Dielectric Properties of Organic Solvents from Non-Polarizable

ARTICLE pubs.acs.org/JPCB

Dielectric Properties of Organic Solvents from Non-Polarizable Molecular Dynamics Simulation with Electronic Continuum Model and Density Functional Theory Sanghun Lee and Sung Soo Park* Corporate R&D Center, Samsung SDI Co. Ltd., Yongin, Gyunggido, 446-577, South Korea ABSTRACT: Dielectric constants of electrolytic organic solvents are calculated employing nonpolarizable Molecular Dynamics simulation with Electronic Continuum (MDEC) model and Density Functional Theory. The molecular polarizabilities are obtained by the B3LYP/ 6-311++G(d,p) level of theory to estimate high-frequency refractive indices while the densities and dipole moment fluctuations are computed using nonpolarizable MD simulations. The dielectric constants reproduced from these procedures are evaluated to provide a reliable approach for estimating the experimental data. An additional feature, two representative solvents which have similar molecular weights but are different dielectric properties, i.e., ethyl methyl carbonate and propylene carbonate, are compared using MD simulations and the distinctly different dielectric behaviors are observed at short times as well as at long times.

1. INTRODUCTION The lithium ion battery (LIB) is the most widely used rechargeable storage device of electric energy for portable electronics. In recent years, highly efficient LIBs emerged in the electricvehicle industry due to rapid advancement in the battery technology. There are still features that need to be improved, however, the c-rate could be higher, lifetime could be longer, and the operation temperature could have a wider range. In particular, the electrolyte in LIBs plays a major role in influencing battery characteristics, which is typically prepared on the basis of organic solvents such as carbonates, propionates, and acetates. The dielectric constant, one of the basic properties representing solvent characteristics, is an important factor to control solvation, mobility, and conductivity of lithium ions. Therefore, it is needed to predict the dielectric constants of organic solvents for developing electrolytes in LIBs. Molecular dynamics (MD) simulation is a very useful method for understanding the behaviors of liquids at the molecular level. In many MD simulation studies, the dipole moment fluctuation formula shown by Neumann1,2 has been employed to estimate the dielectric constants of water,3
10 methanol,10,11 and small organic molecules.12 Even though the dielectric constants with conventional nonpolarizable force fields such as AMBER,13,14 CHARMM,15 GROMOS,16 OPLS,17 and COMPASS18 can be simply calculated by MD simulations and Neumann’s formula, it is not always possible to accurately describe the dielectric response of organic molecules in condensed phase and the results are often not reliable.19
21 This is due to the fixed (additive) partial charges in the nonpolarizable force fields cannot describe the screening effect by the electronic continuum and electronic polrarization. Thus, the dielectric constants of r 2011 American Chemical Society

organic solvents have rarely been performed using nonpolarizable force fields and, furthermore, much effort has been made to develop polarizable force fields.22
28 Nevertheless, the nonpolarizable models are still widely used due to the relatively simple potential functions and, consequentially, cheap computational costs. Meanwhile, there is an increasing interest in capturing a relationship between nonpolarizable force fields and the effects of electronic polarization and screening in conventional MD simulations. Leontyev et al. set up a simple model that combines a nonpolarizable force field MD with an electronic continuum (MDEC) model for electronic polarization29 which predicts dielectric constants of alkanes and alcohols in good agreement with the experiments and the polarizable MD simulations.12 In their procedure, high-frequency dielectric constants (ε∞, the scaling factors for screening effect), which are required to employ the MDEC model, were parametrized from the ClausiusMossotti equation with experimental polarizabilities.30 The MDEC model extended to biological molecules showed that the charge scaling procedure results in more accurate interactions with different environments.31 In addition, theoretical highfrequency dielectric constants were derived from the Lorentz
Lorenz equation involving the molecular polarizability obtained on the basis of ab initio methods.32
34 In this study, combining the MDEC and density functional theory (DFT) calculations, we estimated the dielectric constants of electrolytic organic solvents for LIBs including cyclic Received: August 9, 2011 Revised: September 20, 2011 Published: October 03, 2011 12571

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Equation 2 makes a connection between the total dielectric constant ε, the high frequency dielectric constant ε∞, and the dielectric constant obtained by the nonpolarizable MD εMD. If we consider a case of spherical ion or a pair of spherical ions, then the solvation energies are proportional to the corresponding Born factors, i.e., ΔG ≈ (1
1/ε), ΔGel ≈ (1
1/ε∞), and ΔGMD ≈ (1
1/εMD).35 From these relations and eq 2, we obtain, ε ¼ ε∞ 3 εMD

