Assessing Ionic Liquids Experimental Data Using Molecular Modeling

Article pubs.acs.org/jced

Assessing Ionic Liquids Experimental Data Using Molecular Modeling: [Cnmim][BF4] Case Study Felix Llovell*,† and Lourdes F. Vega†,‡ †

MATGAS, Campus de la UAB, 08193 Bellaterra, Barcelona, Spain Carburos Metálicos/Air Products Group, C/Aragón, 300, 08009 Barcelona, Spain

‡

ABSTRACT: A theoretical tool, the soft-SAFT equation of state combined with the Free-Volume Theory (FVT), is used for the calculation of thermodynamic and transport properties to (1) discriminate among discrepancies observed in different experimental data sets, (2) evaluate its capacity of extrapolation and predictability by comparing to experimental data, and (3) explore phase diagram regions where no experimental measurements are available. The well-known [Cnmim][BF4] ILs family is chosen as a case study. Once a simple but reliable molecular model is proposed for this family, the density of several [Cnmim][BF4] compounds is predicted using correlations of the molecular parameters as a function of the molecular weight. A comparison with different data sets showing discrepancies is addressed from the modeling results. The density of these compounds at high pressures is predicted and compared to the available data. The exploration of the phase diagram region is given by the study of immiscibility gaps in CO2 + [Cnmim][BF4] mixtures. Finally, the viscosity of these fluids is addressed using the equation in a systematic way, and the behavior of ILs mixtures is predicted in agreement with experimental measurements. It is intended to demonstrate the synergy between experimental and modeling work.

1. INTRODUCTION Ionic liquids (ILs) are not new materialsthey have been known for over 100 yearsbut have gained substantial growth and attracted a renewed interest over the last 15 years. In fact, they have become very popular and have been largely studied from an experimental perspective owing to their particular properties. These organic salts are liquid over a wide range of temperatures near and at room temperature, have a very low vapor pressure and high thermal stability, and can be tailored by an appropriate choice of the cation−anion pair, making them suitable for a wide variety of processes. Among these processes, energetic applications have attracted the interest of many researchers, including the use of ILs as entrainers in extractive distillation in order to separate closeboiling or azeotropic mixtures,1−3 supercritical fluid applications,4 liquid−liquid extraction using ILs to remove certain chemicals from a liquid mixture, 5 or gas separation applications.6,7 To select an appropriate IL for a particular application, there is a need for a rational characterization of the physicochemical properties of these fluids. Experimental work is essential in the area, although the amount of cation−anion combinations and properties of interest make experiments costly and timeconsuming. Conversely, the combination of molecular modeling techniques, such as molecular simulations8 and theoretical approaches,9 can be used as a fast tool to screen ILs families or to identify potential candidates for specific applications, just by estimating their thermophysical properties based on their molecular structure. © 2014 American Chemical Society

The purpose of this work is to assess the capabilities of a relatively simple molecular model for the [Cnmim][BF4] family to reproduce several properties of these ILs, including density, compressibility, gas solubility, and viscosity. The model, developed in the context of the soft-SAFT equation of state in a previous work,10 is used here to show some examples of how a coarse-grained theoretical model can be used, either to discriminate about different sets of data, or to predict the behavior of thermophysical properties in a wide range of conditions, becoming a useful tool that can be combined with the experimental work. Soft-SAFT11,12 is a refined version of the molecular-based Statistical Associating Fluid Theory (SAFT),13,14 characterized by the use of a Lennard-Jones (LJ) intermolecular potential to account for the interactions in the reference term of the equation. The performance of SAFT-type equations is known to be very successful in predicting the behavior of a wide variety of industrial relevant mixtures. The equation uses a set of molecular parameters that characterize a molecular model based on statistical mechanics concepts, having physical meaning and independent of the thermodynamic conditions. In particular, soft-SAFT represents an optimal platform for this type of calculations because, as any other SAFT-type equation, it explicitly accounts for the hydrogen-bonding and short-range Special Issue: Modeling and Simulation of Real Systems Received: March 14, 2014 Accepted: July 10, 2014 Published: July 17, 2014 3220

dx.doi.org/10.1021/je5002472 | J. Chem. Eng. Data 2014, 59, 3220−3231

Journal of Chemical & Engineering Data

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separated and quantified. For a mixture of associating chain molecules, the total Helmholtz energy A can be written in the usual SAFT-type form as

