DOI: 10.1002/cphc.200800102

On the Validity of Stokes–Einstein and Stokes–Einstein–Debye Relations in Ionic Liquids and Ionic-Liquid MixturesThorsten Kçddermann,[a] Ralf Ludwig,*[a, b] and Dietmar Paschek[c]

1. Introduction

The uses of ionic liquids (ILs) range from environmentallyfriendly solvents for novel chemical synthesis to electrolyte de-vices, such as batteries and photochemical cells.[1–3] For any ap-plication of ionic liquids, their physical properties are a key fea-ture. In particular, the transport properties are crucial, affectingthe reaction kinetics in a synthesis process or the ion transportin electrochemical devices. Despite extensive research on ionicliquids, the correlation between structure and transport prop-erties is still far from being fully understood. This is particularlytrue for the Stokes–Einstein (SE)[4] and Stokes–Einstein–Debye(SED)[5] relations, which gave pillar insight into the moleculardynamics of liquids for nearly a century. These relations derivefrom classical hydrodynamics (Stokes law) and simple kinetictheory (e.g. the Einstein relation).

In the Stokes–Einstein relation [Eq. (1)]:

D ¼ kBT6phreff

ð1Þ

D is the translational diffusion constant, kB is the Boltzmannconstant, T is the temperature, h is the coefficient of shear vis-cosity, and reff is the effective hydrodynamic radius of a mole-cule.

In the Stokes–Einstein–Debye relation [Eq. (2)]:

tl ¼6Veffh

kBTlðl þ 1Þ ð2Þ

tl is the reorientational diffusion correlation time, which de-scribes the relaxation of spherical harmonics of order l, and Veff

is closely related to the effective molecular volume. In theBrownian (large) particle limit, the conversion between reff andVeff—and the size, shape, and boundary conditions of the dif-fusing particles—can be specified. The validity and limitations

of such determinations have been an active field of experimen-tal and theoretical research during the last years.[6–15]

For normal liquids, the measurement of translational diffu-sion coefficients, rotational correlation times, and viscositiesallows the estimation of an effective hydrodynamic radius oran effective volume by using Equations (1) or (2), respective-ly.[16,17] Deviation from both relations are reported for atomic[18]

and molecular supercooled liquids.[12] Those deviations are de-noted as “breakdown of the SE and SED relations”, “decouplingof D and t from viscosities” or “decoupling between transla-tional and rotational diffusion”.[19,6] These systems are charac-terized by dynamical heterogeneities (DHs). This phenomenonis often called “spatially heterogeneous dynamics”, becausethere are spatial regions in which the structural relaxation timecan differ by orders of magnitude from the average over theentire system. In the “dynamical heterogeneities” view, themotion of atoms, molecules or ions is highly spatially correlat-ed. The presence of DHs has been argued to give rise to thebreakdown of the SE relation. These spatially heterogeneousdynamics have been reported for glass-forming materials, in-

Stokes–Einstein (SE) and Stokes–Einstein–Debye (SED) relations inthe neat ionic liquid (IL) [C2mim]ACHTUNGTRENNUNG[NTf2] and IL/chloroform mix-tures are studied by means of molecular dynamics (MD) simula-tions. For this purpose, we simulate the translational diffusion co-efficients of the cations and anions, the rotational correlationtimes of the C(2)�H bond in the cation C2mim+ , and the viscosi-ties of the whole system. We find that the SE and SED relationsare not valid for the pure ionic liquid, nor for IL/chloroform mix-tures down to the miscibility gap (at 50 wt% IL). The deviations

from both relations could be related to dynamical heterogeneitiesdescribed by the non-Gaussian parameter a(t). If a(t) is close tozero, at a concentration of 1 wt% IL in chloroform, both relationsbecome valid. Then, the effective radii and volumes calculatedfrom the SE and SED equations can be related to the structuresfound in the MD simulations, such as aggregates of ion pairs.Overall, similarities are observed between the dynamical proper-ties of supercooled water and those of ionic liquids.

