Estimation of the Viscosity of Ionic Liquids Containing Binary Mixtures

Article pubs.acs.org/jced

Estimation of the Viscosity of Ionic Liquids Containing Binary Mixtures Based on the Eyring’s Theory and a Modified Gibbs Energy Model Saeid Atashrouz, Mohammad Zarghampour, Shiva Abdolrahimi, Gholamreza Pazuki,* and Bahram Nasernejad Department of Chemical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, 15875, Iran S Supporting Information *

ABSTRACT: Eyring’s absolute rate theory was applied for evaluation of the viscosity of ionic liquids (ILs) containing binary mixtures. Considering the mathematical simplicity of the two-suffix-margules model, the Gibbs energy model was further modified. Furthermore, for viscosity evaluation, the proposed Gibbs energy model was coupled with Eyring’s theory. To validate the accuracy of the proposed model, a large set of data containing the binary mixtures of 122 ILs with a total number of 5512 experimental data points was collected from the literature. Moreover, the average absolute relative deviation (AARD %) was obtained as 2.07 %. Also, the capability of the Eyring−MTSM model was tested for the prediction of viscosity for binary and ternary systems. Additionally, comparison of the proposed model with the Eyring−NRTL model indicated a higher accuracy for our model. Finally, the Eyring−UNIFAC model was also checked, and it was found that this model is not accurate enough in its present form.

1. INTRODUCTION In recent years, ionic liquids (ILs) have attracted the attention of scientific communities owing to their unique and highly tunable properties1 such as nonvolatility, ease of recycle, high solvating capacity for polar and nonpolar compounds, and high thermal, chemical, and electrochemical stability.2 Various combinations of cations and anions result in the creation of tunable ILs.3 Consequently, they have been widely employed in various industrial sectors such as chemical synthesis and catalytic reactions,2 biotechnology, nanofluids,4 extraction, and separation technology.5−7 Insight into the physicochemical properties of IL mixtures is essential for their industrial applications. Among those, the estimation of viscosity and its temperature and composition dependence is crucial for equipment modeling and engineering design.8,9 Moreover, their applications have been restricted because of their high viscosities.10 Consequently, many experimental studies have been dedicated to the estimation of the viscosity of IL + cosolvent mixtures.3,11−13 To model viscosity of the systems containing ILs, empirical equations14 and semitheoretical8,9 and neural network2 models have been proposed. In this regard, Seddon et al.14 introduced an empirical equation with a simple mathematical form. They have considered the effect of the cosolvent’s composition on the viscosity. Recently, Wang et al.9 checked the accuracy of the Seddon’s equation in estimation of the viscosities the binary mixtures of 35 ILs. The results demonstrated that there is a high deviation for the Seddon’s model. Moreover, Fang and He15 developed a new © XXXX American Chemical Society

model on the basis of Eyring’s theory and the modified Flory− Huggins model for correlating the viscosities of 527 binary systems with a total of 63 ILs containing mixtures. This model has manifested fair accuracy for those noncontaining IL mixtures. In another study, Wang et al.9 coupled the Eyring theory with the UNIQUAC model for estimation of the viscosity of 35 IL mixtures and consequently reported an agreeable AARD % value of 2.61 %. However, the main drawback of this model is the addition of two structural parameters in the combinatorial term of the UNIQUAC model, namely the surface and volume parameters, which are scarce and hard to obtain for some ILs.8 Tariq et al.16 have evaluated the applicability of four commonly used viscosity mixing rules for 100 binary liquid mixtures containing ionic liquids. Although the accuracy of these models was not adequate in some cases, their advantage was that they had only one adjustable parameter. In addition to these studies, artificial intelligence systems, such as artificial neural networks (ANN), have been applied for the estimation of viscosity of pure ILs and their mixtures. Fatehi et al.2 developed an ANN system based on the structural groups of the ILs. For this purpose, they collected a large data set of 1996 experimental points which was then divided into two parts: 88 % of data points were used for training and 12 % for testing. Despite Received: June 24, 2014 Accepted: October 23, 2014

A

dx.doi.org/10.1021/je500572t | J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

the fact that the reported AARD % of 0.6 % is agreeable, the complexity of the ANN model makes it tedious for experimental correlations.17 On the other hand, the ANN has no physical basis and is a mere mathematical model.18 The aim of the present study is application of Eyring’s theory and an excess Gibbs energy model with a simple mathematical form. The two-suffix-margules has the simplest relation among excess Gibbs energy models. In this regard, we have modified the two-suffix-margules model in order to obtain a modified excess Gibbs energy model with a simple accurate form. The modified model was coupled with the Eyring’s theory and, consequently, energy parameters were obtained based on the 5512 experimental data points for 122 ILs containing binary systems. The accuracy of the modified model was compared with the Eyring−NRTL model. Also, the proposed model was applied for prediction of the viscosity of binary and ternary systems. Finally, the predictability of Eyring−UNIFAC model for the viscosity calculation of the binary system of the IL was checked.

ln(ηmix ) = x1 ln(η1) + x 2 ln(η2) +

GE = Ax1x 2

Eyring’s theory. According to this theory, Viscosity of a liquid mixture can be obtained from the following equation:19,20

gE = α12X12X 21 RT

(1)

where η is dynamic viscosity, V is the molar volume, and G * is the excess free energy of activation for the flow process. Moreover, “ideal” and “mix” subscripts refer to the real mixture and ideal solution properties, respectively. According to the study of Fang et al.,15 the volume change of mixing can be neglected. So for simplification of eq 1, the following assumptions have been considered: E,

Videal ≅1 Vmix

(5)

where A is the characteristic of components 1 and 2, an empirical constant with units of energy, which is dependent on the temperature but not on the composition. This model has a relatively suitable performance for ideal liquid mixtures.21 However, the complicated structure of ILs and their subsequent capability to form hydrogen-bond interactions with solvents9 makes the two-suffix-margules model inappropriate for the description of nonideality of IL + cosolvent mixtures. Accordingly, modification of this model to obtain a new simple model with higher accuracy is essential. The two-liquid theory considers the IL + cosolvent system consisting of two different cells. As depicted in Figure 1, one cell contains a molecule of IL at its center and the other contains a molecule of cosolvent as its center, and they are closely surrounded by other molecules.22 Thus, the model can be modified as the following form:

Figure 1. Schematic of the two-liquid theory for a binary mixture. Hypothetical liquid (1) has a molecule 1 at the center. Hypothetical liquid (2) has a molecule 2 at its center.

