The Phenomenological Account for Electronic Polarization in Ionic Liquid
Vitaly V. Chabana, Iuliia V. Voroshylovab and Oleg N. Kaluginb
a Department of Chemistry, University of Rochester, Rochester, New York 14627, USA, email: [email protected]
b Department of Inorganic Chemistry, V.N. Karazin Kharkiv National University, Kharkiv 61077, Ukraine
Although a great variety of classical force fields (FFs) for room-temperature ionic liquids (RTILs) have been recently suggested, no systematically derived non-polarizable model is able to reproduce their transport properties, i.e. diffusion constants, conductivities and viscosities. In the present paper, we show that modern FFs greatly overestimate pairwise electrostatic interaction energies in the RTILs systems leading to extremely hindered ionic motions. Based on the results of our ab initio molecular dynamics calculations using explicit ionic environment, we modified the electrostatic interaction potential of 1-ethyl-3-methylimidazolium tetrafluoroborate. Starting from FF, originally derived by Liu et al., we further refined it, so that it reproduces experimental transport properties without compromizing either structure or thermodynamics. The main advantages of the suggested phenomenological account for electronic polarization are its consistency, clearness, simplicity and the possibility to improve functionality of the existing FFs by modifying exclusively atomic partial electrostatic charges.
Introduction
Room-temperature ionic liquids (RTILs) are a specific class of solvents that are now often positioned as a green alternative to toxic and volatile organic solvents (methanol, acetonitrile, etc)2-9. A number of potential applications have been suggested for RTILs2,5-9 that require detailed knowledge about their physical chemical properties.
Partially because of high prices for RTILs and partially because of their own robustness, molecular dynamics (MD) simulations are now widely applied1,10-23 to derive structural, dynamical and thermodynamical properties of these compounds, their mixtures and solutions. Central to accurate and reliable prediction of such properties from classical MD simulations is an adequate and universal force field (FF). So far, many FFs have been derived for RTILs1,15,24-31 covering almost all known species. Recently, Borodin15 and Maginn16 provided brilliant critical reviews of this hot area.
Hereby, we concentrate our attention only on one representative of imidazolium-based RTILs, 1-ethyl-3-methylimidazolium tetrafluoroborate ([EMIM][BF4] that is recently widely referred to5-9,32,33. In spite of their formal difference, all universal non-polarizable FFs for [EMIM][BF4] experience similar problems in reproducing its experimental transport properties, e.g. diffusion coefficients (DCs), ionic conductivities (ICs) and shear viscosities (SVs). For instance, diffusion coefficients of 1.11, 1.534, 2.135,
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0.5136 for cation, EMIM+, and 0.91, 1.134, 2.035, 0.2336 (×10-11 m2/s) for anion, BF4–, were
simulated at 298 K, whereas the experimentalists report 5.037 and 4.237, respectively. Hence, the difference between simulations and experiments ranges from 100 to more than 1 000 % and is absolutely unsatisfactory for studying practical systems. The situation is even worse for ionic conductivity and shear viscosity. Because of their unrealistic values (extremely small in the case of ICs and extremely large in the case of SVs), nobody reported the corresponding computational results at room temperatures. So far, the only available simulated ICs are reported38 at 400 K being 3.5 and 2.2 Sm/m, whereas experimental ICs are 8.2837, 9.7739 and 9.79 Sm/m40. In turn, at 298 K the ICs derived from experiment are 1.3637, 1.541, 1.3141, 1.5841, 1.441, 1.3942 Sm/m. In the present work, we estimate both SVs and ICs of [EMIM][BF4] for a wide temperature range. Rather interestingly, density, structural and sometimes thermodynamical properties can be reproduced properly by means of the same FFs1,25,28-32,34. We believe, such huge underestimation of transport properties is mainly conditioned by the total neglect of electronic polarization in the condensed phase of RTILs that leads to systematically overestimated interaction energy, and consequently, extremely sluggish ionic transport.
