doi.org/10.26434/chemrxiv.8217152.v2

Role of Viscosity in Deviations from the Nernst-Einstein RelationYunqi Shao, Keisuke Shigenobu, Masayoshi Watanabe, Chao Zhang

Submitted date: 18/06/2019 • Posted date: 18/06/2019Licence: CC BY-NC-ND 4.0Citation information: Shao, Yunqi; Shigenobu, Keisuke; Watanabe, Masayoshi; Zhang, Chao (2019): Role ofViscosity in Deviations from the Nernst-Einstein Relation. ChemRxiv. Preprint.

Deviations from the Nernst-Einstein rela- tion are commonly attributed to ion-ion (cross)correlation andion-pairing. Despite the fact that these deviations can be quantified by either experimental measurements ormolecular dynamics simulations, there is no rule of thumb to tell the extent of deviations. Here, we show thatdeviations from the Nernst-Einstein relation scale linearly with the inverse viscosity by exploring the finite-sizeeffect in periodic boundary conditions. This conclusion is in accord with published experimental results of ionicliquids.

File list (3)

download fileview on ChemRxivsupport_20190618.pdf (535.99 KiB)

download fileview on ChemRxivtoc_NEdev.png (140.54 KiB)

download fileview on ChemRxivNE_deviation_20190618.pdf (829.02 KiB)

Role of Viscosity in Deviations from the Nernst-EinsteinRelation

Yunqi Shao†, Keisuke Shigenobu‡, Masayoshi Watanabe‡, Chao Zhang†∗

†Department of Chemistry - Angstrom Laboratory, Uppsala University, Lagerhyddsvagen 1,BOX 538, 75121, Uppsala, Sweden

‡Department of Chemistry and Biotechnology, Yokohama National University,Yokohama 240-8501, Japan

chao.zhang@kemi.uu.se

Supporting Information

A Calculated shear viscosities η of NaCl electrolyte solutions incomparison with experiments

Shear viscosities were computed by integrating the time autocorrelation function of the off-diagonal components of the stress tensor Pij , according to the Green-Kubo formula. [1, 2]

η = Ωβ

∫ ∞0〈Pij(t0)Pij(t0 + t)〉t0dt (1)

where Ω is the system volume and β = 1/kbT is the inverse temperature.The upper limits in the integral were set to be 10 ps to avoid long-tail oscillations [3]. Each

trajectory was divided to 4 segments with length of 5 ns, the viscosity was evaluated for each ofthe segments and standard deviation was used to estimate the error.

We computed the viscosity using simulation boxes with equivalent 64 and 512 H2O andthe results were compared to the experimental data. As shown in Fig.S1, the shear viscosity ofNaCl does not show strong system-size dependence, similar to what was observed in Ref. [4].Comparing to experimental values, the SPC/E water with Joung-Cheatham ion model underes-timates the shear viscosity at lower concentrations and overestimates it at higher concentrations.

B Finite size correction to self-diffusion coefficients of ions andwater

The self-diffusion coefficients were computed from the mean squared displacement (MSD) ofions and oxygen atoms, according to the Einstein relation (Eq. 2) [6]. The MSD was calculatedand fitted in a time windows of 5 to 20 ps. Each trajectory was divided to 4 segments withlength of 5 ns, the diffusion coefficients were evaluated for each of the segments and standarddeviation were used to estimate the error.

1

Figure S1: Computed viscosities of NaCl solutions in comparison with experimental values [5].

Ds = limt→∞

1

6tN

N∑i

〈[ri(t)− ri(0)]2〉 (2)

where t is the time, N is the the number of cations, anions or oxygen atoms and ri is thecorresponding position.

The diffusion coefficients were calculated for each salt concentration with five differentbox sizes, with equivalent 64, 512, 1024, 2048, and 4096 H2O, respectively. The diffusioncoefficient at infinite box size Ds

∞ was estimated by extrapolating a linear fit of Ds against theinverse box length. We then examined the finite size correction Eq. 3 by comparing it to Ds

∞.Here the finite size corrections were computed using the viscosities obtained in Section A with64 H2O.

