Diffusion and structure of propylene carbonate–metal salt electrolyte solutions for post-lithium-ion batteries: From experiment to simulation

Currently, the needs of society for personal electronics and electric vehicles are met by Li-ion batteries.¹ The decarbonization of transportation vehicles in 2035 will be an essential milestone in mitigating air pollution, greenhouse gas (GHG) emissions, global warming, and other critical issues associated with climate change. In Europe, numerous gigafactory projects targeted by 2030 are, therefore, emerging to secure the future electric transportation market, as overall battery demand in the EU is expected to reach 1 TWh by 2025 and exceed 2.6 TWh by 2030.² Thus, energy storage will face new technical and cost-related challenges that will require cheaper batteries, better supply chains, faster charging rates, discharge over longer periods, improved safety, and longer lifetimes.

The expected growth in the battery market and the existing challenges in conventional technologies are causing large manufacturers to start evaluating complementary or alternative chemistries to existent conventional lithium-ion technology.³ In this context, among the most popular potential alternatives are sodium-ion batteries, which charge fast, cost less, are more sustainable, and are able to mitigate the supply risks associated with Li-based batteries.^4,5 The major areas of improvement for Na-ion batteries are their energy density and cycle life, where they still lag behind Li technology.⁶ Furthermore, research has increasingly shifted toward next-generation batteries that are based on earth-abundant minerals^7–15 and work with multivalent ions. For example, the Mg-ion cells have attracted extensive attention due to their high capacity, high safety, and significantly low cost.^16,17 Nevertheless, until now, the energy density and rate capability of Mg battery prototypes have not been attractive enough to commercialize them.¹⁶ Another alternative is associated with calcium-ion batteries, which have also emerged as a promising next-generation electrochemical energy storage system.¹⁸ 

One of the key factors in all metal-ion rechargeable systems is the electrolyte solution. As it serves to transport positively charged ions between the cathode and anode, the diffusion of the respective ions and their drift velocity in the electric field become extremely important to the battery’s performance.¹⁹ Various nuclear magnetic resonance (NMR) techniques highlight different aspects of ion dynamics, such as structure and local dynamics by spectroscopy,²⁰ local ion dynamics by spin relaxation,²¹ diffusion by pulsed field gradients,^22,23 and ion drift velocities by electrophoretic NMR.²⁴ Other aspects of ion dynamics and transport²⁵ can be studied by impedance spectroscopy,²⁶ the rotating disk electrode method,²⁷ or via computer simulations, using density functional theory (DFT) calculations or ab initio molecular dynamics (MD) simulations, in particular.^28–31 

One of the experimental techniques to measure the self-diffusion coefficients of molecules and ions is pulsed field gradient nuclear magnetic resonance (PFG-NMR).^32,33 This NMR method has been widely used to measure the diffusivity of ions and solvents in various Li-ion battery electrolyte systems.^34–41 For instance, the experimental study by Hayamizu⁴² determined the self-diffusion coefficients of Li⁺, PF₆⁻, and various solvents, such as diethyl carbonate (DEC), propylene carbonate (PC), and ethylene carbonate (EC).⁴³ The order of diffusivities was found to be D_solvent > D_anion > D_cation,^43–46 and the self-diffusion and activity coefficients of Li⁺ cation depended greatly on temperature.⁴³ The Li⁺ ion had a smaller diffusion coefficient than the PF₆⁻ anion, although it had a smaller size (the van der Waals radii of Li⁺ and PF₆⁻ are 0.076 and 0.254 nm, respectively⁴⁷). This result is explained by the fact that Li⁺ was solvated; thus, its solvodynamic radius was larger than that of the PF₆⁻ anion.⁴⁸ While ⁷Li is a suitable nucleus for PFG-NMR diffusion measurements, the available isotopes of post-Li cations often lack a nuclear spin, or suffer from short spin relaxation times, which prevent the application of spin echos to observe the transport over the required time scales of at least several milliseconds. Thus, NMR diffusion studies have rarely been performed on such electrolytes. Furthermore, accurate computational modeling of such complex solutions remains challenging.

The ab initio and classical MD simulations of liquid electrolytes were found to correspond more closely with the experimental diffusivity values than other calculation methods applied.⁴⁹ The majority of the publications report the MD simulations of the electrolytes composed of various Li salts and polar organic solvents. In one of the first simulation reports by Soetens et al.^50,51 (calculations were accomplished at 473, 323, and 298 K) along with later calculations performed by Takeuchi⁴⁵ (at 298 K) and Mynam et al.^52,53 (at 373 and 313 K), large deviations in both Li⁺ and PF₆⁻ diffusivities vs experimental measurements were observed in the modeling of 1M LiPF₆ and 1M LiBF₄ PC solutions. These deviations resulted from the overestimation of the ion–ion correlations by the non-polarized force fields, which led to a stronger binding of cations to anions, thus hindering their diffusion coefficients. Many attempts have been performed subsequently to improve the results of simulations and compare them with values measured experimentally. Both polarized⁵⁴ and non-polarized^52,55,56 force fields with and without charge-scaling factors have been applied. Generally, for low Li salt concentrations in PC (≤0.05M), the simulated results were in agreement with experimental values,⁵⁵ while for high concentrations (≥1–2M), a discrepancy in PC diffusion was observed.^52,54,57 Chaudhari and Rempe⁵⁵ simulated a single Li⁺, PF₆⁻ pair in 249 PC molecules, which corresponded to a 0.05M LiPF₆ PC solution. The implementation of either ab initio or classical MD simulations led to a good agreement between the computed Li⁺ diffusion⁵⁵ and experimental measurements at 300 K,⁴³ although the experimental concentration referred to a 1M PC solution. Seki et al.⁵⁴ performed ab initio molecular orbital calculations for a lithium bis(trifluoromethylsulfonyl)imide (LiTFSI)/PC solution in wide concentration (0.25–2.00M) and temperature (283–353 K) ranges. The observed increase in PC diffusion discrepancy with the increase in concentration⁵⁴ can be explained by the fact that at high concentrations, contact ion pairs and multi-ion complexes are formed,⁵⁸ impacting the diffusivity;^45,57,59 thus, the charge-scaling factor of ions needs to be adjusted accordingly.⁵⁷ 