⎛ GE, * ⎞ (ηV )ideal exp⎜ ⎟ Vmix ⎝ RT ⎠

(4)

Eq 4 has two parts. The first two terms in the right-hand side are considered to be the average logarithmic viscosity, which is in fact the “ideal term”, whereas the third term represents the “non-ideal term” of a binary system. The appearance of this term in the model can be attributed to the components’ interactions and the differences in the molecular size and shape. Application of eq 4 requires a proper term to be substituted for GE,*. As a matter of fact, Eyring’s theory relates transport properties to thermodynamic models. In this regard, the excess free Gibbs energy of activation for the flow process (GE,*) can be expressed by a thermodynamic excess free Gibbs energy model.8,9,15 In the next section, we elucidate the procedure of developing a modified excess Gibbs energy model for substitution in eq 4. 2.2. Development of a Modified Gibbs Energy Model. Our main objective is the development a modified model with a simple mathematical form. The two-suffix-margules model is well-known and a simple excess Gibbs energy model. Consequently, we considered this model with some modifications as the basis of the study. The two-suffix-margules model has the following relation for a binary system.21,22

2. THEORETICAL BACKGROUND 2.1. Viscosity Calculation Based on the Eyring Theory. Viscous flow can be regarded as an activated process based on

ηmix =

GE, * RT

(6)

where α12 is an empirical constant similar to constant A in eq 5 and Xij is the local mole fraction of component i around j. The relation between local mole fractions and bulk mole fractions can be written as23 X12 x = 1 G12 X 22 x2

(7)

and (2)

X 21 x = 2 G21 X11 x1

Also for the term of ideal mixing, the following form was considered: ηideal = exp(∑ xi ln(ηi)) i

(8)

Therefore, the above equations can be rearranged to get the local mole fractions of the molecules in the cells:

(3)

X12 =

where xi is the mole fraction of compound i. By replacing eqs 2 and 3 into eq 1, the following relation can be written for estimation of the viscosity for binary mixtures:

x1G12 x1G12 + x 2

(9)

and B



Article

Table 1. List of selected binary mixtures of ILs and temperature ranges no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

component 1: ionic liquid

component 2

1-butyl- 3-methylimidazolium chloride 1-butyl-1-methylpiperidinium thiocyanate 1-butyl-1-methylpyrrolidinium thiocyanate 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)-imide 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)-imide 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)-imide 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)-imide 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)-imide 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)-imide 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)-imide 1-butyl-3-methylimidazolium dimethylphosphate 1-butyl-3-methylimidazolium dimethylphosphate 1-butyl-3-methylimidazolium dimethylphosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium methylsulfate 1-butyl-3-methylimidazolium methylsulfate 1-butyl-3-methylimidazolium nitrate 1-butyl-3-methylimidazolium nitrate 1-butyl-3-methylimidazolium nitrate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolium thiocyanate 1-butyl-3-methylimidazolum alanine acid salt 1-butyl-3-methylimidazolum alanine acid salt 1-butyl-4-methylpyridinium thiocyanate 1-ethyl-3-methylimidazolium acetate 1-ethyl-3-methylimidazolium acetate 1-ethyl-3-methylimidazolium dicyanamide 1-ethyl-3-methylimidazolium dicyanamide 1-ethyl-3-methylimidazolium dimethylphosphate 1-ethyl-3-methylimidazolium dimethylphosphate 1-ethyl-3-methylimidazolium dimethylphosphate 1-ethyl-3-methylimidazolium ethyl Sulfate C

dimethyl sulfoxide water water TFEa acetonitrile dichloromethane methanol 1-butanol t-butanol water methanol ethanol water tetrahydrofuran methanol dimethyl sulfoxide acetonitrile dimethylethanolamine monoethanolamine TFE methyl methacrylate cyclopentanone ethyl acetate pentanone ethanol water ethanol butanol propanol dimethyl sulfoxide water dimethylformamide 2-butanone dichloromethane ethanol acetonitrile dichloromethane acetone ethyl formate methyl acetate methyl Formate TFE 1-octanol 1-nonanol 1-decanol water 1-hexanol 1-pentanol 1-butanol 1-heptanol methanol benzyl alcohol water water ethanol water ethanol methanol ethanol water methanol

temp range (K)

ref

298.15 298.15 298.15 278.15 298 298 298 298 298 298 293.15 293.15 293.15 298.15 298.15 298.15 298.15 288.15 288.15 278.15 283.15 298.15 298.15 298.15 298.15 298.15 283.15 283.15 283.15 293.15 303.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 278.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 293.15 293.15 293.15 298.15

27 28 28 29 30 30 30 30 30 30 31 31 31 11 11 11 11 32 32 29 33 34 34 34 35 35 36 36 36 13 37 38 38 38 38 38 38 39 39 39 39 29 40 40 40 28 41 41 41 40 42 42 28 43 43 43 43 31 31 31 44

to to to to

328.15 348.15 348.15 333.15

to 333.15 to 333.15 to 333.15

to to to to

323.15 323.15 333.15 353.15

to to to to to to to

328.15 328.15 333.15 333.15 333.15 353.15 353.15

to to to to to to to to to to to to to to to to to to to to

333.15 348.15 348.15 348.15 348.15 348.15 348.15 348.15 348.15 313.15 313.15 348.15 343.15 343.15 343.15 343.15 333.15 333.15 333.15 328.15



Article

Table 1. continued no.

