Theoretical Analysis of Carrier Ion Diffusion in Superconcentrated

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Theoretical Analysis of Carrier Ion Diffusion in Superconcentrated Electrolyte Solutions for Sodium-Ion Batteries Masaki Okoshi, Chien-Pin Chou, and Hiromi Nakai J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10589 • Publication Date (Web): 12 Feb 2018 Downloaded from http://pubs.acs.org on February 16, 2018

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The Journal of Physical Chemistry

Theoretical Analysis of Carrier Ion Diffusion in Superconcentrated Electrolyte Solutions for Sodium-Ion Batteries

Masaki Okoshi,a,b Chien-Pin Chou,c and Hiromi Nakai*a,b,c,d

a

b

Department of Chemistry and Biochemistry, Waseda University, Tokyo 169-8555, Japan Elements Strategy Initiative for Catalysts & Batteries (ESICB), Kyoto University, Kyoto

615-8245, Japan c

Research Institute for Science and Engineering (RISE), Waseda University, Tokyo 169-8555,

Japan d

CREST, Japan Science and Technology Agency, Saitama 332-0012, Japan

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Abstract

Superconcentrated electrolyte solutions are receiving increasing attention as a novel class of liquid electrolyte for secondary batteries because of their unusual and favorable characteristics, which arise from a unique solution structure with a very small number of free solvent molecules. The present theoretical study investigates the concentration dependence of the structural and dynamical properties of these electrolyte solutions for Na-ion batteries using large-scale quantum molecular dynamics simulations. Microscopic analysis of the dynamical properties of Na+ ions reveals that ligand (solvent/anion) exchange reactions, an alternative diffusion pathway for Na+ ions, are responsible for carrier ion diffusion in the superconcentrated conditions.


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Introduction

The electrolyte solution is one of the main components of secondary batteries and has a

significant

impact

on

battery

performance.1,2

Among

electrolyte

solutions,

superconcentrated solutions3-5 are receiving growing and intense attention as a novel class of liquid electrolyte with unusual, but favorable, characteristics compared to conventional dilute electrolyte solutions. For example, these electrolytes have

high electrochemical

(oxidative/reductive) and thermal/chemical stabilities and low volatilities. However, a major drawback is the low ionic conductivity arising from high viscosity at superconcentrated conditions, although the decrease in ionic conductivity can be minimized by the appropriate choice of solvent and anion species. Lithium-ion batteries (LIBs) suffer a severe problem: limited lithium resources; thus, sodium-ion batteries (SIBs) are regarded as a potential alternative.1,2 Recent experimental efforts have shed light on the use of superconcentrated electrolyte solutions for SIBs. Zhang and coworkers6 demonstrated that solid sodium metal could work as a reversible anode in combination with a 4 M dimethoxyethane solution of sodium bis(fluorosulfonyl)amide (NaFSA-DME). Freunberger and coworkers7 showed that a NaFSA-DME solution with a molar ratio of 0.5 allows the reversible plating/stripping of a Na-metal anode without dendritic plating. Choi and coworkers8 reported detailed data on the physicochemical and 3 ACS Paragon Plus Environment


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electrochemical properties of 1 to 5 M NaFSA-DME solutions. Theoretical analyses have the potential to provide insights at the molecular, atomistic, and electronic levels. For instance, ab initio molecular dynamics (AIMD) simulations based on density functional theory (DFT) have been successfully applied to superconcentrated electrolyte solutions for LIBs.4,5 AIMD simulations have revealed the unusual solution structure with no free solvent molecules of superconcentrated electrolyte solutions, which results in their characteristic high redox resistance. Similar analysis concerning SIBs would be helpful for the development of superconcentrated electrolyte solutions for SIBs. Furthermore, the conduction mechanism of carrier ions in superconcentrated solutions is of great importance. Because there are few or no free solvent molecules, the conventional and usual vehicular-type diffusion mechanism, in which a carrier ion solvated by several solvent molecules migrates in the form of [Li/Na-(solvent)n]+, is impossible. Instead, a ligand-exchange-type mechanism, in which dissociation/association reactions between the carrier ion and solvent/anion take place repeatedly, is assumed to be the only means of carrier ion conduction.4 Solvation properties and dynamics of carrier ions in (super)concentrated electrolyte solutions, ionic liquid-based electrolyte solutions, and/or solvate ionic liquids have attracted much attentions.9-16 In theoretical studies, classical molecular dynamics (MD) simulations have been widely employed. Dynamics of the solvate formation was analyzed by using MD 4 ACS Paragon Plus Environment

