Literature examples for the modeling of phase equilibria in systems containing ILs
Ionic liquids (ILs) are a class of salts with a melting temperature below 100 °C, and the study of these compounds is considered priority by the U.S. Environmental Protection Agency. Due to their specific properties, which can be adjusted by changing either the cation or the anion, ILs have received great attention by the scientific community as potential replacements for volatile organic solvents (VOCs), and nowadays, ILs are starting to leave academic labs and find their way into a wide variety of industrial applications . For example, ILs are used for the dispersion of nano-materials at IOLITEC, Air Products uses ILs instead of pressurized cylinders as a transport medium for reactive gases, ION Engineering is commercializing technology using ILs and amines for CO2 capture and natural gas sweetening, and many others.
In order to apply these new compounds in different processes, the study of their physical properties, pure or mixed with other solvents, and phase equilibria (vapor-liquid, liquid-liquid, and solid-liquid equilibria) is crucial from a technological point of view. For example, density is fundamental to develop equations of state and it is also required for the design of different equipments, while viscosity is necessary for the design of processing units and pumping systems, and to study heat and mass transfer processes . On the other hand, the refractive index can be used as a measure of the electronic polarizability of a molecule and can provide useful information when studying the forces between molecules or their behavior in solution .
As known, ILs also show an interesting potential to be used in separation processes and extraction media. Therefore, the knowledge of the mutual solubilities of molecular solvents and ILs prior to their industrial applications is also of primary importance. Moreover, many factors that control the phase behavior of these ionic salts with molecular solvents may be described from the phase equilibrium data.
However, as the number of possible ILs is enormous, this cannot be accomplished via experimental determination. Thus, it is very important to obtain models or empirical equations able to describe satisfactorily the experimental data.
In this chapter, a revision of the different equations applied for the modeling of physical properties of pure ILs and their mixtures, and phase equilibria of binary and ternary mixtures containing ILs, is presented and discussed. Future trends regarding the use of new models, namely equations of state accounting for association effects, are also focused.
2. Physical properties
2.1. Pure ionic liquids
Since ILs are relatively new compounds, experimental data on physical properties, such as density, viscosity, or refractive index of pure ILs and its mixtures with other solvents are required for the design of different equipment and processing units and very useful for developing accurate theoretical models.
Due to innumerable number of ILs that can be synthesized, experimental measurements are impractical for selection of a suitable IL for a specific application. Therefore, development of correlations and theoretical approaches allowing accurate modeling of IL-based systems is essential. This section shows the most common empirical equations used to correlate the temperature dependence of some of the physical properties of ILs.
For pure ILs, temperature dependence of physical properties such as density, speed of sound, or refractive index is very important for the successful and large-scale use of these compounds. Usually, this dependence is described using simple polynomial expressions, mainly equations of first, second and third order [4-6].
Several papers were also published concerning the experimental densities of pure ILs as a function of temperature and pressure [6-9]. The Tait equation  with four adjustable parameters is commonly used to fit these experimental data [6,8]. This equation is an integrated form of an empirical equation representative of the isothermal compressibility behavior versus pressure, and it can be expressed as:
Regarding the variation of viscosity with temperature for pure ILs, a large number of empirical equations for correlating this property of pure fluids and mixtures can be found in literature [4,5,11-14]. The most commonly used equation is an Arrhenius-like law:
where the viscosity at infinite temperature (
According to Seddon
Another empirical equation to correlate viscosity data with temperature was proposed by Litovitz :
This equation is used at ambient pressures and has the advantage of containing only two fitting parameters. Comparing all these equations, in general, the best fits for the variation of viscosity with temperature for pure ILs are obtained with the VFT equation .
As reflected by Harris
A hybrid Tait–Litovitz equation at elevated pressures (up to 126 MPa) was also presented in the literature to correlate the viscosity data for a series of room-temperature ILs . The Litovitz equation is firstly used to correlate the data at ambient pressure, and then the Tait parameters are fitted for the higher pressures. This equation has the advantages of containing fewer fitting parameters than other models and simplicity of data analysis. The results show a good fit between the experimental data and those predicted by this equation.
2.2. Binary and ternary mixtures
In order to better understand the nature of ILs and design any future technological processes, detailed knowledge on the physical properties of ILs mixed with other solvents is required. During the last few years, the number of studies on thermophysical and thermodynamic properties of pure ILs and their mixtures with molecular solvents has increased significantly [4,20-24].
