Modeling ionic liquids and the solubility of gases in them: Recent advances and perspectives

Fluid Phase Equilibria 294 (2010) 15–30

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Modeling ionic liquids and the solubility of gases in them: Recent advances and perspectives Lourdes F. Vega a,b,c,∗ , Oriol Vilaseca a,b , Fèlix Llovell b , Jordi S. Andreu a,b a b c

MATGAS Research Center, Campus de la UAB, 08193 Bellaterra, Barcelona, Spain Institut de Ciència de Materials de Barcelona, Consejo Superior de Investigaciones Científicas (ICMAB-CSIC), Campus de la UAB, 08193 Bellaterra, Barcelona, Spain Carburos Metálicos – Air Products Group, C/Aragón, 300 Barcelona, Spain

a r t i c l e

i n f o

Article history: Received 19 November 2009 Received in revised form 7 February 2010 Accepted 9 February 2010 Available online 13 February 2010 Keywords: Ionic liquids CO2 solubility Weak complexation Molecular-based models Classical equations Lattice models Soft-SAFT tPC-PSAFT SWCF

a b s t r a c t The fascinating properties of ionic liquids, their versatility for different applications and their highly non-ideal behavior have promoted the study of these systems from different perspectives. This article provides an overview of the different approaches that have been applied to describe the thermodynamic behavior of ionic liquids and the solubility of selected compounds in them, including carbon dioxide, hydrogen, water, BF3 and other compounds. The paper deals with some of the most recent and refined approaches involving physical models developed to characterize the ionic liquids. Emphasis is put on the models based on statistical mechanics, highlighting the advantages of these models versus classical ones. New modeling results involving the chemical association of BF3 in ionic liquids and interfacial properties of selected ionic liquids within the framework of soft-SAFT are also presented. It is seen that the great advance in the refined modeling tools allows not only quantitative agreement with known experimental data, but also a guide to some of the physics governing the behavior of these systems, a step forward into developing ad hoc ionic liquids for specific applications. © 2010 Published by Elsevier B.V.

1. Introduction Ionic liquids, also known as liquid electrolytes, ionic melts, ionic fluids, liquid salts, or ionic glasses, is a term generally used to refer to salts that form stable liquids. Nowadays it is considered that any organic salt that is liquid below 100 ◦ C falls into this category. They are usually formed by a large organic cation like quaternary ammonium, imidazolium or pyridinium ions combined with an anion of smaller size and more symmetrical shape such as [Cl]− , [Br]− , [I]− ,

Abbreviations: AMQs, additive molar quantities; COSMO-RS, COnductor like Screening MOdel for Realistic Solvents; EoS(s), equation(s) of state; GC, group contribution method; IFP, Institut Franc¸ais du Pétrole; IL, ionic liquid; LJ, Lennard–Jones; LLE, liquid–liquid equilibrium; LLV, liquid–liquid–vapor; NRTL, nonrandom twoliquid model; NRTL-SAC, nonrandom two-liquid segment activity coefficient model; PC, perturbed chain; PCM, polarizable continuum model; PR, Peng–Robinson; pVT, pressure–volume–temperature; RK, Redlich–Kwong; RST, Regular Solution Theory; SAFT, Statistical Associating Fluid Theory; SLE, solid–liquid equilibrium; SWCFs, square-well for chain fluids equation; tPC-PSAFT, truncated Perturbed Chain Polar Statistical Associating Fluid Theory; UNIFAC, universal functional activity coefficient model; UNIQUAC, universal quasi-chemical approach; vdW, van der Waals; VLE, vapor–liquid equilibrium. ∗ Corresponding author at: MATGAS Research Center, Campus de la UAB, 08193 Bellaterra, Barcelona, Spain. Tel.: +34 935 929 950; fax: +34 935 929 951. E-mail addresses: [email protected], [email protected] (L.F. Vega). 0378-3812/$ – see front matter © 2010 Published by Elsevier B.V. doi:10.1016/j.fluid.2010.02.006