[a] T. Kçddermann, Prof. Dr. R. LudwigInstitut f;r Chemie, Abteilung Physikalische ChemieUniversit>t RostockDr.-Lorenz-Weg 1, 18059 Rostock (Germany)Fax: (+49)381-498-6524E-mail : ralf.ludwig@uni-rostock.de

[b] Prof. Dr. R. LudwigLeibniz-Institut f;r Katalyse, an der Universit>t RostockAlbert-Einstein-Str. 29a, 18059 Rostock (Germany)

[c] Dr. D. PaschekFakult>t Chemie, Physikalische Chemie IIITechnische Universit>t DortmundOtto-Hahn-Str. 6, 44221 Dortmund (Germany)

Supporting information for this article is available on the WWW underhttp://dx.doi.org/10.1002/cphc.200800102.

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cluding polymers, metallic glasses, and pharmaceuticals.[7] Re-cently, Mazza et al.[8] showed the complete breakdown of theSE and SED relations for water under supercooled conditions.Dynamical heterogeneities are also reported for ionic liquids.[20]

Herein, we check the validity of the SE and SED relations forimidazolium-based ILs, that is, for [C2mim] ACHTUNGTRENNUNG[NTf2] (1-ethyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl)imide) and[C2mim] ACHTUNGTRENNUNG[NTf2]/chloroform mixtures, by means of molecular dy-namics simulations (MD). We focus on the occurrence of dy-namical heterogeneities (DHs) and on how they influence theabove-given equations. MD simulations are particularly usefulfor studying DHs because they have direct access to the detailsof the molecular motion. The findings of the MD simulationsmay be supported by reproducing the transport and dynami-cal properties, such as the diffusion coefficients, the correlationtimes, and the viscosities, for the ILs under investigation.[21] Im-provements of some of these properties by using a refinedflexible all-atom model were reported by Zhao et al.[22]

Computational Methods

We performed constant-pressure (NPT) MD simulations at 303 Kand at a pressure of 1 bar[23] using our force field for the imidazoli-um-based ILs[21] and the force field for chloroform reported by Cie-plak et al.[24] Our improved (nonpolarizable) fully flexible all-atomforce field has been parameterized for imidazolium-based ILs ofthe type [Cnmim]ACHTUNGTRENNUNG[NTf2] . The model is composed of atom-centeredpoint charges and Lennard–Jones interactions for nonbonded in-teractions.[21] The proper stereochemistry and conformational rep-resentations of the molecules are guaranteed by appropriately par-ameterized bond-angle and dihedral interactions taken from theLopes et al. force field.[25] This force field for [C2mim]ACHTUNGTRENNUNG[NTf2] hasbeen refined in such a way that the simulated properties mostlyagree well with the experimental densities, the accurately mea-sured self-diffusion coefficients for cations and anions in neat[C2mim] ACHTUNGTRENNUNG[NTf2] , and the NMR rotational correlation times for bothcations in [C2mim]ACHTUNGTRENNUNG[NTf2] and water molecules in a 1 w% solution ofwater in [C2mim] ACHTUNGTRENNUNG[NTf2] (at 303 K).[26] For this purpose, the TIP4P-Ewwater model was employed.[27] This kind of parameterization seemsto be unusual, but has significant benefits. The diffusion coeffi-cients can be fitted separately to cations and anions, independent-ly of whether the ions exist as single particles, ion pairs or largeraggregates. The same is true for the NMR rotational correlationtimes. Unfortunately, these reorientation dynamics are availableonly for the CH(2) vector of the imidazolium cations. Thus, weused NMR reorientational correlation times of water in [C2mim]-ACHTUNGTRENNUNG[NTf2] to fit the anion parameters. This unusual procedure is basedon recent experimental findings showing that the structure anddynamics of single water molecules are sensitive probes for anionproperties.[28] We show that the single water behavior is mainly de-termined by anions in the ionic liquid; thus, the water dynamicsmay serve as a good indicator for anion properties and can beused for deriving force-field parameters.