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simulations in the early work by Henderson and coworkers10 for LiBF4-CH3CN systems. Zhang and Maginn11 systematically investigated the lifetimes of ion pairs for a wide variety of ionic liquids. Forsyth et al.12 observed exchange of coordination environments of Na+, which is specific at higher concentration condition, for NaFSA in ionic liquid system via NMR measurements and MD simulations. Abu-Lebdeh and coworkers16 proposed a conceptual model for changes in solution structure and carrier ion diffusion mechanisms with respect to the concentration of salt. One of main issues to treat electrolyte solutions using MD simulations, is the adequacy of classical force fields, since the molecular interactions are highly complicated:15 Coulombic (electrostatic), polarization, charge-transfer, and covalent interactions are taking place. In classical MD simulations, polarization, charge-transfer and covalent interactions are often incorporated into consideration by means of an effective way, i.e., static scaling of charges assigned to atoms in molecules. Recently, Ishizuka and Matubayasi17,18 proposed a sophisticated way to obtain the scaling factors to fit AIMD simulations. However, the results of MD simulations are quite dependent on the charge fitting technique and the best choice is not achieved so far. This may reflect the complicated behavior of electronic structures in electrolyte solutions, which are dynamically fluctuating along with atomic motions of solvents, cations, and anions.15 Quantum molecular dynamics (QMD) simulation, as represented by AIMD, in which 5 ACS Paragon Plus Environment


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the electronic structure of the system is treated quantum mechanically, is a suitable tool to investigate the dynamical processes associated with chemical reactions, as well as dynamical changes of quantum chemical interactions. The only remaining (and the most serious) problem is that QMD is computationally expensive; thus, the system size and simulation time are strictly limited. The use of semi-empirical methods instead of DFT reduces the computational costs drastically, but the accuracy of these methods is severely dependent on the parametrization methodology and system of interest. The density-functional tight-binding (DFTB) method, which is an approximation of DFT, has attracted much attention because of its superior accuracy among the semi-empirical methods. Although DFTB allows an approximately three-orders of magnitude speed-up compared to DFT, the computational costs scale cubically with the system size prohibiting QMD simulations of large-scale systems. However, adopting a linear-scaling technique, such as the divide-and-conquer (DC) method, for instance, is a solution. The recently developed DC-DFTB-MD method9 allows the routine QMD simulation of systems with several thousand (or more) atoms for several tens of picoseconds; this is possible with the aid of massively parallelized computations.19-22 In the present work, we report the structural and dynamical properties of superconcentrated NaFSA-DME solutions, as well as those of dilute solutions, via DC-DFTB-MD simulations. The radial distribution functions (RDFs) of the solvent and anion 6 ACS Paragon Plus Environment

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around the Na+ ion, as well as the integrated RDFs (coordination numbers), and the number of solvent/anion, classified with the number of coordinating Na+ ions, were analyzed. Finally, the diffusion properties of the carrier ions were investigated regarding diffusion coefficients and time-correlation analyses of the coordination structure.

Computational Details

NaFSA-DME solutions with molar concentrations of NaFSA of 5% and 10% were employed as the dilute (conventional) solution, while the 40% solution employed as the superconcentrated solution. The densities of the solutions were determined using classical MD simulations, as follows. The geometries of the FSA- anion and DME molecule were optimized using DFT calculations. The B3LYP hybrid functional,23 consisting of the Hartree-Fock

exchange,

Slater

exchange,24

Becke

exchange,25

Vosko-Wilk-Nusair

correlation,26 and Lee-Yang-Parr correlation functionals,27 was employed in combination with Dunning’s correlation-consistent triple-zeta valence plus polarization and diffuse type basis sets (aug-cc-pVTZ).28-30 The point charges on each atom in FSA- and DME were evaluated at the same level of theory (B3LYP/aug-cc-pVTZ) by fitting to reproduce the electrostatic potentials according to the Merz-Sing-Kollman scheme.31,32 All-atom type molecular fields for Na+, FSA-, and DME were constructed by using the topolbuild utility program,33 based on 7 ACS Paragon Plus Environment


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the general Amber force field34,35 in combination with the obtained point charges. MD simulations were performed for 16 ns with a constant time step of 0.25 fs under three-dimensional periodic boundary conditions (3D-PBC) for 1000 molecules in total, i.e., (NaFSA)nDME1000-n, where n equals 50, 100, and 400 for the 5%, 10%, and 40% solutions, respectively. The Berendsen thermostat and barostat36 with a target temperature and pressure of 298.15 K and 1.013 bar, respectively, were applied to achieve the NPT ensemble. The volumes and densities of the solutions were averaged over the last 2 ns of the simulations, and convergence was confirmed. We also performed similar investigations for the pure DME solution system (n = 0). The density of pure DME solution was evaluated to be 873.8 g/L, which was in reasonable agreement with the experimental value, i.e., 866.5 g/L (293.15 K, 0.10 MPa).37 The DC-DFTB-MD simulations were performed as follows. Simulation boxes were set up by randomly arranging Na+ and FSA- anions and DME molecules with the densities obtained from classical MD simulations. The numbers of DME and NaFSA molecules, the total number of atoms, and the number of basis sets are summarized in Table 1. The densities and estimated computational times are also shown. The DFTB3 method was employed with the 3OB parameter set,38-40 which has been optimized for organic and biological molecules with modifications for the S-N and S-F repulsive potentials using the recent developed parameterization toolkit.41 The modification of the repulsive potentials was carried out not 8 ACS Paragon Plus Environment

