Establishing a unified framework for ion solvation and transport in liquid and solid electrolytes

Solid-state batteries are arguably the most intensely pursued area in the modern battery space; academic and national research laboratories, established battery producers, and numerous battery start-ups are working to commercialize these systems.This article represents a start toward developing a unified understanding of the relationship between the atomic composition of the solvation structure and ion transport in all electrolytes, from liquids to solids.The foundational concept is that of solvation shells and cages: shells are mobile and they translate with the migrating ions before breaking up, while cages cannot translate with the migrating ions.We show the importance of the mobility of the entities that make up shells and cages in liquids, polymers, inorganic crystals, and glasses.We posit that the development of new and improved electrolytes will be accelerated by understanding both similarities and differences across distinct classes of ion conductors. Electrolytes used in rechargeable batteries must enable rapid translation of the working ion between macroscopically separated electrodes. These electrolytes are, however, usually designed and synthesized using atomic-level insights. Whether the ideal electrolyte for a particular battery is a solid or a liquid remains an important unresolved question, especially as solids with conductivities comparable with liquids are discovered. To help resolve such questions, we present the first steps toward a unified framework for relating atomic and continuum scale phenomena. Solvation shells in liquids are entities that translate with the working ion for a short while before they break up due to Brownian motion. By contrast, solvation cages in classical solids and polymers cannot not translate with the working ion. Mobility of the entities that make up the cages and shells, which is quantified by an order parameter, is shown to influence translation of the working ion on continuum length scales. Electrolytes used in rechargeable batteries must enable rapid translation of the working ion between macroscopically separated electrodes. These electrolytes are, however, usually designed and synthesized using atomic-level insights. Whether the ideal electrolyte for a particular battery is a solid or a liquid remains an important unresolved question, especially as solids with conductivities comparable with liquids are discovered. To help resolve such questions, we present the first steps toward a unified framework for relating atomic and continuum scale phenomena. Solvation shells in liquids are entities that translate with the working ion for a short while before they break up due to Brownian motion. By contrast, solvation cages in classical solids and polymers cannot not translate with the working ion. Mobility of the entities that make up the cages and shells, which is quantified by an order parameter, is shown to influence translation of the working ion on continuum length scales. The performance of electrochemical systems such as batteries, hydrogen and solid oxide fuel cells, and electrochemical reactors for production of materials such as aluminum and chlorine depend on the transport of specific working ions across an electrolyte [1.Newman J. Balsara N.P. Electrochemical Systems.4th edn. Wiley, 2021Google Scholar]. In the relatively simple case of a lithium-ion battery, a single working ion, Li+, participates in the redox reactions at both electrodes [2.Goodenough J.B. Park K.-S. The Li-ion rechargeable battery: a perspective.J. Am. Chem. Soc. 2013; 135: 1167-1176Crossref PubMed Scopus (5746) Google Scholar]. Other systems such as the lead acid battery are more complex and both cations and anions (H+ and SO42–) participate in the redox reactions. It has long been recognized that the electrolyte can, in principle, be either a liquid or a solid [3.Mizushima K. et al.LixCoO2 (0, is given by<Δri2t>=∑β=1Niriβt−riβ0.riβt−riβ0Ni.[1] The time dependence of < Δr+2 > of the working ion is qualitatively similar for all the electrolyte classes depicted in Figure 1. The distinction between the different classes becomes clear when we examine the time dependence of < Δri2(t) > of other entities. Different species exhibit different power laws,<Δri2t>=Ditαi,i=0+or–.