A Group Contribution Method for the Correlation of Static Dielectric Constant of Ionic Liquids*

ZHOU Ying (周穎), LIN Zhen (林真), WU Kejun (吳可君), XU Guohua (徐國華) and HE Chaohong (何潮洪)**

State Key Laboratory of Chemical Engineering, Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China

A Group Contribution Method for the Correlation of Static Dielectric Constant of Ionic Liquids*

ZHOU Ying (周穎), LIN Zhen (林真), WU Kejun (吳可君), XU Guohua (徐國華) and HE Chaohong (何潮洪)**

State Key Laboratory of Chemical Engineering, Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China

Static dielectric constant is a key parameter to estimate the electro-viscous effect which plays important roles in the flow and convective heat transfer of fluids with ions in microfluidic devices such as micro reactors and heat exchangers. A group contribution method based on 27 groups is developed for the correlation of static dielectric constant of ionic liquids in this paper. The ionic liquids considered include imidazolium, pyridinium, pyrrolidinium, alkylammonium, alkylsulfonium, morpholinium and piperidinium cations and various anions. The data collected cover the temperature ranges of 278.15-343.15 K and static dielectric constant ranges of 9.4-85.6. The results of the method show a satisfactory agreement with the literature data with an average absolute relative deviation of 7.41%, which is generally of the same order of the experimental data accuracy. The method proposed in this paper provides a simple but reliable approach for the prediction of static dielectric constant of ionic liquids at different temperatures.

ionic liquid, static dielectric constant, group contribution method

1 INTRODUCTION

Ionic liquids are molten salts that are liquid at or near room temperature with specific organic cations and organic or inorganic anions [1]. They have received an increasing amount of interest due to their unique characteristics, i.e., negligible vapor pressure at normal temperature and pressure, wide liquidus ranges, tunable physicochemical properties and high solvating capacity for both organic and inorganic compounds. Ionic liquids have emerged as a promising class of solvents for a wide variety of applications including organic synthesis, extraction separation, biocatalysis, material preparation, electrochemistry as well as heat transfer and heat storage fluid, and so on [2-7]. Most efforts in ionic liquids have been focused on the investigation of their potential applications, the preparation of novel ionic liquids, and measurement of basic physical properties of ionic liquids, while the characterization of the structure-property relationship of ionic liquids that is equally important has been lagged behind. The experimental data available are mainly for melting point, density, viscosity, electrical conductivity and surface tension, while few efforts have been paid to the determination of polarity of ionic liquids, especially the static dielectric constant of ionic liquids.

Static dielectric constant represents the polarity and solvating capability of liquids, and it is an important input parameter in many solvation simulation processes [8]. Moreover, it is crucial in the study of flow and convective heat transfer of fluids with ions in micro devices, because when fluids with ions flow in micro devices, such as micro pumps, micro reactors and micro heat exchangers, the electro-viscous effect exists, which will greatly influence the flow characteristics and heat transfer of fluids [9-11]. The flows of incompressible Newtonian liquids incorporating electro-viscous effect through a circular tube are governed by the Poisson equation relating the electrical potential to the charge distribution and the Navier-Stokes equation with an electrical body force term, as described by Eqs. (1) and. (2) in dimensionless form [11]:

ε and ε0are the static dielectric constant of the fluid and vacuum permittivity, respectively. t, K, B, ρ, η, kb, e, z, R, T, n0, P, Re, V, U, n+and n?are the time, dimensionless electro-kinetic parameter, dimensionless physical property parameter, density, viscosity, the Boltzmann constant, elementary charge, valence of the ion, radius of the tube, temperature, bulk ionic number per unit volume, pressure, Reynolds number, velocity, electrical potential, the number of cations per unit volume and the number of anions per unit volume, respectively. The last term in the right side of Eq. (2) represents an electrical body force due to electro-viscouseffect. From the above equations, it is obvious that the static dielectric constant of fluid is a key parameter in the calculation of additional drag caused by this electro-viscous effect. Since ionic liquids are entirely composed of ions, the electro-viscous effect may exist when ionic liquid flows in micro channels, and the above three equations may still hold for the flow of ionic liquid in micro channels. Therefore, it is a must to get the static dielectric constant of ionic liquid to study the flow characteristics and convective heat transfer of ionic liquid in micro channels, which will be important for the design and optimization of the above mentioned micro chemical equipment. Despite of its significant importance in chemical industry, the static dielectric constant of many ionic liquids is not known to date. Conventional capacitance method fails for static dielectric constant measurement of ionic liquids due to the short-circuited sample cell by the electrical conductance of ionic liquids. The main method for ionic liquid static dielectric constant measurement is microwave dielectric relaxation spectroscopy (DRS) [12], which demands expensive device and sophisticated data processing. Furthermore, there are approximately 1018combinations of ions that could lead to useful ionic liquids [13], and it is impossible to measure the static dielectric constant of all ionic liquids experimentally. Development of simple and rapid methods for the estimation of the static dielectric constant of ionic liquids is important, which will be helpful for the design of task-specific ionic liquids and speed up their industrial applications.