By now, three different solutions of the above problem have been proposed, although each one has its principal weakness. The most universal approach is incarnated by Borodin15 using the explicit account of electronic polarizability. The Borodin’s results for [EMIM][BF4] are much closer to experimental ones than obtained before (4.80 and 3.62 × 10-11 m2/s for DCs of cation and anion, respectively, and 1.62 Sm/m for IC). Unfortunately, the author has not report shear viscosity for [EMIM][BF4] yet. The only nasty drawback of polarizable FF is their significantly bigger computational price. Generally, the usage of polarizable models leads to at least four times increase of computer time. For RTILs this issue is especially crucial since these compounds require long production stages. Another approach is demostrated by Bhargava and Balabramanian43 who did not constrain cationic and anionic partial charges to be +1 and -1, respectively. Based on their Car-Parrinello molecular dynamics (CPMD) simulations of radial distribution functions (RDFs), the authors scaled atomic partial charges of RTILs, so that RDFs derived from CPMD and classical MD coincide. The same idea of non-integral charges was successfully incarnated by Cadena et al.44. The proposed procedure is undoubtedly useful but somewhat tricky, because a lot of manual manipulations are needed to achieve the best RDF fit. Also, below we show that structural properties of RTILs are rather insensitive to electrostatic charges scaling. Another way relies on adjusting Lennard-Jones (LJ) parameters. Koeddermann et al.45 changed LJ parameters on some selected sites of 1-alkyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]amide to achieve correct density, heat of vaporization, Hvap, ionic transport and SV of this particular RTIL. Similarly, Micaelo et al.46 optimized fluorine-fluorine and oxygen-oxygen LJ parameters for hexafluorophosphate and nitrate anions to achieve proper matches for 1-alkyl-3-buthylimidazolium salts. They succeeded by assuming very low (and pretty unrealistic) epsilon values (0.001-0.01 kJ/mol) for the phosphorus and oxygen atoms. Reproducing density and transport properties very well, this FF model, however, gives a considerably lower (about 40 kJ/mol) Hvap. Unfortunately, the authors did not describe any particular procedure to be applied to other RTILs. Undoubtedly, tuning sigmas and epsilons can be useful for single compound although is not transferrable and lacks universality. In this case, it is also necessary that reliable experimental data are available before such tuning is applied.
The objective of our present work is to develop the non-polarizable FF which is able not only to reproduce structure, dynamics and thermodynamics properties, but also based
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on the clear derivation procedure. A special attention should be paid in order to check the FF performance over a wide temperature range. Hereby, such FF is derived and thoroughly tested using 1-ethyl-3-methylimidazolium tetrafluoroborate, [EMIM][BF4] (Figure 1), as a sample popular RTIL. In turn, the general FF which is able to reproduce properties of the imidazolium-based RTILs successfully still does not exist.
Based on the FF parameters of Liu et al.1 and our own density functional theory (DFT) calculations, we developed a new model for [EMIM][BF4]. Its behavior was thoroughly tested for simulations of density, enthalpy of vaporization, diffusion coefficients, ionic conductivities, shear viscosities as well as their temperature dependences (from 298 to 400 K). Also, radial distribution functions (RDFs) and corresponding cumulative numbers (CNs) were reported.
Computational Details
All molecular dynamics simulations were performed using the GROMACS package47 in the constant temperature constant pressure (NPT) ensemble. The systems of the considered ionic liquid, consisted of 128 ion pairs, were placed in the cubic MD box with periodic boundary conditions in all three directions. The long-range electrostatic forces were treated by the classical Ewald method48, whereas cutoff distance for the real-space part the Ewald sum was set to 1.5 nm. The LJ part was treated by shifted force technique with a switch region between 1.1 nm and 1.2 nm.