[BF4 ]− , [PF6 ]− , [Tf2 N]− , etc., although some symmetric cations are also combined with asymmetric anions to form ionic liquids. In spite of their strong charges, their asymmetry frustrates them from being solid below 100 ◦ C and this is why they remain liquid at these low temperatures. It is believed that the first synthesized ionic liquid reported in the literature is ethanolammonium nitrate, published by Gabriel [1]. However, one of the earlier known truly room-temperature ionic liquids was [EtNH3 ]+ [NO3 ]− , the synthesis of which was published in 1914 [2,3]. Much later, different ionic liquids based on mixtures of 1,3-dialkylimidazolium or 1-alkylpyridinium halides and trihalogenoaluminates, initially developed for their use as electrolytes, were to follow [4,5]. Ionic liquids remained unused for years, mostly because of their moisture sensitivity and their acidity/basicity (the latter can sometimes be used as an advantage). However, when in 1992, Wilkes and Zawarotko [6] reported the preparation of ionic liquids with a new set of alternative, ‘neutral’, weakly coordinating anions such as hexafluorophosphate ([PF6 ]− ) and tetrafluoroborate ([BF4 ]− ), a much wider range of applications for ionic liquids were envisioned, and this has been a field of continuous growth since then. There are some key properties of these compounds that make them particularly attractive for different applications: in fact, their extremely low volatility has become one of their most important

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Fig. 1. Number of publications about ionic liquids published in journals included in the Web of Science List, year by year in this decade. Fig. 2. Distribution of topics related to the presentations about ionic liquids at the 17th International Symposium on Thermophysical Properties. See text for details.

benefits compared to volatile organic solvents used as traditional industrial solvents. High thermal and electronic stability, high ionic conductivity, a wide liquid temperature and good solubility characteristics complete the list of advantages versus other traditional compounds for different applications. As already mentioned, a huge amount of different ionic liquids can be synthesized through the combination of an organic or inorganic anion and an organic (or inorganic) cation, and the number of new synthesized ionic liquids continues to grow incessantly. For instance, just regarding the imidazolium family, there are over 30,000 1,3-functionalized entries recorded in the CAS database. Further scope for derivatization beyond ramification of linear alkyl-substituents, for example with branched, chiral, fluorinated, or an active-functionality, can yield further useful materials. The degree and type of substitution renders the salts low melting, largely by reducing cation–anion Coulombic interactions and disrupting ion–ion packing [7]. In this way, the specific properties of an ionic liquid can be almost selected ad hoc, in order to have a compound with the most appropriate characteristics for a specific application. Among an uncountable amount of different possibilities [8], the use of ionic liquids as media for CO2 /gas separations appears especially promising, as CO2 is more highly soluble than the rest of the gases. Process temperature and the chemical structures of the cation and the anion have significant impacts on gas solubility and gas pair selectivity. In addition to their role as a physical solvent, ionic liquids might also be used in supported ionic liquid membranes as a highly permeable and selective transport medium. Overall, they are considered environmentally friendly compounds (even if it depends on the selection of the cation and the anion) that can be used in catalytic reactions [9], gas and liquid separations, cleaning operations, electrolyte or fuel cells or even as lubricants and heat transfer fluids [10,11]. A summary of capabilities and limitations of ionic liquids in CO2 -based separations respect to a variety of materials is provided in a very recent and detailed contribution by Bara et al. [12]. Fig. 1 depicts the number of contributions in scientific Journals included in the Web of Science related to ionic liquids versus the year of publication in the last decade. The blooming of publications in ionic liquids in the last years is due, in part, to the popularization of these compounds done by Rogers and Seddon (see, for instance, Ref. [13]) and the review paper published by Welton 10 years ago [10]. It should be mentioned that, in spite of their potential for several industrial applications, the list of published works is more related to synthesis, characterization and modeling of these materials, than to applications, being the list of industrial applications still very short. As an example of the distribution of works, Fig. 2 shows the allocation of topics devoted to different aspects of ionic liquids presented in the last International Symposium on

