Laying a basis for molecularly specific theory for the mobilities of ions in solutions of practical interest, we report a broad survey of velocity autocorrelation functions (VACFs) of Li^{+} and PF_{6}^{−} ions in water, ethylene carbonate, propylene carbonate, and acetonitrile solutions. We extract the memory function, *γ*(*t*), which characterizes the random forces governing the mobilities of ions. We provide comparisons controlling for the effects of electrolyte concentration and ion-pairing, van der Waals attractive interactions, and solvent molecular characteristics. For the heavier ion (PF_{6}^{−}), velocity relaxations are all similar: negative tail relaxations for the VACF and a clear second relaxation for $\gamma t$, observed previously also for other molecular ions and with *n*-pentanol as the solvent. For the light Li^{+} ion, short time-scale oscillatory behavior masks simple, longer time-scale relaxation of $\gamma t$. But the corresponding analysis of the *solventberg* $Li+H2O4$ does conform to the standard picture set by all the PF_{6}^{−} results.

## I. INTRODUCTION

Here we report molecular dynamics results for single-ion dynamics in liquid solutions, including aqueous solutions. We provide comparisons controlling for the effects of solvent molecular characteristics, electrolyte concentration, and van der Waals attractive forces. We choose LiPF_{6} for our study because of its importance, with ethylene carbonate (EC), to lithium ion batteries. But our comparisons include several solvents of experimental interest, specifically water, EC, propylene carbonate (PC), and acetonitrile (ACN). We obtain the memory function *γ*(*t*), defined below,^{1} which characterizes the random forces governing the mobilities of ions in these solvents.

A specific motivation for this work is the direct observation^{2} that *γ*(*t*) relaxes on time scales longer than the direct collisional time scale, behavior that was anticipated years earlier in the context of *dielectric friction.*^{3} Nevertheless, this longer time-scale relaxation is not limited to ionic interactions (Fig. 1).^{4} The results and comparisons below provide a basis for molecularly specific theory for the mobilities in liquid mixtures of highly asymmetric species, as are electrolyte solutions of practical interest.^{5–8}

## II. METHODS

We perform simulations (Table I) of dilute and 1M solutions of LiPF_{6} using the GROMACS molecular dynamics package with periodic boundary conditions. A Nose-Hoover thermostat^{9,10} and a Parrinello-Rahman^{11} barostat were utilized to achieve equilibration in the *NpT* ensemble at 300 K and 1 atm pressure. A 10 ns simulation was carried out for aging and then a separate 1 ns simulation with a sampling rate of 1 fs was carried out to calculate the velocity autocorrelation and the friction kernel.

### A. Forcefield parameters and adjustments

The interactions were modeled following the OPLS-AA forcefield^{13} with parameters as indicated below for bonded and non-bonded interactions. Li^{+} parameters were obtained from Soetens *et al.*^{16} Partial charges of EC and PC were scaled^{14} to match transport properties of Li^{+} with experiment. In the case of acetonitrile and water, standard OPLS-AA and SPC/E parameters were used.^{15}

The PF_{6}^{−} ions were described initially with parameters from Sharma *et al.*^{17} In initial MD trials, however, we observed PF_{6}^{−} ions that deviated significantly from octahedral geometries, particularly in the case of 1M LiPF_{6} in EC, where substantial ion-pairing was observed. These PF_{6}^{−} displayed extreme bending of the axial F–P–F bond angles.

The possibility of exotic non-octahedral PF_{6}^{−} configurations in ion-paired (EC)_{3}Li^{+}$\u2026$PF_{6}^{−} clusters was investigated with electronic structure calculations. Gaussian09 calculations^{12} employed the Hartee-Fock approximation with the 6-311+G(2d,p) basis set. Initial configurations were sampled from MD observations. The stable and lowest-energy clusters obtained were consistent with octahedral PF_{6}^{−} geometries (Fig. 2). We therefore increased the axial F–P–F (180°) bond-angle parameter by a factor of four in further MD calculations. The modified forcefield parameters for PF_{6}^{−} are provided in the supplementary material.

### B. Solution structure

For Li^{+} in water, the oxygen coordination number is 4,^{19–21} with the inner-shell O atoms positioned at 0.18 nm. Similar Li^{+} coordination is observed in 1M solutions of LiPF_{6} in PC and ACN.

In the case of 1M solutions in EC, the nearest Li–P peak centered at 0.33 nm (Fig. 3) indicates distinct but modest ion-pairing with PF_{6}^{−} at this concentration. The Fuoss/Poisson approximation^{18} is accurate here and that further supports the ion-pairing picture. Reflecting F atom penetration of the natural EC inner shell (Fig. 2), the Li^{+}–O atom inner shell distribution is broader in EC than in water.

We re-emphasize that previous work^{14} scaled partial charges of the solvent EC molecules to match *ab initio* and experimental results for Li^{+} solvation and dynamics. Nevertheless, van der Waals interactions are a primary concern for the description of realistic ion-pairing.

