Ion Association Constants for Lithium Ion Battery Electrolytes from First-Principles Quantum Chemistry

We provide a quantum chemical computational framework to calculate ion association constants relevant to lithium ion battery electrolytes. We compare our method to reported experimental values as the solvent, cation, and anion are varied. For solvent, anion, and cation variations, the standard errors are respectively 0.2 eV, 0.12 eV, and 0.11 eV for the chosen data set, where Pearson correlation values are all above 0.92. © The Author(s) 2019. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI

The degree to which a salt is associated directly affects the solvation structure, the electrochemical stability and the transport properties of electrolytes.In this article we present a simple methodology to estimate the ion association constants K A of 1-1 non-aqueous electrolytes using ab initio computational methods, which we then compare against experimentally reported values relevant to lithium ion battery (LIB) systems.
Conventional wisdom holds that ion association is deleterious as it reduces the conductivity of LIB electrolytes. 1,2However, newly emerging LIB electrolyte formulations, such as those described as low permittivity [3][4][5][6] or superconcentrated, 7,8 are promising alternative candidates.In these active research areas, associated salt plays unconventional roles.For example, neutral associated salt complexes may aid in surface passivation 7 or help dissociate salt via their contributions to the overall electrolyte permittivity. 9A concise and validated methodology to calculate the K A s of a salt is therefore of practical and renewed interest.
1][12] Work toward performing analogous computations for nonaqueous electrolytes with cations other than H + , 13 however, are far less established.While binding energies have been computed for a variety of battery-relevant electrolytes, 2,14,15 the methods used typically focus on trends in the electronic energy of the gas-phase species often omitting the entropic contributions or solvation effects necessary for comparison to experimental K A values. 14o the best of the authors' knowledge, no previous work reports computed ionic K A values comparable to available experimental values and relevant to LIB systems.Herein, we provide a computational methodology for directly calculating such K A s.
We provide a straightforward method that is inexpensive and can be used for predictive screening for electrolyte selection.We simplify the possible equilbria via the assumption of contact-ion pairs as the relevant ion associated species.We evaluate this approximation by comparing the results to experimental K A s.Although simplistic, we find that this assumption provides reliable trends in the prediction of association behavior.
First, we introduce the relevant equilibria relations for ionassociation.We then discuss the computational methods used.The choice of experimental data set is motivated, to which we compare computed K A s. Finally, we discuss limitations of the model, possible improvements, and example LIB electrolyte systems where knowledge of the extent of ion association may be particularly relevant.

Theory
For a cation M + and anion N − in solution, an equilibrium may be established with the neutral contact-ion pair MN: The change in Gibbs free energy of the above reaction can be denoted as G A , and for infinite dilution as G 0 A .G 0 A can be related to the association constant at infinite dilution K 0 A via the following relationship: 16 If ideality is assumed, then K 0 A = K A , and K 0 A can be directly related to the fraction of free ions α at a given concentration c via the mass action law: 16 We note that that for appreciable concentrations, e.g. up to 1 M, there exist semi-empirical expressions for the activity coefficients of the various salt species, which would allow construction of speciation diagrams, 17 the details of which are outside the scope of this paper.