component 1: ionic liquid

component 2

62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

1-ethyl-3-methylimidazolium ethyl Sulfate 1-ethyl-3-methylimidazolium ethyl Sulfate 1-ethyl-3-methylimidazolium ethyl Sulfate 1-ethyl-3-methylpyridinium ethylsulfate 1-ethyl-3-methylpyridinium ethylsulfate 1-ethylpyridinium ethylsulfate 1-ethylpyridinium ethylsulfate 1-hexyl-3-methylimidazolium tetrafluoroborate 1-hexyl-3-methylimidazolium tetrafluoroborate 1-hexyl-3-methylimidazolium tetrafluoroborate 1-hexyl-3-methylimidazolium tetrafluoroborate 1-methyl-3-butylimidazolium perchlorate 1-methyl-3-octylimidazolium chloride 1-methyl-3-octylimidazolium chloride 1-methyl-3-octylimidazolium chloride 1-methyl-3-octylimidazolium tetrafluoroborate 1-methyl-3-octylimidazolium tetrafluoroborate 1-methyl-3-octylimidazolium tetrafluoroborate 1-methyl-3-octylimidazolium tetrafluoroborate 1-methyl-3-octylimidazolium tetrafluoroborate 1-methyl-3-pentylimidazolium hexafluorophosphate 1-methyl-3-pentylimidazolium tetrafluoroborate 1-methylimidazolium acetate 1-methylimidazolium acetate 1-methylimidazolium acetate 1-methylimidazolium acetate 1-n-butyl-3-methylimidazolium hexafluorophospate 1-n-butyl-3-methylimidazolium hexafluorophospate 1-n-butyl-3-methylimidazolium hexafluorophospate 1-n-butyl-3-methylimidazolium tetrafluoroborate 1-n-hexyl-3-methylimidazolium tetrafluoroborate 1-n-hexyl-3-methylimidazolium tetrafluoroborate 1-n-hexyl-3-methylimidazolium tetrafluoroborate 1-n-hexyl-3-methylimidazolium tetrafluoroborate 1-n-octyl-3-methylimidazolium tetrafluoroborate 1-n-octyl-3-methylimidazolium tetrafluoroborate 1-n-octyl-3-methylimidazolium tetrafluoroborate 1-nutyl-3-methylimidazolium tetrafluoroborate 1-octyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl)-imide 1-octyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl)-imide 2-ethoxy-1-ethyl-1,1-dimethyl-2-oxoethanaminium ethyl sulfate 4-methyl-N-butylpyridinium tetrafluoroborate bis(2-hydroxyethyl)ammonium propionate bis(2-hydroxyethyl)ammonium propionate bis(2-hydroxyethyl)ammonium propionate diethyl ammonium hydrogen sulfate n-butylammonium acetate n-butylammonium acetate n-butylammonium acetate n-butylammonium acetate n-butylammonium nitrate n-butylammonium nitrate n-butylammonium nitrate n-butylammonium nitrate n-octylisoquinolinium bis(trifluoro methylsulfonyl)-imide n-octylisoquinolinium bis(trifluoro methylsulfonyl)-imide n-octylisoquinolinium bis(trifluoro methylsulfonyl)-imide pyrrolidinium nitrate tetradecyltrihexylphosphonium bis(2,4,4-trimethylpentyl)phosphinate tris(2-hydroxyethyl) methylammonium methylsulfate tris(2-hydroxyethyl) methylammonium methylsulfate

1-propanol 2-propanol ethanol water ethanol ethanol propanol butanone butylamine ethyl acetate tetrahydrofuran ethanol methanol ethanol 1-propanol ethanol 1-propanol 2-propanol ethanol methanol PEG-methyl etherb polyethylene glycol methanol ethanol 1-propanol 1-butanol acetone butanone [BMIM][BF4]c [BMIM][MEOSO3]d butanone ethyl acetate [EMIM][BF4]e [BMIM][BF4] butanone ethyl acetate methyl acetate ethylene glycol ethyl acetate ethanol poly(ethylene glycol) methanol ethanol 1-propanol methanol dimethyl sulfoxide methanol ethanol n-propanol n-butanol methanol ethanol 1-propanol 1-butanol 1-butanol 1-hexanol 2-phenylethanol propylene carbonate dodecane ethanol methanol

D

temp range (K)

ref

298.15 298.15 298.15 298.15 298.15 298.15 298.15 298 298 298 298 283.15 298.15 298.15 298.15 283.15 298.15 298.15 298.15 298.15 293.15 293.15 293.15 293.15 293.15 293.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 293.15 298.15 298.15 293.15 298.15 293.15 293.15 293.15 298.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 298.15 298.15 298.15 283.15 298.15 298.15 298.15

44 44 45 3 3 46 46 47 47 47 47 12 48 48 48 12 49 49 49 49 50 51 52 52 52 52 53 53 54 54 53 53 54 54 53 53 53 13 55 55 56 57 58 58 58 52 59 31 31 31 60 60 60 60 61 61 61 62 63 64 65

to 328.15 to 328.15 to to to to

328.15 328.15 328.15 328.15

to to to to to

343.15 328.15 328.15 328.15 343.15

to to to to to to

353.15 353.15 313.15 313.15 313.15 313.15

to 308.15 to 308.15

to 308.15 to 308.15

to 353.15

to to to to to to to to to to to to to to to to to to to

353.15 323.15 323.15 323.15 323.15 328.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 338.15 338.15 338.15 353.15 318.15



Article

Table 1. continued a Tetrafluoroethylene. bPoly(ethylene glycol) methyl ether. c1-n-Butyl-3-methylimidazolium tetrafluoroborate. methylsulfate. e1-Ethyl-3-methylimidazolium tetrafluoroborate.

X 21 =

x 2G21 x 2G21 + x1

(10)

τ12 = g12 − g22

τ21 = g21 − g11

⎛ ⎛ V ⎞⎞ V ln(γiC) = 1 − Vi + ln Vi − 5qi⎜⎜1 − i + ln⎜ i ⎟⎟⎟ Fi ⎝ Fi ⎠⎠ ⎝

(11)

Fi = (12)

qi =

rx i i ∑j rjxj

(21)

∑ vk(i)Q kri = ∑ vk(i)R k

(22)

k

v(i) k

In eq 22, corresponds to the number of groups of type k in a molecule. Qk and Rk, given in Table S1 in the Supporting Information, are determined using van der Waals group surface area and volumes: Qk =

Ak 2.5·10

9

Rk =

;

Vk 15.17

(23)

Moreover, the last term of eq 19 is the residual part of the activity coefficient of the UNIFAC model written as

(14)

ln(γi R ) =

∑ vki[ln Γk − ln Γik]

(24)

k

(15)

where Γk is the group residual activity coefficient and Γik is the residual activity coefficient of group k in a reference solution containing only molecules of type i. In eq 24 ln Γk has the following form:

2.3. Nonrandom Two-Liquid (NRTL) Model. Substitution of NRTL excess Gibbs energy in the Eyring model leads to an equation of the following form:24

⎡ ln Γk = Q k ⎢1 − ln(∑ θmψmk) − ⎢⎣ m

ln(ηmix ) = x1 ln(η1) + x 2 ln(η2) ⎛ τ G τ12G12 ⎞ + x1x 2⎜ 21 21 + ⎟ x 2 + x1G12 ⎠ ⎝ x1 + x 2G21

Vi =

;

k

The model can be simplified as ln(ηmix ) = x1 ln(η1) + x 2 ln(η2) + α12X12X 21

∑j qjxj

(13)

Eq 13 is known as the “modified two suffix margules” (MTSM) model with three adjustable parameters namely α12, τ12, and τ21. On the basis of eqs 4 and 13, the Eyring−MTSM takes the following form: ln(ηmix ) = x1ln(η1) + x 2 ln(η2) x1x 2G12G21 + α12 (x1G12 + x 2)(x 2G21 + x1)

qixi

⎛ θmψ ⎞⎤ km ⎟⎟⎥ ⎝ ∑n θnψnm ⎠⎥⎦

∑ ⎜⎜ m

(16)

where τ12 = (Δg12/(RT)) and τ21 = (Δg21/(RT)). Also G12 and G21 have the following definitions: ln(G12) = −α12τ12

(17)

ln(G21) = −α12τ21

(18)

(20)

where qi and ri are molecular surface area and molecular van der Waals volume, respectively. These parameters can be calculated by using group area parameters and group volume, Qk and Rk:

where gij is the potential energy between components i and j. Applying eqs 6 to 12, the modified model takes the following form: gE x1x 2G12G21 = α12 RT (x1G12 + x 2)(x 2G21 + x1)

(19)

The first term of the above equation refers to the combinatorial part written as

and ⎛ τ ⎞ G21 = exp⎜ − 21 ⎟ , ⎝ RT ⎠

1-Butyl-3-methylimidazolium

ln(γi) = ln(γiC) + ln(γi R )

where ⎛ τ ⎞ G12 = exp⎜ − 12 ⎟ , ⎝ RT ⎠

d

θm =

Q mX m ∑n Q nX n

;

Xm =

(25)

∑i vm(i)xi ∑i ∑k vk(i)xi

(26)

where Xm is the fraction of group m in the mixture. Also ψnm, the group interaction parameter, is expressed by ⎡ ⎛ a ⎞⎤ ψnm = exp⎢ −⎜ nm ⎟⎥ ⎣ ⎝ T ⎠⎦

Moreover, Δg12 = g12 − g22 and Δg21 = g21 − g11 are the interaction parameters between (1−2) and (2−1) pairs, respectively. 2.4. Universal Functional Activity Coefficient (UNIFAC) Model. The UNIFAC model is a group contribution method which is widely used for prediction of phase equilibria in binary and multicomponent systems. Recently, the UNIFAC model has been applied for prediction of phase equilibria in IL containing systems. In this regard, Lei et al. estimated group parameters of the UNIFAC model for IL containing systems.25 Furthermore, Hector and Gmehiling developed the modified UNIFAC model for prediction of excess enthalpies in IL-based systems.26 The activity coefficient for the UNIFAC model has the following form:

(27)

where anm is defined as the interaction between group n and m. There are two interaction parameters for every group−group interaction: anm ≠ amn. Parameters anm and amn are obtained from the experimental activity coefficient of solute at infinite dilution.25 With minimization of the following objective function, Lei et al.25 have obtained interaction parameters: N

OF = min ∑ |γi∞ ,exp − γi∞ ,cal| i=1

E

(28)



Article

Table 2. Interaction Parameters for the Eyring−MTSM Model and AARD % for the Eyring−MTSM and the Eyring−NRTL Models for All of the Systems no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

α(0) 12 0.0739 5.1944 −3.1149 6.6108 2.8081 4.0413 3.9224 0.5472 0.9132 2.6684 −0.1593 1.8549 −5.5221 2.4949 2.4163 1.1885 0.6993 −5.5638 −0.1707 3.9395 6.2948 1.6094 1.5769 0.9119 3.1301 −0.2996 2.801 6.2975 62.0728 −0.7089 0.7312 0.4782 1.6919 2.9716 0.8954 1.2669 3.011 2.0712 2.1886 1.988 1.9069 4.9846 4.9846 9.9305 6.3467 −1.886 1.8402 3.3074 2.47 2.7916 −1.7469 3.9858 25.9595 −2.6426 4.0323 0.1447 1.8443 0.6817 5.115 −1.9976

τ12

τ21

α(1) 12

AARD %a MTSM

AARD % NRTL

data point

−0.0356 0.0453 −0.0025 −0.0013 −0.0166 −0.0157 −0.0156 −0.0912 −0.0764 −5.0024 −0.0347 −0.0233 −0.0515 −0.0279 −0.0285 −0.0122 −0.0491 0.024 −18.2366 −0.0577 −0.0173 −0.0088 −0.0222 −0.0245 −0.0312 −0.0626 −0.0739 −0.0162 0.0428 −0.0083 −0.0362 −0.0547 −0.019 −0.0301 −0.0347 −0.0345 −0.0197 −0.0209 −0.0208 −0.0211 −0.0337 0.0084 0.0084 0.0252 0.0141 0.0001 −0.0223 −0.008 −0.0181 −0.0098 −0.1091 −0.0451 0.056 −0.0564 −0.0297 −0.0482 −0.044 −0.0339 −0.0155 −6.0153

0.0017 −0.0054 −0.0477 0.0019 0.0017 0.0137 0.0156 −0.01 −0.0129 0.01 −0.0108 −0.006 −0.0048 −0.0125 0.0285 −0.0279 −0.0402 −0.0222 −0.0569 −0.0204 0.0162 0.0087 0.0222 0.0245 −0.0114 −0.0205 −0.0594 −0.0269 0.0196 0.0085 0.0356 −0.0428 0.0189 0.0301 0.0347 −0.0035 0.0197 0.0208 0.0208 0.0211 0.0094 −0.0087 −0.0087 0.0048 −0.0021 −0.0458 −0.0424 −0.0315 −0.0388 −0.0303 −0.0263 0.0299 0.0037 −0.0005 −0.0085 −0.0014 −0.0151 −0.0108 0.0016 −0.0288