association physical interactions, and has proven to be very accurate in describing the thermophysical properties of different ionic liquids systems.10,15−21 In addition, the Free-Volume Theory proposed by Allal and co-workers22,23 has been recently integrated into soft-SAFT for the evaluation of transport properties.24,25 A very recent paper with some preliminary calculations for [Cl], [MeSO4], and [Me2PO4] imidazolium pure ionic liquids show promising results.21 Also, successful results for n-alkanes, n-hydrofluorocarbons, 1-alkanols, fatty acid esters, and water make us confident that this methodology is valid for application to ILs.24−27 The [Cnmim][BF4] family of ILs has been chosen because of the extensive amount of available experimental data, which will help to show the purpose of this work. In fact, several authors have used SAFT-type equations to study the thermodynamic properties of [Cnmim][BF4]. A summary of the published papers using SAFT-type and other molecular modeling approaches for the density and the gas solubility of this family can be found in a review published in 2010.9 Some additional more recent contributions of authors who have studied [Cnmim][BF4] ILs with SAFT equations can be found in the additional references.18,28−35 However, the study of their transport properties and, in particular, viscosity is almost nonexistent. Apart from correlations obtained from the fitting to experimental measurements, there are very few contributions in which a viscosity approach has been integrated into an equation of state to model the viscosity of some ionic liquids. Aside from our preliminary results on the viscosity of [Cl], [MeSO4], and [Me2PO4] imidazolium ILs,21 only Polishuk and co-workers have used the Modified Yarranton−Satyro Correlation and the Free-Volume Theory coupled into the SAFT + Cubic Equation of State (EoS) to describe the transport properties of some [Cnmim][BF4] and [Cnmim][PF6] ILs.36,37 Hence, the work presented here has a major novelty: the modeling of the viscosity of ILs and their mixtures, completing the density, derivative properties, and gas solubility calculations. However, we remind the reader that the main goal of this work is not to provide a full characterization of this known system, but to show how a molecular-based equation of state as softSAFT can provide helpful insight when experimentally working with these fluids. The rest of the paper is organized as follows: the next section briefly summarizes the different modeling approaches used in this work: the soft-SAFT equation of state, for thermodynamic calculations, and the Free-Volume Theory for the calculation of the viscosity. Then, a variety of examples focused on the description of some themophysical properties of the [Cnmim][BF4] family of ILs will be highlighted in order to show the capabilities of the soft-SAFT + FVT as an accurate modeling tool, able to be a complementary ingredient to assess the experimental work. Some concluding remarks will be given in the last section.

A = Aid + Aref + Achain + Aassoc + Apolar

(1)

where the superscripts id, ref, chain, assoc, and polar represent, respectively, the ideal, the reference (segment), the chain, the association, and the polar contributions to the Helmholtz energy. The reference term in the soft-SAFT equation takes into account the monomer−monomer interactions by a LJ intermolecular potential as the reference fluid, characterized by two molecular parameters: the segment diameter σii and the dispersive energy between segments εii. In soft-SAFT, the EoS of Johnson et al.,42 fitted to computer simulation data, is used to evaluate this term. The extension to mixtures requires the application of the van der Waals one-fluid theory with generalized Lorentz−Berthelot mixing rules for the size σii and energy εii parameters:

⎛ σii + σjj ⎞ σij = ⎜ ⎟ ⎝ 2 ⎠

(2)

εij = ξij(εiiεjj)1/2

(3)

where ξij is an energy binary adjustable parameter to account for differences in the cross-dispersive energy in complex systems. ξij is equivalent to (1 − kij), which is commonly used in other SAFT-type equations. ξij is fitted, when necessary, to phase equilibrium binary data. The chain term, Achain, and the association term, Aassoc, are formally identical in the different versions of SAFT, only differing in the reference fluid model used in each version. They are based on Wertheim’s TPT1 theory and they account for chain formation from the monomers of the reference fluid and for short-range highly directional attractive interactions, respectively: n

Achain = RT ∑ xi(1 − mi) ln gLJ i=1 n ⎡ Mi ⎛ Xα⎞ M⎤ Aassoc = RT ∑ xi⎢ ∑ ⎜ln Xiα − i ⎟ + i ⎥ ⎢ ⎝ 2 ⎠ 2 ⎥⎦ i=1 ⎣ α=1

(4)

(5)

where R is the constant of gases and T is the temperature. In the chain term expression (eq 4), mi is the number of segments in component i forming the chain, xi is the corresponding mole fraction of each compound, and gLJ is the radial distribution function of the reference fluid at contact. This function can be obtained through the expression provided by Johnson et al.43 for LJ chains. In the associating term (eq 5), Mi is the number of associating sites of component i, and Xαi is the mole fraction of component i not bonded at site α, which accounts for the contributions of all associating sites in each species. Xαi can be obtained from a solution of the following mass-action equation:

2. COMPUTATIONAL METHODS 2.1. Soft-SAFT. The soft-SAFT equation of state11,12 is a successful SAFT-version coming from the original SAFT equation,13,14 and it is characterized by the use of a LennardJones (LJ) intermolecular potential in the reference term. As all SAFT equations, it is based on the first order thermodynamic perturbation theory (TPT1) of Wertheim,38−41 and it is expressed as a sum of contributions to the total Helmholtz energy of the system, where different intermolecular effects are

Xiα =

1 1+

n m ρ ∑ j = 1 xj ∑b = 1 M jbX jbΔab , ij

(6)

The specific site−site function, Δαβ,ij, includes two additional parameters describing the size and strength of the association interactions: Kαβ,ij, which is the site−site bonding-volume, and εαβ,ijHB/kB, which defines the site−site association energy. 3221



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moment has been approximated by that of [C2mim][BF4] following the approximated procedure suggested by Izgorodina et al.51 Owing to the lack of dilute-gas experimental data, the value of κ has been taken from the value of water proposed in the original work of Chung et al.48 The purpose of this calculation is not to have an exact value but to see its order of magnitude compared to the total viscosity of the system and estimate the possible errors associated with ignoring it or taking it into account. Results (not included here) have shown that the influence of the dilute-gas term in the range of temperatures of interest is negligible, contributing less than 0.05 % to the total viscosity at the highest temperature with the rest coming from the dense-gas term. This conclusion was somewhat expected because of the very high viscosities of ionic liquids, which mainly depend on the dense-state term, and because of the near-zero vapor pressure of ILs at the working temperature range. As a consequence, to simplify the problem, we have omitted this term. The omission of the dilute term had already been done in the first paper of Allal and co-workers22 and has also been successfully applied in a recent work.21 The dense-state term is obtained through an expression that relates the viscosity to the microstructure of the fluid and, in parallel, to the free space among the molecules, defined as a free-volume fraction through an exponential relation originally proposed by Doolittle:52

