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In Silico Predictions of the Temperature-Dependent Viscosities and Electrical Conductivities of Functionalized and Nonfunctionalized Ionic Liquids Philipp Eiden,† Safak Bulut,† Tobias Ko¨chner,† Christian Friedrich,†,# Thomas Schubert,‡,⊥ and Ingo Krossing*,†,§ Freiburger Materialforschungszentrum (FMF), Albert-Ludwigs-UniVersita¨t Freiburg, Stefan-Meier-Strasse 21, 79104 Freiburg, IoLiTec Ionic Liquids Technologies GmbH, Salzstrasse 184, D-74076 Heilbronn, Germany, and Institut fu¨r Anorganische und Analytische Chemie, Albert-Ludwigs-UniVersita¨t Freiburg, Albertstrasse 21, 79104 Freiburg ReceiVed: August 25, 2010; ReVised Manuscript ReceiVed: NoVember 7, 2010
The viscosity (η) and electrical conductivity (κ) of ionic liquids are, next to the melting point, the two key properties of general interest. The knowledge of temperature-dependent η and κ data before their first synthesis would permit a much more target-oriented development of ionic liquids. We present in this work a novel approach to predict the viscosity and electrical conductivity of an ionic liquid without further input of experimental data. For the viscosity, only some basic physical observables like the Gibbs solvation energy ,∞ (∆G*solv ), which was calculated at the affordable DFT-level (RI-)BP86/TZVP/COSMO, the molecular radius, calculated from the molecular volume Vm of the ion volumes, and the symmetry number (σ), according to group theory, are necessary as input. The temperature dependency (253-373 K) of the viscosity (4-19000 mPa s) was modeled by an Arrhenius approach. An alternative way, which avoids the deficits of the Arrhenius relation by a series expansion in the exponential term, is also presented. On the basis of their close connection, the same set of parameters is suitable to describe the electrical conductivity as well (238-468 K, 0.003-193 mS/cm). Nevertheless, more elegant alternatives like the usage of the Stokes-Einstein/Nernst-Einstein relation or the Walden rule are highlighted in this work. During this investigation, we additionally found an approach ,∞ to predict the dielectric constant ε* of an ionic liquid at 298 K by using Vm and ∆G*solv between ε* ) 9 and 43. Introduction In the past decade, ionic liquids (ILs) came more and more into the focus of chemical research and chemical/technical applications.1,2 The number of already available cation and anion combinations is extraordinary high (some say up to 1012).3-5 Thus, it is impossible to synthesize all of them in order to find a suitable ion combination with the desirable key properties, for example, melting point, viscosity, conductivity, and so forth.5,6 Besides chemical intuition, it would be helpful to have a computational tool for predicting or at least estimating the physical properties of any chosen combination in advance. Some words of caution are necessary here: The viscosity range of common ionic liquids lies between 10 mPa · s to several 10000 mPa · s5,7 and is very sensitive to impurities, especially water, as well as the kind of measurement (rotational for dynamic and the capillary (Ubbelohde) method for kinematic viscosity8). Also, most ILs show a great change in their viscosity at room temperature because this temperature is close to their melting point. Thus, great care has to be ensured when tempering the sample during the measurement. * To whom correspondence should be addressed. Fax: (+49) 761 203 6001. E-mail:
[email protected]. Homepage: http://portal.unifreiburg.de/molchem. † Freiburger Materialforschungszentrum (FMF), Albert-Ludwigs-Universita¨t Freiburg. ‡ IoLiTec Ionic Liquids Technologies GmbH. § Institut fu¨r Anorganische und Analytische Chemie, Albert-LudwigsUniversita¨t Freiburg and FRIAS Fellow of the Section Soft Matter Science. # Fax: +49(0)761-203-4709. E-mail:
[email protected]. ⊥ Fax: (+49) 7131 898 39-109. E-mail:
[email protected]. Homepage: http://www.iolitec.de.