Partial explicit solvation shell.-In order to calculate the free energy upon association G
we account for the various terms for a given species.We can relate the total Gibbs free energy G 0 species to the electronic energy E electronic , the solvation free energy δG solv , the thermal energy E thermal , and the entropy of the species S via the following expression: The terms in Equation 4 are here calculated using quantum chemistry.E electronic includes the electronic energy for the species with the geometry optimized in the liquid phase using the implicit solvent model PCM (polarizable continuum model). 18The PCM models the liquid phase via a structureless dielectric medium surrounding a solute species, and also allows for the approximative calculation of δG solv .The values for the static solvent permittivities used here were taken from the literature. 19,20Figure 1a) shows the scheme used for the calculation of E electronic + δG solv , where an explicit solvent molecule is added for the electronic structure calculations of MN and M + to more accurately reflect the solvation environment. 12For N − simply the implicit solvent model is used, with no explicit solvent molecule (Figure 1b).Here we assume that explicit solvent molecules are not necessary to adequately represent the solvation environment of the anion, an assumption which we discuss in the Results section.In order to obtain S and E thermal for each respective solvated species, implicit solvent calculations were undertaken for each salt species (M + , N − , and NM) with no explicit solvent molecule (Figure 1b).The explicit solvent molecule is avoided in the calculation of S since entropic contributions related to solute-solvent interactions (such as librational modes) are, in principle, already accounted for by δG solv .S includes a translational contribution for the free monoatomic cations M + , and for N − and NM, a translational, vibrational, and rotational contribution as well. 21,22We note that the translational contribution calculated from the Sackur-Tetrode equation 23 is here evaluated for the appropriate standard concentration (1 mol/L), and that vibrational frequencies below 100 cm −1 are adjusted to 100 cm −1 to avoid spurious entropy contributions. 13,24,25Furthermore, all calculations to obtain S and E thermal were undertaken using the PCM with a single dielectric constant, that of 2-methoxyethanol ( = 16.96), as variations calculated for S and E thermal across various are negligible.A detailed example for the application of equation 4 is provided in the supplementary information (SI).
All quantum chemistry calculations were undertaken with Gaus-sian16 software. 22The wb97xd hybrid functional with Grimme's empirical 2D dispersion 26 was employed.It has been shown to provide adequate geometries and electronic energies at a reasonable cost. 26The def2tzvp basis set was employed, 27 and it was preferred to Pople-type basis sets since def2 basis sets by default include effective core potentials, which are necessary for heavy alkali atoms (Rb and Cs).For solvation effects, IEFPCM 18,28 was used with the default cavity construction, 28 which utilizes the UFF force field.Corrections to the thermodynamic data were undertaken with the Good-Vibes analysis code. 25 Initial configurations were found via conformational analysis [29][30][31][32] and energy minimization using the MacroModel package 33 with the semi-empirical OPLS forcefield. 34ll explicit solvation shell.-Electrostaticenergies calculated with the above presented model are only accounted for by a single explicit solvent molecule and a structureless dielectric medium.Increasing the number of explicit solvent molecules in the electronic structure calculations may be helpful, [35][36][37][38] although significantly more expensive computationally.The treatment of explicit solvent molecules S may lead to release or capture upon association of x solvent molecules upon association.Consequently, the following equilibria relation can be established for K A : The number of primary solvation shell solvent molecules n for M + S n and n − x for MNS n−x can be found variationally.Here, since the primary solvation shell is treated explicitly, S and E thermal for each species can be found simply via the Sackur-Tetrode equation (e.g. with translational, vibrational and rotational terms), with similar vibrational frequency corrections as those mentioned in the previous subsection.Here, S for the solvent S is calculated with the standard concentration of the neat solvent.Validation.-In order to evaluate the accuracy and validity of the method presented in the Partial explicit solvent shell section, we compared our calculated K A s to reported K A s, which have been measured via conductometric or dielectric techniques 16 at room temperature.Such techniques allow estimation of the concentration of salt species and relevant equilibrium constants.Conductivity measurements allow inference of thermodynamic data such as the ion-pairing association constant due to the measurable decrease in conductivity arising from the presence of neutral ion-pairs. 16Dielectric spectroscopy measurements also allow the inference of association constants since the frequency dependant dielectric response is a function of the 'cooperative motions of all dipolar species', 16 e.g.solvent and associated salt complexes. 64The chosen K A s are all representative of systems at infinite dilution, and as such our computational model does not add corrections for non-idealities: henceforth K 0 A = K A and G 0 A = G A .LiClO 4 was chosen as the salt for the data set with varying solvent due to the numerous available measured K A s. 19 Since variations in cation or anion can result in K A differences sometimes smaller than the systematic error of the experimental method, our choices favor studies where the relevant K A s were found in the same study.All references and experimental values are listed in Table I.We note that there exists a large body of experimental work studying ion-pairing in battery relevant electrolytes at concentrations ∼1 M [50][51][52][53][54] but that these do not report infinite dilution equilibrium constants: in fact the activity coefficients of the salt species at the concentrations studied in those studies are significant, can vary widely from system to system 16,17 and would not be comparable to results from quantum chemistry alone.Development of a methodology to account for the effects of finite concentration on our computed equilibrium constants, for example using activity coefficients from Debye-Huckel theory, 16,17 is the subject of further work.