[2] The parameter Di reduces to 6Dself,i in the limit t → ∞ if the species i exhibits diffusive motion and Dself,i is the self-diffusion coefficient. For all of the electrolytes in Figure 1, α+ = 1. In liquids, the solvation structure can be ‘carried’ by the ion as it translates across space. However, the random Brownian motion of all entities results in frequent alterations of the solvation structure. Solvation shells, as we define them here, are structures that are capable of breaking up and reforming. In Figure 2B we show <Δri2(t)> based on molecular dynamics (MD) simulations [29.Yu Z. et al.Asymmetric composition of ionic aggregates and the origin of high correlated transference number in water-in-salt electrolytes.J. Phys. Chem. Lett. 2020; 11: 1276-1281Crossref PubMed Scopus (34) Google Scholar,30.Han K.S. et al.Origin of unusual acidity and Li+ diffusivity in a series of water-in-salt electrolytes.J. Phys. Chem. B. 2020; 124: 5284-5291Crossref PubMed Scopus (15) Google Scholar] for a mixture of LiTFSI and water. In liquid electrolytes, all three species exhibit diffusive behavior with αi = 1. The ions find themselves in a multitude of environments but traverse rapidly through them to yield trajectories that are averaged over these environments. In the other electrolytes in Figure 1, however, we obtain different relationships between mean-squared displacement and time. In classical solid electrolytes (both crystals and glasses), the solvation structures do not translate, hence <Δri2> of the cage entities is independent in the limit t → ∞. Solvation cages are thus defined as structures that exhibit subdiffusive motion on the time scale of working ion diffusion. We show the time dependence of <Δr–2> for the negatively charged cage elements of glassy LPS obtained by AIMD simulations in Figure 2C. The simulations were carried out at 500 K so that diffusive regimes could be accessed within the simulation time scale. On the t > 0.5 ns time scale, where the cations show diffusive behavior (α+ = 1), the anions in the cages are characterized by α– = 0. The case of a polymeric ion conductor is intermediate between classical liquids and solids. We limit our discussion to amorphous polymers that are well above their glass transition temperature. On long-enough time scales, polymer chains in amorphous melts exhibit diffusive motion. However, the diffusion of polymer chains is much slower than that of small molecules (such as water) and the solvation cage remains essentially intact on the time scale of cation motion. In Figure 2D we show MD simulation results for a mixture of LiTFSI and PEO [48.Fang C. et al.Salt activity coefficient and chain statistics in poly(ethylene oxide)-based electrolytes.Macromolecules. 2021; 54: 2873-2881Crossref Scopus (6) Google Scholar]. The TFSI– anions exhibit diffusive motion on the t > 5 ns time scale, while time scales greater than 1000 ns are necessary for Li+ cations to exhibit diffusive motion; this time scale is often outside the range of MD simulations. On these time scales, the PEO monomers exhibit a power law with α0 = 0.5. At a coarse-grained level, polymer chains can be modeled as beads connected by springs and the collective motion of the springs on short time scales give an exponent of 0.5 [46.Doi M. Soft Matter Physics. Oxford University Press, 2015Google Scholar]. We define an order parameter, M, to characterize the mobility of the entities in the solvation cages or shellsM=αiα+=αi,i≠+,[3] where αi is the exponent of the power law relationship between < Δri2 > and t. The mobility order parameter lies between 0 and 1: liquid electrolytes exhibit M values of 1, inorganic solids exhibit M values of 0, and polymer electrolytes exhibit M values of 0.5. Other forms of matter such as liquid crystalline electrolytes may exhibit other values of M. In addition to translational displacement of the cage entities, recent work indicates that the diffusion of the cations in both crystalline and glassy LPS is facilitated by the rotation of the anions in the cages. For example, in Figure 2C we show the mean-squared angular displacement, < Δθ−2(t) > of the anions in glassy LPS as a function of time, obtained by AIMD simulations. The linear dependence of < Δθ−2(t) > on time indicates free rotational diffusion of the anions in the cage. It is, in this case, appropriate to expand our definition of M to include rotational dynamics,<Δθi2t>=Dθ,itαθ,i,[4] and we define the rotational order parameter asMθ=αθ,iα+=αθ,i.