Thermophysical Properties [14]. As observed in the figure, 42% of the presentations were dedicated to modeling (22% to theory and 20% to molecular simulations), while 33% of the contributions dealt with the measurement of some specific properties and the remaining 25% dealt with applications. This average share of topics is also found in other similar conferences and in the open literature. As inferred from the literature, there is a gap between the amount of published works and the academic groups working in ionic liquids, and the amount of processes used at industrial scale. Before analyzing the causes for this gap, we will briefly revise the main industrial processes already available. The first industrial process involving ionic liquids, the BASIL process (Biphasic Acid Scavenging utilizing Ionic Liquids), was announced in 2003, by BASF. The process is used for the production of the generic photoinitiator precursors alkoxyphenylphosphines [15]. As stated by BASF, the BASIL process economically avoids the problems resulting from solids generation by making use of ionic liquids to scavenge acids. Instead of using a tertiary amine, a 1-alkylimidazole is used to scavenge the produced acids. As the imidazole reacts with the acid, an alkylimidazolium salt is formed which is an ionic liquid at the reaction temperature. As a liquid, the alkylimidazolium salt can be easily removed by liquid–liquid phase separation. In addition, economic reclamation of the 1-alkylimidazole through deprotonation is possible. Another industrial application of great success is the use of ionic liquids for cellulose processing [16]: cellulose is insoluble in water and most common organic liquids; before ionic liquids proved to work, CS2 was used to dissolve cellulose for processing. As 0.2 billion tones of cellulose are used as feedstock for further processing, the amount of CS2 to be used is far from being an eco-friendly technology. However, it has been proved that with a solution up to 25 wt% with [bmim][Cl] at 100 ◦ C the cellulose fibers can be cracked or modified. The presence of water in the ionic liquid significantly decreases the solubility of cellulose. After processing, water is added to the cellulose ionic liquid solution and the modified cellulose can be separated. The water is evaporated and the ionic liquid is recovered and can be re-used, making it a green process. Some other uses of industrial applications involve the Dimersol–Difasol process, proposed by the Institute Franc¸ais du Pétrole (IFP): the Dimersol process is a traditional way to dimerize short chain alkenes into branched alkenes of higher molecular weight. The IFP developed an ionic liquid-based add-on to this process called the Difasol process, which is operating at a pilot plan scale (see Ref. [17] for more details on this and other industrial processes). However, the list of successful industrial applications is still too short, compared to the amount of research work done in the area. As mentioned, there is a clear gap between the academic

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

use of ionic liquids and the few industrial applications. This fact was already addressed in a recent report entitled “Accelerating ionic liquid commercialization” [18]. Six barriers were identified to be circumvented for the commercialization of ionic liquids. These barriers include: (1) process engineering, as most of the ionic liquid studies to date have been performed at the laboratory scale under conditions that do not adequately represent operating conditions for full-scale industrial applications. Further work is required to know the industrial processing equipment for reactors or separators incorporating ionic liquid processes; (2) environmental, safety and health issues: although ionic liquids have no detectable vapor pressure and this provides a basis for clean manufacturing – “green chemistry” –, each ionic liquid is likely to be classified as a “new chemical” and may require significant environment, safety and health studies prior to widespread use; (3) economic benefit analyses: ionic liquid applications that have broad chemical industry impact only become a reality when the economies of these applications are favorable and well understood. Detailed economic benefit analyses of ionic liquid processes are necessary and must be based on process engineering data (barrier 1); (4) fundamental understanding of compositional structure versus performance: ionic liquid commercialization requires discovery researchers to not only develop fundamental understanding of ionic liquid synthesis and properties, but to impart these liquids with the chemical processing features needed for important industrial applications and markets. Developing an understanding of the reactions that form ionic liquids, ionic liquid chemical and physical properties, mechanisms/functions in catalysis and separation systems, and interactions with other materials (e.g., container vessels) is essential to their usefulness in industrial applications. Since the combinations of ions for potential ionic liquids are virtually infinite, this is a needed but expensive and time-consuming task; (5) ionic liquid manufacturing: as manufacturing of ionic liquids to date has primarily been done on a small scale, the ability to technically and economically scale up the manufacturing process needs to be demonstrated before ionic liquids will be used in industrialscale volumes. Industry is unlikely to adopt this technology if the expenses of producing bulk ionic liquids remain at their current level; (6) institutional issues: in general, industry has been slow to focus on the application of ionic liquids and slow to recognize their potential economic value. Also, ionic liquid researchers frequently lack insight into the specific details of industrial processes and applications. This fundamental disconnect between discovery researchers and industry, also common to many new product and processes, contributes significantly to slow the ionic liquid technology growth and route into commercialization. Hence, steps should be taken towards overcoming these barriers and make ionic liquids readily available for industrial applications. In order to overcome some of the aforementioned barriers it is important to develop tools that will allow for fundamental understanding of compositional structure versus performance (to surmount barrier (4) on the list). The vast amount of possible ionic liquids and the type of required information needed make the modeling tools excellent candidates to advance in this field. However, these are extremely non-ideal systems, with charged and asymmetric compounds and, hence, most classical equations will fail in capturing their physico-chemical properties, unless the specific interactions are taken into account since the inception of the model. Great advance in the understanding of ionic liquids, and the relationship between their structure and the properties, has been done thanks to the molecular simulation work done by several researchers, including the works of Maginn and co-authors [19–24], Padua and Canongia-Lopes with co-workers [25–30] and ReyCastro et al. [31,32], among others (see the book published by Maginn [23], the recent review of the same author [24] and the work of Bhargava et al. [33] for an extensive bibliography on molec-