When Li⁺ diffusivity was computed by atomistic MD simulations using the COMPASS II force field for various salt concentrations in PC at 298 K, it corresponded well with experimental data.⁶⁰ However, a rather small simulation cell was used with only 200 solvent molecules.⁶⁰ When 1M LiPF₆ PC solutions were modeled atomistically applying the General Amber Force Field (GAFF)^61–65 or all atom optimized potentials for liquid simulation (OPLS-AA)^61,66 force fields, the computed PC and ions diffusivities were in accordance with the experimental measurements performed by Hayamizu⁴³ for a wide temperature range (353–253 K). The observed trend, namely D_PC > D_PF6⁻ > D_Li⁺, was in full agreement with the experimental values.⁴³ Similarly, the appropriate charge scaling of ions in Refs. 55 and 62 allowed both GAFF and OPLS-AA to perfectly predict, in comparison with experimental measurements, the PC and ion diffusivities for a broad range of temperatures. It can be further concluded that the non-polarizable, scaled-charge force fields can effectively predict the diffusion and conductivity in PC electrolyte solutions, while being less computationally demanding than polarizable^29,49,67,68 or ReaxFF models.^28,49,69

A significantly smaller number of published reports were devoted to the simulation of other salt solutions based on monovalent (Na⁺, K⁺) or divalent (Ca²⁺, Mg²⁺) metals. In principle, similar approaches to those described above for the solutions of Li salts are applicable for the simulation of other metal salt solutions.^54,70–77 In classical MD simulations, the polarization, charge-transfer, and covalent interactions in solutions of metal salts other than Li were often considered by the static scaling of charges assigned to atoms in molecules. Seki et al.⁵⁴ analyzed the interactions of various metal cations (Li⁺, Na⁺, Ca²⁺, and Mg²⁺) with PC and TFSI⁻ anions by ab initio molecular orbital calculations. It was demonstrated that Mg²⁺ formed a more stable complex with PC and the TFSI⁻ anions than Ca²⁺. Thus, the diffusion coefficients of PC and TFSI⁻ were lower for Mg²⁺ than for the Ca²⁺ cation.⁵⁴ Moreover, the simulations performed revealed that the interaction energy between the metal cations, PC solvent, and TFSI⁻ anion increased in the order of Na⁺ < Li⁺ < Ca²⁺ < Mg²⁺. Accordingly, the self-diffusion coefficients of PC and TFSI⁻ anion followed the reversed order: Na⁺ > Li⁺ > Ca²⁺ > Mg²⁺. However, the simulated self-diffusion coefficients for sodium and divalent cations were not supported experimentally due to the complexity of carrying out such studies.⁵⁴ Okoshi et al. used classical MD simulations to show that the divalent Mg²⁺ cations had a stronger solvation structure in different organic electrolytes than monovalent Li⁺ and Na⁺ cations.^75,76 Likewise, Mg²⁺ and Ca²⁺ in the mixture of PC and EC solutions consisted of stronger cation–carbonate interactions than Li⁺ and Na⁺ in the same solutions.^77,78 It was demonstrated by DFT calculations that the Na⁺ cation was more easily solvated by the carbonate ester solvents (EC, PC, DEC, etc.) than the ether solvents (DOL, THF, and DME).⁷⁹  Ab initio and classical MD simulations of Mg²⁺ and Ca²⁺ in a variety of solvents such as EC, VC, PC, BC, DMC, EMC, and DEC, as well as in their mixtures (i.e., in EC:PC, EC:EMC, and EC:DEC) revealed that Mg²⁺ and Ca²⁺ were more soluble in such electrolytes than Li⁺ and Na⁺ cations.^70–73 

Although there are extensive studies in the literature regarding the solvation structure of Li⁺ metal cations^80,81 and their desolvation energies^82,83 in various aprotic solvents, reports related to the investigation of cation diffusion coefficients in battery electrolytes containing sodium, potassium, and divalent metal salts are rather limited. This is not only due to the difficulty of measuring the diffusion coefficients of Na⁺, K⁺, Mg⁺, etc., cations experimentally, but also due to the lack of appropriate atomistic computational models, since it is well known that non-polarizable force fields underpredict diffusion coefficients^61,84 by orders of magnitude. Thus, a more comprehensive understanding (by the synergy of experimental and computer simulation methods with the appropriate charge-fitting technique) of ion diffusion in electrolyte solutions is needed.