477.0317 2216.284 1735.329 −1612.6

2.05 2.55 2.23 1.91 0.87 1.12 1.67 1.42 0.14 2.01 1.71 1.37 5.48 2.44 4.15 1.74 2.33 2.21 1.14 0.97 1.88 0.94 2.14 1.74 4.18 2.39 4.42 3.5 3.36 0.93 3.73 1.72 1.37 5.53 6.1 0.52 2.25 1.45 1.21 2.76 1.35 1.28 1.28 1.51 1.02 2.76 1.12 1.55 1.19 1.32 5.23 2.24 2.65 2.65 0.83 1.03 0.87 0.78 1.74 8.41

2.27 2.3 2.04 1.81 0.43 1.13 1.72 2.08 2.8 2.08 3 0.99 12.64 2.73 2.97 1.84 3.08 1.75 2.64 3.09 1.75 0.93 1.73 1.02 4.71 5.9 5.78 3.47 3.13 0.92 3.19 2.4 1.26 4.3 4.59 0.88 2.01 1.29 1.18 2.42 1.45 5.79 1.32 1.51 0.99 2.07 1.28 1.64 1.48 1.33 7.09 2.99 3.01 6.06 1.38 1.88 1.36 1.57 1.37 10.69

75 36 60 72 9 7 11 9 6 8 36 36 36 13 13 14 15 88 88 60 104 15 15 15 39 36 88 88 88 77 88 15 15 13 15 15 15 15 15 15 15 72 66 60 72 72 72 78 72 78 36 36 90 48 48 48 48 36 36 36

848.3001 184.3075 2900.332

2204.045 −0.2802 −1282.86 −1322.72

−586.625 726.6358 −741.8 −1830.58 −17856.3 776.6649 758.1769

−1391.86 −1391.86 −2728.81 −1650.59 1217.908 −501.948 −917.911 −655.95 −771.296 792.8292 −1275.52 −2547.78 1820.651 −917.509 458.0402 −358.5 523.5865 −693.133 1314.96 F



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Table 2. continued no. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121

α(0) 12

τ12

τ21

α(1) 12

AARD %a MTSM

AARD % NRTL

data point

4.262 3.4366 8.3469 0.8366 −1.0894 7.2253 2.1377 2.4618 1.9829 1.3715 1.3486 1.9192 1.2467 10.1881 6.261 10.1773 1.2577 0.6638 0.4547 0.8674 1.8963 −3.068 −8.7027 −0.6876 2.626 1.7884 0.5473 1.5986 1.227 −0.0004 −1.3288 1.9216 1.3486 −1.3407 −0.0956 2.2129 2.8967 2.3198 −1.9858 2.3549 2.3985 29.508 4.9055 −17.9092 −295.637 −21.7325 −0.9728 0.9428 4.1947 7.0775 3.8138 2.5993 3.0382 0.9116 0.6486 −0.2222 0.0498 0.666 −0.7308 17.7203 2.6156

−0.0259 −0.0287 −0.0153 −0.0149 −0.0549 −0.0204 −0.0473 −0.042 −0.0134 −0.0497 −0.0399 −0.0408 −17.159 −0.0375 −0.0518 −0.0376 −1.5 −0.0441 −0.0593 −0.047 −0.0356 −0.0056 0.045 −0.0405 −0.0108 −0.083 −0.0814 −0.0177 −0.02 −5.0721 −0.0088 −0.0143 −0.0399 0.0007 0.0187 −0.0195 −0.0191 −0.0264 0.0277 −0.027 −0.0353 0.0044 −0.0368 −0.0846 −0.0817 −0.0006 −0.0319 −0.0333 −0.0243 −0.0206 −0.0308 −0.0091 −0.0217 −0.0326 −0.0441 −0.0755 −16.8791 −0.0635 −0.0302 0.0017 −0.0393

0.0016 −0.0121 0.0052 −0.0435 0.0074 0.0164 −0.0166 −0.0227 0.013 −0.01 −0.0038 −0.0128 0.0016 −0.007 −0.0236 −0.0071 −0.0614 0.0043 −0.0167 −0.0042 0.0093 0.0064 0.0001 −0.004 0.0112 0.0872 −0.0106 0.0173 0.02 −0.0575 0.0093 0.0122 −0.0038 −0.0007 −0.0188 0.0195 0.0191 0.009 0.045 −0.0028 0.0058 0.0296 0.0061 −0.0038 −0.0075 0.0865 0.0015 −0.0031 −0.0001 0.0205 0.0302 0.014 0.0055 −0.0039 −0.0025 −0.0271 −0.0333 −0.0459 0.0024 0.0445 −0.0081

−674.863 −817.13 −2099.22

4.5 1.02 0.79 1.59 2.36 2.35 1.56 1.47 0.87 0.6 0.63 0.71 3.51 3.24 2.47 3.24 3.36 0.52 0.8 0.6 1.22 4.64 0.43 0.67 1.16 1 0.67 0.65 1.13 1.14 0.88 0.87 0.63 1.17 1.01 2.11 1.07 0.69 1.39 0.94 0.7 3.32 1.29 6.61 4.48 6.8 3.08 1.46 1.41 3.3 3.24 1.59 1.09 1.33 2.14 1.74 1.4 3.62 3.04 3.18 0.94

4.53 1.14 0.7 1.29 2.84 2.27 2.45 1.82 0.87 1.88 1.26 2.08 3.9 3.7 2.86 3.3 6.56 1.31 2.07 1.27 1.77 1.01 1.07 1.36 1.1 2.02 2.66 0.66 0.95 1.58 0.88 0.87 1.26 1.17 0.99 1.74 0.94 0.74 2.32 0.97 0.77 3.21 0.9 19.23 12.57 26.18 3.24 1.62 1.34 2.84 2.7 1.58 1.1 1.39 2.14 2.99 1.88 3.74 2.08 7.25 1.7

39 33 33 28 39 37 33 36 15 15 15 15 70 45 33 39 78 13 13 13 13 143 143 45 45 45 45 15 15 135 135 15 15 135 117 15 15 15 55 11 11 56 39 52 52 52 75 50 50 50 50 50 50 50 50 55 45 55 80 100 15

