Ion Dynamics in Concentrated Electrolyte Solutions: Relating Equilibrium Fluctuations of the Ions to Transport Properties in Battery Cells

In recent years, the interest in the development of highly concentrated electrolyte solutions for battery applications has increased enormously. Such electrolyte solutions are typically characterized by a low flammability, a high thermal and electrochemical stability and by the formation of a stable solid electrolyte interphase (SEI) in contact to electrode materials. However, the classification of concentrated electrolyte solutions in terms of the classical scheme “strong” or “weak” has been controversially discussed in the literature. In this paper, a comprehensive theoretical framework is presented for a more general classification, which is based on a comparison of charge transport and mass transport. By combining the Onsager transport formalism with linear response theory, center‐of‐mass fluctuations and collective translational dipole fluctuations of the ions in equilibrium are related to transport properties in a lithium‐ion battery cell, namely mass transport, charge transport and Li+ transport under anion‐blocking conditions. The relevance of the classification approach is substantiated by showing that i) it is straightforward to classify highly concentrated electrolytes and that ii) both fast charge transport and fast mass transport are indispensable for achieving fast Li+ transport under anion‐blocking conditions.


Introduction
State-of-the-art lithium-ion batteries employ typically 1 M electrolyte solutions, since the ionic conductivity becomes maximal in this concentration regime.However, in recent years, the interest in developing highly concentrated electrolyte solutions for battery applications has increased enormously.[19] These electrolyte solutions are being developed for several reasons: i) In highly concentrated solutions, a major part of the solvent molecules is involved in the solvation of the ions, so that only few "free" solvent molecules exist.This leads to a low chemical potential and low vapor pressure of the solvent molecules and thus to a low flammability of the electrolyte.ii) The low chemical potential of the solvent molecules improves the electrochemical and thermal stability of the electrolyte solution.iii) A strong binding of solvent molecules to the Li + ions leads to the formation of a so-called "anionderived" solid electrolyte interphase (SEI) at the negative electrode of a lithium-ion battery, which is mainly composed of anion decomposition product. [20,21]Such anion-derived SEIs have been shown to exhibit a high long-term stability.iv) It has been observed that highly concentrated electrolytes suppress the formation of Li dendrites in Li metal batteries. [2,5]hese advantages of highly concentrated electrolytes for battery applications are often counterbalanced by slower charge and mass transport as compared to 1 M solutions.Therefore, it is important to obtain an improved understanding of transport processes in concentrated electrolyte solutions and to optimize their transport properties.[29][30][31][32] In a recent Frontier Outlook, [33] we have suggested a more general classification, which is based on a comparison of charge and mass transport in the electrolyte.To this end, we defined a molar mass transport coefficient Λ mass , which is the mass transport analogue of the molar ionic conductivity Λ charge and which, according to linear response theory, is related to center-ofmass fluctuations of the ions in thermal equilibrium. [33]Based on this, we distinguished "weak charge transport electrolytes" with Λ charge ≪ Λ mass from "weak mass transport electrolytes" with Λ mass ≪ Λ charge .In "weak charge transport electrolytes," the strong interactions between cations and anion as compared to ion/solvent interactions leads to charge transport limitations, see Figure 1a.An example in the field of Li + electrolyte solutions are strong interactions between Li + ions and trifluoroacetate anions, see Figure 1b.In contrast, in "weak mass transport electrolytes," the strong interactions between cations and solvent molecules as compared to cation/anion interactions reduces the momentum exchange between ions and solvent molecules and thus the center-of-mass fluctuations of the ions and the mass transport, see Figure 1a.An example are the strong interactions between Li + ions and triglyme molecules (G3) in solvate ionic liquids containing In recent years, the interest in the development of highly concentrated electrolyte solutions for battery applications has increased enormously.Such electrolyte solutions are typically characterized by a low flammability, a high thermal and electrochemical stability and by the formation of a stable solid electrolyte interphase (SEI) in contact to electrode materials.However, the classification of concentrated electrolyte solutions in terms of the classical scheme "strong" or "weak" has been controversially discussed in the literature.In this paper, a comprehensive theoretical framework is presented for a more general classification, which is based on a comparison of charge transport and mass transport.By combining the Onsager transport formalism with linear response theory, center-of-mass fluctuations and collective translational dipole fluctuations of the ions in equilibrium are related to transport properties in a lithium-ion battery cell, namely mass transport, charge transport and Li + transport under anion-blocking conditions.The relevance of the classification approach is substantiated by showing that i) it is straightforward to classify highly concentrated electrolytes and that ii) both fast charge transport and fast mass transport are indispensable for achieving fast Li + transport under anion-blocking conditions.
In Roling et al. [33] we gave a short sketch of the theoretical background of this classification in the case of 1-1 electrolytes (univalent cation, univalent anion).In this paper, a comprehensive theoretical framework for this classification is presented, which is valid for electrolytes with a single dissolved salt with arbitrary ionic charge numbers.The Onsager transport formalism is combined with linear response theory for relating equilibrium center-of-mass fluctuations and collective translational dipole fluctuations of the ions to the transport properties in battery cells.The classification approach is substantiated by showing that both fast charge transport and fast mass transport are indispensable for fast Li + transport under anion-blocking conditions in lithium-ion batteries.Furthermore, we consider the influence of cross correlations between center-of-mass and collective translational dipole fluctuations on the transport properties, which were neglected in Roling et al. [33]