Results
Partial explicit solvation shell.-Figure 2 shows the computed K A as a function of experimentally reported K A for the LiClO 4 salt dis- solved in various solvents, using the methods described in the Partial explicit solvent shell methods section.The experimental and computed trends agree, with a Pearson correlation value of 0.96, while the standard error of G A is 0.21 eV.The dielectric constant is generally a reliable descriptor for relative trends in association between solvents which are of considerably different polarity, 16,19 and its effect is directly encompassed by the PCM used in our model.However, the permittivity as a descriptor may fail between solvents of similar permittivity.A notable example is glyme ( = 7.0) and tetrahydrofuran ( = 7.4), where glyme, despite its lower permittivity, exhibits a lower K A for LiClO 4 .6][57] Consistent with this rationale, the inclusion of an explicit solvent molecule in the calculation allows for the relative K A ordering between both solvents to be predicted.In Figure 2, we note one striking outlier, dimethyl carbonate (DMC), which, although accurately reported to exhibit the highest K A among the studied solvents, presents an overestimated calculated G A by 0.6 eV.This may be due to an imprecise calculation of δG solv for the species, which we discuss in the following section.
Figure 3 shows the computed K A as a function of experimentally reported K A for salts of varying anion dissolved in propylene carbonate (PC), glyme and tetrahydrofuran (THF) solvents.The relative K A ordering is fairly reproduced -in this case between varying anions for a single solvent.However, K A (LiPF 6 ) is calculated to exhibit lower association as compared to K A (LiTFSI) and K A (LiAsF 6 ) in PC as compared to experimental results, where K A (LiAsF 6 ) < K A (LiTFSI) < K A (LiPF 6 ) < K A (LiClO 4 ) < K A (LiBF 4 ) < K A (LiTriflate).For glyme and THF, the relative trend of K A (LiAsF 6 ) < K A (LiClO 4 ) < K A (LiBF 4 ) is accurately predicted.We note that binding energies calculated by previous computational studies also generally agree with this trend. 14lthough it is likely that explicit hydrogen bonding plays a role for ClO − 4 in the 2-methoxyethanol solvent, the computed K A is comparable to the other cases in the studied data set.For the data in Figure 3, the standard error of G A is 0.12 eV and the Pearson correlation value is 0.96.We note that previous computational work, e.g. on pK A s, involves errors of similar magnitude. 10,12,13PF − 6 exhibits a much larger volume than BF − 4 , resulting in a lower volumetric charge density, and both properties prompt a weakened electrostatic interaction between oppositely charged ions for LiPF 6 over LiBF 4 .However, the argument invoking size fails to explain why triflate leads to a higher K A than the smaller BF − 4 .In general, quantum chemistry accounts for charge density as well as specific interatomic interactions, the latter being evidently important for LiTriflate.Regarding solvent-anion interactions, we note the following: conventional organic solvents typically exhibit a well-defined negative end of the dipole, but a less well-defined positive end of the dipole, i.e. the charge is less localized. 58,59This would entail that anions interact in a weaker fashion with solvents than cations do, and hence differences in K A between various anions are unlikely to be dominated by solvent-anion interactions.
Figure 4 shows the computed K A as a function of experimentally reported K A for ClO − 4 salts of varying alkali metal cation dissolved in acetonitrile (AN) and 2-methoxyethanol solvents.Here the experimental trend is well reproduced by our computational model.As previously reported, 47 increasing alkali metal cation size results in decreasing solvent-cation electrostatic interactions, which in turn correlates with higher K A s. Two minor deviations from the relative experimental association strengths are noted.The experimental K A s indicate that Li + -AN is more associated than Na + -AN, in disagreement with the computed values.However, as shown in Table I, the experimental data point for Li + -AN is taken from a different study than the other M + -AN values: between different experimental reports there are often errors on the order of 0.02 eV, 19,48 which is comparable to the measured difference in the K A s of Li + -AN and Na + -AN.Previous computational work by Jonsson and Jonhannsen 14 found that Na salts typically bind more strongly than Li salts, in agreement with the present work.Secondly, the computed K A s for Rb + -AN and K + -AN suggest that the Rb + analog is less associated than the K + counterpart, in disagreement with experimental values.Overall, for the data shown in Figure 4, the standard error between experimental and computed K A s is 0.11 eV and the Pearson correlation value is 0.93.Here, the differences between varying cations, although correct in their relative ordering (besides the above mentioned cases), are exaggerated.This may be due to inaccurate solvation energies of the various cations, which we discuss in the Full explicit solvation shell.-Asnoted in the previous section, the DMC-LiClO 4 K A showed significant disagreement with experiment (e.g.0.6 eV).To investigate the role of solvent number we employed a full explicit solvation shell of solvent molecules for the calculation of the DMC-LiClO 4 K A , as detailed in the Methods section.By minimizing the E electronic + δG solv , solvent numbers of n = 6 and n − x = 5 were found for the free ion and the CIP, respectively.Using additional solvent molecules, the overestimation error in G A decreased from 0.6 eV to 0.2 eV.Previous work has shown that the PCM alone is not adequate for modeling DMC due to significant but unaccounted quadrupolar interactions. 60To some extent, explicit solvents (sometimes referred to as cluster-continuum), can help correct the solvation energy errors. 37,53,61We note that in the present work's methodology, the preparation of initial structures followed a conformational analysis with a semi-empirical force field (see Methods section), and that further corrections may be made via a conformational analysis with a more accurate method (e.g.first principles) and more advanced statistical sampling. 61ence, we surmise that for very low permittivity media (e.g.< 5), the PCM may require enhancement via explicit solvent shells, where otherwise solute-solvent interactions may not be well accounted for. 62