In the present work, we combine experimental (PFG-NMR studies) and theoretical (MD simulations) approaches to examine 1M solutions of various bis(trifluoromethylsulfonyl)imide salts in propylene carbonate (PC), to understand the influence of the cation’s nature (Li⁺, Na⁺, K⁺, Mg²⁺, Ca²⁺, Ba²⁺, Sr²⁺) and valency (monovalent Li⁺, Na⁺, K⁺ vs divalent Mg²⁺, Ca²⁺, Ba²⁺, Sr²⁺) on the ion diffusion and transport processes in liquid battery electrolytes. Experimentally, the electrolytes are characterized by the anion and solvent diffusion coefficients, which shed light on the structuring in solution depending on the type of cation employed. Temperature-dependent diffusion coefficients yield the corresponding activation energies. In simulations, a large cell was used to model these PC-based electrolytes. The diffusion coefficients were calculated within a broad temperature range of 313–283 K. In the next step, the MD simulations were employed to extract the radial distribution functions (RDFs) of different cations, anions, and PC solvents over a wide range of temperatures. This demonstrated that, depending on the size of the cation, the solvation shell has a distinct influence on the diffusion of each species.

Lithium bis(trifluoromethylsulfonyl)imide (LiTFSI, 99%, Sigma-Aldrich), sodium bis(trifluoromethylsulfonyl)imide (NaTFSI, 99.5%, Solvionic), calcium(II) bis(trifluoromethylsulfonyl)imide [Ca(TFSI)₂, 99.5%, Solvionic], magnesium(II) bis(trifluoromethylsulfonyl)imide [Mg(TFSI)₂, 99.5%, Sigma-Aldrich], and potassium hexafluorophosphate (KPF₆, ≥99%, Sigma-Aldrich) were unpacked inside an argon-filled glovebox (GS Glovebox MEGA Line, H₂O and O₂ content <0.5 ppm) and were applied as received for the formation of propylene carbonate solutions. Strontium(II) carbonate (SrCO₃, 99.9%, Sigma-Aldrich), barium(II) carbonate (BaCO₃, >99%, Sigma-Aldrich), propylene carbonate (99%, Iolitec), and trifluoromethanesulfonimide (HTFSI, 95.0%, Sigma-Aldrich) were used without further purification.

Trifluoromethanesulfonimide (4.00 g, 0.014 mol) was slowly dissolved in 50 ml of milli-q H₂O at room temperature. BaCO₃ (1.58 g, 0.008 mol) was added gradually to the acidic solution in small portions (caution: active gas release). The resultant suspension was stirred at room temperature for 12 h, then the unreacted BaCO₃ was filtered off, and the solvent was removed under reduced pressure at 80 °C. The residual white powder was washed with anhydrous diethyl ether, dried at 150 °C/0.1 mbar for 12 h in the B-585 oven (Buchi Glass Drying Oven, Switzerland) filled with P₂O₅, and then transferred under vacuum to an argon-filled glovebox (MBRAUN MB-Labstar, H₂O and O₂ content <0.5 ppm). Yield: 4.58 g (92%); m.p. = 310 °C (with decomposition); ¹³C NMR (100.6 MHz, DMSO-d₆): 119.5 (q, J_CF = 322 Hz); ¹⁹F NMR (376.5 MHz, DMSO-d₆): δ = −81.0 (s); IR (ATR-mode): 1605 (w), 1314 (s, v_asSO₂), 1296 (m, vCF), 1194 (vs, vCF), 1129 (vs, v_sSO₂), 1053 (s, vCF), 801 (m), 747 (m), 645 (s), 598 (s), 572 (vs), 514 (s); Calc. for BaC₄F₁₂N₂O₈S₄ (697.60): C, 6.89%; N, 4.02%; found: C, 6.74%; N, 4.13%.

Trifluoromethanesulfonimide (4.00 g, 0.014 mol) was dissolved in 50 ml of milli-q H₂O, and SrCO₃ (1.18 g, 0.008 mol) was added gradually to the acidic solution in small portions (caution: active gas release). The resultant suspension was stirred at room temperature for 12 h, then the unreacted SrCO₃ was filtered off, and the solvent was removed under reduced pressure at 80 °C. The residual white powder was washed with anhydrous diethyl ether, dried at 150 °C/0.1 mbar for 12 h in the B-585 oven (Buchi Glass Drying Oven, Switzerland) filled with P₂O₅, and then transferred under vacuum to an argon-filled glovebox (MBRAUN MB-Labstar, H₂O and O₂ content <0.5 ppm). Yield: 4.08 g (90%); m.p. = 260 °C (with decomposition); ¹³C NMR (100.6 MHz, DMSO-d₆): 119.7 (q, J_CF = 322 Hz); ¹⁹F NMR (376.5 MHz, DMSO-d₆): δ = −80.9 (s); IR (ATR-mode): 1609 (w), 1317 (s, v_asSO₂), 1301 (m, vCF), 1194 (vs, vCF), 1120 (vs, v_sSO₂), 1050 (s, vCF), 803 (m), 749 (m), 645 (s), 598 (s), 572 (vs), 516 (s); Calc. for SrC₄F₁₂N₂O₈S₄ (647.89): C, 7.42%; N, 4.32%; found: C, 7.36%; N, 4.81%.