In order to investigate the temperature dependence of the transport properties, a series of MD simulations were carried out at 298, 323, 348, 373, and 400 K. The constant temperature was maintained by means of Bussi rescaling thermostat (often referred as V-rescale)49 with a relaxation time of 1.0 ps. The constant pressure of 1 bar was maintained applying Parrinello-Rahman pressure coupling50 with a relaxation time of 4.0 ps. The leap-flog integration algorithm was applied to propagate equations of motion with a time-step of 0.001 ps. The relaxation stages were performed during 10,000 ps, and the production stages included 100,000 ps of trajectory for temperatures below 348 K, and 50,000 ps for higher ones. Note, such an extensive trajectory sampling is extremely important for obtaining reliable shear viscosities and ionic conductivities. We calculated the transport properties writing down the atomic coordinates and energy terms every 0.02 ps (20 time-steps). The standard deviations for ICs and SVs were computed using the simulation data obtained from the consequent trajectory parts of 25 ns each. In the case of DCs, shorter time periods (5,000 ps) were analyzed since they are known to converge relatively well after 1,000-2,000 ps.
The diffusion coefficients were computed via Einstein relation through plotting mean square displacements of the atoms, since it is faster than Green-Kubo method exploiting velocity autocorrelation functions. In turn, the ionic conductivity is estimated via Einstein-Helfand fit of the continuous translational dipole moment. Importantly, in order to use this method, the atomic coordinates, stored in the trajectory database, should be preliminarily unfolded to allow free diffusion. Shear viscosities can also be derived from equilibrium MD simulation if the Einstein equation is used. It should be noted, that this Einstein method converges very slowly and is much dependent on the cut-off value for electrostatic interactions, so special care should be taken to get reliable results.
The heats of vaporization, Hvap, have been estimated at 298 K. Hereby, the vapor phase of RTILs is assumed to contain neutral ionic pairs, [EMIM][BF4], which do not interact with one another. The radial distribution functions, gij(r), were calculated using its classical definition on the trajectory parts of 1,000 ps with atomic coordinates saved
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every 0.02 ps. The corresponding cumulative numbers (CNs), nij(r), are integrals of gij(r) taken from 0 to r.
Density functional theory calculations were carried out in the pseudopotential/plane-waves approximation by means of the CPMD code51. The gradient-corrected exchange and correlation functionals prescribed by Becke52, Lee, Yang and Parr53 (BLYP) were employed. The Goedecker pseudopotentials54 were selected to take into account the effects of core electrons and nuclei of all atoms (carbon, nitrogen, hydrogen, boron and fluorine) of these RTILs. The plane-wave basis set was expanded as provided by the cut-off radius equal to 80 Ry. This value was selected after plotting energies and forces of systems using different cut-off radii (from 40 to 160 Ry).
The equilibrated configurations of 16 [EMIM][BF4], obtained from the previous classical MD trajectory, were placed in the cubic supercell with a simple cubic symmetry applied to get a bulk behavior of the considered systems. To calculate the supercell vectors, experimental density values of [EMIM][BF4] was utilized37. A short additional equilibration using 1.0 ps Car-Parrinello MD run at 400±50 K (ionic kinetic energy was controlled using Nose-Hoover chain thermostat55) was conducted, and electrostatic charge distributions were obtained via electrostatic potential fit. The calculated charges were summed on the cation and anion separately and averaged out over the whole system.
In order to sample a few ionic configurations, the short consequent CPMD runs with a total length of 5.0 ps have been performed. The time-step of about 0.0001 ps was used in conjunction with a ficticious electron mass of 400 atomic units to treat electronic degrees of freedom. Unfortunately, the duration of these CPMD runs is predictably very low to observe any considerable conformational changes. It is particularly true for RTIL, the dynamics of which is anomalously sluggish. We believe, at least several tens or even hundreds of ps are needed to get relevant sampling in this case.