Several approaches to describe or predict physical properties of ILs are known in the literature.3,9-21 The most popular are molecular dynamics (MD) and qualitative structure property relationships (QSPR), which are often carried out by the CODESSA code.22 They yield good results; especially, Gardas et al.21 demonstrated recently the great potential of the group additivity method. However, these methods have the disadvantage of high computational efforts in the case of MD and a limitation in the application of ions which are not present in their data set in the case of common QSPR approaches. Here, we present the results of our investigations to describe the temperature-dependent viscosities (72 ILs; 655 data points) and conductivities (69 ILs; 565 data points) of a diverse set of ILs based on their correlation with the underlying molecular volume and additional physical observables that remove the anion dependence of the original model23 and include temperature dependence. Results and Discussion In 2007, we found that the molecular volumes of ILs are correlated with the viscosity and electrical conductivity of ionic liquids.23 This observation followed the Arrhenius relation, that is
η ) A · e∆E/RT
10.1021/jp108059x 2011 American Chemical Society Published on Web 12/07/2010
and
κ ) B · e-∆E/RT
(1)
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SCHEME 1: Cation Types and Their Shortcuts Used for the Parameterization in This Worka
SCHEME 2: Anion Types and Their Shortcuts Used for the Parameterization in This Worka
a (a) Sulfonium ([SR1,R2,R3]+), (b) ammonium ([NR1,R2,R3,R4]+), (c) phosphonium ([PR1,R2,R3,R4]+), (d) imidazolium ([R1R2R3Im]+), (e) pyrrolidinium ([R1R2Pyr]+), (f) piperidinium ([R1R2Pip]+), (g) picolinium ([R1Pic-2/3/4]+), (h) pyridinium ([R1Py]+), (i) morpholinium ([R1R2Morph]+).
The experimental molecular volumes Vm were correlated with experimental viscosity and conductivity values at 295 K according to
pi ) ai · ebiVm
(2)
where p1 ) η, p2 ) κ is the viscosity and ai and bi are the regression parameters. This approach necessitates for each anion type to have its own regression line, and it only works for nonfunctionalized cations. Cation functionalization, for example, in the side chain, creates its own series in this correlation. Thus, this model supports the understanding of viscosity and conductivity at the molecular level but is not applicable to diverse types of new ionic liquids. To eliminate these disadvantages and to describe all ionic liquids with one parameter set, it is obviously necessary to introduce descriptors for the Arrhenius activation energy of the viscous flow and the pre-exponential factor in the Arrhenius relation. (Experimental data were fitted normally with the VogelFulcher-Tamman equation. The problems which occur with this approach are discussed later.) To ensure a sufficient ratio between experimental data and regression parameters for the least-squares fit (usually the ratio Ndata/Nparameter should exceed 10), we had to expand our data set. For this, we measured the viscosity and conductivity of several ionic liquids; other values were provided by IoLiTec, the group of Professor Buchner from the University of Regensburg, as well as from the literature7,8,23-48 (see the Supporting Information for a complete listing). The selection criteria for inclusion in this study were low water content (273 K), we recommend the use of eq 10 because two parameters fewer are needed. In the course of the examination of the viscosity, we found a simple way to predict the dielectric constant ε* of an ionic liquid at 298 K in silico as well. The relation therefore is based only on the knowledge of the ,∞ calculated with (calculated) molecular volume Vm and ∆G*solv the classical COSMO model and εr ) ∞.
(27)
the temperature-dependent specific conductivity as well. The regression parameters in eq 26 were calculated also by correlating all available data. They are in very good agreement with the work of Gardas et al.21 The regression parameters are also shown in Table 6 (rmse ) 0.23, R2 ) 0.90; cf. Supporting Information for plots). All three parametrizations were subjected to a cross-validation according to the LMO method (100 data points left out for NData × 3 cycles). All of them are robust in this validation (cf. Supporting Information), and hence, the Walden-like approach should be preferred as it has the minimal parameter set. The only disadvantage of this approach is that all ILs have the same slope and the same ordinate in the Walden plot. The former is acceptable because many ILs have a slope around 0.9 in the Walden plot.47,48,66 The latter varies stronger in the experiment, depending on the ILs’ nature. Therefore, the model according to eq 25 describes this finding in a better way (see Figure 9 for a comparison of the Walden plots). Since all three models give similar results, we used every model for ILs which were not used for their parametrization
Figure 9. Walden plots of all ILs used in the parametrization. f represents the values calculated with eq 25 (Stokes-Einstein-like approach). s represents the values calculated with eq 26 (Walden-like approach). · · · represents the “ideal” KCl line.