Due to the high salt concentration employed here, the density of the electrolyte deviates substantially from that of the solvent and needs to be considered. We targeted a final anion concentration of 1M, implying 1 mol/l of M⁺TFSI⁻, or 0.5 mol/l of M²⁺(TFSI⁻)₂, according to the valency of the metal cation. In some cases, the required salt to solvent mass ratio could be estimated accurately, and the concentration of 0.5M is confirmed by a subsequent density measurement. In other cases, the required salt to solvent mass ratio was determined by the density measurements of a concentration series; see the procedure described in Sec. II B 3. Finally, salt solutions with an anion concentration of 1M were prepared by weighing the appropriate mass of salt and solvent into a 20 ml vial. The procedure was performed inside an argon-filled glovebox. For NMR measurements, samples were filled into NMR tubes in the glovebox. Finally, the tubes were either closed for immediate measurement or evacuated and then flame-sealed.

NMR spectra of newly synthesized Ba(TFSI)₂ and Sr(TFSI)₂ were obtained with an AMX-600 spectrometer (Bruker, Germany) at 25 °C in DMSO-d₆ solvent and are listed in ppm. The signal corresponding to the residual protons of the deuterated solvent was used as an internal standard for ¹³C NMR, while for ¹⁹F NMR, C₆F₆ (−164.9 ppm) was added to the solution and applied as an external standard.

IR spectra were acquired on a Tensor 27 (Bruker, Germany) Fourier IR-spectrometer using ATR technology (128 scans, resolution is 2 cm⁻¹).

Density measurements were conducted on a DDM 2910 automatic density meter (Rudolph Research Analytical, Germany) at 25 °C. To prevent the electrolytes from the influence of water or oxygen, the instrument was flushed with dry nitrogen gas before each measurement. Density data served to determine the actual molar concentration of an electrolyte sample prepared with a known mass ratio of salts to solvents. For monovalent salts, the desired concentration of 1M was obtained easily. For salts with divalent cations, a calibration curve of the density vs salt concentration dependence was established. Calibration curves for Mg(TFSI)₂, Ca(TFSI)₂, Sr(TFSI)₂, and Ba(TFSI)₂ are shown in Fig. S1 of the supplementary material. There, the desired anion concentration of 1M was approximated stepwise by adjusting the required mass ratio. A linear relationship between density and concentration was obtained, from which a linear fit (Fig. S1) enabled the calculation of the mass ratio of salts to solvents to reach a concentration of 1M anions. The resulting densities and concentrations are summarized in Table S1.

NMR spectra and diffusion coefficients in salt solutions were measured either on a 400 MHz Avance III HD spectrometer or a 400 MHz Avance Neo spectrometer (Bruker, Rheinstetten, Germany). Diffusion measurements were taken with a gradient probe head with selective frequency inserts for ¹H and ¹⁹F, respectively, and a maximum gradient strength of 28 T/m (“Diff50,” Bruker), or a tunable broad band probe head with gradient coils providing a maximum gradient strength of 17 T/m (“BBODiff,” Bruker). The temperature was controlled using a GMH 3710 controller with a PT100 thermocouple (Greisinger electronics, Germany) and calibrated by inserting a 5 mm tube containing a thermocouple in ethylene glycol. All NMR measurements were performed within a range of 10–40 °C.

The gradient pulse length δ was set to values between 1 and 3 ms, and the observation time Δ was set between 50 and 300 ms, depending on the expected diffusion coefficient. The signal decay as a function of the gradient strength was well described by a single exponential in all cases; see Eq. (1).

The classical molecular dynamics (MD) technique integrates the classic Newton equation that governs the motion of particles.^25,61,86,87 The GAFF was used to model the PC solvent, using a combination of Lennard-Jones and point charge potentials. The potential parameters of PC were taken from Ref. 63. The potential parameters of Li⁺,⁵⁰ Na⁺,⁸⁸ K⁺,⁸⁹ divalent cations,⁸⁹ and PF₆⁻⁹⁰ and TFSI⁻⁹¹ anions were used. The PME method was used for the incorporation of long-range electrostatics, with a PME order of 4, a Fourier spacing of 0.12 nm, and an Ewald tolerance of 10⁻⁵. Intermolecular interactions were calculated using Lennard-Jones and Coulomb potentials. The cross-interactions between two kinds of molecular species were calculated based on the Lorentz–Berthelot mixing rules. The Lennard-Jones potential represented the van der Waals interaction, with a cutoff distance of 1.2 nm. Energy-pressure correction was incorporated into the dispersion part of the potential. MD simulations were performed at various temperatures with the GROMACS 2020.1 package. The leap-frog algorithm was used to integrate the equations of motion with a time step of dt = 1 fs, and all bonds of the PC molecule were constrained to its equilibrium distance. The Parrinello–Rahman barostat and Nosé–Hoover thermostat with constants of τ_p = 2 ps and τ_T = 2 ps, respectively, were used. All simulated electrolytes represented 1M salt solutions in PC. The PACKMOL package was used to generate the initial structures.