Force Field Derivation We started with the FF for [EMIM][BF4] described in1. At 298 K, this force field reproduces experimental thermodynamical properties and densities quite satisfactorily1. However, the self-diffusion coefficients at 298 and 313 K are not so realistic, underestimating the experimental values in 3-4 times1. Unfortunately, no ICs and SVs were discussed in that work, although they are extremely important for electrochemical applications of RTILs. Using the same MD parameters, we calculated diffusion coefficients, ionic conductivities, shear viscosities and liquid densities at a number of temperatures (from 298 up to 600 K). Importantly, the only sufficient difference between our simulations and Liu’s was the production phase duration was enlarged in 1,000 times to be 100,000 ps instead of 100 ps used in the original calculations. The simulation results are depicted in Figure 2.
The DCs of both cations and anions as well as ICs of [EMIM][BF4] are very much underestimated, and SVs are greatly overestimated as compared to the experimental values. Noticeably, the calculated diffusion coefficients appear even lower than those reported in1 (0.3 and 0.11 vs 1.1 and 0.9 (×10-11 m2/s)). The observed divergence can be easily understood because of very small production stage duration (100 ps) in the original work. In fact, the previously reported DCs appear to be calculated on the sub-diffusion part of the trajectory. Thereby, they were considerably overrated in a number of earlier publications and cannot be considered reliable.
To summarize, the simulated diffusion coefficients using the FF1 for cations and anions are more than 10 times lower than experimental values. The similar situation is
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observed in the case of ionic conductivities where the difference is also very large, especially at room temperatures (0.09 vs 1.4 Sm/m at 298 K). Consequently, the simulated shear viscosities are much higher than they should be in reality. Generally, the derivation of viscosities of RTILs from computer simulations is still a rare practice because they require very extensive sampling in order to obtain reasonable values. In the recent paper15, Borodin reports some calculated viscosities using his excellent polarizable FF, but the values for [EMIM][BF4] are unfortunately absent there.
The demostrated situation with the Liu’s FF is common for all FFs that were derived using standard force field derivation procedures without further refinement. All such models are developed by considering partial electrostatic charge distributions on the single ion (cation and anion separately) in vacuum. The Lennard-Jones parameters are usually transferred from the well-known force fields or tuned selectively. As practice shows, this approximation is unable to provide realistic ionic motions in the condensed phase, since electrostatic potentials of the systems are systematically overestimated.
To phenomenologically account for electronic polarization effects we apply DFT calculations as described above. The average electrostatic charge was found to be +0.814±0.005 on EMIM+ and -0.814±0.005 on BF4
–. Interestingly, the electrostatic potential derived from the wavefunction corresponds to significantly lower partial charges than are usually used in molecular mechanical FFs. Thermal ionic motions brought by ab initio molecular dynamics lead to rather small fluctuations of the obtained numerical results. This fact also supports important conclusion claiming that energetically favorable configurations of [EMIM][BF4] obtained by classical and ab initio MD simulations are very similar.
Based on the mentioned ab initio findings, the original electrostatic charges were uniformly scaled using the scaling factor of 0.814. To be precise, all partial charges on both ions were simply multiplied by this value. The Lennard-Jones (12,6) parameters (Table 1) and intramolecular (bonds, angles, proper and improper dihedrals) force constants were directly transferred from the AMBER FF56 without any refinements.
The DCs, ICs, SVs and densities at temperature range between 298 and 400 K are summarized in Figure 2. It is clear that the new charge distributions work with a fine-resolution for the transport properties of [EMIM][BF4] not only at room temperatures, but also at much higher temperatures. Almost in all cases, the difference between simulated and experimental data are less or comparable to the standard uncertanties of the calculations. Moreover, they are sometimes smaller than the difference among experimental values reported by different authors37,39,40. Meanwhile, the density with a new FF is somewhat lower at 298 and 400 K17,37,38 but exactly coincides at 323 K57. At the same time, the results from different experiments sometimes differ by more than 2%. The general decrease of density, as compared to the original FF, is predictable because electrostatic part of the interactions decrease always leads to lower attraction energies in the simulated system. Considering scattering among the available experimental data, we suppose that the simulated data are generally in good agreement with the experiment.