The following number of ion pairs and solvent molecules wereused for the [C2mim]ACHTUNGTRENNUNG[NTf2]CHCl3 mixtures: a) 100 wt%: 173/0,b) 90 wt%: 173/70, c) 80 wt%: 173/141, d) 67 wt%: 173/283,e) 55 wt%: 173/454, f) 50 wt%: 173/567, g) 1 wt%: 16/5246, at303 K and at a pressure of 1 bar. The MD simulations are per-formed in the NPT ensemble using a NosN–Hoover thermostat[29,30]

and the Parrinello–Rahman barostat[31,32] with coupling times

tT =0.5 ps and tp =2.0 ps. The electrostatic interactions are treatedby particle-mesh Ewald summation[33] with a cutoff of 0.9 nm and amesh spacing of approximately 0.12 nm (with 4th order interpola-tion). Lennard–Jones cutoff corrections for energy and pressurewere considered. A 2 fs time step was used. Distance constraintswere solved by using the SHAKE procedure.[34] For calculating theviscosities, addition constant-volume (NVT) MD simulations of simi-lar lengths were performed at the obtained average densities.

Simulations with a length of at least 10 ns were performed foreach state point. Overall, the simulation conditions were the sameas those in ref. [21] All simulations were performed with the Gro-macs 3.2 simulation program[35] All our IL topology—and configura-tion—files are available from the Gromacs web site(www.gromacs.org).

2. Simulated Properties

For the imidazolium-based ionic liquid [C2mim] ACHTUNGTRENNUNG[NTf2] and[C2mim] ACHTUNGTRENNUNG[NTf2]/chloroform mixtures, we simulate the transla-tional diffusion coefficients of the cations and anions, the rota-tional correlation times of the C(2)�H bond in the cationC2mim+ , and the viscosities of the whole system.

2.1. Self-Diffusion Coefficients

The self-diffusion coefficients D are calculated from the Ein-stein relation [Eq. (3)]:

Di ¼16

limt!1

ddt

riðtÞ � rið0Þð Þ2h i ð3Þ

where riðtÞ � rið0Þð Þ2 is the mean-square displacement of themolecular centers of mass of species i at time t. As pointed outby Maginn,[36] a typical ionic liquid can have such a low diffu-sivity that the ions only move 1 O over 1 ns, which is muchsmaller than the size of the ions themselves. In our case, theself-diffusion coefficients for the cation and anion are about6P10�11 m2 s�1 at 303 K. Thus, the mean-square displacementof the ions over 10 ns is 60 O, which is more than ten times oftheir size. In the mixtures with chloroform, the IL is more fluidand the length of the simulations is less of a problem. InFigure 1, the mean-square displacements are shown for boththe cations and anions in IL/chloroform mixtures containing100, 80, and 67 wt% IL. The nice slopes of the mean-squaredisplacements within the first 2 ns of the MD simulation under-line the observation of a diffusive behavior. If force fields areused which give diffusion coefficients that are at least a factorof ten below the experimental values, the diffusive regime willbe only reached with longer trajectories.

2.2. Reorientational Correlation Times

The rotational correlation times were calculated only for theC(2)�H vector of the cation in [C2mim+] ACHTUNGTRENNUNG[NTf2

�] . For the ex-treme narrowing case, the NMR relaxation is determined bythe rotational correlation time t2 [Eq. (4)]:

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R. Ludwig et al.

t2 ¼Z10

C2ðtÞdt ð4Þ

where C2(t) is the time correlation function [Eq. (5)]:

C2ðtÞ ¼ hP2½cos qð0Þ� P2½cos qðtÞ�i ð5Þ

which can be expressed by P2(t), the second Legendre poly-nomial of the angle of the C(2)�H vector with a fixed externalaxis. In the rotational diffusion limt, C2(t) exhibits a single expo-nential decay. The MD results for C2(t) for IL/chloroform mix-tures with 100, 80, and 67 wt% IL are shown in Figure 2. It isclearly seen that the reorientational dynamics can be describedby two relaxation times, namely, a relatively fast relaxation of

the small amplitudes for t�1 ps, followed by a slow decay. Weare mainly interested in the slow component. However, for areliable comparison with the reorientational correlation timesobtained from NMR relaxation-time experiments, we calculatet from the total integral including both the fast and the slowcontributions of C2(t) (Figure 2). As discussed by Cumminset al. ,[37] the reorientational correlation times from the NMR re-laxation-time experiments always include vibrational contribu-tions that lead to a fast initial decay.