Commonly, the solvent used for the battery electrolyte preparation should meet the following demands: (a) a high polarity (dielectric constant > 15) should enable the dissolution of the metal salts, (b) it should be thermally stable (at least up to 120–130 °C) to prevent the overheating of the battery during fast charge/discharge processes, (c) it should maintain low viscosity (0.30–3.00 cP at 25 °C) for fast cycling, and (d) it should demonstrate sufficient electrochemical stability (at least 3.8–4.0 V vs Li/Li⁺).⁹⁶ Propylene carbonate (PC) was selected as it fulfills all the criteria mentioned, and electrolytes based on PC have been studied extensively in recent years. In turn, bis(trifluoromethylsulfonyl)imide (TFSI) salts were found to be a potentially good alternative to PF₆⁻-derived compounds since they could improve the chemical and thermal stability of the resultant electrolyte.⁹⁷ Therefore, for this work, 1M solutions (referring to the anion concentration) of Li⁺, Na⁺, K⁺, Mg²⁺, Ca²⁺, Ba²⁺, and Sr²⁺ were chosen in combination with TFSI⁻ or PF₆⁻ anions in PC as the objects of investigation.

An additional advantage of the TFSI salts is that the majority of them can be obtained from commercial sources. Not only lithium and sodium salts are available, but KTFSI, Ca(TFSI)₂, and Mg(TFSI)₂ can also be easily purchased. However, neither Ba(TFSI)₂ nor Sr(TFSI)₂ was available, and both were synthesized in the course of this work.

The metal TFSI salts can typically be synthesized through the reaction of a metal compound, such as a metal oxide, hydroxide, chloride, carbonate, or acetate, with trifluoromethanesulfonimide (HTFSI) acid in solution. However, when organic solvents such as alcohols, acetonitrile, or esters are used, the problem, connected with the fact that metal TFSI salts favor the formation of very stable solvates, complicates their isolation and purification procedures. In this case, the purification includes the evaporation of the solvent and the subsequent sublimation of the residual HTFSI from the salt. The process is complicated by the parallel sublimation of the main product, as metal TFSI salts possess relatively low sublimation points. The synthetic pathway can be improved significantly by conducting the reaction between HTFSI and metal carbonates in water, as shown in Scheme 1.^98–101 In this example, the formation of the solvates is avoided and the sublimation of unreacted HTFSI can be substituted with the simple utilization of a slight excess of metal carbonate, which is insoluble in water and can be filtered off at the end of the reaction. By using this protocol, both Ba(TFSI)₂ and Sr(TFSI)₂ salts were obtained in good yields (≥90%) as white powders. The structure and purity of Ba(TFSI)₂ and Sr(TFSI)₂ were confirmed by NMR and IR spectroscopy, as well as by elemental analysis.

Since most of the cations are not directly accessible for NMR measurements, we focus here on the transport of the solvent as well as the anions, in order to obtain information about the influence of molecular scale structuring on the ion transport properties of electrolytes. The differences, depending on the nature of the cation, will then be interpreted in terms of local molecular structuring obtained from MD simulation results.

At first, the temperature dependence of the PC and anion diffusion coefficients was determined by ¹H PFG-NMR and ¹⁹F PFG-NMR, respectively. All data are summarized in Fig. 1 in the form of Arrhenius plots. The temperature-dependent diffusion coefficients of all the species obey the Arrhenius law with rather similar activation energies, as shown by the straight lines, which all have a similar slope. The diffusion coefficient values, however, show distinct differences depending on the position of the respective cation in the main group series. For the monovalent cations (Li⁺, Na⁺, K⁺), the PC solvent diffusion was enhanced with the increase in cation size: D_1H (LiTFSI) < D_1H (NaTFSI) < D_1H (KPF₆). We note here that KTFSI led to turbid solutions in PC; thus, we employ KPF₆ for comparison. For the divalent cations (Mg²⁺, Ca²⁺, Sr²⁺, Ba²⁺), the trend is opposite, i.e., the diffusion coefficients decreased with the increase in cation size, with only Ba²⁺ being an exception: D_1H [Mg(TFSI)₂] > D_1H [Ca(TFSI)₂] > D_1H [Sr(TFSI)₂] ≈ D_1H [Ba(TFSI)₂]. For the anion diffusion, a similar picture emerged, as anion diffusion coefficients in electrolytes with divalent cations decreased with the increase in cation size: D_19F [Mg(TFSI)₂] > D_19F [Ca(TFSI)₂] > D_19F [Ba(TFSI)₂] ≈ D_19F [Sr(TFSI)₂]. For the monovalent cations, no clear trend was identified since anion diffusion in LiTFSI and NaTFSI solutions was found to be similar. At the same time, the PF₆⁻ diffusion in 1M KPF₆ PC solution was much faster, which can be attributed to the smaller size of the PF₆⁻ anion compared to TFSI⁻.

The calculated activation energies of solvent and anion diffusion, respectively, are provided in Fig. 2. As already visually evident from Fig. 1, the values of the activation energies are in a very similar range. Interestingly, distinct trends can also be identified here: in correlation with the increase in the solvent diffusion coefficient with the rise in the monovalent cation size, the activation energies of solvent diffusion decreased. Similarly, the opposite trend was observed for activation energies of solvent diffusion in the solutions of salts with divalent cations (as shown for Mg²⁺ and Ca²⁺ salts in Fig. 2). For the activation energies of anion diffusion, no clear trends were identified.

Thus, a distinct effect of the cation nature on both the solvent and anion diffusion was shown via experimental (¹H and ¹⁹F PFG-NMR) methods. The trends observed in dependence on the cation size differed distinctly for monovalent and divalent cations, respectively. These opposing trends are likely connected to the solvation structures of the cations, which will be analyzed and discussed further in Secs. III E and III F.