General Theoretical Framework Based on Onsager Formalism and Linear Response Theory
An electrolyte solution is considered containing cations with mass of m þ and charge q þ ¼ z þ Á e as well as anions with mass of m À and charge q À ¼ z À Á e.Here, z AE and e ¼ 1:6 Á 10 À19 C denote the ionic charge numbers and the elementary charge, respectively.The stoichiometric factor of cations and anions, ν þ and ν À , are related to the charge numbers via The mass ratio of the ions is defined as: In thermal equilibrium, movements of an ion i within a time interval t can be described by a displacement vector The time interval t is chosen such that the ion dynamics is dif- i , while the time-dependent displacement vector of an anion i in x direction is denoted by Δ R !À i .

Definition of Center-of-Mass Fluctuations and Collective Translational Dipole Fluctuations in Equilibrium
The ionic displacements lead to center-ofmass fluctuations Δ R ð Þ of the ions, which are given by: We note that the center-of-mass Δ R !CM t ð Þ refers to the system of all individual ions in the sample.3) and ( 4) for

Transport Coefficients and Mobility-Based Cation Transference Number Derived from Linear Response Theory
Now we use Equations ( 3) and ( 4) to calculate the Onsager coefficients σ þþ , σ ÀÀ , and σ þÀ in the framework of linear response theory: [34] σ þþ ¼ lim (5) In Equations ( 5)-( 7), 〈 〉 denotes the ensemble average.The ionic conductivity σ ion is then given by: As well-known, [35,36] the ionic conductivity is exclusively determined by collective translational dipole fluctuation of the ions in equilibrium.
Considering Equations ( 5)-( 8) we define two additional transport coefficients: As will be shown below, σ mass describes mass transport in the electrolyte, that is, the transport of neutral salt in the presence a chemical potential gradient of the salt.Consequently, σ mass is clearly distinct from the so-called Nernst-Einstein conductivity calculated from the self-diffusion coefficients of the ions. [37]The quantity σ corr describes the influence of cross correlations between collective translational dipole fluctuations and center-of-mass fluctuations on transport.This influence will be analyzed in Section 4.
Using Equations ( 8)-( 10), the Onsager coefficients can be rewritten as: The mobility-based cation transference number is then given by:

Onsager Reciprocal Relations for Cation and Anion Flux
According to the Onsager reciprocal relations, the molar fluxes of the cations, J þ , and of the anions, J À , can be written as: The gradients of the electrochemical potential of the ions are the driving forces and are defined as: Here, φ is the electric potential, and denotes the mean activity of the ions.a þ and a À are the individual activities of cations and anions, respectively.