Equations 4 and 5 are written in a general form for multicomponent mixtures. The already tested numerical routine proposed by Tan et al.44 is employed in this work to solve the association term. A multipolar term is also added for dealing with fluids where polarity affects the physical interactions in a significant way, as it is the case of carbon dioxide. This expression is obtained using an extension of the theory of Gubbins and Twu45 to chain fluids, following the ideas of Jog et al.46 The perturbative approach is described in detail in the work of Stell and coworkers.47 This treatment requires the inclusion of an effective quadrupole moment, Q, and the fraction of segments in the chain that contains this effective quadrupole, xp. These two parameters are correlated by the following equation:

Q = Q exp·xp

(7)

where Qexp is the experimental quadrupole moment of the molecule (C·m2). For additional details of the soft-SAFT terms, the reader is referred to the original soft-SAFT articles.11,12 2.2. Free-Volume Theory. The viscosity of ionic liquids is reproduced using the Free-Volume Theory (FVT) approach of Allal and co-workers.22,23 This methodology has recently been coupled into soft-SAFT,24,25 and the reader is referred to these papers and the original ones of Allal et al. for a full description. Only the main relevant details are included here. This approach expresses the viscosity as a sum of two contributions:

η = η0 + Δη

⎛ B⎞ Δη = 10−14ρNaL2ζo exp⎜⎜ ⎟⎟ ⎝ fv ⎠

(8)

where ρ is the density (mol/L), Na is Avogadro’s number (mol−1), ζ0 is a friction coefficient related to the diffusion process and the mobility of the molecules (kg/s), L2 is an average quadratic length related to the size of the molecules (Å2), B is a parameter related to the free-volume overlap among the molecules, and f v is the free-volume fraction. The ζo friction coefficient is proportional to the energy of dissipation for a given length of dissipation bf (Å) as

where η (mPa·s) is the total viscosity of a system, ηo is the viscosity of a dilute gas, where the intermolecular effects are neglected, and Δη is a dense-state correction term, related to the density and the microstructure of the fluid. The dilute gas term is obtained using a modification of the Chapman-Enskog kinetic theory proposed by Chung et al:48 ηo = 40.785·10−2

M wT vc

2/3

Ω*(T *)

Fc

(10)

(9)

ζo = 1010

where Mw is the molecular weight (g/mol), vc is the critical volume (L/mol) and Ω* is a reduced collision integral, whose expression can be found in the bibliography for the LJ potential.49 Fc is an empirical factor introduced by Chung et al.48 that includes three different variables accounting for nonsphericity (ω), polarity (μr), and hydrogen bonding (κ), respectively. In summary, the calculation of the dilute gas term requires the evaluation of the critical parameters, the acentric factor and the dipole moment of the molecule. In the case of ionic liquids (and heavy molecular fluids), there is a significant degree of uncertainty associated with these data. Furthermore, it is necessary to add an empirical parameter κ in associating systems. This parameter is usually fitted to dilute-gas viscosity data, which is not available for ionic liquids. With all these aspects in mind, we have decided to evaluate the effect of the dilute gas term (in an approximate way) on the total value of the viscosity of the fluid, in order to determine how important it is in the final calculation. For this purpose, a preliminary viscosity calculation for [C6mim][BF4] at atmospheric pressure in the temperature range 293 K to 393 K has been performed. The critical parameters and the acentric factor of [C6mim][BF4] have been taken from the work of Valderrama and Robles,50 who determined those properties using an extended group contribution method. The dipole

1/2 −3 E ⎛ 10 M w ⎞ ⎟ ⎜ Nbf ⎝ 3RT ⎠

(11)

where E (J), is the potential energy of interaction, considered as a sum of two terms, an ideal gas term and a second term αρMw related to the density, with α being the energy barrier that the molecule has to cross to diffuse (J·m3/mol·kg). The combination of eqs 10 and 11 leads to a final expression for the dense fluid term, in mPa·s: 10−3M w 3RT 3/2 ⎤ ⎡ ⎛ 3 2 10 P + αρ M w ⎞ ⎥ ⎟ exp⎢B⎜ ⎢ ⎝ ρRT ⎠ ⎥⎦ ⎣

Δη = Lv (0.1P + 10−4αρ2 M w )

(12)

where Lv (Å) = L2/bf. This methodology characterizes each system by means of three parameters: α describes the proportionality between the energy barrier and the density, B corresponds to the free-volume overlap, and Lv is a length parameter related to the structure of the molecules and the characteristic relaxation time.23 These parameters are fitted to available experimental viscosity data and related, when possible, to the molecular weight of members of the same chemical family. The thermodynamic variables, such as the pressure P 3222