2.3. Shear Viscosities

The shear viscosities h were determined from the Green–Kuboformula according to Equation (6):

h ¼ VkBT

Z10

Pabð0ÞPabðtÞ� �

dt ð6Þ

where V is the volume, kB is Boltzmann’s constant, and Pab rep-resents the three independent off-diagonal components of thepressure tensor, Pa¼6 b. The integral in Equation (6) is then evalu-ated numerically, multiplied by V/kBT, and averaged over thethree off-diagonal pressure tensor components to estimate theviscosity. Maginn[36] discussed the problems to calculate a col-lective property, such as the viscosity, by using the Green–Kubo formula. In previous attempts, too short simulations runsgave overestimated values of the viscosities—by an order ofmagnitude and even more.[38]

The running integral for the shear viscosity of [C2mim]ACHTUNGTRENNUNG[NTF2]at 303 K is represented in Figure 3. The inset shows the auto-correlation function of the xy components of the pressuretensor. The calculated value of about 21.7 mPas for neat[C2mim+] ACHTUNGTRENNUNG[NTf2

�] at 303 K is close to the measured viscosity(26 mPas, 303 K) obtained by Crosthwaite et al. ,[39] but lower

Figure 1. Mean-square displacements of cations C2mim+ (c) and anionsNTf2

� (a) in [C2mim] ACHTUNGTRENNUNG[NTf]/chloroform mixtures containing 100, 80, and67 wt% IL (at 303 K).

Figure 2. Reorientational time correlation function of the vector C(2)�H ofthe cations C2mim+ in [C2mim] ACHTUNGTRENNUNG[NTf]/chloroform mixtures with 100, 80, and67 wt% IL (at 303 K).

Figure 3. Shear viscosity of [C2mim] ACHTUNGTRENNUNG[NTf] at 303 K, represented by the aver-age of the integrated correlation function for the off-diagonal pressuretensor components multiplied by V/kB T. The inlay shows the correlationfunction of the xy component of the stress tensor.

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Stokes–Einstein Relations in Ionic Liquids

than the simulated values reported by Kelkar and Maginn[40,36]

(of about 42 and 30 mPas at 293 and 313 K, respectively).

3. Results and Discussion

The simulated translational diffusion coefficients of the cations,C2mim+ , and anions, NTf2

� , the rotational correlation times ofthe C(2)�H bond in the cation C2mim+ , and the viscosities areshown in Figure 4a,b (see also the Supporting Information).Overall, the dynamical properties show the expected behavior.With decreasing IL concentration, the mixtures become lessviscous, which is indicated by the increasing diffusion coeffi-cients, the decreasing correlation times, and the decreasingviscosities. If these dynamical properties are plugged into theSE and SED relations, we obtain effective hydrodynamic radii

and volumes for the neat IL and most of the IL/chloroformmixtures (down to 50 wt%) which are significantly too small. InFigure 5a,b, these values are shown in comparison to radii andvolume values that have been calculated by using DFT meth-ods for B3LYP/6-31+G* optimized particles in the mixtures:CHCl3, C2mim+ , NTf2

� , and the ion pairs [C2mim] ACHTUNGTRENNUNG[NTf2]n withn=1,2,3,11.[41] The value of Veff is much smaller than the small-est particles in the neat IL, which are the cations and anions.Also, reff is smaller than these particles, but to a less extent.Later on in this article, we will give some arguments that al-though reff is closer to the calculated reff values of cations andanions, the SE relation is not valid either. For improving theSED model, several approaches are proposed which refine theassumption made in this relation. For example, the Gierer–

Figure 4. a) Self-diffusion coefficients of the cations C2mim+ and anionsNTf2

� in [C2mim] ACHTUNGTRENNUNG[NTf]/chloroform mixtures at 303 K. b) Rotational correlationtimes of the vector C(2)�H of the cations C2mim+ in [C2mim] ACHTUNGTRENNUNG[NTf]/chloro-form mixtures at 303 K. In both figures, the calculated viscosities of IL/chloroform mixtures are shown. The available experimental data for all prop-erties of the neat ionic liquid are given as filled symbols.[26, 45]

Figure 5. Effective radii reff (a) and effective volumes Veff (b) of IL structures inIL/chloroform mixtures, calculated by using the SE and SED relations fromthe diffusion coefficients, rotational correlation times, and viscosities. For theeffective volumes in (b) the Gierer–Wirtz-corrected values are also shown. Itis clearly seen that both reff and Veff, and even the correct effective volumes,are smaller than the ions C2mim+ and NTf2

� . For comparison, the reff and Veff

values are also given for chloroform, ion pairs (IP), IP dimers, and IP trimers.