As for pure ILs, the dependence of the physical properties with temperature and composition is also correlated using empirical equations. In general, the change of density, speed of sound, refractive index and viscosity with composition is typically fitted to a polynomial expression although other more specific equations can be also found in literature. As example, the Connors and Wright equation  is employed to describe the variation of density with composition:
As it is known, the above mentioned physical properties can be used to obtain the corresponding excess properties, which are generally fitted to a Redlich-Kister type equation :
An extended version of the Redlich-Kister equation, which takes into account the dependence on composition and temperature simultaneously, is also used to fit the excess properties :
In order to take into account the influence of temperature on the excess properties, all the coefficients
Density, refractive index, and viscosity data for ternary mixtures containing ILs are also common in the literature [27-29] and they are usually fitted to polynomial expressions. In these cases, weight fractions are often used instead of mole fractions, due to the large difference of molar mass between ILs and most organic solvents.
For the modeling of the excess properties, such as excess molar volumes, viscosity deviations, refractive index deviations or excess free energies of activation of viscous flow, the use of empirical equations is commonly adopted. Although the Redlich-Kister equation is also applied to correlate the excess properties for ternary systems containing ILs , the equations more widely employed are those proposed by Cibulka , Singh
Singh et al. equation:
Nagata and Sakura equation:
3. Phase equilibria
Despite the large number of published articles and the broad fields of applications, there has not been a model explicitly derived for phase equilibria of ILs. This lack of models intended for systems containing ILs has forced researchers to use the equations available. But these equations were intended for ionic solutions, or not intended for ions at all. Thus, a model to account for a medium which is composed of ions, without a molecular solvent, is still needed. Nevertheless, the phase equilibria of ILs and their mixtures are being modeled in the literature. The models used will be described ahead. In general, those based on the excess Gibbs energy such as Wilson, NRTL or UNIQUAC were the first to appear. Lately, models with modifications for association (UNIQUAC ASM, NRTL1) and for electrolytes (PDH, e-NRTL) were also applied for this kind of systems in order to improve the results obtained with the initial models. Besides, Equations of State (EoS) have also been applied, especially for mixtures with gases and for broad ranges of pressure.
3.1. gE-based models
Most of the gE models available in the literature are for non-electrolytes. Thus, many authors have been using these models, or models for electrolyte solutions, for the phase equilibria of systems containing ILs. For example, for binary systems containing ILs it is common to use the model developed by Debye-Hückel (which was derived for small salt concentrations), although it is not recommendable for solutions at high ionic concentration.
The models for the correlation of these experimental data can be split into two main groups:
i. The model developed by Pitzer has created a new generation of theories which use multiparameter regression. In the ion-interaction Pitzer model  the ion-interaction parameters are dependent on temperature and pressure, and it takes into account the Debye-Hückel constant.
The three-parameter Pitzer-ion interaction model has been successfully used for modeling vapor-liquid data of mixtures of ILs with water [34-37] or with alcohol  and has the following form for a binary 1:1 electrolyte solution:
In these equations,
In the last years, the Extended Pitzer model of Archer [39,40], in which the third adjustable parameter in the Pitzer model is replaced by a two-parameter function depending on the ionic strength, has demonstrated its accuracy in modeling binary Vapor-Liquid Equilibria (VLE) of systems containing ILs. In this model, the equation for
and a new equation is introduced:
In the previous equations, the ion interaction parameters of the extended Pitzer model of Archer are
ii. The local composition (LC) models give a better empirical description and have physical meaning for the correlation of osmotic and activity coefficients. There are several LC models reported in literature for the modeling of phase equilibria experimental data; such as UNIversal QUAsiChemical (UNIQUAC) , Non-Random Two Liquids (NRTL) , electrolyte NRTL (e-NRTL) , Non-Random Factor (NRF) , modified NRTL (MNRTL) , Mean Spherical Approximation NRTL (MSA-NRTL)  or Extended Wilson (EW) . Furthermore, models with modifications for association such as UNIQUAC ASM or NRTL 1 are also applied for this kind of systems, although their use is less common [63,64].
In local composition models it is assumed that the activity coefficient is composed by two terms: a long-range contribution (LR) and a short-range contribution (SR):
The well-known models of UNIQUAC and NRTL calculate the activity coefficients as follows:
ii.i In the UNIQUAC model , the long-range contribution term is expressed as:
where the term
and the coordination number, z, is set to 10.