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A successful modeling involves, as a first and primary step, the choice of an adequate molecular model for the molecule of interest. This model needs to be accurate enough to include the main physical features of the molecule, but simple enough to be used for practical applications and provide quick answers. Here, we keep the molecular model chosen for the [Cnmim][BF4] in a previous contribution.10 Alkyl-imidazolium-[BF4] ionic liquids are modeled as homonuclear chainlike molecules with one associating site in each molecule, mimicking the anisotropic interaction occurring when one cation−anion pair approaches another. The model considers that the cation and the anion are together forming a chain. This assumption is based on results obtained from molecular dynamics simulations showing the continuous formation of ionic pairs or ionic clusters of these systems,54−56 and is supported by the accurate results obtained in previous contributions.10,15−21 For more details, the reader is referred to ref 10. 3.1. Assessment of the Published Density Data. In a previous contribution, the temperature-density diagram of the [Cnmim][BF4] family had already been studied with softSAFT.10 The set of molecular parameters for the soft-SAFT equation were fitted to the available density−temperature data at atmospheric pressure. However, since the year of publication of that article (2007), several new experimental density data have been published. Among them, it is interesting to remark a significant discrepancy in the density data of [C2mim][BF4]. In 2007, the only available data for this IL was the pioneering and outstanding work of Seddon and co-workers,57 who provided in 2002 a huge amount of experimental density data for many ionic liquids. However, for the particular case of [C2mim][BF4], the density data falls far from more recent works.58−61 In this case, we will use the soft-SAFT equation of state to discriminate among the different sets of data for this compound. For that purpose, it is necessary to revise the previous correlations presented in ref 10. Those correlations were done by fitting [Cnmim][BF4] (with n = 2, 4, 6, 8) to temperature-density data at atmospheric pressure. To do our test, we will remove the information concerning [C2mim][BF4] and will recalculate the correlations with the remaining three ILs. The recalculated correlations are

(MPa), the temperature T (K), and the density ρ (mol/L), are obtained from soft-SAFT. Equation 12 can be directly used for multicomponent systems, although viscosity mixture parameters are required. A linear compositional mixing rule of the Lorentz type is used for the three parameters: n

Ymixt =

∑ Yxi i i=1

(13)

where Y applies for α, B and Lv. No adjustable parameters for the viscosity treatment of mixtures are included in eq 13, in a similar manner as done for n-alkanes, n-hydrofluorocarbons, and fatty acids esters in previous work.24,25,27 Although these mixing rules failed to model the complex nature of water +1alkanols mixtures,26 it is expected that they can reproduce mixtures of ionic liquids of the same family with a good degree of accuracy in a predictive way.

3. RESULTS AND DISCUSSION In this section, we show several examples where the use of a theoretical approach becomes a helpful tool either to address

Figure 1. Single-phase properties for the [Cnmim][BF4] family. Temperature-density diagram for [Cnmim][BF4] for n = 2 (blue ○,□,◇,△,+),57−61 n = 3 (brown ○),61 n = 4 (red ○,◇),60,62 n = 5 (gray ○),61 n = 6 (green ◇,△),60,63 n = 8 (black +),62 and n = 10 (magenta □),57 at atmospheric pressure. Symbols are experimental while the lines are the soft-SAFT results. Lines for [Cnmim][BF4] with n = 2, 3, 5, and 10 are predictions.

discrepancies on the experimental data, or to explore areas where data are not available. As mentioned in the introduction, the soft-SAFT + FVT equation of state has been chosen on the basis of its successful results when applied to several ILs systems.10,15−21 We focus our efforts on the description of density, compressibility, solubility, and viscosity of the imidazolium family with the [BF4]− anion. This family of compounds has been extensively studied from an experimental point of view and will be used here to show several examples where the application of softSAFT can be very helpful for an experimental design for specific applications. The different sets of experimental data will be based on references included in the Ionic Liquids Database (IL Thermo v2.0) provided by the NIST Standard Reference Database.53 Among them, those with more experimental measurements in a wider range of thermodynamic conditions will be prioritized.

m = 0.0192M w + 0.1548

(14)

mσ 3(Å3) = 1.9591M w − 149.31

(15)

mε /kB(K) = 8.6439M w − 70.198

(16)

Although the correlations are similar to those presented in ref 10, some differences can be noticed. The next step consists of predicting the molecular parameters for [C2mim][BF4] from the correlation and plot them to see if they match the original data of Seddon et al.57 or they are closer to the other more recent references.58−61 This is shown in Figure 1, where a representation of the temperature-density diagram for [C2mim][BF4] is given. As it can be observed, the soft-SAFT deviates more from the original data of Seddon et al. (squares) for [C2mim][BF4]57 (with an average absolute deviation (AAD %) of 4.71 %) compared to the other data sources. The softSAFT prediction matches the more recent data58−61 in good agreement, falling between the data of Gardas et al.58 (crosses) and Seki et al.59 (diamonds) (AAD% around 1 %), and matching very well the set of Taguchi et al.60 (triangles, AAD% of 0.34 %) and Xu and co-workers (circles, AAD% of 0.46 %).61 3223



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Table 1. Optimized Molecular Parameters for the [Cnmim][BF4] ILs Family and CO2, Including the Average Absolute Deviation (AAD%) of Each Compounda σ

ε/kB

εHB/kB

κHB

g·mol−1

m

Å

K

K

Å3

AAD%

ref

197.97 212.00 226.02 240.05 254.08 282.12 310.18 44.01

3.956 4.225 4.495 4.764 5.005 5.570 6.110 1.571

3.913 3.998 4.029 4.069 4.110 4.170 4.230 3.184

412.3 417.1 420.0 420.8 423.0 426.0 427.3 160.2

3450 3450 3450 3450 3450 3450 3450

2250 2250 2250 2250 2250 2250 2250

0.335d 0.936 0.298 0.293 0.412 0.172 0.934 0.50 to 0.90c

pred. pred. 14 pred. 14 14 pred. 68

Mw [C2mim][BF4] [C3mim][BF4] [C4mim][BF4] [C5mim][BF4] [C6mim][BF4] [C8mim][BF4] [C10mim][BF4] CO2b N

a AAD(%) = ((∑i =data1 points|(calci − expi)/expi·100|)/(NData points)). Parameters for [C4mim][BF4], [C6mim][BF4], and [C8mim][BF4] were obtained from fitting and the rest of the compounds were obtained using the correlations presented in eqs 14 to 16. See text for details. References correspond to the original works from which the molecular parameters are taken. bQ = 4.40·10−40 C·m2; xp= 1/3. cThe first number corresponds to the AAD% of the liquid density and the second number corresponds to the AAD% of the vapor pressure. dOnly the set of data with a better match to the model60 has been considered for the calculation of the AAD%.

