Modeling the Viscosity of Ionic Liquids with the Electrolyte Perturbed

Article pubs.acs.org/IECR

Modeling the Viscosity of Ionic Liquids with the Electrolyte Perturbed-Chain Statistical Association Fluid Theory Gulou Shen,*,†,⊥ Christoph Held,‡ Jyri-Pekka Mikkola,§,∥ Xiaohua Lu,⊥ and Xiaoyan Ji*,† †

Division of Energy Science/Energy Engineering, Luleå University of Technology, 97187 Luleå, Sweden Laboratory of Thermodynamics, Department of Biochemical and Chemical Engineering, TU Dortmund, Emil-Figge-Str. 70, 44227 Dortmund, Germany § Technical Chemistry, Department of Chemistry, Chemical-Biological Centre, Umeå University, 90187 Umeå, Sweden ∥ Industrial Chemistry & Reaction Engineering, Process Chemistry Centre, Åbo Akademi University, Biskopsgatan 8, 20500 Åbo-Turku, Finland ⊥ State Key Laboratory of Materials-Oriented Chemical Engineering, Nanjing Tech University, Nanjing 210009, P. R. China ‡

S Supporting Information *

ABSTRACT: In this work, the friction theory (FT) and free volume theory (FVT) were combined with the electrolyte perturbed-chain statistical association fluid theory (ePC-SAFT) in order to model the viscosity of pure ionic liquids (ILs) and IL/ CO2 mixtures in a wide temperature and pressure (up to 3000 bar) range and with viscosities up to 4000 mPa·s. The ePC-SAFT pure-component parameters for the considered imidazolium-based ILs were adopted from our previous work. These parameters were used to calculate the density and residual pressure of the pure ILs. The density and pressure were then used as inputs for pure-IL viscosity modeling using FVT or FT, respectively. The viscosity-model parameters of FT and FVT were obtained by fitting to experimental viscosity data of imidazolium-based ILs and linearized with the molecular weight of the IL-cation. As a result, the FT viscosity model can more accurately describe the experimental viscosity data of pure ILs than the FVT model, at the cost of an increased number of parameters used in the FT viscosity model. Finally, FT and FVT were applied to model the viscosities of IL/CO2 mixtures in good agreement to experimental data by adjusting one binary viscosity-model parameter between the IL-anion and CO2. The application of FT required fitting the viscosity model parameters of pure ILs to experimental viscosity data of pure ILs and of IL/CO2 mixtures. In contrast, the FVT viscosity model parameters were adjusted to the experimental viscosity data of pure ILs only.

1. INTRODUCTION CO2 separation plays an important role in greenhouse gas emission mitigation, in biofuel production via biomass gasification as well as in biogas upgrading.1 In general, CO2 separation requires a lot of energy, and exploring cost-effective CO2-separation technologies is still a hot research topic.2,3 Recent research reveals that ionic liquids (ILs) are promising liquid absorbents for CO2 separation as many ILs provide high CO2 solubility and high selectivity toward CO2 capturing from gas mixtures.3 Considering the potential industrial applications and scientific interests, studying thermo-physical and transport properties of ILs and IL-containing mixtures is important. Viscosity is one of the most important physical properties required in process design of CO2 separation processes because it strongly affects the mass- and heat-transfer rates and flow behavior. It is well-known that the viscosity of pure ILs is relatively high compared to those of common organic solvents, and it varies strongly depending on the type of the IL-cation and IL-anion, temperature, and pressure.4 In addition, the viscosity of pure ILs changes significantly upon CO 2 dissolution. Because of the high number of ILs that can be synthesized and the high CO2 solubility in ILs, measuring the viscosities of ILs and CO2 + IL means a huge experimental effort. Therefore, it is desirable to develop a theoretical model representing the viscosity of pure ILs and their mixtures with © 2014 American Chemical Society

CO2. This is a challenging task as CO2/IL mixtures are highly asymmetric systems in the sense that the viscosity of an IL is extremely high and the viscosity of CO2 is extremely low. Several theoretical approaches have been proposed to model the viscosity of IL systems. For example, Abbott5 used the hole theory to represent the viscosity of pure ILs with limited success since the average relative deviation (ARD) between calculated and experimental data was very high (122.5%). Group contribution methods6,7 have been proposed to represent the viscosity of ILs, and most of them are only applicable at low pressures. Polishuk8 proposed a modified Yarranton-Satyro model combined with a statistical association fluid theory (SAFT)+cubic equation of state (EoS) and correlated the viscosity of imidazolium-based ILs with good accuracy to the experimental data. However, only a few pure ILs were considered. Riva et al.9 refined the experimental viscosity data of 134 ILs statistically and fitted the viscosity to an Arrhenius-type equation. The viscosity of IL-containing mixtures was represented by the models based on Eyring theory in which the viscosity of pure ILs was required and used as an Received: Revised: Accepted: Published: 20258

September 3, 2014 November 26, 2014 December 4, 2014 December 4, 2014 dx.doi.org/10.1021/ie503485h | Ind. Eng. Chem. Res. 2014, 53, 20258−20268