Ds0 = Ds

PBC +ξ

6πβηL(3)

where Ds0 is corrected self-diffusion coefficient, Ds

PBC is the self-diffusion coefficient obtainedfrom Eq. 2, ξ ≈ 2.837297 for cubic cells is a constant [4], and L is the length of a cubicsimulation box.

Figure S2: System-size dependence of the self-diffusion coefficient a) Oxygen atoms, b) Na+

ions and c) Cl− ions at different concentrations. Solid lines are forDsPBC from Eq. 2, and dashed

lines are for Ds0 from Eq. 3. All diffusion coefficients are subtracted by the extrapolated infinite

box size value Ds∞.

As shown in Fig.S2, the finite size corrections agree reasonably well with the extrapolationvalue for pure water. For electrolyte solutions, the corrections deviate slightly from the extrap-olation value at lower concentrations. However, this tendency is relatively small compared to

2

the finite size correction. Therefore, Eq. 3 was used in the main Text to make the finite-sizecorrection.

C The Walden plots of σG-K and σN-E with experimental valuesand the Stokes-Einstein relation of self-diffusion coefficients forNaCl electrolyte solution

As shown in Fig S3 and Fig S4, the molar conductivity and the self-diffusion coefficients fol-low a fractional Stokes-Einstein relation and a fractional Walden relation, respectively. Theexponents obtained with different box sizes are listed in Table S1.

Λ ∝(

1

η

)α(4)

Ds+/− ∝

(1

η

)α(5)

Figure S3: The Walden plot of molar conductivity calculated from the Nernst-Einstein relation(with finite size correction) and the Green-Kubo formula using simulation boxes with 64 H2O,compared to experimental values [5].

3

Figure S4: Self-diffusion coefficient (with finite size correction) of cation and anion versus theinverse viscosity in logarithm scale. The result was obtained using a simulation box with 64H2O.

Box size (H2O) α(ΛN-E) α(ΛG-K) α(Ds+) α(Ds

−)

64 0.79 0.79 0.76 0.82512 0.76 0.84 0.73 0.794096 0.74 0.80 0.72 0.76

Table S1: The Stokes-Einstein and the Walden exponents α with different box sizes.

References

[1] Berk Hess, “Determining the shear viscosity of model liquids from molecular dynamicssimulations”, J. Chem. Phys. 116(1), pp. 209–217 (2001).

[2] Miguel Angel Gonzalez and Jose L. F. Abascal, “The shear viscosity of rigid water models”,J. Chem. Phys. 132(9), pp. 096101 (2010).

[3] Yong Zhang, Akihito Otani, and Edward J Maginn, “Reliable Viscosity Calculation fromEquilibrium Molecular Dynamics Simulations: A Time Decomposition Method”, J. Chem.Theory Comput. 11(8), pp. 3537–3546 (2015).

[4] In-Chul Yeh and Gerhard Hummer, “System-Size Dependence of Diffusion Coefficientsand Viscosities from Molecular Dynamics Simulations with Periodic Boundary Condi-tions”, J. Phys. Chem. B 108(40), pp. 15873–15879 (2004).

[5] J. Rumble, CRC Handbook of Chemistry and Physics, CRC Handbook of Chemistry andPhysics. CRC Press LLC (2018).

[6] Michael P. Allen and Dominic J. Tildesley, Computer Simulation of Liquids: SecondEdition, Oxford University Press (2017).

4

download fileview on ChemRxivsupport_20190618.pdf (535.99 KiB)

download fileview on ChemRxivtoc_NEdev.png (140.54 KiB)

Role of Viscosity in Deviations from theNernst-Einstein Relation

Yunqi Shao,† Keisuke Shigenobu,‡ Masayoshi Watanabe,‡ and Chao Zhang∗,†

†Department of Chemistry-Ångström Laboratory, Uppsala University, Lägerhyddsvägen 1,BOX 538, 75121, Uppsala, Sweden

‡Department of Chemistry and Biotechnology, Yokohama National University, Yokohama240-8501, Japan

E-mail: chao.zhang@kemi.uu.se

AbstractDeviations from the Nernst-Einstein rela-tion are commonly attributed to ion-ion(cross)correlation and ion-pairing. Despite thefact that these deviations can be quantified byeither experimental measurements or moleculardynamics simulations, there is no rule of thumbto tell the extent of deviations. Here, we showthat deviations from the Nernst-Einstein re-lation scale linearly with the inverse viscosityby exploring the finite-size effect in periodicboundary conditions. This conclusion is inaccord with published experimental results ofionic liquids.