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only to remove the discontinuity found in the S-F repulsive potential but improve the description of the S-N and S-F bonds in the FSA- molecule. The resulting S-N and S-F potential files are given in the Supporting Information. The dispersion interactions were empirically incorporated using the so-called DFT-D3 correction method with the Becke-Johnson damping scheme.42,43 The accuracy of the employed parameters, i.e., DFTB/3OB-D3, were confirmed via comparisons with the DFT calculations, as shown below. The DC technique using automatically constructed subsystems was applied to accelerate calculations. Following the equilibrium calculations performed for 10 ps with a time step of 1.0 fs under the NVT ensemble (T = 298.15 K) controlled by the Berendsen thermostat,36 production runs were performed for 20 ps with a time step of 0.5 fs under the NVE ensemble with the velocity Verlet integration scheme. The 3D-PBC was applied throughout. DFT calculations were performed using Gaussian 09,44 while the GROMACS 5.1.4 software45-51 was employed for classical MD simulations. The DC-DFTB-MD simulations were performed by using the DC-DFTB-K program,19 developed in our laboratory, in combination with the Fujitsu FX-10 supercomputer, installed at the Research Center for Computational Science (RCCS), Okazaki Research Facilities, National Institutes of Natural Sciences (NINS) and the Institute for Solid State Physics (ISSP), the University of Tokyo.



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Results and Discussion

Accuracy of the DFTB/3OB-D3 method. – The accuracy of the DFTB/3OB-D3 method was confirmed via comparisons with DFT calculations at the B3LYP/aug-cc-pVTZ level of theory. The desolvation energies were evaluated based on the one-to-one model complexes that have one Na+ ion interacting with one solvent molecule (or FSA- anion) following our previous protocols.52,53 The employed solvent species were butylene carbonate, diethyl carbonate, ethylene carbonate, ethyl methyl carbonate, and propylene carbonate as representative carbonates; dimethyl formamide, N-methyl oxazolidinone, and N-methyl pyrrolidinone as representative amides; adiponitrile, acetonitrile, glutaronitrile, 2-methoxy acetonitrile, and 3-methoxy propionitrile as representative nitriles; 1,4-dioxane, 2-methyl tetrahydrofuran, and DME as representative ethers; and nitroethane. Figure 1 shows the desolvation energies (EDsol) for one-to-one complexes, evaluated from the following energy differences: EDsol = E[Na+] + E[solvent/ FSA-] – E[complex],

(1)

where E[Na+], E[solvent/ FSA-], and E[complex] are the electronic energies of the free Na+ ion, free solvent (or anion), and complex, respectively. Structures optimized at the corresponding levels of theory, DFTB/3OB-D3 and B3LYP/aug-cc-pVTZ, were employed for the structures of the solvents, FSA- anion, and the complexes. Because DFTB neglects the 10 ACS Paragon Plus Environment

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core electrons and only considers the valence electrons, a unique procedure is required to evaluate E[Na+], as follows: E[Na+]DFTB = E[Na]DFTB + E[Na+]DFT – E[Na]DFT,

(2)

where the subscripts represent the electronic energies calculated using DFTB or DFT. We employed the value of 0.088184 Hartree as E[Na+]DFTB, following the procedure used to construct the 3OB parameter set. The desolvation energies evaluated at the DFTB/3OB-D3 level of theory, shown on the vertical axis, show excellent agreement with the reference values evaluated at the B3LYP/aug-cc-pVTZ level of theory and shown on the horizontal axis. A linear relationship with the gradient, intercept, and coefficient of determination (R2 value) of 1.02, 8.8, and 0.974 was obtained.

Concentration dependence of the solution structure of the NaFSA-DME solution. – The solvent/FSA- coordination structures were analyzed from the viewpoints of their RDFs (g(r)) and the integrated RDFs, i.e., coordination numbers (n(r)) around Na+ ions. Figure 2 shows the RDFs for Na+ and O of DME (blue), Na+ and O of FSA- (orange), Na+ and F (green), and Na+ and N (purple) atom pairs for (a) 5%, (b) 10%, and (c) 40% solutions. The horizontal axis shows the distance between the Na-X (X = O(DME), O(FSA-), F, N) atom pairs, while the vertical axis shows the distributions, which were normalized for each element. It should be noted that O(DME) and O(FSA-) were not distinguished in normalization, i.e., 11 ACS Paragon Plus Environment


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O(DME) and O(FSA-) were normalized so that their sum becomes unity at the long-range limit. In the case of salt concentration of 5%, two peaks were observed for the Na-O(DME) and Na-O(FSA-) pairs at the peak positions of 2.35 and 2.45 Å, respectively. These bond distances correspond to covalent/coordination bonds, as compared to the typical bond lengths taken from Ref. 54 (Table 2). F and N atoms showed no clear peak corresponding to the first solvation shell. Concerning the 10% and 40% solutions, there were peaks corresponding to F atoms within the first solvation shell, in addition to those of the O atoms of DME and FSA-. In contrast, the N atoms showed no clear peak in the first solvation shell. All RDFs for the first solvation shell showed minimum values at ca. r = 3 Å. The numbers of coordinating atoms to Na+ (n(r)), defined as the integration of number density weighted RDFs, are shown in Figure 3. The values of n(r) for the individual atom pairs are summarized in Table 3 with a cut-off value of r = 3 Å, reflecting the minimum of RDFs for the first solvation shell. The total coordination numbers, evaluated as the summation of n(r = 3 Å) for each pair, are 5.8, 5.9, and 5.7 for the 5%, 10%, and 40% solutions, respectively, indicating that the Na+ ion was approximately hexacoordinated in all cases. The oxygen atoms of DME and FSA- mainly contributed to solvation, while F atom of FSA- moderately coordinated in 10% and 40% solutions, with coordination numbers of (O(DME), O(FSA-), F) of (5.4, 0.4, 0.0), (3.9, 1.7, 0.3), and (2.6, 2.8, 0.3). The FSAcontribution increased with increasing salt concentration, while that for DME decreased. The 12 ACS Paragon Plus Environment