ð3Þ

Equation 3 can also be obtained by Neumann’s dipole moment fluctuation formula,1,2 ε ¼ ε∞ þ Figure 1. Chemical structures of investigated organic solvents for lithium ion battery.

carbonates (ethylene carbonate; EC, propylene carbonate; PC and fluoroethylene carbonate; FEC), linear carbonates (dimethyl carbonate; DMC, ethyl methyl carbonate; EMC and diethyl carbonate; DEC), propionates (methyl propionate; MP and ethyl propionate; EP), acetates (ethyl acetate; EA and isopropyl acetate; IA), and 1,2-dimethoxyethane; DME (Figure 1). The present contribution is intended to estimate the viability of this approach by comparison between experimental and calculated data of high-frequency refractive indices and dielectric constants for the organic solvents shown in Figure 1. It is of both systematic and theoretical interest to evaluate the capacity of the simplified method proposed here. In addition, we show distinctly different dielectric behaviors of two contrasting solvents (high- and lowdielectric materials, i.e., PC and EMC, respectively) comparing the total dipole moment autocorrelation functions.

εMD ¼ 1 þ

ð1Þ

The nuclear part ΔGnu from the rearrangement of the medium nuclear is modeled by the nonpolarizable MD, whereas the electronic part ΔGel from the pure electronic polarization response of the medium is obtained by the polarizable continuum model. When the interaction of a solute with solvent molecules in the MDEC model is considered, the solute partial charges (i.e., obtained by ab initio calculations in the gas phase or dielectric environment) should be scaled by 1/(ε∞)1/2, otherwise, the free energies obtained by the nonpolarizable MD, ΔGmd, should be corrected by a factor of 1/ε∞, i.e., ΔGnu = ΔGMD/ε∞. Then eq 1 is written as follows: ΔG ¼

1 ΔGMD þ ΔGel ε∞

4π ÆM 2 æ 3VkB T MD

ð2Þ

ð5Þ

Hence, if one has ε∞ of a material, then its dielectric constant can be calculated by the nonpolarizable MD simulation. 2.2. High Frequency Dielectric Constant Calculation. The molecular polarizability, α, is defined as the linear response of a molecular electronic distribution to the action of an external electric field. The experimental polarizability is measured as follows: αave ¼ Æαæ ¼ ðαxx þ αyy þ αzz Þ=3

2.1. MDEC Model. The MDEC model considers point charges moving in electronic continuum of known dielectric constant (εel = ε∞). In this case, all electrostatic interactions are scaled by the factor 1/ε∞, while the electronic polarization energy of the solvated charges is calculated explicitly using the electronic continuum model.12 Details of the MDEC model are given in Leontyev et al.’s publications.12,29,31 Here, we briefly restate only the main features of the model. In the MDEC model, the total solvation energy is written as follows:

ð4Þ

where ÆM2æ is the mean square fluctuation of the total dipole moment and V, kB, and T are volume, the Boltzmann constant and temperature, respectively. According to the MDEC scaling procedure, the actual dipole moment, μ, is related to the dipole moment obtained by the nonpolarizable MD, μMD, as μ = (ε∞)1/2μMD; therefore ÆM2æ = ε∞ÆM2MDæ, where ÆM2MDæ is the mean square fluctuation of the dipole moment obtained by the nonpolarizable MD, and we find,

2. THEORY AND SIMULATION METHODS

ΔG ¼ ΔGnu þ ΔGel

4π ÆM 2 æ 3VkB T

ð6Þ

While this work is directed at refractive indices in the liquid phase, the molecular polarizability is calculated for the free molecule. Assuming that the structure of the liquid phase is the same as that of the gas phase is, evidently, an approximation. The macroscopic optical susceptibility χ is related to the microscopic molecular polarizability as follows: χ¼

Nα0 1
4π=3 3 Nα0

ð7Þ

where N is the number density of liquid phase and α0 = α/4πε0 (ε0: permittivity of vacuum). Then we find an equation of the refractive index (high-frequency dielectric constant) as follows: n2 ¼ ε∞ ¼ 1 þ 4πχ

ð8Þ

Using eq 8, we recently reported refractive index calculations of liquid-forming organic solvents by DFT, which were in good correlation with experiments.34 2.3. Simulation Details: MD and DFT Calculation. We performed MD simulations with the most recent version of nonpolarizable COMPASS force field18 using FORCITE module in Material Studio 5.5 package.36 Each system includes 100 molecules in a cubic box with the periodic boundary condition. In order to obtain randomized initial coordinates, the systems were equilibrated for 1 ns with the NPT (fixed number of particles, pressure and temperature) ensemble at 298 K (in the case of EC, 313 K) and 0.1 MPa employing Nose’s thermostat37 and Andersen’s barostat.38 After the equilibration, the properties of 12572