Industrial & Engineering Chemistry Research

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the modified Chapman−Enskog theory. In this method, η0 is expressed as

input value. The work representing the viscosity of pure ILs and IL-containing mixtures in the same framework is limited. Very recently Llovell and Vega et al. used FVT combined with softSAFT to represent the viscosity of [Cnmim][BF4] and a binary mixture of [Cnmim][BF4].10 Meanwhile, theoretical models have been developed for calculating the viscosity of fluids other than ILs. Among them, friction theory (FT) and free volume theory (FVT) are promising with the additional advantage of representing thermodynamic and transport properties simultaneously, within the same framework. FT11 yields an expression for the viscosity based on residual pressure resulting from repulsive and attractive interactions between the molecules. In FVT,12 the free volume fraction is related to fluid-density that serves as an input value to express the viscosity. Therefore, both FT and FVT are related to the thermodynamic properties such as pressure and density. These properties are accessible by an EoS. SAFT13 is one of the most promising EoS applied to model the thermodynamic properties of density and residual pressure of fluids. Various versions of SAFT have been developed, such as SAFT-VR,14 soft-SAFT,15 SAFT1,16 SAFT2,17 perturbed chain (PC)-SAFT18 and the further generalized versions,19 and widely used for systems containing nonpolar, polar, ionic and associating substances.20−22 Both FT and FVT have already been combined with a SAFT-based EoS to estimate the viscosity of fluids. For example, Quinones-Cisneros et al. proposed a general FT model coupled with PC-SAFT EoS to represent the viscosity of alkanes, CO2 and their mixtures.23 Llovell et al. used FVT+soft-SAFT to represent the viscosity of n-alkanes, hydrofluorocarbons and their mixtures as well as water/1-alkanol mixtures.24−26 Tan et al. used both FT and FVT combined with SAFT1 and PC-SAFT to estimate the viscosity of n-alkanes.27,28 Burgess et al. combined FT and FVT with PC-SAFT to calculate the viscosity of n-alkanes, branched alkanes, single-ring aromatics, double-ring aromatics, and naphthenic compounds up to 276 MPa.29 Polishuk used the FT8 and FVT30 models combined with a SAFT+cubic EoS to represent the viscosity of various pure fluids8 and mixtures.30 In our previous works, ePC-SAFT has been used to accurately represent the densities and pressures of pure ILs in a wide temperature and pressure range allowing for quantitative predictions of gas solubility in ILs.31,32 Therefore, the goal of this work is to extend our previous work on ePC-SAFT to represent the viscosity of IL-containing systems including pure ILs and mixtures up to high pressures by combining with FT or FVT. To achieve the goal, ePC-SAFT was used to model the density and residual pressure of pure IL; the obtained residual pressure and density were used as inputs for FVT and FT, respectively, allowing a calculation of the viscosity of imidazolium-based ILs. The viscosity models of ePC-SAFT +FT and ePC-SAFT+FVT were further applied to represent the viscosity of IL/CO2 mixtures.

η0 = 40.785

v

Ω*T *

Fc ,

μP (2)

where M (g/mol) is the molecular weight, vc (cm3/mol) is the critical volume, T* is the reduced temperature T* = T/Tc with T (K) and Tc being the absolute and the critical temperature, respectively, and Ω* is the reduced collision integral and given by Ω*

1.16145 0.52487 2.1678 + + T* exp(0.77320T *) exp(2.43787T *)

− 6345 × 10−4(T *)0.14784 sin[18.0323(T *)−0.76830 − 7.27371]

(3)

In eq 2, the Fc factor can be empirically expressed as Fc = 1 − 0.2756ω + 0.059035μr4 + κ′

(4)

where ω is the acentric factor, κ′ is the correction factor for the hydrogen bonding effect of associating substances, and μr is the dimensionless dipole moment estimated by μr = 131.3μ/(v cTc)1/2

(5)

In eq 5, μ (D) denotes the dipole moment. The Dense State Correction Term ηres. The dense state correction term ηres can be obtained from FT or FVT. The description of these two theories is shown in the following section. Friction Theory. According to the general FT proposed by Quinones-Cisneros and Deiters34 the dense state correction term (or residual viscosity) ηres can be written as η res = κ ipid + κ rp r + κ apa + κ ii(pid )2 + κ rr(pr )2 + κ aa(pa )2 id

a

(6) r

where p , p and p are the ideal, attractive and repulsive contributions to the total pressure. The pressure is related to the Helmholtz energy, and the pressure contributions pid, pa, and pr can thus be expressed as pid = kTρ

(7a)

p r = kTρ2

∂a ̃hc ∂ρ

(7b)

pa = kTρ2

∂a disp ∂a ̃ion ̃ + kTρ2 ∂ρ ∂p

(7c)

In eq (7), k is the Boltzmann constant, ρ is the number density. ãhc, ãdisp, and ãion describe the Helmholtz-energy contributions due to hard-chain repulsion, dispersion, and Coulomb interactions in a charged system, respectively. Values for ãhc, ãdisp, and ãion are accessible by an equation of state. The temperature-dependent coefficients κ in eq 6 can be expressed as

2. THEORY The viscosity η of a fluid can be described by the summation of a dilute gas viscosity term η0 and a dense state correction term ηres η = η0 + η res

MT c 2/3

κ r = (a0 + a1Ψ1 + a 2 Ψ2)

(1)

The Dilute Gas Viscosity Term η0. The value of η0 is quite small compared to the viscosity of dense fluids. η0 can be calculated by the model proposed by Chung et al.33 based on

1 T*

κ rr = (A 0 + A1Ψ1 + A 2 Ψ2) 20259

1 T *3

(8a)

(8b)

dx.doi.org/10.1021/ie503485h | Ind. Eng. Chem. Res. 2014, 53, 20258−20268

Industrial & Engineering Chemistry Research κ a = (b0 + b1Ψ1 + b2 Ψ2)

1 T*

κ aa = (B0 + B1Ψ1 + B2 Ψ2) κ i = (c0 + c1Ψ1 + c 2 Ψ2)

Using the free volume fraction derived by Allal et al.12 the final expression is given by

(8c)

1 T *3

η res = ρlE

(8d)

1 T*

κ ii = (C0 + C1Ψ1 + C2 Ψ2)

Article

E = αρ +

(8e)

1 T *3

with (9a)

⎛ 1 ⎞ Ψ2 = exp⎜ 2 ⎟ − 1 ⎝ T* ⎠

(9b)