1190.04 −1655.68 −423.439 −596.004

−230.547 −3199 −1869.01 −3195.58 −233.845

692.3246 2782.059 517.9225 −611.775 −616.29 −211.299

−14.1779 422.7977

491.3429 43.7467

−3201.79

−4358.99 −1249.59 6543.577 125141.5 7943.47 719.026 266.0961 −840.368 −1802.23 −1017.1 21.182 −469.428 −0.6541 0.0347 324.9053 127.8176 −102.947 351.0139 −416.965

G



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Table 2. continued α(0) 12

no. 122 overall a

2.7933

τ12

τ21

−0.0419

−0.0133

α(1) 12

AARD %a MTSM

AARD % NRTL

data point

1.63 2.07

2.78 2.78

14 5512

The AARD % is calculated as follows:

AARD % =

100 N

N

∑ i=1

ηicalcd − ηiexpt ηiexpt

where N is the number of data points and γ∞ i is the activity coefficient of solute at infinite dilution. Table S2 in the Supporting Information gives group interaction parameters reported by Lei et al.25

α12 = α12(0) +

α12(1) T

(29)

As a result, the MTSM model has now four adjustable parameters (1) (α(0) 12 , α12 , τ12 and τ21). For evaluation of these interaction parameters, an objective function (OF) was defined as follows:

3. EXPERIMENTAL DATABASE To check its validity, the Eyring−MTSM theory was applied to a wide range of experimental data and various binary systems. Accordingly, a large data set consisting of 122 IL-based binary systems and 5512 experimental data points was collected from the literature. The vast applicability of hydrophilic ILs have made their binary data more available; however, the potential application of hydrophobic ILs in separation technologies have also made them an interesting candidate in experimental studies. Subsequently, binary data of both hydrophobic and hydrophilic ILs were taken into account. Table 1 demonstrates the list of the components for each binary system along with their respective temperature ranges.

OF =

1 N

ηicalcd − ηiexpt

N

∑

ηiexpt

i=1

(30)

where N is the total number of data points for each binary system, and calcd and expt denote calculated and experimental, respectively. Minimization of this objective function resulted in the calculation of the interaction parameters of every system. Also, the temperature dependency for the Eyring−NRTL model has the following form: Δgij =

4. RESULT AND DISCUSSION The data set under study contains a global number of 5512 data points for a total number of 122 binary systems. The Eyring−

Δgij(0)

+

Δgij(1) (31)

T

Δg(0) ij

Δg(1) ij

where T is temperature in Kelvin. Also, and are temperature independent constants for an IL binary system. The resultant NRTL model now has four parameters. The empirical constant of α in the NRTL model was supposed to be 0.1. According to the Renon and Prausnitz24 aspect, α is the nonrandomness constant which is equal to (1/Z), where Z is the coordination number. Z is of the order 6 to 12 and we considered its value equal to 10 for our study. So, its physical meaning is based on the assumption of the coordination number of 10 for molecules around a reference molecule. It should be noted that the interaction parameters of the NRTL model were also obtained on the basis of the experimental viscosity data and eq 30. Consequently, the interaction parameters of two models were estimated using the available experimental data and were tabulated in Table 2. The results indicated an overall AARD % of 2.02 % and 2.72 % for the Eyring−MTSM and the Eyring− NRTL models, respectively. The interaction parameters for the Eyring−NRTL models were reported in Table S3 in the Supporting Information. For almost all systems at constant temperature, that is, system 1 in Table 1, the Eyring−MTSM model proved to have a relatively higher accuracy. However, in these cases, the Eyring− MTSM and the Eyring−NRTL models had a different number of adjustable parameters. Consequently, by taking into account the temperature variations the attained models had the same number of adjustable parameters. In this regard, comparison of these models was more acceptable as both had four adjustable parameters. The main drawback with the Eyring−NRTL model was that it exerted high errors for some systems, such as systems 60 and 104 to 106 in Table 2. Consequently, the Eyring−MTSM model was a suitable model for estimation of viscosity for a wide range of systems.

Figure 2. Plot of correlated results versus experimental dynamic viscosity data for all of the systems.

NRTL model was considered to be compared with the Eyring− MTSM model. In the present study the interaction parameters for the Eyring−MTSM and the Eyring−NRTL models were considered to be temperature dependent. For the development of a four parameter model based on MTSM, α12 was considered to have the following temperature dependency: H



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Figure 3. Experimental and correlated results for the polyethylene glycol (1) + 1-methyl-3-pentylimidazolium tetrafluoroborate (2) binary system. Experimental data taken from ref 51.

Figure 4. Experimental and correlated results for the dimethyl sulfoxide (1) + diethylammonium hydrogen sulfate (2) binary system. Experimental data taken from ref 52.

parameters of binary mixtures are used in the ternary form of the Eyring−MTSM model. Figure 6 demonstrates the prediction of viscosity for the ternary system of 1-ethyl-3-methylimidazolium acetate (1) + water (2) + ethanol (3) with AARD % of 3.86 %. Additionally, Figure 7 shows the results for the ternary system of 1-ethyl-3-methylimidazolium dicyanamide (1) + water (2) + ethanol (3) with the subsequent AARD % of 7.75 %. As another ternary system,66 the prediction accuracy for the ternary system of 1-butyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl)imide (1) + ethanol (2) + ethyl acetate (3) was evaluated and the AARD % for this system was accessed as 1.89 %. In addition, the ternary form of the Eyring−MTSM model was described in Appendix A. It should be noted that, for systems outside the scope of this study, the model cannot be applied for the viscosity prediction of binary systems. One method for retrieving a predictive model for estimation of viscosity is the application of excess Gibbs energy model based on the group contribution method. Recently, phase equilibria of IL systems have been correlated by the UNIFAC method as a group contribution model.25 In this regard, Lei et al.25 reported that the UNIFAC model could successfully predict the VLE data for the IL−polar systems. However, their predicted results for nonpolar solute−IL systems were not satisfactory as liquid−liquid demixing took place. On the other hand,