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R. Ludwig et al.

Wirtz theory assumes a distance-dependent viscous friction,which increases the effective volume for the case in which thesolute molecules are much smaller than the solvent ones (seeTable ).[42] However, this procedure fails, since the predicted

volumes are still too small to reasonably describe reasonablyions or ion-pairs in ionic liquids (see Figure 5a,b). Othermodels have been propsed by Perrin[43] and Youngren-Acri-vos[44] to account for the shape of the molecules. Despite theirconsiderable latitude, all theoretical estimates are far abovethe experimental values, as shown by a comparative dielectricand magnetic relaxation study.[26] Obviously this is not thereason for the failure of SE and SED.

Herein, we show that dynamical heterogeneities are respon-sible for the failure of SE and SED. This can be shown by usingthe non-Gaussian contribution of the van Hove self-correlationfunction for the IL and the IL/chloroform mixtures. The vanHove self-correlation function is given by Equation (7):

Gselfðr; tÞ ¼1N

XN

i¼1

d rci ðtÞ � rc

i ð0Þ � r� �* +

ð7Þ

which measures the probability of finding the center of massof a molecule at a position rc at time t, assuming that it was atthe orign, rc(0), at time time t=0.[47] The function GselfACHTUNGTRENNUNG(r,t) exhib-its a Gaussian form at very short times, corresponding to afree-particle behaviour, while it shows a form that correspondsto the hydrodynamic limit at longer timescales. Usually, formost of the liquids under normal conditions, Gself ACHTUNGTRENNUNG(r,t) gives onlysmall deviations from its Gaussian form [Eq. (8)]:

Gselfðr; tÞ ¼ ð4pDtÞ�3=2 exp � r2

4Dt

ð8Þ

It has been shown in the past that supercooled liquids ex-hibit deviations from the Gaussian form at intermediate times.These deviations are associated with dynamic heterogeneitiesand can be quantified by the non-Gaussian parameter, whichis given by Equation (9):[48]

aðtÞ ¼ 35

DrðtÞj j4h iDrðtÞj j2h i2

� 1 ð9Þ

Here, Dr(t) is the displacement at time t with respect to theposition at time t=0. For a Gaussian distribution of jDr(t) j , a

must be equal to zero.[48]

At short times, a(t) does not depend on time or concentra-tion, because in this initial temporal regime, the dynamics areballistic at all concentrations. This is indicated by a first should-er of a(t) at 0.5 ps (see Figure 6a,b). At intermediate times, a(t)

develops a distinct maximum which increases in magnitudeand shifts to lower observation times as the IL concentration isdecreased. The maximum occurs at the timescale where thefastest molecules of the dynamical heterogeneities “brake theircages” and enter the corresponding diffusive regimes. For thecations (C2mim+), this maximum occurs earlier and is morepronounced than that for the anions (NTf2

�). In the long-timelimit, the Gaussian form of GselfACHTUNGTRENNUNG(r,t) should be recovered tozero. For the neat IL this decay is very slow (see Figure 6). After1 ns, a(t) still has 50% of its maximum value.

It has been shown for atomic,[18] molecular supercooled,[49]

and ionic liquids[20,50, 51] that non-Gaussian effects are related toa distribution of relaxation processes with different rates. Thefluid shows spatially heterogeneous dynamics characterized byregions, or domains of molecules of enhanced or diminishedmobility. A clear connection between a(t) and the degree ofheterogeneity exists.[49] Thus, a(t) seems to be a good indicatorfor the validity of SE and SED. If a(t) still exhibits significantcontributions, these relations are not valid. Already in the neatIL, the anions, NTf2

� , give larger effective radii than the cations,

Table 1. Simulated rotational correlation times of the vector C(2)�H ofthe cations (C2mim+) and viscosities of [C2mim] ACHTUNGTRENNUNG[NTf]/chloroform mixturesat 303 K.