The general FT involves in total 18 adjustable parameters, that is, ai, Ai, bi, Bi, ci, Ci (i = 0, 1, 2). A preliminary investigation in this work showed that setting κi and κii to zero in eq 6 still allows for satisfying viscosity modeling results. Thus, a 12parameter formula can be used instead of eq 6, which simplifies to η res = κ rp r + κ apa + κ rr(p r )2 + κ aa(pa )2

i

lmix =

i

Bmix =

(12)

a ̃res = a ̃hc + a disp ̃ + a ̃ion

(13) 11

In this work, the following mixing rules were used to calculate the κ parameters of mixtures: rr = ∑ zjκjrr ∑ zjκjr ,κmix

a κ mix =

j=1

(14a)

aa = ∑ zjκjaa ∑ zjκja ,κmix j=1

(14b)

In eq 14 zj was calculated by xj zj = ε x M j ∑i Miε

(15)

j=1

i

where ε is a binary parameter that might be adjusted to the experimental viscosity data of mixtures. Free Volume Theory. In FVT, the residual viscosity ηres is related to the free volume fraction f v via an empirical relation proposed by Doolittle35 η res = A exp(B /fv )

(18b)

(18c)

(19)

On the basis of the Helmholtz-energy contributions calculated by ePC-SAFT, the pressure contributions required for FT are obtained by eq 7. The density required for FVT can be obtained from eq 19 as described in our previous works.1,25 Modeling IL systems with ePC-SAFT31,32 assumes an IL to be composed of IL-cation and IL-anion. In our previous work, this model has been proven to accurately model the density of ILs and quantitatively predict the solubility of CO2 in ILs without any binary parameter. It was further shown that treating IL-ions as associating or polar species did not allow quantitatively predicting CO2 solubility. In order to apply ePCSAFT to model systems with IL-ions, the relative dielectric constant is required. However, the relative dielectric constant is only available for a limited number of ILs. We thus assumed this property to be unity. This assumption makes the model less physical, but more independent of dielectric-constant data and thus more applicable to a broader number of ILs. Each individual IL-ion is modeled as a nonspherical species with repulsive, dispersive, and Coulomb interactions. This requires three pure IL-ion parameters, segment number mseg, segment diameter σ, and dispersion-energy parameter ε/k. The IL-ion specific ePC-SAFT parameters were fitted to the experimental liquid-density data of pure ILs,31 and are summarized in Table 1. The ePC-SAFT parameters for CO2 were taken from the

res r r a a rr r aa a ηmix = κ mix pmix + κ mix pmix + κ mix (pmix )2 + κ mix (pmix )2

j=1

(18a)

The binary parameter kα might be adjusted to the experimental viscosity data of mixtures. Representation of Viscosity by Combining FT or FVT with ePC-SAFT. In this work ePC-SAFT19 was used to calculate the residual pressure and density of IL-containing systems based on the residual Helmholtz energy. The original residual Helmholtz energy expression19 was used as in our previous works31,32

where xi and η0,i are the mole fraction and viscosity of pure fluid i, respectively. The residual contribution ηmixres is given by

r κ mix =

1 x ∑i Bi

i

The dilute gas viscosity term η0,mix can be expressed by the logarithmic mixing rule11 i=1

∑ xili

(10)

(11)

η0,mix = exp[∑ xi ln(η0, i)]

(17b)

αmix = (1 − kα) ∑ xiαi

Next to the viscosity of pure fluids, FT also applies to mixtures. The viscosity of mixtures is given by res ηmix = η0,mix + ηmix

PM ρ

(17a)

where R is the gas constant, 8.3145 J/(mol·K). For each pure fluid, there are three parameters α, l, and B, in which α (J·m3/ (mol·kg)) represents the energy barrier that a molecule has to pass to diffuse, l (Å) is a length parameter that accounts for the molecular size of the fluid, and B is a characteristic parameter for the free volume overlap. Several mixing rules have been proposed to represent the viscosity of mixtures for particular systems.24,36,37 In this work, preliminary investigation showed that the following mixing rules are especially suitable for representing the viscosities of IL/CO2 mixtures

(8f)

⎛ 1 ⎞ Ψ1 = exp⎜ ⎟ − 1 ⎝ T* ⎠

⎛ ⎛ E ⎞3/2 ⎞ 1 ⎟ ⎟⎟ exp⎜⎜B⎜ 3MRT ⎝ ⎝ RT ⎠ ⎠

(16) 20260



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Table 1. Pure-Component ePC-SAFT Parameters of the IL-Ions31 and CO218 seg

m σ, Å ε/k, K

[C2mim]+

[C4mim]+

[C6mim]+

[C8mim]+

[BF4]−

[PF6]−

[Tf2N]−

CO2

1.4872 3.5926 206.49

2.4805 3.6371 218.144

3.4131 3.6781 230.00

4.2977 3.7187 242.00

3.8227 3.5088 496.12

4.2771 3.5889 492.28

6.0103 3.7469 375.65

2.0729 2.7852 169.21

original PC-SAFT publication18 as listed in Table 1. In our previous work we have shown that this approach allows accurate modeling of the densities of pure ILs as well as reliable predictions of the CO2 solubilities in ILs.31,32 To calculate viscosities with FT and FVT, the residual pressure and density at the given temperature and pressure have to be known. In this work, the residual pressure and density were calculated with ePC-SAFT and used as inputs in eqs 6, 7, 10, 13, and 17. Following our previous work, binary parameters between CO2 and ILs have not been applied in ePC-SAFT. The pure and binary viscosity-model parameters required for FT and FVT were obtained from the fitting to experimental viscosity data. The adjusted viscosity-model parameters were further linearized with molecular weight of IL-cations for a series with a constant IL-anion. Since multiple solutions can be obtained in parameter fitting, it is necessary to include as many accurate experimental data of different ILs as possible in parameter fitting in order to obtain the most reliable set of viscosity-model parameters. Experimental Viscosity Data. Due to the limited experimental viscosity data for pure ILs at high pressures, only nine imidazolium-based ILs with three different IL-anions were considered in this work ([BF4]−-, [PF6]−-, [Tf2N]−imidazolium-based ILs). The sources of the experimental data used in this work are summarized in Table 2. Critical and Other Properties. To calculate the dilute gas viscosity η0 and the dense state correction term ηres in FT, the

critical properties such as Tc, Pc, and vc are needed. For the ILs considered in this work, these critical properties are not experimentally accessible. However, they can be estimated based on the group contribution method proposed by Valderrama and Robles.48 The critical properties of the studied imidazolium-based ILs have been estimated by Shen et al.49 as listed in Table 3 and used in this work. Table 3. Critical Properties and Acentric Factor of Fluids Used in This Work49

T, K

P, bar

no. of points

[C4mim][BF4]

273.15−353.15

1−3000

82

[C6mim][BF4] [C8mim][BF4] [C4mim][PF6]

298.15−343.15 273.15−353.15 273.15−353.15

1−1218 1−2242 1−2493

33 81 93

[C6mim][PF6]

273.15−353.15

1−2385

108

refs.