Figure 2. Pressure−density diagram predictions for [C2mim][BF4] (blue), [C4mim][BF4] (red), and [C6mim][BF4] (green), at 312.8 K (□), 392.8 K (○), and 472.8 K (△). The experimental data (symbols) are taken from the work of Taguchi et al.60 Solid lines are the softSAFT calculations.

This reinforces the idea that the pioneering reported data of Seddon et al. for [C2mim][BF4]57 was not accurate enough. Additionally, the correlated ILs [Cnmim][BF4] with n = 4, 6, and 8 have also been included in Figure 1, together with available experimental data.57−63 Finally, we have added the prediction of the temperature-density diagram for three other compounds, [C3mim][BF4], [C5mim][BF4], and [C10mim][BF4]. Their respective molecular parameters are also obtained using eqs 14 to 16. As it can be seen, very good agreement between the soft-SAFT predictions and the available experimental data57,61 is obtained, with an AAD% below 1% in all cases. It is also fair to note that the data for [C10mim][BF4] also comes from the original work of Seddon and co-workers57 and, on this occasion, good agreement with soft-SAFT predictions are achieved. The whole set of molecular parameters for each fluid, as well as the average relative deviations for each compound, are included in Table 1. This family of compounds is described with and overall average relative deviation (OARD %) of 0.49 %. 3.2. Extrapolation of the Model to High Pressures and Prediction of Second-Order Derivative Properties. In the previous example, it has been shown how soft-SAFT can be used as a tool to discriminate among different sets of data. In

Figure 3. Predicted derivative properties for [C4mim][BF4]. (a) Isothermal compressibility kT at 298.15 K (green ○), 313.15 K (blue □), and 323.15 K (red ◇). Symbols are calculated data using the values fitted to experimental densities from Azevedo et al.66 (b) Isobaric expansion coefficient αp at 298.15 K (green ○), 318.15 K (blue □), and 348.15 K (red ◇). Symbols are experimental data.67 In both figures, lines are the soft-SAFT predictions.

this subsection, we highlight the capability of the equation to predict the thermodynamic behavior of these ILs at other conditions different from those at atmospheric pressure (extrapolation), as well as to evaluate additional thermodynamic properties (prediction). 3224



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Figure 4. Solubility of CO2 in (a) [C4mim][BF4] at 298.15 K (blue), 312.15 K (red), 323.15 K (green), and 348.15 K (brown). Symbols are experimental data from Shiflett and Yokozeki (○)68 and Zhou et al. (□).70 (b) [C6mim][BF4] at 307.55 K (blue ○), 312.15 K (red □) and 323.15 K (green △) using a binary parameter ξ = 0.960. Symbols are experimental data from Chen et al.71 In both figures, the full lines are the soft-SAFT calculations, while the dashed-lines are tie-lines indicating the VLLE area.

Figure 5. Correlations with the molecular weight of (a) α and (b) Lv for the [Cnmim][BF4] ILs (eqs 19 and 20, continuous lines). Parameter B is set to a constant value for all ionic liquids. Circles are the parameter values while lines are the fitted correlations.

With the current set of molecular parameters, the isothermal compressibility and the isobaric thermal expansivity can be directly calculated with soft-SAFT using the following thermodynamic relationships:

First, we have focused in reproducing the density behavior at high pressures up to 200 MPa. Using the set of molecular parameters established to describe the density at atmospheric pressure, we have predicted the density of [C2mim][BF4], [C4mim][BF4], and [C6mim][BF4] at different temperatures, ranging from 313 K until 473 K. The available experimental data,60 covers a very wide range of pressure up to 200 MPa. In Figure 2, the density representation of three isotherms as a function of pressure is depicted. The predictions with the softSAFT EoS are in excellent agreement for the different isotherms in the whole pressure range, showing the extrapolation capability of the model to perform accurate calculations at conditions far from those used in the parameters optimization. Additionally, an excellent and very stringent analysis to demonstrate the validity of any molecular model is the calculation of derived properties, as they are very sensitive to errors. This analysis is of particular interest as it has been shown that ionic liquids have an unusual behavior concerning some second-order derivative properties, such as the isobaric thermal expansivity.64,65 This property exhibits a negative temperature dependency on many ionic liquids studied. Apparently, this is related to a very high degree of cohesion of their solid-like structure, dominated by the strength and range of Coulombic interactions.64,65

1 ⎛ δρ ⎞ 1 ⎛ δP ⎞ ⎜ ⎟ = ⎜ ⎟ ρ ⎝ δP ⎠T ρ ⎝ δρ ⎠T

−1

kT =

αP = −

⎛ δP ⎞ 1 ⎛ δρ ⎞ ⎟ ⎜ ⎟ = k ⎜ T ⎝ δT ⎠ ρ ρ ⎝ δT ⎠ P

(17)