Mixture [w%] t2 [ps][a] h [mPas][a] Veff [nm3][b] VeffGW [nm3][c]

100 57.0 21.7 0.00597 0.037390 45.4 14.1 0.01347 0.074080 40.4 11.9 0.01420 0.078067 41.9 7.7 0.02276 0.125155 44.3 6.4 0.02896 0.159150 47.0 7.8 0.02521 0.13851 27.6 0.4 0.28865 1.5860

[a] The possible errors for t2 and h were estimated to be 2 and 20%, re-spectively. [b] The values of Veff were calculated with the SED relation.[c] The Gierer–Wirtz-corrected values Veff

GW were determined using f=

0.16 for the neat IL and f=0.182 for the IL/CHCl3 mixtures.[42, 46]

Figure 6. Non-Gaussian parameters a(t) for a) C2mim+ and b) NTf2� as a

function of time. The maxima of a(t) is given for times t*. The correspondingvan Hove self-correlation function GeffACHTUNGTRENNUNG(r,t) (c) and Gaussian approximationGeff ACHTUNGTRENNUNG(r,t) (a) for the center-of-mass motion of a) C2mim+ and b) NTf2

� in theneat ionic liquid are shown in the insets.

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Stokes–Einstein Relations in Ionic Liquids

C2mim+ , which is in accordance with a smaller and larger a(t)value, respectively.

If we dilute the IL with chloroform up to the miscibility gapat 50 wt%,[52] the non-Gaussian parameter a(t) decreases con-tinuously for both ions—stronger for the cation than for theanion (Figure 7). But even for the 50 wt% mixture, the maxi-

mum values of a(t) are still 0.175 and 0.15 for the cations andanions, respectively. After 1 ns, a(t) has still not decayed tozero. SE and SED are still not valid, as indicated by too smallreff and Veff values (see Fig-ure 5a,b). For the 1 wt% IL/chloroform mixture (at the otherside of the miscibility gap in theIL-poor region), a(t) decreases tovalues close to zero immediatelyafter the initial ballistic increase.No maximum occurs, which indi-cates that there are no fast mol-ecules that have to “brake theircages”. Obviously, for the 1 wt%IL/CHCl3 mixture, no heterogene-ities can be detected. If we nowuse the SE and SED relations forcalculating the effective radiiand effective volumes, weobtain 0.9 nm for reff and1.58 nm3 for Veff, respectively(using the Gierer–Wirtz approxi-mation, see Figure 5a,b). Thesereff and Veff values can be relatedto the calculated cluster sizesfrom our MD simulations. At this

low IL concentration we find a dimer, a trimer and an 11-merof ion pairs (see Figure 8a). Taking the average values of thecalculated reff for a dimer, a trimer, and an 11-mer we obtain7.87 nm3, which is close to the values we obtained from the SErelation (of about 10.00 nm3, see Figure 5a). In this respect, itis also interesting to note that the diffusion coefficients for cat-ions and anions are now the same (see Figure 1b). Both ionsstay for long time within these clusters and diffuse collectively.Obviously, at low IL concentrations, the SE and SED relationsbecome valid. This could be expected because dynamical het-erogeneities do not occur. The “Brownian” particles (here theIL clusters) are large compared to the solvent molecules, andthe conditions for the validity of SE and SED relations are fulfil-led. In the light of the apparent relation between dynamicalheterogeneities and the validity of SE and SED, we would liketo discuss the validity of SE for neat ILs. Here, the reff only dif-fers by about a factor of two (or less) from the calculatedvalues for the anions and cations, and SE is often assumed tobe valid. If reff can be really related to the size of the ions, thenno temperature dependence of reff is possible. However, if wecalculate reff from the experimental viscosities[39] and diffusioncoefficients[45] for temperatures of 283, 303, and 343 K, weobtain values of 1.245, 1.39, and 1.63 O, respectively, for thecation (C2mim�) and 2.059, 2.449, and 2.22 O for the anion(NTf2

�). Both reff show a significant temperature dependence.Interestingly, both effective radii decrease with decreasing tem-perature. Again, this behavior can be related to increasing het-erogeneities at lower temperatures.