Cation Flux under Anion-Blocking Conditions
When an electrolyte is placed between two cation-reversible and anionblocking electrodes with spacing d, a salt concentration gradient and an electric gradient are formed, such that anion migration and anion diffusion cancel out.This implies that J À ¼ 0. With Equations ( 16) and ( 17) it follows that: Integration of Equation ( 18) leads to the following expression for the electric potential drop over the bulk of the electrolyte in the case of anion-blocking conditions (index abc): The Nernst potential drop over the interfaces between the electrolyte and the cation-reversible electrodes is given by: When we now normalize the overall potential drop between the electrodes to the electrode spacing d, we obtain: The cation flux is obtained by combining Equations ( 15) and ( 17): Taking into account Equations ( 18) and ( 21) results in: Consequently, the cation conductivity σ abc under anion-blocking conditions and the related cation transference number t abc þ are given by:

Neutral Salt Transport
Neutral salt transport with zero electrical current flow j ¼ 0 implies that: Taking into account Equations ( 15)-( 17) results in: Inserting Equation (27) into Equation ( 15) combined with Equation (17) leads to the following expression for the salt flux: ÞÁdlna AE , Equation ( 28) can be rewritten as: With the Onsager definition the following expression for the salt transport coefficient σ salt is obtained:

General Relations for 1-1-Electrolytes
For a 1-1-electrolyte containing univalent cations and univalent anions, we have , and z À ¼ À1.In this case, Equations ( 11)-( 14) for the Onsager coefficients and for the mobility-based cation transference number simplify to: Equation ( 31) for the salt transport coefficient σ salt simplifies to: Inserting Equations ( 32)-( 34) into (36) yields: Equations ( 24) and ( 25) for the effective cation conductivity σ abc and for the cation transference number t abc þ under anion-blocking conditions are also valid for 1-1 electrolytes.

Influence of Transport Coefficient σ corr on Mass Transport in Weak Charge Transport Electrolytes and in Weak Mass Transport Electrolytes
When considering Equations ( 35) and ( 37), the question arises whether the mass transport coefficient σ mass and the mass ratio k can be determined for an electrolyte with unknown value for σ corr .Therefore, we analyze in the following the influence of σ corr on the mass transport properties of weak charge transport electrolytes and of weak mass transport electrolytes.

Weak Charge Transport Electrolytes
We consider an electrolyte with weak ion dissociation.When the degree of dissociation of the salt is given by α ≪ 1, the transport coefficients of free ions and ion pairs can be written as: [34] σ with and D Ã AX denoting the self-diffusion coefficients of free cations, free anions, and ion pairs, respectively.α ≪ 1 implies that σ A þ ≪ s and σ X À ≪ s.
In classical weak electrolyte theory, an ion pair exists, if the distance between cation and anion is smaller than the Bjerrum length. [38]From a dynamic point of view, the transport of ion pairs is relevant, if the average distance over which a cation-anion pair is transported is significantly larger than the diameters of the solvated ions.
The Onsager coefficients and the ionic conductivity for such an electrolyte are given by: [34] Now we calculate the salt transport coefficient σ salt : From Equations ( 32)-( 34), it follows for the mass transport coefficient σ mass : Since σ salt and σ mass are virtually identical, the term σ corr ð Þ 2 =σ ion must be negligible.This can also be seen when calculating σ corr from Equations ( 32)-( 34): From this we conclude that for weak charge transport electrolytes, the mass transport coefficient σ mass can be safely identified with the salt transport coefficient σ salt .However, the transport data yield no information about the mass ratio k.

Weak Mass Transport Electrolytes
Weak mass transport electrolytes are characterized by σ mass ≪ σ ion .In order to derive information about σ corr , two weak mass transport electrolytes are considered with inverse mass ratio k 1 and k 2 ¼ 1=k 1 and otherwise identical properties.For symmetry reasons, these electrolytes exhibit the same values for σ ion , σ mass and σ þÀ .With Equation (34)  this implies that: Energy Environ.Mater.2024, 7, e12533 From Equation (49) it follows that: Equation ( 50) implies that for identical ion masses, that is, for k = 1, correlations between dipole and center-of-mass fluctuation are absent, that is, σ corr = 0.
Next, the limits k !∞ and k !0 are considered.In these limits, one type of ion moves much faster than the other type, so that cationanion correlations become negligible, that is, σ þÀ !0. Taking into account Equation (34), k !∞ leads to σ corr ¼ Àσ mass and k !0 leads to σ corr ¼ σ mass .Consequently, the following relation is generally valid.Together with σ mass ≪ σ ion , this implies that the σ corr -containing terms in Equations ( 35) and ( 37) can be safely neglected for weak mass transport electrolytes.