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changes in coordination numbers showed a non-linear behavior: larger changes were observed from 5% to 10% solutions than from 10% to 40% solutions. The coordination number of N atom was zero for all systems. Table 4 summarizes the numbers of solvents/anions classified with the numbers of coordinating Na+ ions. The coordinating numbers of the solvents/anions were evaluated by counting the number of Na+ ions, which were interacting with oxygen atoms in the solvents/anions of interest with bond lengths of 3 Å or shorter. The numbers of Na+ ions, which held another Na+ ion in the second neighbor, are also shown. The values averaged over the 20-ps simulation are shown. The number of free solvent molecules/anions, for which the coordinating number equals zero, monotonically and drastically decreased with increasing salt concentration. In the case of the superconcentrated solution with a salt concentration of 40%, 10.5% and 0.5% of free DME and FSA- molecules remained, respectively. Observations consistent with these findings have been obtained by recent experimental studies; that is, a small amount of free DME remains even in superconcentrated solutions.8 Although the specific characters of superconcentrated solutions would mainly come from the vanished free solvent, the previous experimental studies showed the unusual favorable properties of superconcentrated NaFSA-DME solutions.6-8 This implies that very small amount of remaining free solvent may not have so huge implications. The coordinating number of two or more corresponds to the multi-coordinated 13 ACS Paragon Plus Environment


character

of

solvents/anions.

In

the

5%

solution,

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no

solvents/anions

showed

multi-coordinated character. This is consistent with that no Na+ ion had another one in the second neighbor. In the case of 40% solution, contrastingly, 87.1% (42.1% + 32.0% + 12.6% + 0.4%) of FSA- and 18.4% (17.6% + 0.7%) of DME showed multi-coordinated character. At the same time, 96.4% of the Na+ ions held one or more Na+ ions in the second neighbor. Consequently, the 40% solution possessed the network structure, in which almost all Na+ ions connected to each other mainly via FSA-, while DME also contributed to the network partially. It should be noted that the network structure may prohibit the conventional vehicular-type diffusion, in which every carrier ion moves independently with surrounding solvent molecules forming the first solvation shell. As for the 10% solution, the network structure was partially formed: 43.7% (30.1% + 13.6%) of FSA- showed multi-coordinated character and 74.8% of Na+ ion held another one in the second neighbor.

Carrier-ion diffusion properties of NaFSA-DME solutions. – The diffusion coefficients of Na+, FSA-, and DME were evaluated based on the gradient of the mean-square displacements (MSDs) of the center-of-masses (COMs) of each molecule/atom: 〈∑| − 0| 〉 = 6,

(3)

where i runs over all COMs for molecules, atoms, and ions of concern, ri(t) is the position of ith COM at time t, n is the total number of molecules, atoms, and ions, and D is the diffusion 14 ACS Paragon Plus Environment

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coefficient. Figure 4 shows the correlation between the MSD and the simulation time for (a) DME, (b) FSA-, and (c) Na+ in the 5%, 10%, and 40% solutions; these are shown in blue, orange, and green, respectively. The dotted lines, which were determined by using the least-squares fitting technique for t = 5 – 20 ps, showed quasi-linear correlations with R2 values more than or equal to 0.74. The log-log plots for the same data are given in the Supporting Information. In the case of 5% solution, the slopes of the log-log plots were (DME, FSA-, Na+) = (0.98, 0.98, 1.02), implying the simulation reached to the normal diffusion region, in which the slope becomes one. As for the 10% and 40% solutions, the slopes were (DME, FSA-, Na+) = (0.79, 0.91, 0.84) and (0.62, 0.71, 0.49), respectively. We should mention that the longer simulation might be needed to obtain the highly accurate diffusion coefficients and the better statistical convergence. Table 5 summarizes the diffusion coefficients of DME, FSA-, and Na+ ions for all systems, evaluated from the gradient of the fitting functions. The diffusion coefficient of DME for the most dilute 5% solution, i.e., 30.62 × 10-10 m2/s, agrees with the experimental values for pure DME solution obtained by NMR measurements, ca. 32 × 10-10 m2/s.55,56 The diffusion coefficients monotonically decreased as the salt concentration increased. In the case of dilute solutions with salt concentrations of 5% and 10%, DME, FSA-, and Na+ showed ca. 2 to 3-fold decreases in their diffusion coefficients. For the 40% solution, the diffusion 15 ACS Paragon Plus Environment