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Eiden et al. meter. The samples were measured in an argon-filled glovebox (water and oxygen content < 1 ppm) and tempered with a metal thermostat (accuracy of about (0.1 K). Pictures of the apparatus are deposited with the Supporting Information. Computational Details. The geometry optimization of the gas-phase structures of anions and cations was performed at the RI-BP86/def-TZVP67-76 level with the Turbomole 6.0 package.77 ThefrequencyanalysiswasdonebytheAOFORCE76,78,79 module to prove that a minimum structure was found. Due to performance reasons (parallelization) for large molecules, the frequency analysis was done with NumForce. After a full geometry optimization of the single ions, a run with the activated COSMO module52 and a dielectric constant set to infinity was ,∞,( single ion values. The Gibbs performed to obtain the ∆G*solv solvation energy of the entire IL was then calculated by addition of the single ion values, that is ,∞ ,∞,+ ,∞,∆G*solv ) ∆G*solv + ∆G*solv
Figure 10. Log-log plot of experimental versus calculated conductivities (f, eq 24, ], eq 25; left-pointing triangle, eq 26) of 73 ILs not present in the model set (117 data points in total). The temperature ranges from 293 up to 353 K.
The temperature-dependent specific conductivity was described successfully by an Arrhenius approach also. Due to the close connection of conductivity with viscosity through the Stokes-Einstein and Nernst-Einstein relations, we showed that the predicted viscosity can be used to calculate the specific conductivity. Here, the temperature dependence of the conductivity is induced by the temperature dependence of the viscosity only. The simplest approach to calculate the conductivity is derived from the Walden rule, but this way holds the disadvantage that in a Walden plot, only one averaged line for all ionic liquids can be recalculated. The Arrhenius-like and Stokes-Einstein-like models are better in this respect. Because the former needs twice the number of parameters as the latter, we recommend the use of the corresponding eq 25. The robustness of the parametrizations was proven in all cases by cross-validations. The accuracy of these models is below that of recent QSPR publications, but because it is based on physical observables as input and not on approximate group additivity values, we hope to have shed some light on the molecular origin of the interactions leading to viscosity and conductivity of ionic liquids. Additionally, we are not constrained to group-specific parameters, and therefore, our models are easily applicable to unknown compounds. This is demonstrated nicely by external validations with, compared to the model set, chemically totally different ionic liquids. Because of the simplicity of our relations, we are optimistic to continue in reducing them to very basic physical connections in ongoing work.
(28)
,∞,( where ∆G*solv is the Gibbs solvation energy of the single cation/ anion. The ordinary least-squares estimation was performed with the Octave 3.0.3 program.80 The square of the correlation coefficient (R2) was calculated according to
R )12
∑ (xexp - xcalc)2 ∑ xexp 2 ∑ xexp - N
(
)
(29)
the root-mean-square error was calculated according to
rmse )
∑ (xexp - xcalc)2 N
(30)
while x is the decadal logarithm in the case of viscosity and conductivity. Acknowledgment. This work was supported by the Deutsche Forschungsgemeinschaft DFG (SPP 1191), the Deutsche Bundesstiftung Umwelt DBU, the Fonds der Chemischen Industrie FCI, and the Albert-Ludwigs-Universita¨t Freiburg, Germany. We thank Professor Buchner for the support with highly accurate experimental data of temperature-dependent conductivities. Note Added after ASAP Publication. This article was published on the Web on December 7, 2010. Equation 7 and other text corrections were updated. The corrected version was reposted on December 14, 2010.
Experimental Section The ionic liquids were dried by heating at 333 K and 10-3 mbar for several days. The water content was measured by Karl Fischer titration. The viscosity was measured with a programmable Brookfield rotation viscosimeter (RVDV-III UCP) in an atmosphere of dry air in a specifically home-built glovebox (relative humidity below 0.1%) The samples were tempered with an external cryostat, so that the measurement of the viscosity was performed temperature-dependent between 273 and 353 K (if the samples permitted; accuracy of about (0.1 K). Electrical conductivities were measured with a Metrohm 712 conducto-
Supporting Information Available: Comparison of eperimental results and calculated values, validation of the model, conductivity and viscosity plots, and parameters used. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Endres, F.; Abbott, A.; MacFarlane, D. Electrodeposition from ionic liquids, 1st ed.; Wiley-VCH: New York, 2008. (2) Wasserscheid, P.; Welton, T. Ionic Liquids in Synthesis, 2nd ed.; Wiley VCH: New York, 2008.
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