the Born term, which increases at a higher rate for the electrolyte subjected to a stronger decrease in εr.

In Fig. 3c, we applied the DH+B model to ACN solutions of LiBr, LiPF6 and lithium bis(trifluoromethanesulfonyl)imide (LiTFSI), with the latter two being common salts in Li-ion batteries. The experimental γ± is highest for the salt with the largest anion (i.e. LiTFSI) and lowest for the salt with the smallest anion (LiBr) over the whole concentration range. This experimental trend is correctly reproduced by the DH+B model, and is dictated by both the \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) terms (Supplementary Fig. 5). The LiBr and LiPF6 electrolytes share approximately the same values of \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\), whereas LiTFSI has much higher values for \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) over the whole concentration range. At first glance, this trend of \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) may appear surprising, as LiTFSI, together with LiPF6, displays the lowest εr, whereas one should expect a general decrease of \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) with lower values of εr (Fig. 2). The reason for this unexpected trend is that \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) also decreases with a smaller ionic radius ri. With Br having the smallest (~1.95 Å) and TFSI having the largest (~3.27 Å) radius among the salts considered here, the trend in \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) is partly dictated by the ionic radius (Supplementary Fig. 6). Interestingly, variations in ri on \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) become more pronounced for electrolytes with a lower static permittivity (Supplementary Fig. 7). In practice, this signifies that \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) becomes more sensitive to the choice of ri for lower permittivity electrolytes. The ion-solvent term \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) is again dictated by the static permittivity, with a more pronounced decrease in εr leading to a larger increase in \({\gamma }_{\pm }^{{{{\rm{B}}}}}\).

Temperature dependence

We assessed the performance of the DH+B model with computationally derived εr and RB at higher temperatures T (313 and 353 K) for NaCl in H2O. Figure 4 shows the activity coefficients and the static permittivities as a function of the salt concentration. In Fig. 4a, an increase in temperature results in a decrease of the experimental γ±34; this trend has also been observed for other electrolytes2,35,36. The DH+B model with \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) reproduces this temperature dependence correctly. Furthermore, the computed values for \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) are close to the experimental values of γ±, and the monotonic behavior is correctly captured by the DH+B model for both temperatures.

Fig. 4: Temperature dependence of the mean activity coefficient and static permittivity.
figure 4

Panel (a) shows the mean activity coefficients γ± for NaCl in H2O as a function of the concentration in mol kg−1 at 313 K (blue) and 353 (red) K. Experimental data are obtained from ref. 34 and represented by dots. Solid lines represent the DH+B data. Panel (b) shows the static permittivity as a function of the concentration in mol kg−1 obtained with MD simulations.

In the case of LiCl in H2O, we also computed the activity coefficients as a function of temperature. Despite the overestimation of experimental static permittivity by \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) (see Fig. 1c), the correct trend in γ± is again correctly recovered, see Supplementary Fig. 8.


In general, our analyses demonstrated that the DH+B model is highly sensitive to the quality of the electrolyte static permittivity εr, and this sensitivity increases with a lower value of εr. Therefore, an important point of consideration is the error in \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\), which directly influences the resulting predictions of \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\). The calculated values for \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) are subjected to systematic errors, arising, for example, from force fields (FFs) used for the organic electrolytes, which are not specifically designed to reproduce the static permittivity. We have partly accounted for these systematic errors by introducing a posteriori correction factors to the atomic charges. The atomic charges of the organic solvents were multiplied by a correction factor such that the experimental static permittivities of the pure solvent were correctly reproduced. The same correction factors were used for the solvent molecules in the concentrated electrolytes. The atomic charges of the salts were multiplied by a correction factor such that their dipole moment became equal to the dipole moment as obtained by DFT. These correction factors were only applied after completing the MD simulations and hence did not affect the molecular trajectories. More information is given in Supplementary Methods 3.

The accuracy of \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) may be further improved by parameterizing specific FFs on experimental values of εr of the pure solvents (whose data is widely available), in line with the approach adopted for the TIP4P/ε FF for water37. For the salts, the parameterization of FFs can be achieved at a single salt concentration. An optimized FF is then applied to any salt concentration of interest. We emphasize that this workflow substantially reduces the amount of experimental data necessary (to be used for the force field parameterization) in comparison with using the experimental values of εr directly in the DH+B equations.

An approach worth investigating is to parameterize the salt FFs on available experimental data for εr (e.g. aqueous electrolytes), and later use the same FFs when dissolving the salt in other solvent types. While the parameterization of dedicated FFs is beyond the scope of this study, a similar approach may be applied through Eq. (6). To demonstrate this concept, we utilized the values of β and λ (the parameters of Eq. (6)) optimized for aqueous electrolytes (see Supplementary Table 1) to calculate the static permittivities \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) of the nonaqueous electrolytes. Specifically, we calculated \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) for LiBr, LiCl, and LiNO3 in DMSO, which are the same systems as studied in Fig. 3b and for which experimental values for εr are available in H2O. The results are visualized in Fig. 5, with the activity coefficient γ± plotted on the left y-axis, and the static permittivity \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) plotted on the right y-axis as a function of the concentration in mol kg−1. The y-axes in Figs. 5, 3b share the same scale to enable direct comparison between the two sets of data.