The short-range contribution in this model is expressed as:
The expression for the energy parameter,
This model has been used for VLE of binary systems containing ILs . The equation has also been used successfully for Liquid-Liquid Equilibria (LLE) of ternary systems including an IL [66,67], and even for Solid-Liquid Equilibria (SLE) of binary systems [68-70]. The main problem using the UNIQUAC model is the need for structural parameters
ii.ii In the NRTL model , the activity coefficients are calculated as follows:
This is the local composition model most widely used in literature for binary and ternary systems containing ILs, regarding VLE [71-79] with alcohols or water, LLE of binary [80,81] and ternary systems [67,82-85] with ethanol or hydrocarbons or water and also SLE [68-70].
For the explanation of the following models, the Pitzer-Debye-Hückel (PDH) equation  has been used as the long-range term on a mole fraction scale as proposed by Chen
The short-range contribution calculated using different models, such as electrolyte NRTL (e-NRTL) , non-random factor (NRF) , modified NRTL (MNRTL) , mean spherical approximation NRTL (MSA-NRTL)  or Extended Wilson (EW)  are explained below.
ii.iii In the e-NRTL model , the short-range contribution for the activity coefficient is calculated as:
Examples of the correlation of VLE for binary mixtures [38,73-75] and ternary mixtures [73-75] containing ILs and ethanol or water can be found in literature. Nevertheless, the literature is scarce in examples for SLE  and LLE [80,81].
ii.iv The NRF model  calculates the short-range contribution for the activity coefficient of the solvent as:
ii.vi The equation for calculating the short-range contribution for the activity coefficient of the solvent given by the MSA-NRTL model  is as follows:
taking into account that:
The correlation using this model it is not common for the treatment of the VLE data of systems containing ILs .
ii.vii The EW model was presented by Zhao
|Extended Pitzer of Archer||[46-55]|
A list of representative examples found in literature for the modeling of phase equilibria in systems containing ILs with the above mentioned models is presented in Table 1.
Among these correlation models, those which have demonstrated to give the best results in VLE are the Extended Pitzer model of Archer, the NRTL and the MNRTL models. Regarding LLE and SLE, there are less examples available, specially comparing different models. Nevertheless, it is clear that NRTL is, by far, the most used equation. It is also important to highlight that differences in performance among the models are small.
3.2. Equations of state
Equations of state (EoS) are powerful tools, which can be used to describe the properties of pure fluids or their mixtures. In the last 10 years, this kind of models has been widely applied to describe the properties of pure ILs, as well as to model the phase equilibrium (VLE and LLE) of mixtures containing them.
The Peng-Robinson EoS was developed in 1976 by Peng and Robinson  and can be expressed as:
For mixtures of fluids, mixing rules have to be applied to parameters
The application of the Peng-Robinson EoS to systems with ILs has been mainly focusing on the VLE with CO2 and other gases. For example, Shin
A Soave modification of the Redlich-Kwong EoS  has been frequently applied to systems with ILs. The Soave-Redlich-Kwogn (SRK) EoS was introduced in 1972 by Giorgio Soave, and can be expressed as:
In the works involving systems with ILs, the temperature dependent part of the
Shiflett and Yokozeki  first used this EoS to correlate the solubility of CO2 in ILs l-butyl-3-methylimidazolium hexafluorophosphate ([C4mim][PF6]) and l-butyl-3-methylimidazolium tetrafluoroborate ([C4mim][BF4]). The authors obtained an excellent fit between experimental and calculated solubility data, with standard deviations of 17 and 10.5 kPa for each system, respectively. They used the same model to correlate the VLE of CO2 and other gases (like ammonia [107,108] or SO2 ) in different ILs. Later, the same authors modeled the solubility of water in several different ILs using the same EoS . The four binary interaction parameters were determined using binary VLE data from the literature, having obtained standard deviations lower than 0.6 kPa for all systems. The graphical results presented by the authors show a very good agreement between experimental and calculated solubilities of water in all ILs considered. The same research group has also used the SRK EoS to study the phase behavior of ternary mixtures CO2/H2/[C4mim][PF6] , CO2/SO2/1-butyl-3-methylimidazolium methyl sulfate ([C4mim][MeSO4]) , CO2/H2S/[C4mim][PF6]  and CO2/H2S/[C4mim][MeSO4] . For all the previous studies the authors obtained good agreement between calculated and experimental data. Very recently, Shiflett
3.2.3. Statistical associating fluid theory
The original Statistical Associating Fluid Theory (SAFT) was developed in 1989  based on Wertheim’s first-order thermodynamic perturbation theory [117-120]. Its main advantage over the traditional cubic EoS, is that it takes into account the structure of the molecule, similarly to group contribution models. It regards the molecules as chains of hard-spheres, which contain multiple association sites. Several variations of the SAFT EoS have been used to model the phase behavior of systems containing ILs: tPC-PSAFT [121-125], soft-SAFT [126-128], hetero-SAFT , PCP-SAFT  and PC-SAFT .