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coefficient of DME decreased to ca. 20% (3.25 × 10-10 m2/s) of that of the 10% solution case (16.72 × 10-10 m2/s), while that for Na+ decreased to ca. 40%, i.e., from 3.80 to 1.39 × 10-10 m2/s. A moderate decrease in the diffusion coefficient of Na+ compared with that of DME, as well as the network structure of the solution, prohibiting the conventional vehicular-type diffusion, may indicate an alternative ion diffusion path. The cascade of dissociation/association reactions between the carrier ion and the solvent/anion is expected to be a plausible alternative pathway in the case of superconcentrated solutions.4 Dissociation/association reactions can be characterized as ligand exchange reactions around Na+ ions. Here, we introduce a time-dependent index, Cij(t), which equals one if the ith Na+ ion is coordinated to the jth solvent/anion at time t, otherwise it is zero. The coordinating solvent/anion is defined as a solvent/anion molecule with one or more oxygen atom(s) located around a Na+ ion with a bond length of 3 Å or shorter. Because Cij(t) represents an instantaneous solution structure, the ligand exchange reaction could be monitored as the difference in Cij(t) for two different timings. Figure 5 shows the reaction rate of ligand exchange, defined as follows: =

Na

∑ − 0

(4)

where, τ is the interval to evaluate the difference in the solution structure, nNa is the number of Na+ ions in the system for normalization, and the factor of a half indicates a pair of dissociation and association reactions, resulting in − 0 being equal to two and 16 ACS Paragon Plus Environment

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corresponding to one ligand exchange reaction. Blue, orange, and green colored lines represent the 5%, 10%, and 40% solutions, respectively. All lines show highly oscillating behavior with the short time interval of τ < 10 ps, because of numerical instability arising from the denominator of τ in eq. (4). Table 6 summarizes time averages of reaction rate, , for the range of τ = 0 – 5, 5 – 10, 10 – 15, and 15 – 20 ps. Components of , divided into the contributions of solvents and anions, are also shown. Standard deviations for the total reaction rate are shown in the parentheses. Reflecting the oscillating behaviors in short time interval, the standard deviations showed comparable or even larger values than , in the case of τ = 0 – 5 ps. The standard deviations monotonically decreased with the larger time intervals. For the case of τ = 15 – 20 ps, standard deviations were less than 10% of values except for the 5% solution case, in which equals zero. The inverse of , which corresponds to the reaction time, is also shown for τ = 15 – 20 ps. For the 5% solution, ligand exchange reactions were limited (Figure 5, Table 6). Therefore, the Na+ ion diffusion process in a 5% solution was dominated by the conventional diffusion mechanism. In contrast, in the 10% solution, ligand exchange reactions were observed with (, 1/) = (0.0152 ps-1, 65.9 ps) for τ = 15 – 20 ps. This implies that ligand-exchange reactions might be enhanced in high concentration conditions, as noted in the recent work.12 Because 76.3% of the solvent molecules remained free and 25.2% (100.0% – 74.8%) of Na+ ions were not included in the network structure (Table 4), both conventional 17 ACS Paragon Plus Environment


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and ligand exchange type mechanisms might contribute to the diffusion of Na+ ions. Concerning the 40% solution, the reaction rate and time for ligand exchange were evaluated as (, 1/) = (0.0173 ps-1, 57.9 ps) for τ = 15 – 20 ps, values similar to those in the 10% solution. The ligand exchange reactions in the 40% solution were as frequent as that in 10% solution, maintaining the diffusion coefficient of Na+ ions in superconcentrated system (1.39 × 10-10 m2/s) to about ca. 40% that of a conventional dilute solution (3.80 × 10-10 m2/s), while the diffusion coefficient of solvent decreased by more than 80%, i.e., from 16.72 to 3.25 × 10-10 m2/s. As for the components, divided into solvents and anions, FSA- showed dominant contributions in both 10% and 40% solutions. This would be due to the fact that the network structures were mainly formed by FSA- rather than DME, as discussed above. As an illustrative example, Figure 6 shows screenshots of a ligand exchange reaction, taking place during the 20-ps simulation of 40% solution. Atoms other than the Na+ ion of concern and the surrounding DME molecules and FSA- anions, which are labeled as DME 1 – 3, as well as FSA 1 and FSA 2, are not shown for convenience. Here, the oxygen atoms of DME and FSA- with a separation from the Na+ ion of less than or equal to 3 Å were considered to be ligands interacting with the Na+ ion. Na-O bonds, of 3 Å or less, are shown in thick lines. It should be mentioned that all oxygen atoms were distinguished and treated separately to each other. At t = 0.00 ps (Figure 6(a)), the Na+ ion was coordinated by five oxygen atoms from two DME molecules and two FSA- anions. Namely, DME 1, FSA 1, and 18 ACS Paragon Plus Environment

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FSA 2 had one oxygen atom, coordinating to the Na+ ion, while two oxygen atoms in DME 2 were interacting with the Na+ ion. DME 3 had no oxygen atom, bonding to the Na+ ion. At t = 9.14 ps (Figure 6(b)), an oxygen atom of DME 3 formed a new interaction with the Na+ ion, and the Na+ ion had a hexacoordinated structure. Then, FSA 1 lost the Na-O interaction at t = 10.82 ps, and the Na+ ion became pentacoordinated (Figure 6(c)). The pentacoordinated structure remained until the simulation ended at t = 20.00 ps (Figure 6(d)). Consequently, a ligand exchange reaction occurred around the Na+ ion of concern within 20 ps (or less), as represented by the difference in Cij(t), that is, 20 ps − 0 ps equals two. Dynamical changes of solvation structures with ligand exchange reactions, occurring in a few ten picoseconds, are consistent with the previous studies.10,11 Finally, the lifetime of the solution structure, τ0, was evaluated from the relaxation time of the autocorrelation function (ACF) of the index Cij(t), as follows:

= exp − = 〈∑ # + # 〉 "

# = −

Na solv (anion

〈∑ 〉

(5) (6)

where δCij(t) is the fluctuation of Cij(t) from the average value defined by eq. (6). The second term of the right-hand side of eq. (6) corresponds to the average probability of finding the jth solvent/anion coordinated to the ith Na+ ion, assuming the system is uniform. Figure 7 shows the relationships between the ACFs and the delay time τ for each system. The ACFs for the 10% and 40% solutions decayed monotonically, reflecting the ligand exchange reactions. In 19 ACS Paragon Plus Environment


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the case of the 5% solution, no obvious decay was observed. The corresponding lifetimes, τ0, were evaluated using least-squares fitting and are summarized in Table 7, along with the R2 values. For both 10% and 40% solutions, the ACFs show reasonable fitting to single exponential functions, having R2 values of ca. 0.65. The lifetimes, τ0, of 124.1 and 120.6 ps for the 10% and 40% solutions, respectively, are consistent with the reaction times shown in Table 6. That is, the ligand exchange reactions that take place in the 10% and 40% solutions as frequent as each other on a timescale of ca. 60 – 120 ps, while limited reactions were observed in the case of the 5% solution.

Conclusions

Understanding the diffusion process of carrier ion is one of the most important issues concerning superconcentrated electrolyte solutions. In the present study, using the DC-DFTB-MD method, we performed large-scale QMD simulations of electrolyte solutions composed of DME solvent and NaFSA salt. The solution structures, as well as the diffusion properties, of the components of the electrolyte solutions were analyzed concerning the concentration dependence. For the solution structure, the number of free solvent molecules decreased with increasing salt concentration, as reported in recent experimental studies. The network structure, in which almost all Na+ ions connected to each other via FSA-, was found 20 ACS Paragon Plus Environment

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in the 40% solution. In the case of 5% solution, such a structure was not observed, while the network structure was partially formed in the 10% solution. Concerning the diffusion of carrier ions, we observed the ligand exchange reactions responsible for carrier ion diffusion in superconcentrated solutions. Analyses of the reaction rate and reaction time for ligand exchange and the lifetime of the solution structure show that the ligand exchange reactions take place on a timescale of ca. 60 – 120 ps in the electrolyte solutions. Finally, we should mention that there remain major issues on superconcentrated electrolyte solutions for SIB. For example, the dependences of solution structures and Na+ ion diffusion properties on anion species are definitely important. It is well known that the use of bis(trifluoromethyl)sulfonylamide (TFSA-) leads to lower conductivity of carrier ion, in the case of LIB. Interfacial properties between electrolyte solutions and electrode, such as interfacial carrier ion transportation and solid-electrolyte interphase (SEI) formation, are also attracting and considerable issue. Large-scale QMD simulations by means of DC-DFTB-MD with the improved parameter for the battery system would be a useful technique to tackle to these issues. As for the computational methodologies, larger and longer simulations are still needed for the better statistics, while continuous and extensive improvements may bring more efficient computations.57



Page 22 of 56

Supporting Information

The modified S-N and S-F potential files that are S-N.txt, S-F.txt, N-S.txt, and F-S.txt, are given in the Slater-Koster file (skf) format, which is widely accepted by computational programs for DFTB calculations. Note that the given parameter files are only compatible with 3ob-3-1 DFTB parameter set, which is freely available on www.dftb.org website. Figure S1 shows the correlation between the MSD and the simulation time for DME, FSA-, and Na+ in the 5%, 10%, and 40% solutions in the log-log scale.


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Acknowledgments

Some of the present calculations were performed at the Research Center for Computational Science (RCCS), Okazaki Research Facilities, National Institutes of Natural Sciences (NINS) and the Institute for Solid State Physics (ISSP), the University of Tokyo. This study was supported in part by the “Elements Strategy Initiative for Catalysts & Batteries (ESICB)” project supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan, by the Core Research for Evolutional Science and Technology (CREST) Program, “Theoretical Design of Materials with Innovative Functions Based on Relativistic Electronic Theory” of the Japan Science and Technology Agency (JST), by a Grant-in-Aid for Scientific Research (A) “KAKENHI Grant Number JP26248009” from the Japan Society for the Promotion of Science (JSPS), by MEXT as “Priority Issue 5 on Post-K computer” (Development of new fundamental technologies for high-efficiency energy creation, conversion/storage, and use), and by Waseda University Grant for Special Research Projects (Project number: 2017S-105).