(18)

Unfortunately, no direct experimental data is available for the isothermal compressibility, as this information is usually calculated from derivation of experimental density measurements or using the Tait equation. To illustrate the performance of the isothermal compressibility, Figure 3a displays this property compared to the values proposed by Gomes de Azevedo et al.66 for the case of [C4mim][BF4] at three isotherms (from 298 K until 323 K). The soft-SAFT results are in good agreement with the values proposed in the literature, although soft-SAFT tends to overpredict the isothermal compressibility when increasing the temperature and the pressure. The calculation of the isobaric thermal expansivity can be compared to recent directly measured experimental data from the work of Navia and co-workers.67 The results for [C4mim][BF4] are displayed in Figure 4b for three temperatures ranging from 298.15 K until 348.15 K. As observed in the 3225



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Figure 6. Viscosity of some selected [Cnmim][BF4] ILs at high pressures. (a) Viscosity-pressure diagram of [C2mim][BF4] at a temperature of 282.55 K (black ○), 293.85 K (brown □), 304.05 K (blue ◇), 313.75 K (red △), and 323.75 K (green ∗). (b) Viscosity−pressure diagram of [C4mim][BF4] at a temperature of 298.15 K (black ○), 323.15 K (blue □), and 343.15 K (red ◇). (c) Viscosity−pressure diagram of [C6mim][BF4] at a temperature of 298.15 K (black ○), 323.15 K (blue □), and 343.15 K (red ◇). (d) Viscosity−pressure diagram of [C8mim][BF4] at a temperature of 298.15 K (black ○), 308.15 K (brown □), 323.15 K (blue ◇), 333.15 K (red △) and 343.15 K (green ∗). Symbols represent the experimental data,72−75 while the curves correspond to the soft-SAFT + FVT modeling.

Table 2. Optimized Characteristic Parameters of the Free-Volume Theory for Some Selected Compounds of the [Cnmim][BF4] Family. The Temperature and Pressure Range of Optimization and the Absolute Average Deviation (AAD%) Are Also Included T range K [C2mim][BF4] [C3mim][BF4] [C4mim][BF4] [C6mim][BF4] [C8mim][BF4] [C10mim][BF4]

283 293 298 298 298 293

to to to to to to

MPa 323 363 343 343 343 363

α

P range 0.1 0.1 0.1 0.1 0.1 0.1

Lv

3

J m /mol kg

B·10

Å

AAD%

367.3 423.0 459.3 528.4 587.0 657.1

2.401 2.401 2.401 2.401 2.401 2.401

0.038 56 0.029 02 0.022 81 0.012 71 0.008 264 0.005 420

8.80 5.13 pred 5.80 6.29 7.42 9.11 pred

to 15 to 300 to 120 to 200

figures, the molecular model is not able to predict the unusual behavior of this property, which experimentally decreases with the temperature, while soft-SAFT predicts an increase of αp with the temperature. However, the behavior of this property with the pressure is very well described with excellent agreement at 318.15 K. The analysis of these results suggests that a better ratio between the dispersive energy and the association energy may be found to capture this behavior and indicates a limitation of the current model. In this case, experimental data can help the modelers to fine-tune the parameters of the model, which can then be used to predict other properties. 3.3. Determination of Immiscibility Gaps in Mixtures. Another interesting advantage of theoretical models based on

3

statistical concepts is the capability to explore areas where experimental data is not available. Here, we show an interesting example concerning the solubility of carbon dioxide (CO2) on the [Cnmim][BF4] family. The solubility of CO2 in this ILs family has been experimentally measured by many authors and several equations of state have been used to describe its behavior. In fact, our group modeled these mixtures in a previous contribution (ref 10). However, experiments are always focused at low CO2 concentrations, because the gas solubility decreases very quickly when the composition reaches around 50%. A liquid−liquid like behavior has been observed, indicating the possibility of a vapor−liquid−liquid equilibrium region. The use of a theoretical model can provide insights at any composition range and can help to identify immiscibility 3226



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has been modeled as a nonassociating homonuclear chain of monomers with a specific effective quadrupole moment, affecting approximately one-third of the molecule (xp = 1/3). The molecular model for CO2 and the set of molecular parameters has been taken from a previous work69 and included in Table 1 for completeness. In Figure 4a, the solubility of CO2 in [C4mim][BF4] is calculated with soft-SAFT at four isotherms (283.15 K, 298.15 K, 312.15 K, and 323.15 K) and compared with the experimental data of Shiflett and Yokozeki68 and the recent results of Zhou et al.70 No binary adjustable parameters are needed to predict the behavior of the CO2 solubility at low concentrations, predictions showing an excellent agreement with the experimental data. Once we are confident on the validity of the model, we study the high CO2 concentration region and also predict a VLLE gap, confirming the suggestion that this mixture exhibits liquid−liquid immiscibility. The upper critical solution temperature (UCST) has been found at 319.6 K, which is approximately 10 K higher than the predictions of Shiflett and Yokozeki.68 Also, we have not been able to find a lower critical solution temperature (LCST) in the lower temperature range where the equation of Johnson,42 used in the soft-SAFT reference term, is valid. Shiflett and Yokozeki predicted a LCST at 199 K.68 In Figure 4b, we have extended the calculation to the [C6mim][BF4] + CO2 mixture. In this case, predictions from pure compounds did not provide quantitative agreement with