[C8mim][PF6]

273.15−353.15

1−1759

76

[C2mim][Tf2N] [C4mim][Tf2N] [C6mim][Tf2N] [C4mim] [PF6]+CO2 [C6mim] [PF6]+CO2 [C8mim] [PF6]+CO2 [C4mim] [Tf2N]+CO2 [C6mim] [Tf2N]+CO2

298.15−343.15 273.15−353.15 298.15−343.15 293.15−353.15

1−1255 1−2989 1−1240 100−200

33 65 33 59

38 42 39 40 41 42 39 44 43 40 43 39 44 39 45

293.15−353.15

100−200

60

46

293.15−353.15

100−200

48

46

298.15−343.15

10.1−131.6

20

47

298.15−343.15

7.8−125.2

20

47

Tc, K

Pc, bar

Vc, cm3/mol

ω

[C4mim][BF4] [C6mim][BF4] [C8mim][BF4] [C4mim][PF6] [C6mim][PF6] [C8mim][PF6] [C2mim][Tf2N] [C4mim][Tf2N] [C6mim][Tf2N] CO2

746.1 758.2 770.2 503.7 518.6 533.4 1056.5 1067.1 1077.8 304.1

26.8 23.2 20.4 18.0 16.3 14.9 25.0 22.0 19.6 73.8

624.6 745.2 865.9 653.8 774.4 895.1 927.7 1048.4 1169.0 91.9

0.4160 0.4088 0.4061 0.2633 0.2695 0.2773 0.7970 0.7865 0.7793 0.225

For the calculation of the dilute gas viscosity η0, the correction factor κ′ and μ are needed. In this work, the κ′ of the studied ILs was assumed to be the same as that for ethanol,33 and their dipole moment μ was assumed to be the same as that for methanol which has also been used by other researchers modeling ILs.50 These approximations are accurate enough as the contribution of the dilute gas viscosity to the total viscosity is very small. The dilute gas viscosity of pure CO2 was calculated using the empirical equation proposed by Quinones-Cisneros and Deiters34 that assumes the dilute gas viscosity of CO2 is a polynomial function of reduced temperature only. In principle the dilute gas viscosity of CO2 can also be calculated from eq 2, and similar results of the dilute gas viscosity of CO2 can be obtained compared to the empirical equation. Considering the simplicity, the empirical equation proposed by QuinonesCisneros and Deiters34 was used in this work.

Table 2. Sources of the Experimental Pure-IL Viscosity Data Available in Literature IL

components

3. RESULTS AND DISCUSSION To evaluate the model performance, the absolute relative deviation (ARD) was used and calculated by ARD =

1 N

∑ i

ηical − ηiexp ηiexp

(20)

where η is the calculated viscosity, η is the experimental viscosity, and N denotes the number of data points. 3.1. Model Performance Evaluation. Both FVT and FT were combined with ePC-SAFT to model the viscosity of pure ILs and IL/CO2 mixtures in this work. There are several versions of FT, the original version with 7 viscosity parameters (original FT), the general version with 18 viscosity parameters (general FT), and diverse simplified general versions with less cal

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Figure 1. (a) Viscosity of [C4mim][PF6] at temperatures of 298.15, 308.15, 323.15, 333.15, 343.15 K and pressures up to 2500 bar. Symbols, experimental data. Blue solid line, 12-parameter FT; orange dotted line, FVT; green dashed-dotted-dotted line, original FT; black dashed line, 18parameter FT. (b) Panel a plotted on a logarithmic scale of viscosity.

when considering the inevitable experimental errors embedded in the experimental data. The comparison of the modeling results with the experimental viscosity of [C4mim][PF6] is further illustrated in Figure 1. The experimental viscosity increases with increasing pressure and decreasing temperature. At low temperatures, the experimental viscosity increases exponentially with increasing pressure, for example at 298.15 K. It can be observed in Figure 1 that all the four viscosity models represent the viscosity well at 343.15 K. However, original FT yields high deviations from the experimental viscosity data at high pressure and lower temperatures than 343.15 K. This nonlinear dependence can only be represented using general FT and simplified general FT. As it could have been expected, the model deviation from experimental data is larger with fewer viscosity-model parameters. Considering the number of parameters and the model accuracy, FVT and 12-parameter-FT are preferable models. Therefore, all the viscosity modeling results shown from now on using ePC-SAFT+FVT and 12-parameter-FT (eq 10), and the following results and discussion are based on these two viscosity models. 3.2. Viscosity of Pure-ILs. The viscosity models 12 parameters-FT and FVT were used to represent the viscosity of nine pure imidazolium-based ILs listed in Table 1. Two strategies were applied in the modeling: A molecular-based approach with viscosity-model parameters for each IL (strategy 1) and an approach that uses IL-cation linearized viscositymodel parameters (strategy 2). In strategy 1, the viscosity-model parameters for each IL were obtained by fitting to the pure-IL viscosity data at different temperatures and pressures (Table 2). The fitting accuracy of strategy 1 is expressed in terms of ARD values in Table 4. The fitted viscosity-model parameters for FT and FVT are listed in Table S1 and Table S2 in the Supporting Information, respectively. The FT model shows generally very accurate results. Among the considered ILs, the maximum ARD of 2.41% is observed for [C4mim][BF4]. Compared to the FT model, the deviation using FVT is larger with the maximum ARD (among the considered ILs) of 6.28% for [C4mim][BF4]. This ARD value is higher than the experimental uncertainty that can be estimated to be roughly 2%. It should be mentioned again that only three pure-IL viscosity-model parameters were