The scaling factors derived from the DFT calculations using explicit ionic environment are found to be very efficient in conjunction with an ordinary RESP procedure (originally applied by Liu1 for [EMIM][BF4]). Our present findings may become useful to easily derive electrostatic charges for other RTILs being a possible cheap computational alternative to the modern polarizable FF schemes. We also believe the described procedure can be applied to a number of existing FFs to improve their functionality where necessary. Importantly, the suggested procedure does not use any arbitrarily modified parameters, as well as no experimental data were used to tune the
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model. It is very important that exclusively quantum mechanical calculations can be used to derive a reliable FF.
To explore the overall possibilities of scaling electrostatic charges, we further insignificantly changed the scaling factor for [EMIM][BF4] based on the general considerations with respect to available experimental properties. Finally, the optimal scaling factor for [EMIM][BF4] was selected to be 0.80 that is rather close to 0.814. The resulting properties are shown in Figure 2 and are even closer to the respective experimental values. We believe, the last refinement is not principally necessary. Hereby, it is applied mostly to illustrate the possibility of transport properties control by modifying only electrostatic potential. In the classical MD framework with pairwise interactions, it can be done by modifying partial charges on the corresponding interaction centers.
Heat of vaporization, Hvap, is a basic measure of interaction strength between the single ionic pair and its ionic environment. Hence, electrostatic potential scaling necessarily leads to Hvap decrease. For scaling factor of 0.814 we get Hvap equal to 125±1 kJ/mol, and for scaling factor of 0.80 – 124±1 kJ/mol, whereas original FF exhibits 159±1 kJ/mol. One should understand that experimental estimation of the heats of vaporization is not a trivial task in the case of RTILs due to their specific ionic nature and condensed structure. In fact, contradicting experimental results for the imidazolium-based RTILs have been recently published. Thus, in58 the authors report Hvap of [EMIM][BF4] equal to 255.8 kJ/mol, but Hvap for [BMIM][BF4] equals to 128.2 kJ/mol, referring to surface tension method elsewhere59. Obviously, such incomparable and generally weird results for these similar compounds cannot be considered reliable. Noticeably, our results are in quite satisfactory agreement with Hvap reported by Borodin15 (135.3 kJ/mol). Meanwhile, the corresponding heats simulated with the original FF1 are overestimated. This fact supports the conclusion that described scaling procedure has a strong physical background and improves transport properties of RTIL without compromizing other ones.
To be convinced that the suggested refinement does not also crucially modify structure properties of [EMIM][BF4], we simulated RDFs and cumulative numbers (CNs) between fluorine atom (F) (Figure 1) of anion and hydrogen atom (H5) (Figure 1) of cation. This atom pair is of primary interest for all imidazolium-based RTILs since hydrogen bonding is expected between imidazolium ring containing active H5 of cation and highly electronegative fluorine atom of anion.
The relatively high first peak occur at ~0.22 nm, whereas the second and the third ones can also be resolved at ~0.40 nm and ~0.55 nm, respectively (Figure 3). The scaling procedure insignificantly affects the corresponding positions and heights of these peaks. The maximum observed difference is 0.020 nm (for the second peak) and it can be neglected with a glance at the general peak shape. However, the height of the first peak on gHF(r) decreases from 2.5 (original FF) to ~2.0 as scaling factors of 0.80 and 0.814 are applied.