Graphical TOC Entry

At low salt concentration, the ionic conduc-tivity of 1-1 symmetric electrolyte can be de-scribed by the Nernst-Einstein (N-E) equation(Eq. 1), in which the ionic conductivity is onlylinked to the self-diffusion of ions.1

σN-E = q2ρβ(Ds+ +Ds

−) (1)= σs+ + σs− (2)

where β = 1/kbT is the inverse temperature, qis the formal charge of each ion and ρ = N/Ω isthe number density of electrolyte solution withΩ as the system volume. σs+ and σs− are con-tributions to the ionic conductivity from self-diffusion coefficients Ds

+ and Ds− of cations and

anions respectively.The Nernst-Einstein relation becomes ap-

proximated at high salt concentration, wherethe effect of ion-ion cross-correlation starts toshow up.2–9 In this case, the ionic conductiv-ity can be formally defined by the Green-Kubo(G-K) formula:10

σG-K =β

3

∫dr

∫ ∞0

dt〈J(0)J(t)〉 (3)

= limt→∞

βΩ

6t〈[P(t)−P(0)]2〉 (4)

where J is the total current and P is the itin-erant polarization in ionic solutions11 or theBerry phase polarization in solids.12,13 〈· · · 〉 in-dicates the ensemble average.The difference between σG-K and σN-E can

1

be decomposed into contributions from so-called distinct diffusion coefficients of cationsDd

+, anions Dd− and cation-anion Dd

+−.4,8 Thename “distinct” emphasizes the nature of cross-correlation between different ions either in thesame species or in different species. Subse-quently, this allows to rewrite the Green-Kuboconductivity as

σG-K = q2ρβ(Ds+ +Ds

− +Dd+/2 +Dd

−/2−Dd+−)(5)

= σN-E + σd+ + σd− + σd+− (6)= σs+ + σs− + σd+ + σd− + σd+− (7)

where σd+, σd− and σd+− are contributions to theionic conductivity from corresponding distinctdiffusion coefficients.Deviations from the Nernst-Einstein relation,

i.e. σN-E − σG-K, can be quantified by eitherexperiments or molecular dynamics (MD) sim-ulations. In experiments, they can be obtainedas the difference between pulsed-field gradientNMR and impedance (Imp) spectroscopy mea-surements for the same system in the same con-ditions.14–17 In MD simulations, one can com-pute σG-K using either Eq. 3 or Eq. 4 and σN-Ewith self-diffusion coefficients obtained from ei-ther velocity autocorrelation functions Eq. 8 ormean squared displacement Eq. 9.10

Ds+/− =

1

3N

∫ ∞0

dtN∑i

〈vi,+/−(0)vi,+/−(t)〉

(8)

= limt→∞

1

6tN

N∑i

〈[ri,+/−(t)− ri,+/−(0)]2〉

(9)

where t is the time, N is the the number ofcations or anions in solution, vi,+/− is the ve-locity of ith cation or anion, ri,+/− is the corre-sponding position.Despite that one can measure σN-E − σG-K

in experiments or compute it in MD simula-tions, there is no rule of thumb which can tellthe extent of deviations for a particular sys-tem. In addition, MD results of σN-E obtainedby computing self-diffusion coefficients Ds

+ and

Ds− contain a significant finite-size error because

of the periodic boundary condition (PBC).18,19Thus, the discussion about deviations from theNernst-Einstein relation needs to take this intoaccount as well.20 In this Letter, we showed thatthese two issues are in fact connected and thesystem-size dependence can be used as a the-oretical tool to reveal the role of viscosity indeviations from the Nernst-Einstein relation.