Here, we calculate the diffusion coefficients of ions and solvents by means of MD simulations and compare them to experimental measurements (see Sec. III C) for the 1M PC solutions of LiTFSI, NaTFSI, KPF₆, Mg(TFSI)₂, Ca(TFSI)₂, Ba(TFSI)₂, and Sr(TFSI)₂ salts. It is known that the non-polarizable force fields do not explicitly take into account the electronic polarization.¹⁰² One way to remedy this issue is to incorporate electronic polarization in a mean field way via charge rescaling.¹⁰² Leontyev and Stuchebrukhov proposed an electronic continuum theory¹⁰³ to account for the effects of electronic polarization in non-polarizable force fields. In particular, their theory claims that ions in solutions should have charges scaled by a factor of about 0.7.¹⁰⁴ In parallel, in ionic liquids, quantum calculations have shown that the total charge of monovalent ions should be scaled between 0.6 and 0.8.^105,106 Moreover, it was demonstrated that the simulation of the carbonate solvent molecule diffusion in electrolyte solutions also requires this correction.⁵⁶ By using the ab initio and classical molecular dynamics, Chaudhari and Rempe established that the best fitting between the simulation and experimental diffusion coefficient data is achieved when a 0.9 charge-scaling factor is applied for propylene carbonate solvent.⁵⁶ Finally, as published by our team recently,⁶¹ the optimum charge-scaling factor for monovalent Li⁺ cations was found to be 0.8. The application of the 0.8 charge-scaling factor allowed for the best match between the simulation results for LiPF₆ salt diffusion^57,61 in solution and experimental data.

Taking into account the points mentioned above, the current work started with the application of an 0.8 charge-scaling for the calculation of the Li⁺ and TFSI⁻ diffusion and 0.9 charge-scaling for the simulation of the PC molecules in 1M LiTFSI PC solution. The calculated temperature dependences for the diffusion coefficients of both PC and TFSI anions in 1M LiTFSI PC solution are shown in Fig. 3. The two simulated diffusion coefficients conform very well, within the error margin, with the experimental values for both PC and TFSI⁻ over the whole temperature range.

Then, the scaling factors for the charge of the other cations (and identically the charges of TFSI⁻ anions, in order to have a neutral solution) were varied for the simulated diffusion coefficients, of both solvents and anions, over the whole temperature range, to fit with the respective experimental values (Figs. 4 and S3). For this, the charge-scaling factor for the PC was kept unchanged and is equal to 0.9 (a 0.92 value used for Li systems) as proposed in Ref. 56. In particular, charge-scaling factors equal to 0.8 and 0.82 for Na⁺ and K⁺ cations were used, respectively. The computed diffusion coefficients for PC solvent and TFSI⁻ anions in 1M NaTFSI PC solutions (Fig. S3) and 1M KPF₆ PC solutions (Fig. 4) were found to match perfectly, within the error margin, with the corresponding values of the experimental diffusion coefficients measured via PFG-NMR over the whole temperature range considered.

In addition, we compared the diffusivity in the KPF₆/PC solution to that in LiPF₆/PC solutions. It can be seen from Fig. 4(a) that the PC diffusion coefficient in electrolytes containing K⁺ cations is much faster than in PC solutions with Li⁺ cations.

Switching to the simulations of the salt solutions derived from divalent metals in 0.5M Ca(TFSI)₂ solutions, the best agreement with experimental data was obtained when the charges of both Ca²⁺ and TFSI⁻ were scaled by 0.7. At this point, simulated values for the TFSI⁻ anion and PC diffusion conformed well, within the error margin, with the experimental values in the whole temperature range (Fig. 5), while the calculated temperature dependence for PC diffusion was found to be slightly lower (outside the error margin) than the experimental values. Further on, the scaling factor of Mg²⁺ and TFSI⁻ anions was varied and finally set to 0.6 to yield the best agreement between the simulation and the experiment. The diffusion coefficients calculated for the PC and TFSI anions in 0.5M solution of Mg(TFSI)₂ are presented in Fig. S4. The simulated TFSI⁻ diffusion coefficients ideally coincided with those measured experimentally in the 313–283 K temperature range (Fig. S4). As with the simulation of the Mg(TFSI)₂ solution, the calculation of PC anion diffusion coefficients for 0.5M Mg(TFSI)₂ solution was found to be slightly lower with experimentally obtained values over the whole temperature range.

The application of the 0.75 scaling factor to the simulation of the TFSI⁻ anion diffusion coefficients in 0.5M Sr(TFSI)₂ solution allowed us to achieve an excellent overlap with experimental values (Fig. S5), although the PC diffusion was slightly overestimated [Fig. S5(a)]. For the simulation of the 0.5M and Ba(TFSI)₂ PC solution, a scaling factor of 0.75 conformed well with the experiment within the error margins (Fig. S6). Finally, the charge scaling factors of the ions, which were required to achieve agreement between simulation and experiment, are shown in Fig. S2. While the size of the monovalent cations Li⁺, K⁺, and Na⁺ does not have an influence on the scaling factors, the factors for the divalent cations scaled almost linearly with their crystal ionic radius with a factor of R² = 0.9 (Fig. S2).