### C. The friction kernel

We define the friction kernel $\gamma t$ (or memory function) by

where *m* is the mass of the molecule and *C*(*t*) is the velocity autocorrelation function (VACF),

The friction kernel $\gamma t$ is the autocorrelation function of the *random* forces on a molecule.^{1} The standard formality for extracting $\gamma t$ utilizes Laplace transforms. But inverting the Laplace transform is non-trivial and we have found the well-known Stehfest algorithm^{22} to be problematic. Berne and Harp^{23} developed a finite-difference-in-time procedure for extracting $\gamma t$ from Eq. (1). That procedure is satisfactory, but sensitive to time resolution in the discrete numerical $Ct$ used as the input. An alternative^{4} expresses the Laplace transform as Fourier integrals, utilizing specifically the transforms

Then,

Taking $\gamma t$ to be even time, the cosine transform is straightforwardly inverted. $\Omega 2=F2/3mkBT,$ with $F=|F\u2192|$ being the force on the molecule, provides the normalization *γ*(0) = *m*Ω^{2}. A comparison of these methods is provided in the supplementary material and Ref. 4.

## III. RESULTS

We discuss quantitative simulation results that lay a basis for molecularly specific theory of the friction coefficients of ions in solution. Our initial discussion focuses on dynamics of ions such as Li^{+} and PF_{6}^{−} in water, followed by overall comparisons with common non-aqueous solvents.

### A. Oscillatory behavior of Li^{+} dynamics

The Li^{+} ion has an unusually small mass, and the oscillatory behavior of its dynamics at short times is prominent compared to PF_{6}^{−}. These differences are reflected in the mean squared displacement (Fig. 4) of these ions in water. This short-time behavior has been the particular target of the molecular time scale generalized Langevin theory.^{25} The vibrational power spectrum (Fig. 5) then provides a more immediate discrimination of the forces on the ions by the different solvents. Electronic structure calculations identify the high frequency vibrations that are related to the motion of a Li^{+} trapped within an inner solvation shell. In the case of Li^{+} (aq), this frequency occurs at 650 cm^{−1}. Nevertheless, the low frequency (*ω* ≈ 0) diffusive behavior can be only subtly distinct for different solution cases (Fig. 5), including electrolyte concentrations (Fig. 6).

### B. Solventberg picture

A common view why the transport parameters can depend only weakly on the differences in the molecular-time-scale dynamics (Fig. 4) follows from the appreciation that the exchange time for inner shell solvent molecules can be long compared to the dynamical differences. For Li^{+}(aq), that exchange time is of the order of 30 ps.^{26,27} Then, ion *plus* inner-shell solvent molecules—a *solventberg*^{3}—can be viewed as the transporting species.

The mean-squared displacement of the ion followed over times that are long on molecular time scale but shorter than that exchange time should not differ much from the mean-squared displacement of the solventberg. The oscillations internal to the solventberg, which are reflected in the VACF, are not essential to the transport. Nevertheless, molecular dynamics simulations permit us to check the VACF of the center-of-mass of the solventberg. This VACF is free of oscillations and reveals a negative tail relaxation that is qualitatively similar to PF_{6}^{−} (Fig. 7). Indeed, previous calculations, treating both water^{28,29} and EC,^{30} fixed a Li^{+} ion coordinate for the calculation of the *force* autocorrelation. Those prior studies indeed also observed this second, longer time-scale relaxation that the present calculations highlight.

### C. Overall comparisons

The overall comparisons of these single-ion VACFs and $\gamma t$ for our collection of solvents (Fig. 8) show that these relaxations are similar to each other for the heavier ion PF_{6}^{−}: a clear second relaxation for $\gamma t$, consistent with negative tail relaxations for the VACF. This behavior is similar for other molecular ions considered recently, including *n*-pentanol as the solvent.^{2} Numerical VACF results for PF_{6}^{−} (aq) show that the molecular time-scale relaxation is insensitive to electrolyte concentrations and to van der Waals attractive forces (supplementary material). For Li^{+}, short time-scale oscillatory behavior masks that longer time-scale relaxation of $\gamma t$, as discussed above. Detailed results corresponding to Fig. 8 but for a Li^{+} ion are provided in the supplementary material.

## IV. CONCLUSIONS

We extract the VACF and the memory function, *γ*(*t*), which characterize the mobility of ions in solution. For the heavier PF_{6}^{−} ion, velocity relaxations are all similar: negative tail relaxations for the VACF and a clear second relaxation for $\gamma t$. For the light Li^{+} ion, analysis of the solventberg dynamics conform to the standard picture set by all the PF_{6}^{−} results. These results lay a quantitative basis for establishing a molecularly specific theory of the friction coefficients of ions in solution.

## SUPPLEMENTARY MATERIAL

See supplementary material for a comparison of methods for extracting the friction kernel, a comparison of Li^{+} dynamics in different solvents, forcefield parameters for PF_{6}^{−}, and the effect of removing van der Waals attractions on the dynamics of PF_{6}^{−} (aq).^{31}

## ACKNOWLEDGMENTS

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-NA0003525. This work is supported by Sandia’s LDRD program (M.I.C. and S.B.R.) and by the National Science Foundation, Grant No. CHE-1300993. This work was performed, in part, at the Center for Integrated Nanotechnologies (CINT), an Office of Science User Facility operated for the U.S. DOE’s Office of Science by Los Alamos National Laboratory (Contract No. DE-AC52-06NA25296) and SNL.