Figure 2 demonstrates a comparison between the experimental data and those obtained from the Eyring−MTSM model. As seen, a reasonable conformity was validated as the data points were close to the diagonal line. With application of eq 29, good performance could be achieved for binary systems with temperature variation. Figures 3 and 4 show the correlated results of the proposed model in different temperatures for binary systems of 1-methyl-3pentylimidazolium tetrafluoroborate + polyethylene glycol and diethylammonium hydrogen sulfate + dimethyl sulfoxide, respectively. As can be observed, the Eyring−MTSM model could satisfactorily correlate the effect of temperature and composition on the viscosity of the systems. A challenge for the Eyring−MTSM model would be its temperature extrapolating ability. In this regard, this ability for some binary systems with a relatively wide range of temperature was checked and the results were reported in Table 3. Additionally, Figure 5 manifests extrapolation ability of the Eyring−MTSM model for the binary system of 1-butyl-3-methylimidazolium thiocyanate + 1-hexanol. It is a reasonable deduction that the prediction accuracy of the proposed model is within an acceptable range. Another aspect which should be inspected is the ability of the Eyring−MTSM model in the prediction of viscosity of ILs containing ternary systems. On this subject, interaction I



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Table 3. Results of Extrapolation for the Eyring−MTSM Model binary system pyrrolidinium nitrate + propylene carbonate

1-butyl-3-methylimidazolium tetrafluoroborate + water

1-butyl- 4-methylpyridinium thiocyanate + water

1-butyl-3-methylimidazolium hexafluorophosphate + methyl methacrylate

1-butyl-3-methylimidazolium hexafluorophosphate + monoethanolamine

1-butyl-3-methylimidazolium methylsulfate +1-n-butyl-3-methylimidazolium tetrafluoroborate

1-butyl-3-methylimidazolium thiocyanate +1-hexanol

correlated temp (K)

AARD %

283.15 293.15 303.15 313.15 323.15 303.15 308.15 313.15 318.15 323.15 298.15 308.15 318.15 328.15 283.15 288.15 293.15 298.15 303.15 308.15

3.81 3.10 2.61 2.04 1.77 4.1 3.67 3.94 3.46 4.19 2.81 1.22 1.20 3.12 1.15 1.35 1.52 1.52 1.77 1.67

288.15 293.15 298.15 303.15 298.15 299.15 301.15 302.15 303.15 298.15 308.15 318.15

1.42 0.97 1.03 0.72 0.96 0.90 0.77 0.87 0.71 1.03 1.40 1.29

predicted temp (K)

AARD %

ref

333.15 343.15 353.15

4.35 6.61 10.09

62

333.15 343.15 353.15

5.33 6.22 7.50

37

338.15 348.15

4.35 5.05

28

313.15 318.15 323.15 328.15 333.15 343.15 353.15 308.15 313.15 318.15 323.15 304.15 305.15 307.15 308.15

1.95 2.28 1.66 2.14 2.46 3.21 4.05 1.21 1.27 1.58 3.94 1.10 0.83 0.89 0.92

33

328.15 338.15 348.15

2.10 2.25 1.98

32

54

41

Figure 5. Extrapolation ability of the Eyring−MTSM model for the binary system of 1-butyl-3-methylimidazolium thiocyanate (1) + 1-hexanol (2). Experimental data taken from ref 41.

interaction parameters of various groups, the Eyring−UNIFAC model could only correlate a limited number of binary systems. The predicted results of this model are presented in Table 4. The deviation of the Eyring−UNIFAC model can be attributed to the value of binary interaction parameters which have been derived

predictions of the viscosities of IL-containing binary systems have never been accomplished by the UNIFAC method. Consequently, we coupled the Eyring equation with the UNIFAC excess Gibbs energy model in order to check its performance. However, as the UNIFAC model required J



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Figure 6. Prediction of viscosity for the ternary system of 1-ethyl-3-methylimidazolium acetate (1) + water (2) + ethanol (3) based on the Eyring− MTSM. Experimental data taken from ref 43.

Figure 7. Prediction of viscosity for the ternary system of 1-ethyl-3-methylimidazolium dicyanamide (1) + water (2) +ethanol (3) based on the Eyring− MTSM. Experimental data taken from ref 43.

Consequently, the UNIFAC model in its present form is not accurate for the prediction of viscosity.

Table 4. AARD % for Predicted Results Using the Eyring− UNIFAC Model component 1: ionic liquid 1-butyl-3-methylimidazolium Thiocyanate 1-butyl-3-methylimidazoliumThiocyanate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium methylsulfate 1-methyl-3-octylimidazolium tetrafluoroborate 1-methyl-3-octylimidazolium tetrafluoroborate 1-methyl-3-octylimidazolium tetrafluoroborate

component 2 AARD % 13.14 18.21 20.87

40 40 38

water

21.52

37

ethanol 1-propanol

12.57 8.73

35 49

2-propanol

9.73

49

20.87

12

ethanol

■

ref

1-decanol 1-heptanol ethanol

CONCLUSION We applied the Eyring’s absolute rate theory based on the excess Gibbs energy models for estimation of viscosity of IL containing binary systems. In order to obtain an excess Gibbs free energy model in simple mathematical form and yet accurate, we modified the structure of the two-suffix-margules model. The study showed that coupling of a new model with Eyring’s equation and applying it to a large data set resulted in an AARD % equal to 2.07 %. Moreover, the accuracy of the Eyring−MTSM model was compared with the Eyring−NRTL model, and it was found that the Eyring−MTSM model had a better accuracy. Furthermore, the capability of the Eyring−MTSM was inspected in two aspects. The first one was its competence for prediction of viscosity in extrapolated temperatures, and the other one was its capability in the prediction of viscosity in ternary systems. In both of these aspects, the proposed model has a relatively good

from the equilibrium data. The complementary information concerning the interaction parameters of the UNIFAC model was given in the Supporting Information (Table S2). K



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Notes

performance. Finally, performance of the Eyring−UNIFAC model was checked for viscosity calculations. This model generated high errors, and it was not reliable in its present form. In this regard, correlative methods such as the Eyring− MTSM model were estimated to be more reliable for the calculation of the viscosity of IL-based systems.

The authors declare no competing financial interest.