Figure 1: Ionic conductivities calculated fromGreen-Kubo formula, Nernst-Einstein relationand Nernst-Einstein relation with finite-sizecorrection for NaCl aqueous electrolytes at20C with equivalent a) 4096 water molecules,b) 512 water molecules and c) 64 watermolecules. Finite-size corrections was donewith Eq. 12 in Ref.19 using viscosities computedfor the same system. See the Supporting Infor-mation for details.

Following the spirit of using the ideal potas-sium chloride (KCl) line for the Walden plot,21here we took sodium chloride (NaCl) electrolytesolution as a prototype system to carry out

2

MD simulations. Water molecules were de-scribed by the simple point charge/extended(SPC/E) model22 and Na+/Cl− ions weremodelled as point charge plus Lennard-Jonespotential using the parameters from Joungand Cheatham,23 which is suitable for highlyconcentrated solution.24–26 The molecular dy-namics simulations were performed with theLAMMPS code.27 The size of cubic simu-lation box was determined by experimentaldensities.28 The long-range electrostatics wascomputed using Particle Mesh Ewald (PME)scheme.29 Short-range cutoffs for the Van derWaals and Coulomb interaction in direct spaceare 9.8 Å. For computing ionic conductivitiesand viscosities, NV T (constant number of par-ticles, constant volume and constant tempera-ture) simulations ran for 20 ns with a timestepof 2 fs, partial pressure tensors were saved everystep and trajectories were collected every 0.5ps. The Bussi-Donadio-Parrinello thermostat30was used to keep the temperature at 20C.As shown in Fig. 1, we found that: 1) ionic

conductivities computed using Green-Kubo for-mula show no system-size dependence; 2) De-viation from the Nernst-Einstein relation be-comes negligible for a small simulation box. Re-garding the first observation, since both totalcurrent J in Eq. 3 and supercell polarization PEq. 4 are collective properties, one may not besurprised to see that σG-K is system-size inde-pendent. Viscosity η computed from anothercollective property—pressure is also system-size insensitive.19 Regarding the second obser-vation, we know that σN-E is commonly consid-ered as an upper bound of the ionic conductiv-ity at a given condition and this value decreaseswith the system-size because of the finite-sizeeffect, therefore, the overlapping between com-puted σN-E and σG-K can be anticipated for aminimal simulations box.The implication of Fig. 1 is twofold: First,

since Eq. 6 holds regardless of the system-size, this suggests that ion-ion cross-correlationshould also have a significant system-size de-pendence; Second, system-size dependencewhich is usually considered as the fintie-sizeerror can be instead used as a theoretical toolto investigate the role of viscosity in deviation

from the Nernst-Einstein relation.

Figure 2: System-size dependence of differ-ent contributions to Green-Kubo ionic conduc-tivities for NaCl aqueous electrolytes at 20C:a) Cation contribution; b) Anion contribu-tions; c) Cation-Anion distinct diffusion contri-bution. Solid lines: With equivalent 4096 watermolecules; Dashed lines: With equivalent 512water molecules; Dotted lines: With equivalent64 water molecules. σs+ and σs− are self-diffusioncontributions. σd+, σd− and σd+− are distinct dif-fusion contributions. See Text and Eq. 7 fordetails.

To understand the system-size dependence ofion-ion cross-correlation, we decomposed vari-ous contributions to σG-K according to Eq. 7as shown in Fig. 2. Among three distinct dif-fusion contributions, σd+ and σd− has an appar-ent system-size dependence in contrast to σd+−.Moreover, the system-size dependence of cross-correlation for both cations and anions are smi-lar to that of self-diffusion. Because signs ofσs+/− and σd+/− are opposite, therefore, theirsystem-size dependences cancel out each other.This is the reason why ionic conductivities com-puted from Green-Kubo formula is system-size

3

independent (Fig. 1).What is more interesting is that we can arrive

at the following expression:

(ΛN-E(L→∞)− ΛG-K) · η ' 2NAq2ξ

6πLmin(10)

where Λ is the corresponding molar conductiv-ity (Λ = σ/c with c as the molar concentrationor molarity) and η is the corresponding viscos-ity. NA is the Avogadro constant. ξ is a con-stant and about 2.837297 for cubic simulationbox.18,19 Lmin is determined as the simulationbox length L in which ΛN-E(L = Lmin) ' ΛG-K.Eq. 10 suggests that deviations from Nernst-

Einstein relation has a linear relation with re-spect to 1/η. In other words, the system hasa high viscosity will have a small deviation orvice versa. This is reminiscent of the well-known Walden rule Ληα = k.31 However, it dif-fers from the Walden rule in three aspects: 1)k is a temperature dependent constant17 whileEq. 10 has no explicit temperature dependence.2) The right side of Eq. 10 is non-empirical andLmin can be determined theoretically. 3) Theexponent α in the Walden rule is usually frac-tional.32For the prototype system NaCl used here,

Lmin is about 12.3 Å. We notice that Lmin/2is about 6.1 Å, which is close to the Kirkwoodcorrelation length in bulk liquid water33 andmay be regarded as an effective ion-ion corre-lation length. Although one suspects that Lmin

should depend on the type of electrolyte underinvestigation, it is still curious to see how wellEq. 10 with Lmin/2 = 6.1 Å can be general-ized. For this purpose, we took a few of sem-inal experimental works in ionic liquids (ILs)which introduced the idea of “ionicity” 14–16 andmade the following mapping: ΛG-K ↔ ΛImp andΛN-E(L → ∞) ↔ ΛNMR. This leads to resultsshown in Fig. 3.In Fig. 3a, its contains 13 different types of

ILs measured at 30C16 and the agreement be-tween experimental deviations from the Nernst-Einstein relation and the theoretical “NaCl” lineusing Eq. 10 with Lmin/2 = 6.1 Å is striking, inspite of noticeable outliers.

In Fig. 3b, the temperature dependence ofmolar conductivities and viscosities of 8 types ofILs were measured experimentally and fitted toVogel-Fulcher-Tammann (VFT) equations.14,15This allows us to operate conveniently with fit-ted VFT equations in a wide temperature rangefrom -10C to 100C. It should be pointed outthat the Walden exponent α of ΛNMR − ΛImp

with respect to the inverse viscosity (1/η)α is0.99(5). This is in excellent agreement withEq. 10.

Figure 3: a) Experimental deviation from theNernst-Einstein relation for 13 types of ILs ex-tracted from Ref.,16 in together with the theo-retical line from Eq. 10 with Lmin/2 = 6.1 Å.b) Experimental deviation from the Nernst-Einstein relation for 8 types of ILs at varioustemperatures using fitting coefficients of VFTequations from Ref.,14,15 in together with thetheoretical line from Eq. 10 with Lmin/2 =6.1 Å.

The spread of experimental lines shownin Fig. 3b suggests that Lmin depends onthe nature of individual system. Exper-imental lines below “NaCl” lines impliesthat the correlation length Lmin/2 becomeslarger than 6.1 Å and the lower bound case[MMIM][TFSA] gives Lmin/2 as 9.5 Å. Wenotice that the corresponding Lmin/2 followsthe order [MMIM][TFSA] > [EMIM][TFSI]

4

> [BMIM][TFSI] > [C6MIM][TFSI] >[C8MIM][TFSI] which is in reverse to the alkylchain length. This suggests the correlationlength goes down as the size of cation becomeslarger, following the strength of electrostaticinteractions. Verifying these implications andidentifying physical quantities associated withLmin/2 should be the topics for future worksand this may lead to a new avenue to examine“ionicity” in ILs.In one of a seriers of classic works on dense

ionized matter from Hansen and McDonald,2they commented that “It is also clear that de-viations from the Nernst-Einstein relation arenot necessarily the result of a permanent as-sociation of ions of opposite charge”. Now wecan add that (for systems without permanentcation-anion associations) a low viscosity leadsa strong deviation and a high viscosity leads toa weak deviation, following Eq. 10.