Electrolyte Classification and Assessment of Transport Properties in Batteries
In Roling et al., [33] a general electrolyte classification based on comparing charge and mass transport was suggested.The molar mass transport coefficient Λ mass ¼ σ salt =c salt was plotted versus the molar ionic conductivity Λ charge ¼ σ ion =c salt , and it was shown that the transport data of an ideal strong electrolyte with negligible ion-ion interactions are characterized by For k values close to unity, this can be approximated by Λ mass ≈ 1 4 Á Λ charge .Electrolytes with strong charge transport limitations, that is, 1 4 Á Λ charge ≪ Λ mass , were termed as "weak charge transport electrolytes," whereas electrolytes with strong mass transport limitations, that is, Λ mass ≪ 1 4 Á Λ charge , were termed as "weak mass transport electrolytes." In the following, we relate this classification to the transport properties of a Li + electrolyte in a lithium-ion battery.During the cycling of such a battery, the Li + ions are transported under anion-blocking conditions described by the transport coefficient 36) and (38).In this expression, specific transport coefficients σ ion and σ salt appear and not their molar counterparts.Consequently, in Figure 2, we plot σ salt versus σ ion for different electrolyte solutions, and we add iso-σ abc lines to this plot with two specific σ abc values.The first value σ abc standard ¼ 3:5 mS cm À1 is an estimate for standard carbonate-based battery electrolytes with ionic conductivities in the range of σ ion ¼ 10 mS cm À1 and with Li + transference numbers in the range of 0.35. [39]These electrolytes enable fast Li + ion transport in the battery.The second value is based on the assumption that σ abc should not fall below a value of σ abc min ¼ 1 mS cm À1 in order to ensure sufficiently fast Li + ion transport.Of course, this value can be adjusted to the specific requirements of a battery under consideration.In Figure 2, we use a background color code with green in case of σ abc ≥ σ abc standard , yellow in case of σ abc standard > σ abc ≥ σ abc min and red in case of σ abc < σ abc min .The plot contains also an ideal strong electrolyte line with σ salt ¼ 1 4 Á σ ion .The color code reveals that for achieving sufficiently high value of σ abc , both the ionic conductivity σ ion and the salt transport coefficient σ salt must be sufficiently high.This can also be seen from Equation (38), which in case of k ≈ 1 and σ corr < σ mass can be approximated by: In Figure 2 we show σ salt versus σ ion data points for a number of Li + electrolytes, for which reliable Onsager coefficients either from experiment or from MD simulations are available in the literature.As seen from the figure, only a fraction of the data points is located in the green or in the yellow regime.The data point for a 0.95 M solution of LiPF 6 in a carbonate-based electrolyte (EC:EMC with molar ratio of 27:73) is clearly in the green regime.However, it is important to note that this data point is based on a MD simulation at 333 K. [40] At room temperature, the ion dynamics is slower and the data point is expected to located close to the boundary between the green and the yellow regime.For the water-in-salt electrolytes LiTFSI/H 2 O, the ionic conductivity σ ion as well as the salt transport coefficient σ salt are maximal at a molar salt concentration of 2.3 mol L −1 . [41]With increasing salt concentration, the ionic conductivity σ ion drops, but the salt transport coefficient σ salt drops even more strongly, so that the data points for high concentrations fall into the "weak mass transport electrolyte" regime.This implies that despite the fast ion dynamics in these water-in-salt electrolytes, mass transport limitations at very high salt concentration hinder fast Li + transport under anion-blocking conditions in a lithium-ion battery.The data points for the electrolytes LiTFSI/G4 are close to those of the LiTFSI/H 2 O electrolytes and show a qualitatively similar concentration dependence.However, it is important to note that these data points were obtained from molecular dynamics simulations at 373 K. [42] At room temperature, the ion dynamics and transport in these electrolytes is much slower, as also be seen from the experimental data points of LiFSI/G4. [43]These data points for salt concentrations between 1.72 and 3.23 mol L −1 are in the red regime and are located considerably below the ideal strong electrolyte line.This implies that Li + transport in a battery is strongly hindered by slow mass transport.In contrast, the data points for LiTFSI/sulfolane electrolyte solutions are close to the ideal strong electrolyte lines. [44]However, the overall ion dynamics in these electrolytes is relatively slow, leading to relatively low values of both σ ion and σ salt (red regime).The data points for the two electrolytes LiBF 4 /G4 and LiOTf/G3 are located above the ideal strong electrolyte line despite the high salt concentration of almost 4 mol L −1 and the molar ratio of salt to solvent of 1:1. [45]This gives indication that the interactions between Li + ions and the anions (BF À 4 or OTf À ) are stronger than the interactions between Li + ions and glyme molecules (G3 or G4), resulting in charge transport limitations, which are slightly stronger than the mass transport limitations.Overall, charge transport and mass transport are rather slow, so that the data points of two electrolytes are located in the red regime.
Overall, our analysis reveals the following prerequisites for fast Li + ion transport in a concentrated electrolyte solution under anion-Energy Environ.Mater.2024, 7, e12533 blocking conditions: i) The interactions between Li + ions and anions should be balanced with the interactions between Li + ions and solvent molecules, such that the data point in a σ salt versus σ ion plots is close to the ideal strong electrolyte line.ii) Both interactions should be sufficiently weak for enabling fast ion dynamics.
Regarding the characterization of transport by experiments and simulations, it is important to realize that charge transport and mass transport properties are of equal importance.