The RDFs between either the Li⁺ or Na⁺ cation and carbonyl oxygen atom of PC solvent calculated at 298 K are demonstrated in Fig. 6(a). The nearest neighboring distance of the Li⁺-carbonyl oxygen was found to be 0.20 nm, which is lower than the respective neighboring distance for Na⁺-carbonyl oxygen (0.25 nm). Thus, the Li⁺ cation is more strongly solvated by the solvent’s carbonyl oxygen than the Na⁺ cation.

The interaction between the anion and solvent was then calculated. Figure 6(b) demonstrates the RDFs between the N atom of the TFSI anion and the carbonyl oxygen of the solvent. It can be observed that for both salt solutions, the anion–solvent coordination was very weak and the TFSI anions were hardly solvated by PC. The anion–cation interactions for LiTFSI and NaTFSI solutions can be traced from Fig. 7. While Fig. 7(a) shows the RDF of Li⁺ and Na⁺ cations with N atoms of TFSI anions, Fig. 7(b) demonstrates their RDF with oxygen atoms of TFSI anions. The peak of Na–O(TFSI anion) was found to be slightly higher than the peak of Li–O(TFSI), demonstrating that Na⁺ is slightly more strongly solvated by the TFSI anion in comparison with the lithium cation. Furthermore, comparing Figs. 6(a) and 7(b), the peak of Na-carbonyl (solvent) is higher than that of Na–O(TFSI anion), leading to the conclusion that Na⁺ cations are less solvated by TFSI anions than by PC molecules.

Figure 8(a) shows the RDF between the Li⁺ and K⁺ cations and the P atom of the PF₆ anion [Fig. 8(a)] or the carbonyl oxygen of PC solvent [Fig. 8(b)] calculated at two different temperatures (283 and 303 K). As can be seen from Fig. 8, the change in temperature had practically no impact on the surroundings of either (Li⁺ and K⁺) of the cations in solution. The nearest neighboring distance of the Li⁺-carbonyl oxygen is 0.2 nm, which is lower than that of the neighboring distance of K⁺-carbonyl oxygen (0.3 nm) due to the smaller size of Li⁺ in comparison with K⁺. Figure 8(b) shows that Li⁺ is more strongly solvated by the PC carbonyl oxygen than K⁺. Moreover, the solvation of Li⁺ cations by the P atom of the PF₆⁻ anion is less pronounced than the solvation by the carbonyl oxygen of PC. This leads to the conclusion that Li⁺ is more prone to interaction with the oxygen atoms of the solvent than with the counter anion. In contrast to Li⁺, K⁺ cations showed the opposite behavior, consisting of higher solvation by the anions in comparison with solvation by PC.

Figure 9(a) shows the RDF of Mg²⁺ and Ca²⁺ cations with oxygen atoms of either PC or TFSI⁻ anions at 298 K. It can be seen that Ca²⁺ cations are solvated with the TFSI⁻ anions rather than the Mg²⁺ cation. In Fig. 9(b), we can see the RDF of the N atom of the TFSI⁻ anion with the Mg²⁺, Ca²⁺ cations, for both types of solutions, showing a weak but different layering structure. The nearest neighboring distance of the Mg²⁺-carbonyl oxygen is 0.2 nm, which is lower than the neighboring distances of the Ca²⁺-carbonyl oxygen, due to its smaller size, as seen in Fig. 10(a). The Mg²⁺-carbonyl oxygen RDF appears as a secondary smaller peak at 0.28 nm. The RDF of the N atom of TFSI⁻ and carbonyl oxygen is very similar for the PC solutions, containing either Mg²⁺ or Ca²⁺ cations, showing a weak layering, as can be seen in Fig. 10(b). The Mg²⁺ divalent counterions, due to their smallest size, are only solvated by the PC molecules and are strongly attached to PC, and not to the TFSI⁻ anions; thus, the anions are completely free to move inside the solution, leading to enhanced diffusivity.

The increase in the PC diffusion coefficient with the size of the cation (Fig. 1) can be clearly attributed to the decrease in the cation’s solvation shell. The smallest cations, namely Li⁺, exhibit the largest solvation shell, as the PC solvent molecules coordinate strongly with the cation, thus reducing the overall solvent diffusion coefficient. In solutions with larger monovalent cations, such as Na⁺ or K⁺, and a smaller solvation shell of PC [Figs. 6(a) and 8(a)], the PC diffusion is faster due to a larger fraction of free solvent molecules.

At the same time, the anion diffusion coefficient was found to be independent of the cation size for monovalent cations [Fig. 1(b) compares LiTFSI and NaTFSI and Fig. 4(b) compares LiPF₆ and KPF₆]. While the larger cations experience a reduced solvation shell, thus impacting the cation as well as the solvent transport, the impact on anion transport is negligible. In fact, when comparing the RDFs, the larger cation only has a minor influence: (i) A slightly enhanced anion–solvent first coordination shell is observed [Fig. 6(b) for TFSI⁻ anion solutions and Fig. S8(a) for PF₆⁻ anion solutions], which was possibly enabled by the lower solvation of the larger cations [Figs. 6(a) and 8(b) for TFSI⁻ and PF₆⁻ anion solutions, respectively]. Note that the salt concentration is very high and a reduction of the solvation shell of the cation may slightly shift the solvation equilibrium toward a larger solvation shell of the anion. (ii) The cation–TFSI pair correlation is only slightly different for larger cations, such as Na⁺ compared to Li⁺ [see Figs. 7(a) and 7(b)]. Both minor effects apparently do not alter the anion diffusion coefficient in comparison with smaller monovalent cations.