Page 24 of 56

Author Information

*

Corresponding author: H. Nakai; Phone: +81-3-5286-3452; E-mail: [email protected]


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19

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Page 34 of 56

Table 1. Numbers of DME and NaFSA Molecules, the Total Atom Count, and Number of Basis Functions for 5%, 10% and 40% Solutions. Molar (mol/L) and Mass Densities (g/L), Evaluated from Classical MD Simulations, Are Also Shown. CPU Times (s) for One DC-DFTB-MD Step Were Estimated Using 768 Cores (48 Nodes) of a Fujitsu FX-10 Supercomputer. molar ratio

molar density

mass density

# DME

# NaFSA

# atom

# basis

CPU time

5%

0.50

948.7

209

11

3454

7656

8.0

10% 40%

1.01 4.12

1027.9 1392.3

198 129

22 86

3388 2924

7832 8686

7.3 6.3


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Table 2. Covalent Bond Radii (Å) for Na, N, O, and F Taken from Ref. 54. element

covalent radius

Na

1.57 – 1.75

N O

0.70 – 0.72 0.64 – 0.68

F

0.54 – 0.60



Page 36 of 56

Table 3. Coordination Numbers (n(r)) of Atom Pairs of Na-O(DME), Na-O(FSA-), Na-F, and Na-N (Cut-off Length of r = 3 Å). molar ratio

Na-O(DME)

Na-O(FSA-)

Na-F

Na-N

total

5%

5.4

0.4

0.0

0.0

5.8

10% 40%

3.9 2.6

1.7 2.8

0.3 0.3

0.0 0.0

5.9 5.7


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Table 4. Numbers of Solvents/Anions Classified with the Numbers of Coordinating Na+ Ions. Numbers of Na+ Ions That Hold Another Na+ Ion in the Second Neighbor Are Also Shown. Ratios to Total Number of Solvents/Anions/Na+ Ions Are Shown in Parentheses. coordinating number DME

FSA-

Na+ w/ another Na+

5%

10%

40%

0

175.0

(83.7%)

151.0

(76.3%)

13.5

(10.5%)

1

34.0

(16.3%)

46.9

(23.7%)

91.8

(71.1%)

2

0.0

(0.0%)

0.1

(0.1%)

22.7

(17.6%)

3

0.0

(0.0%)

0.0

(0.0%)

1.0

(0.7%)

0

7.1

(64.2%)

1.5

(6.7%)

0.4

(0.5%)

1

3.9

(35.8%)

10.9

(49.6%)

10.7

(12.4%)

2

0.0

(0.0%)

6.6

(30.1%)

36.2

(42.1%)

3

0.0

(0.0%)

3.0

(13.6%)

27.5

(32.0%)

4

0.0

(0.0%)

0.0

(0.0%)

10.8

(12.6%)

5

0.0

(0.0%)

0.0

(0.0%)

0.3

(0.4%)

0.0

(0.0%)

16.5

(74.8%)

82.9

(96.4%)



Page 38 of 56

Table 5. Diffusion Coefficients of DME, FSA-, and Na+ (10-10 m2/s). molar ratio

DME

FSA-

Na+

5%

30.62

17.41

11.88

10% 40%

16.72 3.25

5.25 1.99

3.80 1.39


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Table 6. Reaction Rate (ps-1) and the Components, Divided into the Contributions of Solvents and Anions, Averaged for the Range of τ = 0 – 5, 5 – 10, 10 – 15, and 15 – 20 ps. The Standard Deviations Are Shown in Parentheses. Reaction Time (ps), 1/, for τ = 15 – 20, Are Also Shown. molar ratio component 5%

DME -

FSA total

10%

DME FSAtotal

40%

DME FSAtotal

reaction rate 0 – 5 ps

5 – 10 ps

10 – 15 ps

15 – 20 ps

0.0000

0.0001

0.0000

0.0000

0.0008 0.0008 (0.0035)

0.0010 0.0010

0.0001 0.0001

0.0000 0.0000

(0.0024)

(0.0006)

(0.0002)

0.0001

0.0000

0.0000

0.0006

0.0885 0.0886 (0.1104)

0.0210 0.0210

0.0195 0.0195

0.0146 0.0152

(0.0033)

(0.0020)

(0.0015)

0.0076

0.0031

0.0029

0.0029

0.0742 0.0818 (0.0537)

0.0269 0.0300

0.0201 0.0230

0.0144 0.0173

(0.0043)

(0.0021)

(0.0017)

reaction time 1/

–

65.9

57.9



Page 40 of 56

Table 7. Lifetimes of the Solution Structures, τ0 (ps), Evaluated by Single Exponential Fitting of the Decay of the ACFs. R2 Values for the Least-Squares Fitting Are Also Shown. molar ratio

lifetime τ0 (ps)

R2 value

5%

–

–

10% 40%

124.1 120.6

0.66 0.64


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Figure Captions

Figure 1. Desolvation energies (EDsol) for one-to-one complexes (kJ/mol).

Figure 2. RDFs of Na-O(DME) (blue), Na-O(FSA-) (orange), Na-F (green), and Na-N (purple) atom pairs for the (a) 5%, (b) 10%, and (c) 40% solutions.

Figure 3. Relationships between the coordination number and bond distances (n(r)) of Na-O(DME) (blue), Na-O(FSA-) (orange), Na-F (green), and Na-N (purple) atom pairs for the (a) 5%, (b) 10%, and (c) 40% solutions.