than 18 viscosity parameters. In this work, different simplifications of the general FT were developed with the aim to reduce the number of viscosity-model parameters: (1) omitting the second order terms in ideal and attractive terms. In this case, ηres = κipid + κrpr + κapa + κrr(pr)2, the number of viscosity-model parameters is reduced from 18 to 12, and the ARD of [C4mim][PF6] is about 3%. (2) omitting the second order terms in ideal and attractive terms as well as the constant term in the temperaturedependent coefficients κ. There are 8 viscosity-model parameters (ηres = κipid + κrpr + κapa + κrr(pr)2) and the ARD of [C4mim][PF6] is about 5%. (3) combining the pressure contributions pid and pr, ηres = κr(pid+pr) + κapa + κrr((pid)2 + (pr)2) + κaa(pa)2 with 10 viscosity-model parameters and an ARD of 2.6% for [C4mim][PF6]; combining the pressure contributions pa and pr, ηres =κidpid + κr(pa + pr) + κi(pid)2 + κrr((pr)2 + (pa)2) with 8 viscosity-model parameters and an ARD of 2.4% for [C4mim][PF6]. (4) removal of the ideal contribution to the viscosity and applying eq 10 yielding 12 viscosity-model parameters and an ARD of 1.87% for [C4mim][PF6]. The investigations (1−4) show that the removal of the ideal contribution to the viscosity (i.e., eq 10) is an optimal option considering the accuracy and the number of viscosity-model parameters. The performance of different models (FVT, original FT, general FT, and simplified general FT (eq 10)) was further discussed by studying the viscosity of [C4mim][PF6]. The deviations (ARDs) between experimental and modeled viscosities are 5.21% and 10.7% for FVT and original FT with seven parameters, respectively. Comparing these two viscosity models shows that the accuracy of FVT is much better although only three pure-IL viscosity-model parameters were used in FVT. The general 18 parameters-FT can represent the viscosity accurately with an ARD value of 1.35%. The simplified general 12 parameters-FT also provides accurate results with an ARD value of 1.87%, which is comparable to general FT. In summary, the viscosity of [C4mim][PF6] can be satisfactorily modeled using FVT (three parameters) and general FT with 12 and 18 parameters. These results are very promising particularly 20262



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several pure ILs. It can be observed that the viscosity of the considered ILs increases with increasing pressure and decreasing temperature. At low temperatures, the pressure strongly influences the viscosity, that is, the viscosity increases rapidly with increasing pressure. It can be further observed that the viscosity increases with increasing length of carbon chain substituents of the IL-cation. [BF4]-Based ILs. FT can be used to represent the viscosity of [Cnmim][BF4] reliably at temperatures and pressures studied. The influence of temperature, pressure, and the length of IL-cation alkyl substituent on viscosity is captured accurately. FVT can be used to represent the viscosity of [C8mim][BF4] at temperatures 273.15−353.15 K and pressures up to 2242 bar. From the comparison of the viscosity modeling results of pure [C4mim][BF4] and pure [C6mim][BF4] shown in Figure 2 it can be concluded that the overall modeling results using FT are more accurate than the FVT modeling results. Especially, the nonlinear experimental viscosity dependence on pressure can be described with FVT only qualitatively, whereas quantitative results are obtained with FT. This can be observed in Figure 2a at 283.15 and 298.15 K above 500 bar as well as in Figure 2b at 298.15 K and above 1000 bar. [PF6]-Based ILs and [Tf2N]-Based ILs. FT can be used to represent the viscosity of [PF6]-based ILs and [Tf2N]-based ILs in a broad temperature and pressure range with good agreement to experimental data. FVT can be used to represent the viscosity of [C6mim][PF6], [C8mim][PF6], and [C6mim][Tf2N] in a broad temperature and pressure range as well. However, FVT underestimates the viscosity of [C4mim][PF6], [C2mim][Tf2N], and [C4mim][Tf2N] at low temperatures and high pressures as shown in Figures 3a and Figure 4, respectively. The effect of the IL-cation alkyl length on the viscosity of [PF6]-based ILs at 323.15 K is illustrated in Figure 3b. It can be observed that both FT and FVT capture the influence of IL-cation alkyl length well and provide reliable results based on the viscosity-model parameters using strategy 2 (parameters linearized in IL−cation alkyl length). It can be seen that FVT-modeled viscosities deviate from experimental data at low temperatures and high pressures. To validate whether this deviation is from the FVT model itself or from the ePC-SAFT calculated densities, experimental density data were used as input to model the viscosity of [C4mim][PF6]. Almost equivalent viscosity modeling results were obtained using either experimental or ePC-SAFT calculated

Table 4. ARD (see eq 20) Values Obtained after Fitting Viscosity-Model Parameters to Experimental Pure-IL Viscosity Dataa ARD of FT, %

ARD of FVT, %

ionic liquid

strategy 1

strategy 2

strategy 1

strategy 2

[C4mim][BF4] [C6mim][BF4] [C8mim][BF4] [C4mim][PF6] [C6mim][PF6] [C8mim][PF6] [C2mim][Tf2N] [C4mim][Tf2N] [C6mim][Tf2N]

2.41 1.05 0.84 1.87 1.79 1.17 0.58 0.79 0.47

2.52 2.72 1.19 2.46 2.74 2.13 1.51 1.33 1.72

6.28 5.28 4.11 5.21 4.22 3.69 4.59 4.42 2.53

7.75 6.19 4.84 7.23 4.48 4.91 7.13 4.38 4.86

a

In strategy 1 the viscosity-model parameters are IL-specific. In strategy 2, the viscosity-model parameters depend linearly on the molecular weight of the IL-cation.