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our interest were averaged for 20 ns, which is sufficient to obtain accurate ensemble averages. For example, the total dipole moment autocorrelation functions rapidly decay to 0 within ∼100 ps and the standard errors of dielectric constants during the simulations from block average method are at most 5% as shown later. The time step was set to 1 fs and the cutoff radius for the van der Waals interactions was 9 Å. In order to account for the charge interactions, the Ewald sum technique was employed. The polarizability calculations were performed by DFT calculations with the B3LYP method39 in combination with diffuse and polarized functions, resulting in a 6-311++G(d,p) basis set.40 Prior to evaluating α, we subject each molecule to full geometry optimization at the same level of theory without any symmetry constraints, followed by frequency analysis to ensure the thermodynamic stability of the converged structure. We point out that polarizabilities of organic molecules have been evaluated at a higher level of accuracy in the correlation effect than is done in this work, involving the MP2 methods.41,42 We thus performed additional calculations for polarizabilitiy of all molecules considered here using the MP2 procedure. Figure 2 shows the comparison between the MP2 and the B3LYP potential, where both sets of results were generated with the 6-311++G(d,p) basis

set, and full optimizations were carried out in both cases. The agreement between the two methods amounts to 99.9%. All geometry optimization and polarizability computations were carried out by the Gaussian09 program package.43

3. RESULTS AND DISCUSSION Table 1 lists densities, polarizabilities (Æαæ), macroscopic optical susceptibilities (χ) and ε1/2 ∞ s of electrolytic organic solvents, which were obtained from DFT, NPT MD, and their combined method. The polarizabilities of these species computed at the B3LYP/6-311++G(d,p) level of theory provide a useful estimation of this quantity because of their dependency on the basis set size and the exchange-correlation treatment.34 The experimental densities and refractive indices of the sodium D line at 589 nm (nD) are also shown for comparison with the simulations. The calculated densities are well reproduced with the experimental values within ∼3% except for FEC (∼ 6% overestimated). Although several thermodynamic properties of various fluoroalkanes were successfully described using the COMPASS force field,18 it needs to be improved for a better description of the fluorinated cyclic carbonates.

Figure 3. Correlation between experimental refractive indices of sodium D line at 598 nm (nD) and calculated high frequency dielectric constant (ε1/2 ∞ ).

Figure 2. Comparison between polarizabilities computed at B3LYP and MP2 procedure.

Table 1. Experimental and Computational Densities, Polarizabilities (Æαæ), Susceptibilities (χ), High Frequency Dielectric a Constants (ε1/2 ∞ ) and Refractive Indices (nD) Æαæ

density (298 K, g/cm3)

a

2
1

X
2

ε∞1/2

nD (293K)

species

(MD)

(exp.)

EC

1.274b

1.321b

43.1

7.26

1.383

1.416c

PC FEC

1.182 1.502

1.204 1.410

55.6 43.4

7.57 7.13

1.397 1.377

1.421 1.384

DMC

1.043

1.069

47.8

6.23

1.335

1.368

EMC

1.002

1.007

60.4

6.64

1.354

1.370

DEC

0.974

0.969

73.0

6.94

1.368

1.383

MP

0.900

0.915

55.8

6.47

1.346

1.376

EP

0.879

0.888

68.4

6.74

1.359

1.384

EA

0.890

0.902

56.4

6.47

1.346

1.372

IA DME

0.882 0.888

0.872 0.867

68.6 61.2

6.79 6.95

1.361 1.369

1.377 1.379

2

(C m J , DFT)

(  10 )

(DFT)

(exp.)

Experimental values are from Refs 46
50. b At 313 K. c At 323 K. 12573

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Figure 4. Dipole moment autrocorrelation function at (a) short times and (b) long times of EMC and PC.

Figure 6. Accumulated average of dielectric constant estimated by MDEC model with nonpolarizable MD of (a) low- and (b) high-dielectric solvents. (c) Comparison of dielectric constants between experiment, standard nonpolarizable MD (without a scaling by ε∞) and MDEC. Inset figure provides fine-scale resolution for low-dielectric solvents.

Figure 5. Instantaneous total dipole moment of (a) EMC and (b) PC.