Fig. 5: Mean activity coefficients γ± and static permittivity \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\) at 298 K as a function of the concentration in mol kg−1 for LiBr, LiCl, and LiNO3 in DMSO.
figure 5

The parameters used in Eq. (6) are the same as those obtained for the corresponding salts in H2O (see Supplementary Table 1). Experimental data obtained from ref. 25. Solid lines: \({\gamma }_{\pm }^{\exp }\). Dashed lines: \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\). Dots: \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\).

Using the static permittivity derived from Eq. (6), we predicted the correct trend in \({f}_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) (i.e. LiNO3 < LiCl < LiBr) over the whole concentration range, as depicted in Fig. 5. Values of \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) are in close agreement with the experimental values \({\gamma }_{\pm }^{\exp }\). This is a significant improvement in comparison with using static permittivities derived from MDs \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\), which gave an incorrect trend in the value of \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) (Fig. 3b).

The lack of experimental data makes it difficult to quantify any remaining errors in \({\varepsilon }_{r}^{{{{\rm{MD}}}}}\) and \({\varepsilon }_{r}^{{{{\rm{Eq.}}}}\,6}\). Furthermore, the kinetic depolarization effect in experimental measurements is complex to estimate, making a direct comparison between experimental and calculated values for εr challenging2,38, see Supplementary Discussion 1. The observation that εr does not only decrease but at some points also increase with an increase in salt concentration has recently been discussed by Yao et al. for several nonaqueous electrolytes31. They show that an increase in εr with higher salt concentrations is caused by the dipole moment of the associated salt, which can be larger than the dipole moment of the solvent molecules. For the electrolytes studied in our work, we also find that the salts have larger dipole moments than the solvent molecules (see Supplementary Fig. 9), which may lead to an increase in εr with increasing salt concentrations. However, to the best of our knowledge, this nonmonotonic behavior in εr has yet to be confirmed by experimental studies.

It is insightful to look at the general effect of a given error in εr on the activity coefficients \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\). We emphasize that the remaining part of the discussion applies to the static permittivity in a general sense, regardless of the method by which its value is obtained. For the DH term, a lower static permittivity gives a lower \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\), and this lowering becomes more pronounced with a lower εr (Fig. 2). Therefore, a given error in εr will have more pronounced effects on electrolytes with low static permittivity. In specific solvents prevalent in battery applications, such as DMC, we demonstrated that even small (i.e. < 1) deviations in εr translate into errors in \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) of multiple orders of magnitude (e.g. 4 orders of magnitude for 1 mol LiCl when εr changes from 4 to 3, see Fig. 2). The increased sensitivity of \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) to the quality of εr is further demonstrated in Supplementary Fig. 10.

For the Born term, the quality of \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) is dictated by both the static permittivity of the solvent (\({\varepsilon }_{r}^{{{{\rm{solv}}}}}\)) and the change in εr upon the addition of the salt (\({\varepsilon }_{r}^{{{{\rm{soln}}}}}\)). The effect of a given error in εr on the value of \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) is visualized in Supplementary Fig. 11. In line with Fig. 2, the uncertainty in \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) increases with larger changes in εr, and this effect becomes more pronounced for electrolytes with lower permittivity solvents. For example, for 2 mol kg-1 LiClO4 in AC, the static permittivity of the solution has a value of 19.7, which corresponds to a change in εr of only 1.0 from pure AC. Given an error of ± 3 in the static permittivity, the lower and upper bounds in \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) (i.e. the values for \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) generated with εr − 3 and εr + 3, respectively) are 0.7 and 10.3, which corresponds to an uncertainty in \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) of 9.6 (see Supplementary Fig. 11). These results underline the increased sensitivity of the Born term for the quality of the static permittivity for lower permittivity solvents.

The resulting inaccuracies on \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) caused by the errors in εr will partly cancel each other out, as an increase in εr leads to a simultaneous decrease in \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and increase in \({\gamma }_{\pm }^{{{{\rm{B}}}}}\). However, the extent to which \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) cancel out is difficult to predict, as this depends not only on the value of εr but also on the change in εr with respect to the pure solvent. An additional factor that affects the performance of the DH+B model is that any change in the ionic parameters ri and \({R}_{i}^{{{{\rm{B}}}}}\) has a more pronounced effect when εr is lower, which introduces an additional (potentially large) uncertainty in \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\). Overall, the increasingly large errors in \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) and \({\gamma }_{\pm }^{{{{\rm{B}}}}}\) introduced with lower static permittivities means that DH+B-type models are less robust for low-permittivity electrolytes. Usually, electrolytes in energy storage devices carry higher permittivity solvents (e.g. EC:DMC, EC:DME, EC:PC:DMC, where PC=propylene carbonate)3,4,5,7 to adequately dissolve the salts.