Several studies have been reported for the estimation of the static dielectric constant of ionic liquids. By combining volume-based thermodynamics (VBT) and quantum chemical calculations, Krossing et al. [14] have proposed a method for the estimation of static dielectric constant (ε) of ionic liquids. The prediction mainly depends on the Gibbs solvation energy and the molecular volume of the ions. This method requires the knowledge of quantum chemical calculations and it assumes that Gibbs solvation energy is independent of temperature, which is not true for most liquids, and reduces the accuracy of the method. Ludwig et al. [15] have attempted to predict ε by Fourier transform infrared spectroscopy (FTIR) based on the association of water molecules in ionic liquids and makes use of shifts in the stretching frequencies of water molecules. Ludwig’s method is limited to the ionic liquids that do not hydrolyze upon contact with water and also limited by the assumption that ionic liquid solutions are homogeneous fluid. Studies have indicated that many ionic liquids are nano-structured materials and will segregate into polar and nonpolar regions with the addition of water [16-18]. Singh et al. [19] predicted ε by means of the ratio of internal pressure and cohesive energy density. This method shows reasonable accuracy for ionic liquids with anions Cl?, [BF4]?, [PF6]?, [NTf2]?and [CF3SO3]?, while ionic liquids having anions alkylsulfate tend to have a large deviation up to 60%. Furthermore, some physical properties which are not easy to obtain, such as heat of vaporization, are required for the determination of static dielectric constant in Singh’s method, which restricts its applications.

In this paper, a simple correlation method for the correlation of the static dielectric constant of ionic liquids with a group contribution method based on 27 groups is proposed. The method developed in this work overcomes the limitations of the previously existing methods and it has a better correlation accuracy and wider application ranges for the prediction of the static dielectric constant of ionic liquids.

2 MODEL DEVELOPMENT

Static dielectric constant is defined as the zero frequency limit of the complex permittivity in the absence of ionic conductivity, and it reflects the contributions from molecular polarization (ionic polarization and electronic polarization) and the orientation polarization caused by the permanent dipole moment of the molecule [20]. A general relationship between the static dielectric constant ε and the microscopic properties, such as molecular polarizability α, is given by B?ttcher theory [21]:

NAis Avogadro’s number, μ is permanent dipole moment, M is the molecular mass and αsis the sum of all forms of molecular polarizability which represents the deformability of the molecule and ions in an external electrical field, and it is made up of electronic polarizability and ionic polarizability. Further, f (a) is the static reaction field factor that depends on the cavity radius, a, and it is expressed as f (a)=2(ε?1)/[a3(2ε+1)]. The first and second term in the parenthesis in the right side of Eq. (4) represent the contribution of molecular polarization and the orientation polarization caused by the permanent dipole moment of the molecule to static dielectric constant, respectively.

Ionic liquids are highly structured and their microscopic behavior is dominated by strong, long-ranged inter-ion Coulomb interactions [22]. The study of Izgorodina’s group [20] has indicated that ionic polarization contributes significantly to the static dielectric constant of ionic liquids, and they also pointed out that the static dielectric constant of ionic liquids may not be mainly influenced by the size of the permanent dipole moment, indicating that the contribution of orientation polarization caused by the permanent dipole moment of the ions to static dielectric constant of ionic liquids may not be significant. Therefore, for ionic liquids, it might be reasonable for us to assume that the second term in the parenthesis in the right side of Eq. (4), which reflects the contribution of orientation polarization caused by the permanent dipole moment to static dielectric constant, can be neglected. Using the Onsager approximation [21], 4πNAρa(bǔ)3/(3M)=1, for the cavity radius and neglecting the contribution oforientation polarization, Eq. (4) can be simplified to Eq. (5), which has the same form as the Clausius-Mossotti equation [21]:

In conventional organic liquids, electronic polarizability is the only form of molecular polarizability, while both electronic polarizability and ionic polarizability predominate in ionic compounds, such as sodium chloride. Ionic liquids are entirely composed of ions, and the molecular polarizability is made up of electronic polarizability and ionic polarizability. At high (optical) frequencies, the electronic polarization is the only polarization that survives [20], hence, it is the only contribution to the dielectric constant at these frequencies. At optical frequencies, Eq. (5) becomes the Lorentz-Lorentz equation, in which the dielectric constant ε is equal to the square of the molar refractivity (n2), and αsequals to electronic polarizability αe. At zero frequency, both electronic polarization and ionic polarization survive [20], therefore the Lorentz-Lorentz equation is not applicable for the calculation of the zero frequency static dielectric constant of ionic liquids due to the contribution of ionic polarization.