[5] The values of M (and Mθ when relevant) for the four examples in Figure 1 are presented in Table 1.Table 1Solvation structure, mobility order parameter, and conductivity of exemplar liquid and solid electrolytesSystemsSolvation structureMobility order parameterConductivity (S/cm)LiquidsLiTFSI/water0.2 M, 298 KShellM = 13x10–2Polymers LiTFSI/PEO1.6 M, 363 KCageM = 0.52x10–3Crystals Li3.25[P3/4Si1/4]S4298 KCageM = 0, Mθ = 11x10–3Inorganic glassesLPS298 KCageM = 0, Mθ = 12x10–4 Open table in a new tab The connection between displacement of charged species and ion transport in isotropic electrolytes comprising a solvent and two mobile charged species (Figure 1A,D) was developed formally by Onsager [49.Onsager L. Theories and problems of liquid diffusion.Ann. N. Y. Acad. Sci. 1945; 46: 241-265Crossref PubMed Scopus (582) Google Scholar]. In this framework, ion transport is governed by transport coefficients that we now refer to as Onsager coefficients, Lij. While several approaches for determining these coefficients have been published [50.France-Lanord A. Grossman J.C. Correlations from ion pairing and the Nernst-Einstein equation.Phys. Rev. Lett. 2019; 122: 136001Crossref PubMed Scopus (57) Google Scholar, 51.Molinari N. et al.General trend of a negative Li effective charge in ionic liquid electrolytes.J. Phys. Chem. Lett. 2019; 10: 2313-2319Crossref PubMed Scopus (41) Google Scholar, 52.Harris K.R. Relations between the fractional Stokes−Einstein and Nernst−Einstein equations and velocity correlation coefficients in ionic liquids and molten salts.J. Phys. Chem. B. 2010; 114: 9572-9577Crossref PubMed Scopus (140) Google Scholar, 53.Dong D. et al.How efficient is Li+ ion transport in solvate ionic liquids under anion-blocking conditions in a battery?.Phys. Chem. Chem. Phys. 2018; 20: 29174-29183Crossref PubMed Google Scholar, 54.Wheeler D.R. Newman J. Molecular dynamics simulations of multicomponent diffusion. 1. Equilibrium method.J. Phys. Chem. B. 2004; 108: 18353-18361Crossref Scopus (82) Google Scholar], a particularly transparent approach was recently proposed by Fong and colleagues [55.Fong K.D. et al.Transport phenomena in electrolyte solutions: nonequilibrium thermodynamics and statistical mechanics.AIChE J. 2020; 66e17091Crossref Scopus (13) Google Scholar], who derived the following expression for Lij,Lij=16kBTVlimt→∞∂∂t<∑β=1Niriβt−riβ0.∑γ=1Njrjγt−rjγ0>,ij=+or−[6] where kB is the Boltzmann constant, T is the temperature, and V is the volume of the system. The extent to which the different approaches [50.France-Lanord A. Grossman J.C. Correlations from ion pairing and the Nernst-Einstein equation.Phys. Rev. Lett. 2019; 122: 136001Crossref PubMed Scopus (57) Google Scholar, 51.Molinari N. et al.General trend of a negative Li effective charge in ionic liquid electrolytes.J. Phys. Chem. Lett. 2019; 10: 2313-2319Crossref PubMed Scopus (41) Google Scholar, 52.Harris K.R. Relations between the fractional Stokes−Einstein and Nernst−Einstein equations and velocity correlation coefficients in ionic liquids and molten salts.J. Phys. Chem. B. 2010; 114: 9572-9577Crossref PubMed Scopus (140) Google Scholar, 53.Dong D. et al.How efficient is Li+ ion transport in solvate ionic liquids under anion-blocking conditions in a battery?.Phys. Chem. Chem. Phys. 2018; 20: 29174-29183Crossref PubMed Google Scholar, 54.Wheeler D.R. Newman J. Molecular dynamics simulations of multicomponent diffusion. 1. Equilibrium method.J. Phys. Chem. B. 2004; 108: 18353-18361Crossref Scopus (82) Google Scholar, 55.Fong K.D. et al.Transport phenomena in electrolyte solutions: nonequilibrium thermodynamics and statistical mechanics.AIChE J. 2020; 66e17091Crossref Scopus (13) Google Scholar] in the literature are consistent with each other remains to be established. On the continuum scale, coefficients Lij relate the flux of species i to gradients in the chemical potentials of species j [49.Onsager L. Theories and problems of liquid diffusion.Ann. N. Y. Acad. Sci. 1945; 46: 241-265Crossref PubMed Scopus (582) Google Scholar]. While the three Onsager coefficients can be calculated using simulations as described earlier, they are difficult to measure by direct experimentation. Newman developed strategies for measuring three different transport properties: ionic conductivity, salt diffusion coefficient, and the cation transference number [1.Newman J. Balsara N.P. Electrochemical Systems.4th edn. Wiley, 2021Google Scholar,56.Mongcopa K.I.S. et al.Relationship between segmental dynamics measured by quasi-elastic neutron scattering and conductivity in polymer electrolytes.ACS Macro Lett. 2018; 7: 504-508Crossref Scopus (52) Google Scholar]. Relationships between the experimentally measured transport parameters and the Onsager coefficients can be found in the literature (e.g., conductivity, κ=F2∑i∑jzizjLij; F is the Faraday constant and zi is the charge number on species i) [1.Newman J. Balsara N.P. Electrochemical Systems.4th edn. Wiley, 2021Google Scholar,55.Fong K.D. et al.Transport pheno