Figure 7. Predictions of the viscosity−density diagram for [Cnmim][BF4] for n = 2 (black □),76 n = 3 (red ○),80 n = 4 (blue ◇),75 n = 6 (green □,◇),57,72 n = 8 (brown ○,+,◇),57,77,78 and n = 10 (gray △),57 at atmospheric pressure. Curves are the soft-SAFT predictions.

areas. In fact, Shiflett and co-workers used the Peng−Robinson to identify this region for the [C4mim][BF4] + CO2 mixture.68 Here, we use soft-SAFT to predict the VLLE region of [C4mim][BF4] + CO2 and to determine a similar behavior for the [C6mim][BF4] + CO2 in an easy and straightforward manner. In Figure 4a,b, the phase equilibrium of these two mixtures is depicted, including a predicted VLLE region. CO2

Figure 8. Influence of the alkyl chain length of the cation on the imidazolium [BF4]. Viscosity−composition diagram of the mixtures (a) [C2mim][BF4] + [C3mim][BF4] at 298.15 K (red ○) and 343.15 K (blue ◇), (b) [C4mim][BF4] + [C6mim][BF4] at 298.15 K (red ○) and 308.15 K (blue ◇), (c) [C3mim][BF4] + [C6mim][BF4] at 298.15 K (red ○), 323.15 K (blue □), and 343.15 K (green ◇), and (d) [C2mim][BF4] + [C6mim][BF4] at 298.15 K (red ○), 308.15 K (blue □), and 343.15 K (green ◇). Symbols are experimental data,79,80 while the lines are the softSAFT predictions. No binary interaction parameters are used. 3227



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the available experimental data from Chen and co-workers.71 Because of the increase of the alkyl chain length and the asymmetry of the system, a binary parameter ξ is needed for quantitative agreement. This parameter has been fitted to an intermediate isotherm, and it has been used to predict the rest of them. Using ξ = 0.96, a very good description at 307.55 K, 312.55 K, and 323.15 K is achieved. Then, the high CO2 composition region has been explored. Again, a very similar diagram to that exhibited for the CO2 + [C4mim][BF4] mixture was found, with a VLLE region slightly bigger than before. It is believed that the immiscibility gap will increase with the alkyl chain length of the cation in the IL. The UCST is found at an almost identical temperature as in the previous case (T = 319.7 K). The additional information obtained from these predictions is very helpful in order to guide the experiments on ILs for which not much data are available. 3.4. Assessment of the Published Viscosity Data. The use of a molecular-based equation of state such as soft-SAFT is very helpful to deal with a wide variety of thermodynamic properties. The integration of the Free-Volume Theory (FVT) for the evaluation of transport properties is particularly useful to assess the viscosity of highly viscous fluids, such as ionic liquids. Using the [Cnmim][BF4] family as a case study, we have found important discrepancies in the experimental viscosity data reported for this family, particularly at low temperatures. Once again, a semipredictive theoretical model can provide some insights and guide the experimental work. The first attempt to model the viscosity of the [Cnmim][BF4] family was done by selecting the viscosity-temperature data at atmospheric pressure of [C2mim][BF4], [C4mim][BF4], [C6mim][BF4], and [C8mim][BF4]. The purpose was for one side to reproduce the experimental viscosity behavior with the FVT and, for the other side, to identify trends between the viscosity parameters α, B, and Lv and the molecular weight. However, a quick test showed important parameter degeneracy, with a massive amount of possible solutions offering minimal deviations with respect to the experimental data. In addition, the discrepancies in the viscosity data at low temperatures were providing too much noise in the optimization process. At this point, the strategy was changed and a search for viscosity data at high pressures was performed. Some authors had published a significant amount of viscosity data at different isotherms in a wide range of pressures for [C2mim][BF4], [C4mim][BF4], [C6mim][BF4], and [C8mim][BF4].72−75 These data were then used to fit the viscosity parameters. This second approach significantly reduced the parameters degeneracy and allowed the finding of a set of optimal values. Still with that, as our interest is in developing a tool with predictive and extrapolation capabilities, we refined the fitting to identify clear tendencies between the viscosity parameters and the molecular weight. The barrier energy α parameter was found to linearly increase with the molecular weight. This result was expected as it had already been noticed for the n-alkanes, the 1-alkanols, and the fatty acids esters families.24−27 Also, it is interesting to notice that the values of α are significantly high compared to the values of the previous families. This is logical as it is expected that the size of the molecule, the cation−anion asymmetry, and the effect of the short-range interactions will increase the barrier energy of diffusion. The value of the freevolume overlap B was seen to change very slightly from one compound to another. As a consequence, a constant value (B = 2.401 × 10‑3) was set up for all compounds. Finally, the Lv parameter was found to decrease following a potential function.