Eq. (5) can be rewritten as Eq. (6):

where A=4πNAα/3, a parameter that is related to the structure of the molecule and reflects the contributions of electronic polarization and ionic polarization to the static dielectric constant of ionic liquids. We tried to obtain the value of A by a group contribution method according to Eq. (7):

The group division principle is identical to that of Valderrama’s [23, 24] article for the prediction of critical properties of ionic liquids. The values of the 27 groups used in this paper are summarized in Table 1. Different from the big group division method adopted by Gardas’s group [25, 26] for the prediction of thermophysical and transport properties of ionic liquids, both the cation and anion are divided into smaller groups in this paper, which enables our model to be applicable to more kinds of ionic liquids.

The following ionic liquids were considered in this paper: imidazolium, pyridinium, pyrrolidinium, alkylammonium, alkylsulfonium, morpholinium and piperidinium cations with tetrafluoroborate ([BF4]?), hexafluorophosphate ([PF6]?), bis(trifluoromethanesulfonyl) amide ([NTf2]?), trifluoromethanesulfonate ([CF3SO3]?), ethylsulfate ([EtSO4]?), butylsulfate ([BuSO4]?), dimethyl phosphate ([Me2PO4]?), diethyl phosphate ([Et2PO4]?), dicyanamide ([DCA]?), nitrate ([NO3]?), formate ([HCOO]?), hydrogensulfate ([HSO4]?), acetate ([Ac]?), lactate ([Lac]?) as well as tetra(hexafluoroisopropoxy)aluminate ([Al(hfip)4]?) anions. 128 data points for 58 ionic liquids available in literature [8, 12, 27-37] covering temperature ranges of 278.15-343.15 K were used in the correlation. The optimum values for group contribution parameters Aiin Eq. (7) are obtained by minimizing the objective function f in Eq. (8).

The relative deviation (rRD) and average absolute relative deviation (rAARD) are defined as:

3 RESULTS AND DISCUSSION

All the static dielectric constant data were taken from the published literature [8, 12, 27-37]. Most density data covered in this paper were taken from experimental values in the published literature [34, 38-63], while for the substances of which experimental density data were not available, the density data used were calculated according to the group contribution method of Ye’s group [64] and Krossing’s group [65].

Huang’s study [36] has indicated that static dielectric constant is rather insensitive to the water content, and the experimental static dielectric constant data currently available is quite limited. The rejection of some experimental data is limited in the development of the proposed correlation method. Therefore, all experimental static dielectric constant data points available in literature is employed in this study. Taking all the available data points into Eqs. (6) and (7), the values of parameters Aiare obtained, as shown in Table 1. During the correlation, some data, for example the static dielectric constants of [C2mim]BF4[8, 35] and [C4mim]BF4[8, 31, 35] are excluded since the data given in these literature are later revised to new values [12, 27] by the authors with the adoption of the symmetrical Cole-Cole (C-C) model.

The uncertainty in the experimental determination of the static dielectric constant of ionic liquid is typically of the order of 0.3-1.5 [8, 12, 27?34], which means the experimental accuracy of ε is about 5%. By the correlation of Eq. (6) using the objective function described in Eq. (8), an average absolute relative deviation of 7.41% is obtained as shown in Table A1 in Appendix, which is generally of the same order of the experimental data accuracy.

Table 1 Group contribution parameters Aiin Eq. (7)

Figure 1 Linear relationship between experimental and correlated static dielectric constants with the method developed in this paper

For 128 data points of 58 ionic liquids available in literature, the maximum relative deviation of 37.47% is observed for [C2mim][Al(hfip)4] at 343.15 K. Among these data points, 14.9% of the estimated static dielectric constants are within relative deviations of 10%-20%, 20.3% within 5%-10%, 53.9% within 0-5%, and only 10.9% of the estimated static dielectric constants have relative deviations larger than 20%. Moreover, about 74.2% of the data present relative deviations smaller than 10%. As most of the experimental static dielectric constants lie in the range of 9-20, Fig. 1 (a) displaying all the data points employed in this study and Fig. 1 (b) presenting the data points with experimental static dielectric constants less than 20 are used to make a comparison between the correlated static dielectric constants with the method developed by us and experimental values. As observed in Fig. 1 and Table A1 in Appendix, a satisfactory agreement is observed between the correlated values and the experimental static dielectric constant data for both aprotic and protic ionic liquids based on imidazolium, pyridinium, pyrrolidinium, alkylammonium, alkylsulfonium, morpholinium and piperidinium cations with various anions.