Figure 4. Time-course changes of MSDs of (a) DME, (b) FSA-, and (c) Na+ for the 5%, 10%, and 40% solutions, denoted in blue, orange, and green lines, respectively.

Figure 5. Relationships between reaction rate, R(τ), and time interval, τ, for each system.

Figure 6. Screenshots of a ligand exchange reaction.

Figure 7. Relationships between ACFs and delay time, τ, for each system. 41 ACS Paragon Plus Environment


600 EDsol (DFTB-D3/3ob)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 56

y = 1.02 x + 8.8 2 = 0.974 R 400 FSADME 200

0

0 200 400 600 EDsol (B3LYP/aug-cc-pVTZ)

Figure 1. of Okoshi et al.


Page 43 of 56

20

(a) 5%

Na-O(DME)

15 10 Na-N 5

Na-O(FSA-)

0 20

Na-F

(b) 10% Na-O(DME)

15 RDF g(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

Na-F Na-N

5

Na-O(FSA-)

0 20

(c) 40%

15 10

Na-O(DME)

Na-O(FSA-)

5 Na-F 0

0

1

2 3 Na-X length (Å)

Na-N 4

5




6

(a) 5%

5

Na-O(DME)

4 3 2 Na-O(FSA-)

1 Coordination number n(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 56

0 6

(b) 10%

5 Na-O(DME)

4 3

Na-O(FSA-)

2 1

Na-F Na-N

0 6

(c) 40%

Na-O(FSA-)

5 4 3

Na-O(DME)

2 Na-F

1 0

Na-N 0

1


4

5



Page 45 of 56

40

5% (R2 = 0.99)

(a) DME

30 10% (R2 = 0.98)

20 10

40% (R2 = 0.96) 0 40

MSD (Å2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(b) FSA-

30 5% (R2 = 0.91)

20 10

10% (R2 = 0.85) 40% (R2 = 0.91)

0 40

(c) Na+

30 20 5% (R2 = 0.95)

10 0

0

5 10 15 Simulation time t (ps)

10% (R2 = 0.74) 40% (R2 = 0.78) 20



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Reaction rate R(τ) (ps-1)


Page 46 of 56

0.05 0.04 0.03 0.02

40%

0.01

10% 5%

0.00 0

5 10 15 Time interval τ (ps)

20



Page 47 of 56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





Page 48 of 56



Page 49 of 56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table of Contents Graphic


600 EDsol (DFTB-D3/3ob)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28


Page 50 of 56

y = 1.02 x + 8.8 2 = 0.974 R 400 FSADME 200 0

0 200 400 600 EDsol (B3LYP/aug-cc-pVTZ) ACS Paragon Plus Environment

Page 51 of 56


20

RDF g(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a) 5%

Na-O(DME)

15 10 Na-N 5

Na-O(FSA-)

0 20

Na-F

(b) 10% Na-O(DME)

15 10

Na-F Na-N

5

Na-O(FSA-)

0 20

(c) 40%

15 10

Na-O(DME)

5 0

Na-O(FSA-)

Na-F 0

1


ACS Paragon Plus Environment

Na-N 4

5


6

Coordination number n(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 52 of 56

(a) 5%

5

Na-O(DME)

4 3 2 Na-O(FSA-)

1 0 6

(b) 10%

5

Na-O(DME)

4 3

Na-O(FSA-)

2 1

Na-F Na-N

0 6

(c) 40%

5

Na-O(FSA-)

4 3

Na-O(DME)

2

Na-F

1 0

0

1



Na-N 4

5

Page 53 of 56


40

MSD (Å2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

5% (R2 = 0.99)

(a) DME

30 10% (R2 = 0.98)

20 10

40% (R2 = 0.96) 0 40

(b) FSA-

30 5% (R2 = 0.91)

20 10

10% (R2 = 0.85) 40% (R2 = 0.91)

0 40

(c) Na+

30 20 5% (R2 = 0.95)

10 0

0

5 10 15 Simulation time t (ps)


10% (R2 = 0.74) 40% (R2 = 0.78) 20

Reaction rate R(τ) (ps-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.05


Page 54 of 56

0.04 0.03 0.02

40%

0.01

10% 5%

0.00 0

5 10 15 Time interval τ (ps) ACS Paragon Plus Environment

20

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(a) t = 0.00 ps


(b) t = 9.14 ps

FSA 1

FSA 2 DME 1

FSA 1 FSA 2

DME 1 Na

Na

DME 2

DME 2

DME 3 DME 3

(c) t = 10.82 ps

(d) t = 20.00 ps

FSA 1

FSA 1 FSA 2

DME 1

DME 1

FSA 2 Na

Na

DME 2

DME 2

DME 3 ACS Paragon Plus Environment

DME 3

ACF G(τ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

1.00 0.99


Page 56 of 56

10%: G(τ)=exp(-τ/124.1) R2 = 0.66 0.97 0.95 40%: G(τ)=exp(-τ/120.6) R2 = 0.64 0.93 0 2 4 6 8 Delay time τ (ps) ACS Paragon Plus Environment

10

Theoretical Analysis of Carrier Ion Diffusion in Superconcentrated

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