Previous studies mostly report the appearance of dynamicalheterogeneities (DHs) for the supercooled liquid state ap-proaching the glass transition temperature. This has beendemonstrated for molecular liquids,[12,49] weakly interactingmodel-liquids,[18] as well as for network-forming liquids such aswater.[8] It is noteworthy that for the ILs, a pronounced appear-

Figure 7. Non-Gaussian parameters a(t) for C2mim+ (c) and NTf2� (a)

in [C2mim] ACHTUNGTRENNUNG[NTf]/chloroform mixtures (given in wt%). It can be clearly seenthat the peak of a(t) continuously decreases with increasing IL concentra-tion. The slope of the curves is discussed in the text.

Figure 8. Snapshots of simulation cells for the [C2mim] ACHTUNGTRENNUNG[NTf]/chloroform mixtures with increasing IL concentrationsof 1, 50, 55, 67, 80, and 100 wt%. The cations are colored red, the anions orange, and the chloroform moleculesblue. All snapshots were taken at the end of the simulation runs. The snapshot of the neat IL is enlarged for abetter view of the microstructure.

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R. Ludwig et al.

ance of DH and and the breakdown of the SE and SED rela-tions is apparently observed already for normal liquid condi-tions. However, the supercooled state is identified only by itsrelative thermodynamic (in)stability with respect to the solidstate as determined by the location of the melting line. In thecase of ILs this feature might be regarded as a conseqence ofa (low-lying) melting temperature Tm, being situated ratherclose above the glass transition temperature Tg, and suggest-ing a signifcantly smaller Tm/Tg ratio as for other kinds of net-work forming liquids. Smaller Tm/Tg ratios were also found fortypical fragile glass formers.[53]

4. Conclusions

We studied the Stokes–Einstein (SE) and Stokes–Einstein–Debye (SED) relations for the neat ionic liquid [C2mim] ACHTUNGTRENNUNG[NTf2]and [C2mim] ACHTUNGTRENNUNG[NTf2]/chloroform mixtures by means of MD simu-lations. We found that these relations are not valid for the neationic liquid—nor for IL/chloroform mixtures down to the misci-bility gap (at 50 wt% IL). This non-validity of SE and SED canbe related to dynamical heterogeneities, as is indicated by thenon-Gaussian parameter a(t). If a(t) is close zero at concentra-tions of 1 wt% IL in chloroform, both relations become valid.The effective radii and volumes calculated from the SE andSED equations can be related to the structures found in theMD simulations. Overall, similarities are observed between thedynamical properties of supercooled water and those of ionicliquids.

Acknowledgements

This work was supported by the German Science Foundation(DFG), priority programme SPP 1191, and the State of Mecklen-burg-Vorpommern, Germany.

Keywords: computer chemistry · ionic liquids · moleculardynamics · Stokes–Einstein relations · Stokes–Einstein–Debyerelations

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[42] The rotational microviscosity correction factor f for the Gierer–Wirtztheory depends on the hydrodynamic boundary conditions and theshape of the rotator. The parameter f is given byf ¼ 6 rm

r

� �þ 1þ rm

r

� ��3� ��1

where rm is the radius of the solvent molecule (chloroform) and r thatof the solute molecule (the cation C2mim+). For pure liquids (where rm/r=1), the microviscosity factor has a value of f=0.16. For the IL/CHCl3mixture (with rm/r=1.125), f will be 0.182, assuming that the cationC2mim+ is the solute molecule. The assumption of larger solute struc-tures in the mixtures leads to even large f values and smaller vol-umes.[46]

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Phys. Chem. B 2006, 110, 19593–19600.

ChemPhysChem 2008, 9, 1851 – 1858 @ 2008 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.chemphyschem.org 1857

Stokes–Einstein Relations in Ionic Liquids

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[53] The ratio Tm/Tg is 1.29 for [C2mim] ACHTUNGTRENNUNG[NTf] ,[39] compared to 1.64 for etha-nol, 2.01 for water, and 2.58 for glycerol. For the fragile glass formersalol, this ratio is 1.43.

Received: February 21, 2008Revised: June 28, 2008Published online on August 27, 2008

1858 www.chemphyschem.org @ 2008 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2008, 9, 1851 – 1858

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