Conclusions
We have presented a comprehensive theoretical framework for relating the equilibrium ion dynamics in an electrolyte solution to the transport properties of the electrolyte solution in a lithium-ion battery cell.By combining the Onsager transport formalism with linear response theory, we have shown that center-of-mass fluctuation of the ions govern the mass transport properties, while it is well known that collective translational dipole fluctuations of the ions govern the charge transport properties.
Based on this theoretical framework, we have suggested a general classification of electrolyte solutions by comparing the ionic conductivity σ ion with the salt transport coefficient σ salt .Ideal strong electrolytes with negligible ion-ion interactions are characterized by σ salt ¼ k 2 1þk ð Þ 2 Á σ ion , which can be approximated by σ salt ≈ 1 4 Á σ ion for ion mass ratios k close to unity.Electrolyte solutions with 1 4 Á σ ion ≪ σ salt are termed as "weak charge transport electrolytes," while electrolyte solution with σ salt ≪ 1 4 Á σ ion are termed as "weak mass transport electrolytes." Our data analysis revealed that many highly concentrated electrolyte solutions for lithium-ion batteries are "weak mass transport electrolytes."This is the case, when the interactions of the Li + ions with the solvent molecules are stronger than the Li + /anion interactions.Due to the strong Li + /solvent interactions, the fraction of "free" solvent molecules, which are not bound to Li + ions, is small.This hinders momentum exchange between the ions and the solvent molecules, and thus the equilibrium center-of-mass fluctuations of the ions are small.
The most relevant electrolyte transport coefficient in a lithium-ion battery is the Li + conductivity under anion-blocking conditions, σ abc .We have shown that high values of σ abc can only be achieved, if both charge and mass transport are sufficiently fast.To this end, the Li + /solvent interactions should be balanced with the Li + /anion interactions, and both types of interactions should be sufficiently weak for allowing for fast ion dynamics.In experiments and simulations, the characterization of charge transport and mass transport is of equal importance.
collective translational dipole fluctuations e Á Δ R !DP t

Figure 1 .
Figure 1.a) Schematic illustration of the interactions between Li + ions, anions X − and solvent molecules S in weak charge transport electrolytes as compared to weak mass transport electrolytes for lithium-ion batteries.b) Examples for weak mass transport and weak charge transport electrolytes with illustration of interactions.

Figure 2 .
Figure 2. Plot of the salt transport coefficient σ salt versus the ionic conductivity σ ion for different Li + electrolyte solutions.The salt concentration of the respective solution in mol L −1 is given as a number at the data point.If not specified otherwise, the data points originate from experiments or simulations at room temperature.The blue line refers to the transport properties of an ideal strong electrolyte with negligible ion-ion interactions.Two iso-σ abc lines (black lines) with σ abc values of 3.5 and 1 mS cm −1 , respectively, are sketched for assessing the Li + transport in the electrolyte under anion-blocking conditions in a lithium-ion battery.