Acknowledgement CZ is grateful to Upp-sala University for a start-up grant. Fundingfrom the Swedish National Strategic e-Scienceprogram eSSENCE is also gratefully acknowl-edged. The simulations were performed on theresources provided by the Swedish National In-frastructure for Computing (SNIC) at UPP-MAX and PDC. We thank helpful discussionswith M. Hellström.

Supporting Information Avail-ableCalculated viscosities η of NaCl electrolyte so-lutions in comparison with experiments; Finite-size correction to self-diffusion coefficient Ds

+,Ds− and that of water oxygen atoms; The

Walden plots of σG-K and σN-E with experi-mental values and the Stokes-Einstein relationof self-diffusion coefficients for NaCl electrolytesolution.

References(1) Nitzan, A. Chemical Dynamics in Con-

densed Phases: Relaxation, Transfer and

Reactions in Condensed Molecular Sys-tems ; Oxford Graduate Texts; OxfordUniversity Press: Oxford ; New York,2006; OCLC: ocm62118341.

(2) Hansen, J. P.; McDonald, I. R. StatisticalMechanics of Dense Ionized Matter. IV.Density and Charge Fluctuations in a Sim-ple Molten Salt. Phys. Rev. A 1975, 11,2111–2123.

(3) Altenberger, A. R.; Friedman, H. L. The-ory of Conductance and Related Isother-mal Transport Coefficients in Electrolytes.J. Chem. Phys. 1983, 78, 4162–4173.

(4) Zhong, E. C.; Friedman, H. L. Self-Diffusion and Distinct Diffusion of Ions inSolution. J. Phys. Chem. 1988, 92, 1685–1692.

(5) Padró, J.; Trullàs, J.; Sesé, G. ComputerSimulation Study of the Dynamic Cross-Correlations in Liquids. Mol. Phys. 1991,72, 1035–1049.

(6) Chowdhuri, S.; Chandra, A. MolecularDynamics Simulations of Aqueous NaCland KCl Solutions: Effects of Ion Con-centration on the Single-Particle, Pair,and Collective Dynamical Properties ofIons and Water Molecules. J. Chem. Phys.2001, 115, 3732–3741.

(7) Harris, K. R. Relations between theFractional Stokes-Einstein and Nernst-Einstein Equations and Velocity Corre-lation Coefficients in Ionic Liquids andMolten Salts. J. Phys. Chem. B 2010,114, 9572–9577.

(8) Kashyap, H. K.; Annapureddy, H. V. R.;Raineri, F. O.; Margulis, C. J. How IsCharge Transport Different in Ionic Liq-uids and Electrolyte Solutions? J. Phys.Chem. B 2011, 115, 13212–13221.

(9) Hansen, J.-P.; McDonald, I. R. Theory ofSimple Liquids ; Elsevier, 2013; pp 403–454.

5

(10) Allen, M. P.; Tildesley, D. J. ComputerSimulation of Liquids: Second Edition;Oxford University Press, 2017.

(11) Caillol, J.-M. Comments on the numer-ical simulations of electrolytes in peri-odic boundary conditions. J. Chem. Phys.1994, 101, 6080–12.

(12) King-Smith, R.; Vanderbilt, D. Theoryof polarization of crystalline solids. Phys.Rev. B 1993, 47, 1651.

(13) Resta, R. Quantum-mechanical positionoperator in extended systems. Phys. Rev.Lett. 1998, 80, 1800–1803.

(14) Tokuda, H.; Hayamizu, K.; Ishii, K.; Su-san, M. A. B. H.; Watanabe, M. Physic-ochemical Properties and Structures ofRoom Temperature Ionic Liquids. 2. Vari-ation of Alkyl Chain Length in Imida-zolium Cation. J. Phys. Chem. B 2005,109, 6103–6110.

(15) Tokuda, H.; Ishii, K.; Susan, M. A. B. H.;Tsuzuki, S.; Hayamizu, K.; Watanabe, M.Physicochemical Properties and Struc-tures of Room-Temperature Ionic Liquids.3. Variation of Cationic Structures. J.Phys. Chem. B 2006, 110, 2833–2839.