used in FVT, which is much fewer than that in FT model. This explains the better accuracy of FT compared to FVT. In strategy 2, the viscosity-model parameters were assumed to be molecular-weight-dependent; that is, the viscosity-model parameters linearly depend on the molecular weight of ILcation for a series of ILs with a constant IL-anion. Applying strategy 2, the total number of the parameters for one series of ILs with the same IL-anion (e.g., in the series [C2mim][Tf2N] [C4mim][Tf2N] [C6mim][Tf2N]) was reduced from 36 to 24 for FT and from 9 to 6 for FVT. New viscosity-model parameter sets of 24 parameters and 6 parameters were obtained by simultaneously fitting to the available experimental data for all studied ILs (Table 2). The deviations in terms of ARD are listed in Table 4 (strategy 2), whereas the parameters fitted are listed in Table S3 and Table S4 in the Supporting Information, respectively. When comparing the ARDs in strategies 1 and 2, the ARDs in strategy 2 are only slightly larger than those obtained in strategy 1 for both viscosity models and for all the ILs studied. This is an unexpected result as the number of adjustable viscosity-model parameters differs significantly between strategies 1 and 2. To further illustrate the accuracy of modeling viscosity of pure ILs, Figures 2−4 show a comparison of the modeling results in strategy 2 with the experimental viscosity data of

Figure 2. Viscosity of (a) [C4mim][BF4] and (b) [C6mim][BF4]. Symbols, experimental data. Solid lines, results of FT. Dashed line, results of FVT. 20263



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Figure 3. (a) Viscosity of [C4mim][PF6]. (b) Viscosity of [C4mim][PF6], [C6mim][PF6], and [C8mim][PF6] at 323.15 K. Symbols, experimental data. Solid lines, results of FT. Dashed line, results of FVT.

Figure 4. Viscosity of (a) [C2mim][Tf2N] and (b) [C4mim][Tf2N]. Symbols, experimental data. Solid lines, results of FT. Dashed line, results of FVT.

densities (results not shown). Thus, it can be stated that the deviation originates mainly from the FVT model. Based on the viscosity-model parameters using strategy 2 (parameters linearized in IL-cation alkyl length), it is possible to predict the viscosity of ILs that were not considered in the parameter estimation. Figure 5 compares the model prediction for the viscosity of [C8mim][Tf2N] at atmospheric pressure with experimental data.51 Figure 5 shows that both models, ePC-SAFT+FVT and ePC-SAFT+FT reliably predict the viscosity of [C8mim][Tf2N], with ARDs of 7.24% (FT) and 25.3% (FVT), respectively. In summary, the accuracy of the FT viscosity model is better than that of the FVT model. This is reasonable as the number of adjustable parameters is higher using FT. FVT cannot be used to quantitatively model the viscosity of some pure-ILs at low temperature and high pressure. However, FT allows for quantitatively modeling the viscosity of all pure-ILs in a broad temperature and pressure range. 3.2. Viscosity of Pure CO2 and IL/CO2 Mixtures. In a first step the viscosity of pure CO2 was studied using FT and FVT. The viscosity-model parameters were fitted to the recommended viscosity data of pure CO2 by Fenghour et al.52 between 280 and 360 K. The viscosity-model parameters are listed in Tables S1−S2 in the Supporting Information. Figure 6 shows the results at three temperatures. The modeling

Figure 5. Viscosity of [C8mim][Tf2N] at atmospheric pressure. Symbols, experimental data. Solid lines, prediction of FT. Dashed line, prediction of FVT.

results accurately agree with the experimental data with ARD values of 0.43% using FT and 0.73% using FVT, respectively. The viscosity of pure ILs is extremely high and even about hundreds to thousands times higher than the viscosity of pure CO2. The mixtures of CO2 and IL thus show a highly asymmetric viscosity behavior, which makes it difficult to model 20264



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modeling: a molecular-based approach with viscosity-model parameters for each IL (strategy 1) and an approach that uses IL-cation linearized viscosity-model parameters (strategy 2). Applying strategy 1 requires one binary viscosity parameter ε between CO2 and each IL. A comparison of the results in Table 5 strategy 1 with those in Table 4 strategy 1 shows that a worse result is obtained compared to fitting viscosity-model parameters to pure-IL viscosity data only. Because of the limited available experimental data, only five CO2/IL mixtures were investigated in this work. The maximum ARD of all considered mixtures is 16.8% for the viscosity of [C8mim][PF6]/CO2, and the corresponding ARD for pure [C8mim][PF6] is 8.5%. Applying strategy 2 requires only one binary viscosity parameter ε between CO2 and ILs with the same IL-anion. The parameters used are listed in the Supporting Information Table S6. The results for strategy 2 in terms of ARD values are listed in Table 5 (strategy 2). The maximum ARD of all considered mixtures is 19.97% for the viscosity of [C8mim][PF6]/CO2. Comparing the ARDs obtained with strategy 1 and strategy 2 listed in Table 5 reveals similar model accuracy for the viscosity of CO2/IL mixtures, which implies that using the linearized parameter is effective and reasonable. The comparison of the experimental data with the modeling results obtained by strategy 2 for [Cnmim][PF6]/CO2 at 293.15 K with only one ε for one series of ILs with the same IL-anion is shown in Figure 7. At a constant CO2 mole fraction the experimental viscosity increases linearly with increasing pressure; at a constant pressure the experimental viscosity decreases significantly with increasing CO2 mole fraction. FT provides generally good modeling results for most of the considered system conditions (temperature, pressure, CO2 mole fraction); and ARD values are higher at rather low CO2 mole fractions in the mixtures [Cnmim][PF6]/CO2. Figure 8 illustrates the viscosity modeling results of the mixture [C6mim][Tf2N]/CO2 at 323.15 K. It can be observed from Figure 8a that the CO2 solubility increases with increasing pressure, while Figure 8b shows that the viscosity decreases dramatically at the same conditions. This can be described by the suggested FT model with sufficient accuracy. Modeling Results Using FVT. As strategy 2 (IL-cationlinearized parameters) is efficient for the description of the viscosity of CO2/IL mixtures, this strategy was also used in FVT. Using strategy 1 and strategy 2 in FVT, similar results were obtained, for example, [C4mim][PF6]/CO2 was modeled with an ARD of 24.36% using strategy 1 and 23.07% using strategy 2, respectively. In FVT, it is possible to model the viscosity of CO2/IL mixtures on the basis of the pure-IL and pure-CO2 viscosity-model parameters obtained exclusively from pure viscosity data. Additionally, one binary parameter (kα in eq 18a) was adjusted to the viscosity data of CO2/IL mixtures, and was kept constant for ILs with the same IL-anion. The binary kα