Andreu and Vega  used the soft-SAFT EoS to describe the solubility of CO2 in ILs. They modeled the families of ILs [Cnmim][BF4] and [Cnmim][PF6] as Lennard-Jones chains with one associating site in each molecule. The chain length, size and energy parameters of the ILs were obtained by fitting the model predictions to available density data, obtaining AAD values lower than 0.2%. For the association parameters of ILs, values previously used for alkanols were adopted. The authors found that the model correlations and experimental data for VLE are in good agreement. They later used the same model to describe the solubility of hydrogen, CO2 and xenon in ILs of the family [Cnmim][Tf2N] . In this case, the ILs were modeled as Lennard-Jones chains with three associating sites in each molecule. The pure component parameters for ILs were obtained as in the previous work, and a good description of experimental solubilities was obtained. Recently, Llovell
Ji and Adidharma  used the heterosegmented SAFT (hetero-SAFT) to describe the solubility of CO2 in the families of ILs [Cnmim][BF4], [Cnmim][PF6] and [Cnmim][Tf2N]. The molecules of ILs were divided into groups representing the alkyl chain, the cation head and the anion. To account for the electrostatic/polar interactions between cation and anion, the spherical segments representing the cation head and the anion were assumed to have one association site each, which can only associate to each other. The parameters for the alkyl chains were obtained from those of the corresponding n-alkanes and the parameters for groups representing the cation head and the anion, including the two association parameters, were fitted to experimental IL density data. The model was capable of satisfactorily describing the solubility of CO2 in the ILs studied.
3.2.4. Other EoS
Several other EoS have been used to model systems with ILs. For example, Tsioptsias et al.  used the Non-Random Hydrogen-Bonding (NRHB) model  to describe the phase behavior of binary systems containing ILs of the family [Cnmim][Tf2N], obtaining good agreement between model correlations and experimental data. Wang et al.  used the square well chain fluid (SWCF) EoS  to model the solubilities of gases such as CO2, C3H6, C3H8 and C4H10 in several ILs. Breure et al.  used a group contribution EoS to study the phase behavior of binary systems of ILs of the families [Cnmim][PF6] and [Cnmim][BF4] with CO2, also obtaining good agreement between model predictions and experimental data. Very recently, Maia et al.  applied the Cubic Plus Association (CPA) EoS , which combines the SRK EoS with an advanced association term similar to that of the SAFT type models, to describe the VLE with CO2 and the LLE with water of ILs [C2mim][Tf2N] and [C4mim][Tf2N]. Good agreement was obtained between calculated and experimental data, even though smaller AAD percentage values were obtained for the VLE than for the LLE.
Regarding the treatment of physical properties of pure ILs, temperature dependence of physical properties such as density, speed of sound, or refractive index is described using simple polynomial expressions, mainly equations of first, second and third order. For the viscosity, usually the VFT or mVFT equations are strongly recommended. For binary mixtures containing ILs, the dependence of the physical properties with temperature and composition is also correlated using empirical equations, and their excess properties are generally fitted to a Redlich-Kister type equation. For the fitting of the excess properties of ternary systems containing ILs, the most widely used equation is that proposed by Cibulka.
For the correlation of experimental data concerning phase equilibria of mixtures containing ILs, several gE-based models have been applied in literature (Pitzer, Extended Pitzer model of Archer, UNIQUAC, NRTL, e-NRTL, NRF, MNRTL, MSA-NRTL, EW), being the NRTL model the one that unifies simplicity and satisfactory results for the treatment of vapor-liquid, liquid-liquid and solid-liquid equilibria.
The use of EoS for the modeling of phase equilibria involving ILs is frequent. Unlike what happens with gE models, most of the literature with EoS involve VLE data, rather than LLE or SLE. Nevertheless, many authors have proved that excellent results can be obtained in data correlation or, for some cases, even prediction. The main difficulty with the application of EoS is the calculation of pure component parameters for ILs. Up to date, a general procedure has not yet been defined. Very recently, a complete review on the use of EoS with ILs, with special emphasis on the obtention of model parameters, has been published by Maia and co-workers . The interested reader is directed to that work for further details.