Figure 3 shows the correlation between ε1/2 ∞ from DFT calculations and nD from the experimental data of refs 41
45. The values of ε1/2 ∞ are smaller than those of nD because of the refractive index at the high-frequency limit as we described in our previous work, where we covered a large number of solvents.34 This clearly demonstrates that our procedure generates a reliable prediction for the refractive indices, i.e., nDs, from ε1/2 ∞ s with a uniform scaling. Even though there are highly rigorous and accurate methods to predict the refractive indices over a wide variety of small organic molecules (Truchon et al.’s electronic polarization from internal continuum (EPIC) model, for example32,33), it is noticeable that our relatively simple and economic procedure is also able to calculate the refractive indices with moderate accuracy. Figure 4 shows the total dipole moment autocorrelation functions (DMAF), which are defined as DMAF = (M(t) 3 M(0))/(M2), where M(t) is the time dependent total dipole moment, of two representatives of the low- and high-dielectric 12574

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Table 2. Dielectric Constants of Organic Solvents Computed with Different Modelsa

a

ε∞

εMD

εMDEC

εexp

species

(DFT)

(MD, 298 K)

(MDEC, 298K)

(exp., 293 K)

ECb PC

1.913 1.952

33.8 23.0

64.7 44.9

89.8 66.1 ∼ 110

FEC

1.896

48.0

91.0

DMC

1.782

1.8

3.2

3.1

EMC

1.833

1.6

2.9

3

DEC

1.871

1.5

2.8

2.8

MP

1.812

3.3

6.0

5.4

EP

1.847

2.9

5.4

5.7

EA IA

1.812 1.852

3.3 3.0

6.0 5.6

6.0 6.3

DME

1.874

5.1

9.6

7.2

Experimental values are from Refs 46
50. b at 313 K.

solvents, namely, EMC (C4H8O3) and PC (C4H6O3). Even though the DMAFs of both EMC and PC exhibit the typical oscillatory behaviors at short times of a Debye dielectric44,45 (Figure 4a), the amplitude of oscillation of the PC strongly attenuates compared with that of the EMC. At long times, however, the DMAFs show the exponential-decay, which is a characteristic of the Debye dielectric (Figure 4b). The decay of the PC is slower than that of the EMC, i.e., the Debye relaxation times (τD) of the DMAFs of the EMC and PC, which are obtained by fitting them to the exponential-decay function (DMAF ≈ exp(
t/τD)), are ∼5 and ∼27 ps, respectively. Because the EMC and PC have similar molecular weights and partial charges in the COMPASS force field, these differences between the DMAF characteristics of the EMC and PC seem to be related to the dipole
dipole interactions. The magnitude of instantaneous total dipole moment of the PC is much noticeably larger than that of the EMC as compared in Figure 5 and the fluctuation of these total dipole moments is used to determine the dielectric constant in eq 5 In Figure 6(a),(b), the accumulated average values of dielectric constants of the low- and high-dielectric solvents are shown, respectively. Unlike density, energy, and other properties, it is necessary to use a sampling time longer than at least 10 ns in order to obtain a reliable value of the dielectric constant. The uncertainty from a simulation of 20 ns is estimated to be at most, in the case of FEC, ∼5% of the values. In Figure 6(c) and Table 2, the dielectric constants obtained by standard MD (without a scaling by ε∞), MDEC and experiment are compared. For the low-dielectric solvents, the MDEC simulations are in good agreement with the experiments within ∼10% whereas for the high-dielectric solvents, the MDEC results are underestimated by ∼30%. The large discrepancy between the experimental and simulated dielectric constants of the cyclic carbonates may be partly attributed to the force field. In order to find out reasons for the relatively large disagreement, comparison with other nonpolarizable force fields (CHARMM or AMBER, for example) is required, which is out of the scope in this study. Nevertheless, it is noticeable that the simulation results become comparable with the experimental values by introducing the MDEC concept.

4. CONCLUSIONS We report that a simplified computational approach describes the dielectric properties of electrolytic organic solvents. Leontyev et al.’s MDEC model combined with the DFT calculations was

employed to calculate the dielectric constants. For each of the considered species, this procedure is subdivided into B3LYP/6311++G(d,p) method of polarizability in gas phase equilibrium and classical MD simulation with a nonpolarizable force field that yields the respective number density and dipole moment fluctuations. From the calculated polarizabilities, the refractive indices at the high-frequency limit (ε∞) are predicted. The high frequency refractive indices have a role of scaling factors in the MDEC model to compute the dielectric constants. The calculated values yield highly satisfactory agreement with experimental data as statistical deviations less than ∼10% in the low-dielectric range whereas ∼30% in the high range. Consequently, the MDEC model predicts dielectric constants of organic solvents with relatively economic computational costs and significantly improves the accuracy with respect to the standard MD simulation. In the analysis of total dipole moment autocorrelation functions, the EMC shows the typical oscillatory decaying behavior at short times, followed by the exponential decay at long times. Meanwhile, the PC, which has a stronger permanent dipole moment, shows the much attenuated oscillation at short times and slower decay at long times.

’ AUTHOR INFORMATION Corresponding Author

*e-mail: [email protected].

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