It is noted that Ludwig’s group [15] has only predicted the static dielectric constant of two ionic liquids and the comparison between their results and our method is skipped. Krossing et al. [14] have correlated the static dielectric constant of 40 ionic liquids, and based on the Gibbs solvation energy and the molecular volume of the ions given as well as the regression parameters, the static dielectric constant of another 6 ionic liquids can be predicted. An average absolute relative deviation of 15.66% and maximum deviation higher than 60% were obtained for all the data points which include both of the correlation and predicted values. Moreover, 30% of the data points present deviations higher than 20% in Krossing et al.’s method. Based on a theoretical model, Singh et al. [19] have predicted the static dielectric constants of 17 aprotic ionic liquids at 298.15 K with the maximum deviation of 61.65% and the average absolute relative deviation observed is 22.23%, which is not satisfactory for engineering practicality. Compared to the above two methods, the method proposed in this paper has a bettercorrelation accuracy with an average absolute relative deviation of 7.41% and maximum deviation of 37.47%, which is satisfactory for engineering practicality. Therefore the method developed in this paper will be quite useful in chemical process design. The only physical property needed in this method is density, with the availability of density data, the static dielectric constant of ionic liquids can be calculated with ease. The density data of many ionic liquids are available. Ye et al. [64], Krossing et al. [65] and several other groups [66-68] have developed group contribution methods for the prediction of density of ionic liquids at different temperatures with excellent accuracy, which facilitates the prediction of static dielectric constant of ionic liquids by means of the method proposed in this paper. Moreover, the adoption of smaller group division makes this method have wider ranges for application since it is applicable to both aprotic and protic ionic liquids with Aρ/M being less or equal to 0.97 at different temperatures.

4 CONCLUSIONS

A group contribution method based on 27 groups was successfully applied for the correlation of the static dielectric constant of 58 aprotic and protic ionic liquids with an average absolute relative deviation of 7.41%, which is generally of the same order of the experimental data accuracy, and will be quite useful in chemical process design. The method shows a satisfactory agreement between the calculated and experimental static dielectric constants for imidazolium, pyrrolidinium, pyridinium, ammonium, morpholinium and piperidinium based ionic liquids with various anions. Since the only physical property needed in this method is density, with the availability of density data, the method proposed in this paper provides a simple but reliable approach to predict the static dielectric constant of ionic liquids with Aρ/M being less or equal to 0.97 at different temperatures.

NOMENCLATURE

Aicontribution value of group i, cm3·mol?1

a cavity radius, m

e elementary charge, C

k total number of different groups in a molecule

kbBoltzmann constant, J·K?1

M molecular mass, g·mol?1

NAAvogadro’s number, mol?1

Npnumber of data points

n0bulk ion concentration, m?3

ninumber of ith group

n+number of cations per unit volume, m?3

n?number of anions per unit volume, m?3

P pressure, Pa

R radius of the tube, m

Re Reynolds number

T temperature, K

U total electrical potential

V velocity, m·s?1

z valence of the ion

α polarizability, cm3·mol?1

αsthe sum of forms of polarizability, cm3·mol?1

ε static dielectric constant of the fluid

ε0vacuum permittivity, C·V?1·m?1

η viscosity of the fluid, Pa·s

μ permanent dipole moment, 3.33564×10?30C·m

ρ density of the fluid, g·cm?3

NOMENCLATURE

cal calculated property

exp experimental property

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APPENDIX

Table A1 that contains the correlated and experimental static dielectric constants of 58 ionic liquids at different temperatures, the correlation value of A and molecular mass for each ionic liquid, density data employed in this study and the correlation results comparison with Krossing’s method and Singh’s method.

Table A1 Comparison of correlated and experimental static dielectric constants①

Table A1 (Continued)

Table A1 (Continued)

Table A1 (Continued)

10.1016/S1004-9541(14)60009-4

2012-08-16, accepted 2012-11-06.

* Supported by the National Natural Science Foundation of China (21176206) and the Project of Zhejiang Key Scientific and Technological Innovation Team (2010R50017).

** To whom correspondence should be addressed. E-mail: chhezju@zju.edu.cn