Figure 6. Viscosity of pure CO2: symbols, experimental data;52 solid lines, results of FT; dashed line, results of FVT (three parameters).

the viscosity of ILs/CO2 mixtures. In the following sections, FT and FVT were used to model the viscosity of such mixtures. Modeling Results Using FT. The most straightforward way to model mixtures is to apply the viscosity-model parameters obtained from the pure-component fitting (Supporting Information Table S1 for the pure ILs and pure CO2), directly with mixing rules. Such a mixing rule was presented above (eq 15), containing additionally the binary parameter ε. However, the pure-component viscosity-model parameters combined with the mixing rule (eq 15) yielded viscositymodeling results with ARD values greater than 100%. This number could be reduced by adjusting the binary parameter ε in eq 15; however, this also did not allow for quantitative modeling results. This occurs because the fitted pure viscositymodel parameters are not optimal owing to the large number of adjustable parameters in FT. This can be solved by including more ILs in the parameter estimation procedure. However, because of the limitation of the available experimental viscosity data up to high pressure for pure ILs as well as the ePC-SAFT model parameters for ILs, this cannot be achieved in this work. In preliminary investigations within this work we could show that the accuracy of the viscosity modeling of CO2/IL mixtures could only be improved by refitting the pure-IL viscosity-model parameters. This was achieved by simultaneously adjusting the viscosity-model parameters for pure ILs and the binary parameter ε to the experimental viscosity data of ILs and CO2/IL mixtures. Thus, the CO2 viscosity-model parameters were used as listed in Table S1 in the Supporting Information. However, for the ILs the parameters listed in the Supporting Information Table S5 were used. The ARD values between modeled and experimental viscosity data for pure ILs and CO2/IL mixtures are listed in Table 5. Two strategies were applied in the

Table 5. ARD Values and Binary Viscosity-Model Parameters of FT Correlations for the Pure ILs and CO2/IL Mixtures ARD in strategy 1, % ionic liquid

pure IL

mixture

[C4mim][PF6] [C6mim][PF6] [C8mim][PF6] [C2mim][Tf2N] [C6mim][Tf2N]

4.81 5.54 8.50 1.59 3.80

10.86 12.48 16.8 3.51 10.22

ARD in strategy 2, % binary parameter 0.855 0.905 0.611 −0.0274 −0.500 20265

pure IL

mixture

5.68 2.52 1.90 2.15 1.46

9.29 12.20 19.97 4.14 11.89

binary parameter 0.775

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Figure 7. Viscosity of (a) [C4mim][PF6]/CO2, (b) [C6mim][PF6]/CO2, and (c) [C8mim][PF6]/CO2 at 293.15 K: symbols, experimental data; solid lines, results of FT with strategy 2; dashed line, results of FVT with strategy 2.

Figure 8. (a) Saturated concentration of CO2 at 323.15 K: symbols, experimental data; lines, model prediction. (b) Viscosity of [C6mim][Tf2N]/ CO2 at 323.15 K: open circles, experimental data of viscosity; solid lines, results of FT with strategy 2; dashed line, results of FVT with strategy 2.

parameters and the corresponding ARD values are listed in Table 6. A comparison of the results listed in Tables 5 and 6, shows that the ARD values are higher for FVT (Table 6) compared to those for FT (Table 5). Especially, modeling the viscosity of [C6mim][Tf2N]/CO2 yields high ARD values (∼32%). The modeling results using FVT are illustrated in Figures 7 and 8, respectively, as dashed lines. As shown in Figure 7, the deviation between the results of FVT and experimental data is remarkably high at low CO2 mole fractions in the mixtures [Cnmim][PF6]/CO2 at 293.15 K. As shown in

Table 6. ARD Values and Binary Viscosity-Model Parameters of FVT Ccorrelation for CO2/IL Mixtures

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ionic liquid

ARD, %

binary parameter

[C4mim][PF6] [C6mim][PF6] [C8mim][PF6] [C2mim][Tf2N] [C6mim][Tf2N]

23.07 20.63 25.03 15.8 32.31

0.132

0.0318



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Centre, a center of excellence funded by the Åbo Akademi University.

Figure 8, FVT can represent the viscosity of the mixtures [Cnmim][Tf2N]/CO2, and under the most conditions (temperature, pressure, CO2 mole fraction) the model accuracy is acceptable for mixtures. Considering the ARD values listed in Tables 5 and 6 and the results shown in Figures 7 and 8, it is obvious that the modeling viscosity of CO2/IL mixtures is promising using FT at various conditions. Compared to FT, the accuracy of the FVT-modeled viscosities at low CO2 mole fractions is worse. However, the higher accuracy of 12-parameters-FT is at the cost of a more complex model with a higher number of model parameters; FVT is simpler with lower number of parameters and nevertheless yields acceptable estimation of viscosity of mixtures. In summary, it is sufficient to use FVT combined with ePC-SAFT to represent the viscosity of IL-containing systems. To obtain highly accurate viscosity modeling results, the use of 12-parameters-FT is recommended.