Here, the values of Lv are lower than that for the n-alkanes,24 a factor again related to the correlation length among the molecules, which form a more ordered structure due to the association, as it happened for the 1-alkanols family.26 The equations describing the behavior of α and Lv are shown here and plotted in Figure 5: ⎛ J·m 3 ⎞ α⎜ ⎟ = 2.376M w − 79.91 ⎝ mol·kg ⎠

(19)

Lv (Å) = 5.258· 108M w −4.409

(20)

The results of the fitting for [C2mim][BF4], [C4mim][BF4], [C6mim][BF4] and [C8mim][BF4] are plotted in Figures 6a to 6d, while the final set of selected parameters is included in Table 2. As it can be seen, good agreement is found at all temperatures and pressures, with the main deviations being systematically observed at the lowest temperature and highest pressure, but with a reasonable average relative deviation between 5 % and 9 % in all cases. Once this parametrization is done, the viscosity-temperature diagram at atmospheric pressure is predicted, not only for these compounds, but also for those not included in the fitting. The results for [C2mim][BF4], [C3mim][BF4], [C4mim][BF4], [C6mim][BF4], [C8mim][BF4], and [C10mim][BF4] are plotted in Figure 7. The comparison between the experimental data and the predictive model for [C3mim][BF4] is very good, with an AAD% of only 5.13 %. This is particularly encouraging as all the parameters of this compound have been taken from correlations and no experimental data has been used to fit any of them. The performance of [C10mim][BF4] is also in good agreement with the available data although the AAD% increases up to 9.11 %. In addition, it is interesting to check the performance of softSAFT as it can be used as a tool to discriminate between different sets of data.57,76−80 For instance, three sets of data have been plotted for [C8mim][BF4].57,77,78 The soft-SAFT model fits in fair agreement the data of Seddon et al.57 (brown +, AAD% = 10.3 %) and Mokhtarani et al.78 (brown circles, AAD% = 11.5 %), while it deviates significantly from the data of Sanmamed et al.72 (brown diamonds, AAD% = 28.4 %). For the case of [C6mim][BF4], two different data sources have been compared, with the soft-SAFT model falling in between the two sets of data.57,72 If one considers only the set of data closer to the soft-SAFT model for each compound, the Overall Relative Deviation for all the compounds studied here is 8.74 %. The FVT treatment can be easily extended to model the viscosity of binary mixtures. When dealing with compounds of the same family, no viscosity binary parameters are needed. We have used this approach to evaluate the viscosity of four [BF4]− imidazolium mixtures differing in the length of the alkyl chain of the imidazolium ring. In Figures 8 panels a, b, c, and d, the viscosity of the mixtures [C2mim][BF4] + [C3mim][BF4], [C4mim][BF4] + [C6mim][BF4], [C3mim][BF4] + [C6mim][BF4], and [C2mim][BF4] + C6mim[BF4] at atmospheric pressure is plotted, respectively. Very good agreement is found for all cases, with some overestimation for the most asymmetric mixture [C2mim][BF4] + [C6mim][BF4]. Also, more discrepancies are observed at the lowest temperature (298.15 K), where the values of the viscosity are higher. The results are compared to the data of Navia et al.78 and Dong et al.,79 finding better agreement with the data of the latter reference. 3228



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Funding

4. CONCLUSIONS In this work, the soft-SAFT equation of state combined with the Free-Volume Theory was used as a predictive tool to describe a group of thermophysical properties to provide additional information to that obtained from experiments. In addition, it has also been shown how accurate experimental data can help modelers to refine their parameters. The [Cnmim][BF4] family of ionic liquids has been used as a case study. Several tests have been made to show the advantages of using a molecular-based EoS, either to assess discrepancies found among the experimental data or to explore areas where these data are unavailable. In a first example, the molecular parameters of soft-SAFT have been fitted to temperature−density data at atmospheric pressure for several [Cnmim][BF4] ionic liquids. Several linear trends with the molecular weight have been identified and correlations have been established. Using these correlations, the [C2mim][BF4] density behavior has been predicted allowing discrimination between different sets of data showing significant discrepancies. In a second example, the density behavior at high pressures up to 200 MPa has been predicted in excellent agreement with the experimental data, demonstrating a relevant capacity of extrapolation without a loss of accuracy. In addition, the prediction of derivative properties has also been addressed by estimating the isothermal compressibility and the isobaric thermal expansivity of [C4mim][BF4]. It has been shown that the current model was not able to capture the negative temperature dependency of αp, indicating a limitation of the current parametrization. This highlights the possibility of considering the nonregular behavior of some properties for obtaining molecular parameters. In a third example, the possibility of exploring regions where data are not available has been checked by determining the VLLE regions in the CO2 + [Cnmim][BF4] mixtures in a predictive way. Finally, the viscosity of this family has been addressed by using the Free-Volume Theory. To reduce the parameter degeneracy of the method, high pressure viscosity data have been used to find an optimal set of viscosity parameters for each compound. Trends of these parameters with the molecular weight have also been identified and correlated. This information has been used to predict the viscosity-temperature diagram at atmospheric pressure for a wide variety of [BF4] ILs. In this way, the viscosity at low temperatures can be calculated in a straightforward manner, providing some information which is valid to discriminate among the different values that are found in the literature. In a last exercise, the viscosity of binary mixtures between ILs of the same family has been predicted in good agreement with the experimental data. The main goal of this paper was not to fully characterize the properties of the [Cnmim][BF4] ILs, but to highlight the advantages of using refined molecular-based equations of state as a complementary tool to experimental work or even to assess data. The procedure applied here can be used for any ionic liquid or fluid, as far as accurate structural information is accounted for when building the molecular model.

■

F. Llovell acknowledges a TALENT fellowship from the Catalan Government. This work was partially financed by the Spanish Government under project CENIT SOST-CO2, CEN2008-1027. Additional support from Carburos Metálicos, Air Products Group, and the Catalan Government was also provided (2009SGR-666 and 2014SGR-1582). Notes

The authors declare no competing financial interest.

■

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Assessing Ionic Liquids Experimental Data Using Molecular Modeling

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