(16) Tokuda, H.; Tsuzuki, S.; Susan, M. A.B. H.; Hayamizu, K.; Watanabe, M. HowIonic Are Room-Temperature Ionic Liq-uids? An Indicator of the Physicochem-ical Properties. J. Phys. Chem. B 2006,110, 19593–19600.

(17) MacFarlane, D. R.; Forsyth, M.; Izgoro-dina, E. I.; Abbott, A. P.; Annat, G.;Fraser, K. On the concept of ionicity inionic liquids. Phys. Chem. Chem. Phys.2009, 11, 4962–6.

(18) Dünweg, B.; Kremer, K. Molecular dy-namics simulation of a polymer chain insolution. J. Chem. Phys. 1993, 99, 6983.

(19) Yeh, I.-C.; Hummer, G. System-Size De-pendence of Diffusion Coefficients and

Viscosities from Molecular Dynamics Sim-ulations with Periodic Boundary Con-ditions. J. Phys. Chem. B 2004, 108,15873–15879.

(20) Borodin, O. Polarizable Force Field Devel-opment and Molecular Dynamics Simula-tions of Ionic Liquids. J. Phys. Chem. B2009, 113, 11463–11478.

(21) Yoshizawa, M.; Xu, W.; Angell, C. A.Ionic Liquids by Proton Transfer: VaporPressure, Conductivity, and the Relevanceof ∆pKa from Aqueous Solutions. J. Am.Chem. Soc. 2003, 125, 15411–15419.

(22) Berendsen, H. J. C.; Grigera, J. R.;Straatsma, T. P. The missing term in ef-fective pair potentials. J. Phys. Chem.1987, 91, 6269–6271.

(23) Joung, I. S.; Cheatham, T. E. Determina-tion of Alkali and Halide Monovalent IonParameters for Use in Explicitly SolvatedBiomolecular Simulations. J. Phys. Chem.B 2008, 112, 9020–9041.

(24) Zhang, C.; Raugei, S.; Eisenberg, B.; Car-loni, P. Molecular Dynamics in Physiolog-ical Solutions: Force Fields, Alkali MetalIons, and Ionic Strength. J. Chem. TheoryComput. 2010, 6, 2167–2175.

(25) Yllö, A.; Zhang, C. Experimental andmolecular dynamics study of the ionicconductivity in aqueous LiCl electrolytes.Chem. Phys. Lett. 2019, 729, 6–10.

(26) Nezbeda, I.; Moučka, F.; Smith, W. R.Recent Progress in Molecular Simulationof Aqueous Electrolytes: Force Fields,Chemical Potentials and Solubility. Mol.Phys. 2016, 114, 1665–1690.

(27) Plimpton, S. Fast Parallel Algorithmsfor Short-Range Molecular Dynamics. J.Comput. Phys. 1995, 117, 1–19.

(28) Rumble, J. CRC Handbook of Chemistryand Physics ; CRC Handbook of Chem-istry and Physics; CRC Press LLC, 2018.

6

(29) Darden, T.; York, D.; Pedersen, L. Parti-cle mesh Ewald: An Nlog(N) method forEwald sums in large systems. J. Chem.Phys. 1993, 98, 10089–10092.

(30) Bussi, G.; Donadio, D.; Parrinello, M.Canonical Sampling through VelocityRescaling. J. Chem. Phys. 2007, 126,014101.

(31) Walden, P. Über organische Lösungs- undIonisierungsmittel. Z. Phys. Chem. 1906,55U, 207–249.

(32) Harris, K. R.; Kanakubo, M. High pres-sure studies of the transport propertiesof ionic liquids. Faraday Disc. 2012, 154,425–438.

(33) Zhang, C.; Hutter, J.; Sprik, M. Comput-ing the Kirkwood g-Factor by Combin-ing Constant Maxwell Electric Field andElectric Displacement Simulations: Appli-cation to the Dielectric Constant of Liq-uid Water. J. Phys. Chem. Lett. 2016, 7,2696–2701.

7

download fileview on ChemRxivNE_deviation_20190618.pdf (829.02 KiB)