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(1) Orr, J. F. M. CO2 Capture and Storage: Are We Ready? Energy Environ. Sci. 2009, 2, 449−458. (2) Shiflett, M. B.; Drew, D. W.; Cantini, R. A.; Yokozeki, A. Carbon Dioxide Capture Using Ionic Liquid 1-Butyl-3-methylimidazolium Acetate. Energy Fuels 2010, 24, 5781−5789. (3) Brennecke, J. F.; Gurkan, B. E. Ionic Liquids for CO2 Capture and Emission Reduction. J. Phys. Chem. Lett. 2010, 1, 3459−3464. (4) Seki, S.; Kobayashi, T.; Kobayashi, Y.; Takei, K.; Miyashiro, H.; Hayamizu, K.; Tsuzuki, S.; Mitsugi, T.; Umebayashi, Y. Effects of Cation and Anion on Physical Properties of Room-Temperature Ionic Liquids. J. Mol. Liq. 2010, 152, 9−13. (5) Abbott, A. P. Application of Hole Theory to the Viscosity of Ionic and Molecular Liquids. ChemPhysChem 2004, 5, 1242−1246. (6) Gharagheizi, F.; Ilani-Kashkouli, P.; Mohammadi, A. H.; Ramjugernath, D.; Richon, D. Development of a Group Contribution Method for Determination of Viscosity of Ionic Liquids at Atmospheric Pressure. Chem. Eng. Sci. 2012, 80, 326−333. (7) Gardas, R. L.; Coutinho, J. A. P. A Group Contribution Method for Viscosity Estimation of Ionic Liquids. Fluid Phase Equilib. 2008, 266, 195−201. (8) Polishuk, I. Modeling of Viscosities in Extended Pressure Range Using SAFT+Cubic EoS and Modified Yarranton-Satyro Correlation. Ind. Eng. Chem. Res. 2012, 51, 13527−13537. (9) de Riva, J.; Ferro, V. R.; del Olmo, L.; Ruiz, E.; Lopez, R.; Palomar, J. Statistical Refinement and Fitting of Experimental Viscosity-to-Temperature Data in Ionic Liquids. Ind. Eng. Chem. Res. 2014, 53, 10475−10484. (10) Llovell, F.; Vega, L. F. Assessing Ionic Liquids Experimental Data Using Molecular Modeling: [Cnmim][BF4] Case Study. J. Chem. Eng. Data 2014, 59, 3220−3231. (11) Quiñones-Cisneros, S. E.; Zéberg-Mikkelsen, C. K.; Stenby, E. H. The Friction Theory (F-Theory) for Viscosity Modeling. Fluid Phase Equilib. 2000, 169, 249−276. (12) Allal, A.; Boned, C.; Baylaucq, A. Free-Volume Viscosity Model for Fluids in the Dense and Gaseous States. Phys. Rev. E 2001, 64, 011203. (13) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709−1721. (14) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. Statistical Associating Fluid Theory for Chain Molecules with Attractive Potentials of Variable Range. J. Chem. Phys. 1997, 106, 4168−4186. (15) Felipe, J. B.; Lourdes, F. V. Thermodynamic Behaviour of Homonuclear and Heteronuclear Lennard-Jones Chains with Association Sites from Simulation and Theory. Mol. Phys. 1997, 92, 135− 150. (16) Adidharma, H.; Radosz, M. Prototype of an Engineering Equation of State for Heterosegmented Polymers. Ind. Eng. Chem. Res. 1998, 37, 4453−4462. (17) Tan, S. P.; Ji, X.; Adidharma, H.; Radosz, M. Statistical Associating Fluid Theory Coupled with Restrictive Primitive Model Extended to Bivalent Ions. SAFT2:1. Single Salt+Water Solutions. J. Phys. Chem. B 2006, 110, 16694−16699. (18) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (19) Cameretti, L. F.; Sadowski, G.; Mollerup, J. M. Modeling of Aqueous Electrolyte Solutions with Perturbed-Chain Statistical Associated Fluid Theory. Ind. Eng. Chem. Res. 2005, 44, 3355−3362. (20) Muller, E. A.; Gubbins, K. E. Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches. Ind. Eng. Chem. Res. 2001, 40, 2193−2211. (21) Economou, I. G. Statistical Associating Fluid Theory: A Successful Model for the Calculation of Thermodynamic and Phase

4. CONCLUSIONS In the present work, two viscosity models (FT and FVT) have been combined with ePC-SAFT in order to successfully represent the viscosity of imidazolium-based ILs. The viscosity model parameters were obtained by fitting to the experimental viscosity data and linearized with the molecular weight of the IL-cation. The linear expression of parameters allows predicting the viscosity of other imidazolium-based ILs in a wide temperature and pressure range. The viscosity models were further extended to IL/CO2 mixtures with only one binary parameter between CO2 and IL that is IL-anion specific. The results indicate that both viscosity models can represent the viscosity of pure ILs and IL/CO2 mixtures with acceptable accuracy. The results of FT are better than FVT, however, at a cost of more parameters; modeling viscosity with FT requires at least 12 parameters, whereas only 3 parameters are required using FVT. FVT can be recommended for mixtures up to elevated pressures, while FT should be the preferred model for highly accurate modeling results in a wide temperature and pressure range and for all considered CO2 mole fractions in the CO2/IL mixtures.

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ASSOCIATED CONTENT

S Supporting Information *

More details of the parameters of viscosity models using different strategies. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS G.S. and X.J. acknowledge the financial support from the Swedish research council. X.L. thanks the Key Project of the National Natural Science Foundation of China (Grant No. 21136004), and National Basic Research Program of China (Grant No. 2013CB733500). J.-P. Mikkola acknowledges the financial support from Bio4Energy and the Swedish Research Council for funding; this work is also associated with the Wallenberg Wood Science Center and Process Chemistry 20267



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