Discussion and Conclusions
In the case of alkali cations, shown in Figure 4, although the relative trend in association strength is generally reproduced, the differences in K A are exaggerated.We believe this is due to limitations of the PCM at accounting for differences in solvation energies between various alkali metal cations.We speculate that the Gaussian 22 default parameters employed here for the PCM cavity construction 28 may require additional tailoring. 63Nonetheless, despite the apparent deviations, the standard error is modest (0.11 eV) with the largest error being 0.24 eV.Finally, we note that there may exist an equilibrium between various types of associated salt, e.g.solvent-separated ion pairs, 16 double solvent-separated ion pairs, 64 triple ions, 19 dimers, 9 etc., which could be included in a more elaborate model than the present one.
There exists a design space for electrolytes utilizing the properties of various salts, where knowledge of preferential association can be informative.Notably, LIB electrolytes comprised of binary mixtures of salt, unavoidably with varying K A , are an active area of research.For example, a combination of LiBF 4 and LiPF 6 has shown promise in improving Coulombic efficiency of Li-ion cells. 65Although nonidealities are left unaccounted for in the present model, preferential association will persist for a certain concentration range above the dilute limit. 66,679][70] In low permittivity electrolytes, where polar associated salt species can help increase solution permittivity via polar contact-ion pairs, 3,5,9 the addition of a preferentially associated salt may be useful, as long as formation of larger but less polar aggregates (e.g.quadrupolar dimers) is minimized.
We presented and evaluated a methodology which was shown to be accurate in its predicted trends between various association constants K A .Although limited in its ability to obtain accurate K A differences across various cations, it was shown to be reliable for varying anions and solvents for a same cation.We hope that the current methodology can help guide future electrolyte salt selection for LIBs, as well as serve as a building block for more advanced computational models.

Figure 1 .
Figure 1.Scheme to calculate each of the components of the free energy of association ( G 0 A ), exemplified using the salt LiBF 4 and solvent PC. a) Electronic energy (E electronic ) and solvation free energy (δG solv ) are calculated using one explicit solvent molecule.b) Thermal energy ( E thermal ) and entropy (S) are calculated in implicit solvent.

Figure 2 .
Figure 2. Computed K A as a function of experimental K A for LiClO 4 based electrolytes.The different solvents studied are shown above.

Figure 3 .
Figure 3. Computed K A as a function of experimental K A for Li + based electrolytes with various anions in PC, glyme and THF.The different anions and solvents studied are shown above.

Figure 4 .
Figure 4. Computed K A as a function of experimental K A for various alkali metal cation based electrolytes.The different solvents and salts studied are shown above.