As discussed above, while Mg²⁺ is only solvated by PC, larger divalent cations experience an enhanced interaction with the anions, which also has an influence on the anion diffusion coefficient. A decrease in the anion diffusion coefficients with the increase in cation size was clearly observed in Fig. 1(b) and confirmed by MD simulations. Such a decrease can be attributed to the enhanced cation–anion coordination with increasing cation size (Mg²⁺ < Ca²⁺ < Sr²⁺ < Ba²⁺), as observed in Figs. 9(a) and 9(b). Especially in the M²⁺-N(TFSI) RDF [Figure 9(b)], a clear increase in the population of the first coordination shell of the cation by anions is evident with the increasing cation size. Larger (such as Ba²⁺, Sr²⁺) and thus softer cations apparently interact more strongly with the anions and lead to the diffusion of anions in pairs or larger clusters. Such clusters or pairs seem to appear only for the salts of divalent cations due to the stronger electrostatic interaction. Meanwhile, the cation size has only a small effect on the cation’s solvation with solvent molecules [Fig. 10(a)]. Nevertheless, the PC diffusion coefficient clearly decreases with the increase in cation size, with the size of the cation solvation shell possibly increasing due to the presence of anions in the latter, inducing an increasing fraction of PC in the solvation shell and thus a decreasing PC diffusion coefficient with the increasing cation size. These trends of diffusion coefficient reduction for both PC and anions end for very large cations (Ba²⁺, Sr²⁺), where the respective diffusion coefficients are identical within the error margin.

The temperature dependence of the diffusion of two most popular anions [PF₆⁻ and (CF₃SO₂)₂N⁻ (TFSI)] and the propylene carbonate (PC) solvent molecules in liquid electrolyte solutions prepared from different salts of monovalent (Li⁺, Na⁺, K⁺) and divalent (Mg²⁺, Ca²⁺, Ba²⁺, Sr²⁺) metals has been systematically investigated. Experimental measurements of the anion and PC diffusion coefficients were successfully combined with MD simulations to provide an understanding of how the cation’s size and anion’s type influence the solvation structures and ion diffusion in liquid electrolytes. A very good agreement between the simulated and experimentally determined diffusion coefficients was observed. The charge-scaling factors exhibit a systematic almost linear dependence on the cation size, making feasible the prediction of diffusion coefficients, even without employing experimental data.

It was found that for the monovalent cations (Li⁺, Na⁺, K⁺), the PC solvent diffusion enhanced with the increase in cation size, while the anion diffusion remained almost unchanged. For the divalent cations (Mg²⁺, Ca²⁺, Sr²⁺, Ba²⁺), the opposite trend was detected, i.e., the PC and the anion diffusion coefficients decreased with the increase in cation size. Comparing anions and solvent diffusion, PC molecules are the faster species. To explain these observations, additional calculations of the radial distribution functions (RDF) between cations, anions, and solvent molecules were performed. It was revealed that the local coordination shell of the cations, which strongly varies with their size, has a distinct implication for the ion’s dynamics and diffusion in the electrolyte. The strong solvation of ionic species increases their solvation radius and thus reduces their diffusion coefficient. The larger divalent cations were found to more strongly interact with anions in comparison with small monovalent cations, thus leading to a decrease in diffusion. The solvent diffusion was affected by the solvation structure of the ionic species, as the solvent molecules in the solvation shell are hindered in their dynamics. Finally, it was demonstrated that the electrolyte solution structural picture obtained from the calculations of radial distribution functions, although describing only static interactions, could serve to provide explanations for the dynamic transport behavior of all species in the system.

The supplementary material (SI file) is available free of charge and comprises the following: densities and concentrations of electrolytes used for diffusion measurements at 25 °C; correlation of the charge-scaling factors with cation ionic radius and electronegativity; comparison of MD simulations with experimental diffusion data for the 1M PC solutions of NaTFSI, Ca(TFSI)2, Ba(TFSI)2, and Sr(TFSI)2; and RDFs of the P atom of PF6 and carbonyl oxygen and the N atom of TFSI− and carbonyl oxygen of PC.

The financial support was provided by the Luxembourg Institute of Science and Technology (LIST) via the internal project “Simulation methodology for electrolyte behavior in modern batteries and ionomers” (SIMBATT). MM was supported by the International Graduate School for Battery Chemistry, Characterization, Analysis, Recycling and Application (BACCARA) funded by the Ministry for Culture and Science of North Rhine Westphalia, Germany. We would like to acknowledge the use of the computational facilities at both the University of Surrey (Eureka) and LIST. We would also like to thank Professor Daniel F. Schmidt (LIST) for useful discussions.

The authors have no conflicts to disclose.

Argyrios V. Karatrantos: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Validation (equal); Writing – original draft (equal). Maleen Middendorf: Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Validation (equal); Writing – review & editing (equal). Daniil R. Nosov: Investigation (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). Qiong Cai: Resources (lead); Writing – review & editing (equal). Stephan Westermann: Validation (equal); Writing – review & editing (equal). Katja Hoffmann: Investigation (equal); Methodology (equal); Validation (equal). Pinchas Nürnberg: Investigation (equal); Methodology (equal); Validation (equal). Alexander S. Shaplov: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (lead); Supervision (equal); Validation (equal); Writing – review & editing (equal). Monika Schönhoff: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.