The topic on the presence or absence of C-H•••F hydrogen bonding is still somewhat controversial60. Our simulations show that the considered C-H•••F interaction becomes a bit weaker with a suggested FF, but the distance between hydrogen and fluorine atoms varies insignificantly (~0.01 nm). The analysis of the mutual orientation of counterions (Figure 3, inset) confirms that BF4
– tends to be coordinated above the imidazolium ring of the cation, and this principal fact does not depend on electrostatics scaling. Interestingly, another calculated RDF, gFN(r), is not affected by scaling electrostatic charges (Figure 3). The cumulative numbers, nFX(r) where X = hydrogen, nitrogen
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(Figure 3, bottom), show the coordination numbers of fluorine with respect to hydrogen and nitrogen as a function of interatomic distances. Note, the imidazolium-based cations contain two nitrogen atoms, so in order to get the coordination number for the cation itself, one should divide the corresponding value of CN by two. For the case of EMIM+ this CN equals to 6. The scaling procedure does not affect this observation anyway (Figure 3, bottom).
The phenomenological account for electronic polarization within classical non-polarizable FF is proved to be successful. Scaling charges was found not to be crucial for thermodynamical and structural properties of [EMIM][BF4]. The temperature dependencies of transport properties and density are well predicted, and the maxima and minima on RDFs are not shifted noticeably. The suggested numerical procedure is believed to be important for other RTILs, for which the traditionally derived force fields exhibit hindered ionic dynamics.
Conclusions
Using methylimidazolium tetrafluoroborate as a sample ionic liquid, we demonstrated that reliable force field of RTIL can be derived by scaling only pairwise electrostatic interaction energies. Based on the recent work1 and own ab initio MD calculations of several ionic pairs in the periodic system, a new FF of methylimidazolium tetrafluoroborate was proposed and thoroughly tested over a wide temperature range. The resulting model realistically reproduces all transport properties of the ionic liquid, whereas structure properties and densities are changed very insignificantly as compared to the original data. Being successful with 1-ethyl-3-methylimidazolium tetrafluoroborate, we believe that the developed procedure should be useful for other classes of RTILs as well as probably other condensed matter systems where electronic polarization plays a key role.
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TABLES Table I. Original (derived using RESP procedure) electrostatic charges and Lennard-Jones (12,6) parameters on the interaction sites of [EMIM][BF4] all-atom FF. In order to obtain scaled-charge FFs, all qi must be multiplied by the respective scaling factor (0.814 or 0.80). (m) designates the atoms that are closer to methyl group, whereas (a) designates atoms that are closer to ethyl group of the [EMIM]+ cation. For ethyl group of [EMIM]+, (a1) designates the carbon atom that is closer to the ring, whereas (a2) designates the carbon atom of the terminal CH3 group. Atomtype σii, Å εii, kJ/mol qi, e
CR 3.400 0.3598 0.0213 NA (m) 3.250 0.7113 0.0618 NA (a) 3.250 0.7113 0.0095
HC 2.650 0.0657 0.0487 B 3.581 0.3975 1.1504 F 3.118 0.2552 -0.5376
ECS Transactions, 33 (28) 43-55 (2011)
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FIGURES
Figure 1. (Color online) The optimized geometries of 1-ethyl-3-imidazolium cation, EMIM+, and tetrafluoroborate anion, BF4
–. The atom types are taken from the all-atom AMBER force field. The Lennard-Jones parameters and Coulombic charges of all atoms can be found in Table 1.
ECS Transactions, 33 (28) 43-55 (2011)
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Figure 2. (Color online) The simulated and experimental properties of [EMIM][BF4], a) density; b) conductivity; diffusion coefficients of c) cation and d) anion, e) shear viscosity. For each property, experimental values (black triangles) are compared with simulated using unmodified Liu’s FF (blue squares), scaling factor of 0.817 (red circles) and scaling factor of 0.80 (green diamonds).
ECS Transactions, 33 (28) 43-55 (2011)
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Figure 3. (Color online) Radial distribution functions, gFX (r), and cumulative numbers, nFX(r), in [EMIM][BF4] derived from molecular dynamics simulations at 298 K. “X” corresponds to H and N atoms, consequently.
ECS Transactions, 33 (28) 43-55 (2011)
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