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(1) Zhu, A.; Wang, J.; Liu, R. A volumetric and viscosity study for the binary mixtures of 1-hexyl-3-methylimidazolium tetrafluoroborate with some molecular solvents. J. Chem. Thermodyn. 2011, 43, 796−799. (2) Fatehi, M.; Raeissi, S.; Mowla, D. Estimation of viscosity of binary mixtures of ionic liquids and solvents using an artificial neural network based on the structure groups of the ionic liquid. Fluid Phase Equilib. 2014, 364, 88−94. (3) Gonzalez, B.; Calvar, N.; Gomez, E.; Macedo, A.; Dominguez, A. Synthesis and physical properties of 1-ethyl 3-methylpyridinium ethyl sulfate and its binary mixtures with ethanol and water at several temperatures. J. Chem. Eng. Data 2008, 53, 1824−1828. (4) Wang, X.; Chi, Y.; Mu, T. A review on the transport properties of ionic liquids. J. Mol. Liq. 2014, 193, 262−266. (5) Welton, T. Room-temperature ionic liquids. Solvents for synthesis and catalysis. Chem. Rev. 1999, 99, 2071−2084. (6) Esperana, J.; Lopes, J.; Tariq, M.; Santos, L.; Magee, J.; Rebelo, L. Volatility of aprotic ionic liquids. A review. J. Chem. Eng. Data 2010, 55, 3−12. (7) Plechkova, N.; Seddon, K. Application of ionic liquids in the chemical industry. Chem. Soc. Rev. 2008, 37, 123−150. (8) He, Y.; Xu, X.; Yang, L.; Ding, B. Viscosity modeling for ionic liquid solutions by Eyring−Wilson equation. Chem. Ind. Chem. Eng. Q. 2012, 18, 441−447. (9) Wang, Y.; Chen, D.; Yang, X. Viscosity calculations for ionic liquidcosolvent mixtures based on Eyring’s absolute rate theory and activity coefficient models. J. Chem. Eng. Data 2010, 55, 4878−4884. (10) Wang, F.; Han, L.; Zhang, Z.; Fang, X.; Shi, J.; Ma, W. Surfactantfree ionic liquid-based nanofluids with remarkable thermal conductivity enhancement at very low loading of graphene. Nanoscale Res. Lett. 2012, 7, 314. (11) Zafarani-Moattar, M.; Majdan-Cegincara, R. Viscosity, density, speed of sound, and refractive index of binary mixtures of organic solvent + ionic liquid, 1-butyl-3-methylimidazolium hexafluorophosphate at 298.15 K. J. Chem. Eng. Data 2007, 52, 2359−2364. (12) Mokhtarani, B.; Mojtahedi, M.; Mortaheb, H.; Mafi, M.; Yazdani, F.; Sadeghian, F. Densities, refractive indices, and viscosities of the ionic liquids 1-methyl-3-octylimidazolium tetrafluoroborate and 1-methyl-3butylimidazolium perchlorate and their binary mixtures with ethanol at several temperatures. J. Chem. Eng. Data 2008, 53, 677−682. (13) Ciocirlan, O.; Croitoru, O.; Iulian, O. Densities and viscosities for binary mixtures of 1-butyl-3- methylimidazolium tetrafluoroborate ionic liquid with molecular solvents. J. Chem. Eng. Data 2011, 56, 1526−1534. (14) Seddon, K.; Stark, A.; Torres, M. Influence of chloride, water, and organic solvents on the physical properties of ionic liquids. Pure Appl. Chem. 2000, 72, 2275−2287. (15) Fang, S.; He, C. A new one parameter viscosity model for binary mixtures. AIChE J. 2011, 57, 517−524. (16) Tariq, M.; Altamash, T.; Salavera, D.; Coronas, A.; Rebelo, L.; Lopes, J. Viscosity mixing rules for binary systems containing one ionic liquid. ChemPhysChem. 2013, 14, 1956−1968. (17) Atashrouz, S.; Pazuki, G.; Alimoradi, Y. Estimation of the viscosity of nine nanofluids using a hybrid GMDH-type neural network system. Fluid Phase Equilib. 2014, 372, 43−48. (18) Atashrouz, S.; Mirshekar, H. Phase equilibrium modeling for binary systems containing CO2 using artificial neural networks. Bulg. Chem. Commun. 2014, 46, 104−116. (19) Eyring, H. Viscosity, plasticity, and diffusion as examples of absolute reaction rates. J. Chem. Phys. 1936, 4, 283−291. (20) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena; McGraw-Hill: New York, 1941. (21) Poling, B.; Prausnitz, J.; O’Connell, J. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2004.

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APPENDIX A The Eyring model for a ternary system has the following form: ln(ηmix ) = x1 ln(η1) + x 2 ln(η2) + x3 ln(η3) +

GE, * RT (A.1)

Based on MTSM model (GE,*/(RT)) is defined as ⎛ GE, * ⎞ ⎛ GE, * ⎞ ⎛ GE, * ⎞ GE, * =⎜ ⎟ +⎜ ⎟ +⎜ ⎟ RT ⎝ RT ⎠12 ⎝ RT ⎠13 ⎝ RT ⎠23 = α12X12X 21 + α13X13X31 + α23X 23X32

(A.2)

Application of eq A.2 requires calculation of local mole fractions. It is evident that the summation of local mole fractions around a reference molecule is equal to 1. Thus, considering molecule 1 as the reference molecule, we have X11 + X 21 + X31 = 1

(A.3)

⎛ X ⎞ X X11⎜1 + 21 + 31 ⎟ = 1 X11 X11 ⎠ ⎝

(A.4)

X11 =

1 1+

X 21 X11

+

X31 X11

(A.5)

Consequently, substitution of the definition of local mole fractions results in the following equations: x X 21 = 2 G21X11 x1 (A.6) X31 =

x3 G31X11 x1

(A.7)

An application of the same procedure, local mole fraction for the other two molecules (as reference), can be obtained. Therefore all of the terms in eq A.2 can be calculated. Consequently, the Eyring−MTSM for the ternary system has the following function: ln(ηmix ) = x1 ln(η1) + x 2 ln(η2) + x3 ln(η3) + α12X12X 21

■

+ α13X13X31 + α23X 23X32

(A.8)

ASSOCIATED CONTENT

S Supporting Information *

Volume and surface area group parameters for 25 main groups; fitted interaction parameters for the UNIFAC model; interaction parameters for the Eyring−NRTL model. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

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