Since the performance of the DH+B model does not only depend on εr but also on the extent of change in εr with increasing salt concentration, the overall accuracy of the DH+B model appears strongly system-specific. Therefore, Fig. 2 provides a useful guideline to estimate the predictive accuracy of the DH+B model for technologically relevant electrolytes.

In summary, we have applied the extended Debye-Hückel model in combination with the Born equation (DH+B) on a series of aqueous and organic electrolytes, which are highly relevant for energy storage applications. The experimentally measurable variables that enter the DH+B equations, namely the Born solvation radius RB and solution’s static permittivity εr, were obtained via computer simulations. This workflow circumvents the need for experimental data, which are difficult to determine and not always available, thereby allowing the use of the DH+B model on any electrolyte at any concentration and temperature of interest.

The DH+B model with computationally obtained parameters performs mostly satisfactorily for aqueous electrolytes and becomes increasingly inaccurate for nonaqueous electrolytes bearing low static permittivities, such as DMC, DME, and DEC. The accuracy of the resulting activity coefficients \({\gamma }_{\pm }^{{{{\rm{DH}}}}+{{{\rm{B}}}}}\) is affected by both the static permittivity εr of the solvent as well as the change in εr at technologically relevant concentrations. Overall, the DH+B model may aid in understanding and predicting the properties of relevant aqueous and organic electrolytes, which remains an important task in paving both scientific and technological progress.


The Extended Debye-Hückel model

We have used the extended Debye-Hückel (EDH) model for describing the chemical potential μ associated with the ion-ion interactions in solution. Via Eq. (1), the activity coefficient \({\gamma }_{\pm }^{{{{\rm{DH}}}}}\) is given by

$$\ln {\gamma }_{\pm }^{{{{\rm{DH}}}}}=\frac{-\zeta k}{{\varepsilon }_{r}{k}_{B}T}\left(\frac{1}{1+ka}\right)\,{{{\rm{with}}}}\,\zeta =\frac{{z}_{i}^{2}{e}_{0}^{2}}{8\pi {\varepsilon }_{0}}$$

where zi, e0, ε0 and kB are the charge number, the elementary charge, the vacuum permittivity, and the Boltzmann constant, εr is the relative static permittivity, and a is the ionic size. k is the inverse of the Debye-Hückel length:

$$k=\sqrt{\frac{{N}_{A}{e}_{0}^{2}}{{\varepsilon }_{0}{\varepsilon }_{r}{k}_{B}T}\mathop{\sum}\limits_{i}{c}_{i}{z}_{i}^{2}}$$

where NA is Avogadro’s constant and ci is the molar concentration. Here, the ionic size a is the distance of closest approach between two ions, which is approximated as the sum of ionic radii of the cation and anion (Supplementary Table 2). εr was set to the static permittivity of the solution, which depends on the salt concentration c and the temperature T. Recently, Sun et al. have shown that the performance of EDH is superior over the full DH equations in predicting the activity coefficients for 14 aqueous electrolytes24.

The Born equation

The ion-solvent interactions were described using the Born equation, which gives the electrostatic energy change when moving a charged, spherical ion from a vacuum to a continuous dielectric medium. Moving from an infinitely dilute regime to a concentrated solution, the associated activity coefficient \({\gamma }_{i}^{{{{\rm{B}}}}}\) is:

$$\ln {\gamma }_{i}^{{{{\rm{B}}}}}(c,T)=\frac{\zeta }{{R}_{i}^{B}(T)}\left(\frac{1}{{\varepsilon }_{r}^{{{{\rm{soln}}}}}(c,T)}-\frac{1}{{\varepsilon }_{r}^{{{{\rm{solv}}}}}(T)}\right)$$

where \({R}_{i}^{{{{\rm{B}}}}}\) is the Born radius, and \({\varepsilon }_{r}^{{{{\rm{soln}}}}}\) and \({\varepsilon }_{r}^{{{{\rm{solv}}}}}\) are the static permittivity of the solution and the solvent, respectively. DFT calculations were used to compute the Gibbs solvation energy, from which the Born radius RB was obtained via the Born equation. A comparison between the calculated Born radii and Born radii derived from experimental Gibbs solvation energies is given in the Supplementary Discussion 2.

The Static permittivity

The static permittivity of the solution εr was predicted with MD simulations by computing the cumulative average of the dipole moment M39, using Eq. (5):

$${\varepsilon }_{r}=1+\frac{\langle {M}^{2}\rangle -{\langle M\rangle }^{2}}{3{\varepsilon }_{0}V{k}_{B}T}$$

where V is the volume of the simulation box. We have also implemented the following relationship between εr and salt concentration c developed in Ref. 40:

$${\varepsilon }_{r}(c)={\varepsilon }_{r}^{{{{\rm{solv}}}}}{e}^{-\beta \cdot \arctan (\lambda {{{\rm{c}}}})}$$

where β and λ were fitted on three experimental data points per electrolyte. Notably, Eq. (6) reliably predicts the correct asymptotic behavior of εr(c) as the concentration of the solute is increased40. Full methodological details are given in Supplementary Methods 14.