17

ular simulation studies done in the field). These simulation results have greatly helped in understanding the local structure of ionic liquids, the solubility of some given compounds in them, and their transport properties. Although very useful from this perspective, the vast amount on computational time required to obtain these properties precludes the use of molecular simulations as standard tools to characterize these systems for screening purposes before selecting an optimum one for a given application. Hence, the use of equations of state or models that reliably provide fast calculations remains an excellent alternative for a quick description of the selected ionic liquids. This contribution aims to review the different modeling approaches that have been used in order to describe the phase behavior of ionic liquids and their mixtures, highlighting the most important efforts done in the last years from the statistical mechanics point of view, as the use of lattice models, group-contribution equations, and SAFT-based approaches, including the soft-SAFT equation [34], the truncated Perturbed Chain Polar SAFT (tPCPSAFT) equation [35] and the square-well for chain fluids (SWCFs) equation [36]. Some new results for the application of soft-SAFT with a relatively simple model to new properties and systems are also presented here. The rest of the paper is organized as follows. Section 2 is devoted to the description of the most recent and refined efforts towards the modeling of ionic liquids, with special attention to the physical model behind them, together with the systems that have been described with them. New results are presented related to the application of the soft-SAFT approach to two challenging cases: (1) the description of interfacial tensions of pure ionic liquids and (2) the description of the weak chemical complexation between a chemical compound and an ionic liquid, focused on the absorption of BF3 in [C4 mim][BF4 ] ionic liquid at different temperatures. Finally, some concluding remarks are provided in last section. 2. Modeling theories for ionic liquids As explained in the previous section, accurate models for the appropriate description of thermodynamic properties of ionic liquids are needed. Several different theoretical approaches, correlations and equations of state (EoS) have been used for that purpose in recent years. Due to the huge amount of contributions published in the last years we have decided to present the information in a tabular form (see Table 1), which shows a summary of all of them, highlighting the equation used, the cations and anions that were considered, and the main properties discussed in the given paper. We have classified them according to the “molecular model” used to describe the ionic liquid in the different approaches: classical cubic equations, activity coefficient and group contribution methods, quantum chemistry calculations and statistical mechanics-based molecular approaches. A sketch of each one of the models for a specific ionic liquid is presented in Table 2. We will summarize next the main approaches, with emphasis on their advantages and limitations versus other approaches. 2.1. Classical models A first effort towards the thermodynamic characterization of ionic liquids was done by the use of relatively simple cubic EoS. In this case the ionic liquid is “modeled” as a whole molecule with a certain volume and cohesive energy, but no specification about its structure nor its association effects are explicitly considered (a sketch of the model is presented in Table 2 – see Table 2b). The Peng–Robinson (PR) EoS [37] was used in a pioneering work of Shariati and Peters to model the solubility of fluoroform in 1ethyl-3-methylimidazolium hexafluorophospate, [C2 mim][PF6 ], in

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Table 1 Summary of published modeling works in ionic liquids, including the references. See text for details. EoS

[BF4 ]

Cation: [Cn mim] Imidazolium PR

[PF6 ]

[Tf2 N]

Other anions

Properties discussed

[38]

[40]

[41]

Solubility of fluroform (CF3 H) and CO2 up to 367 K and 50 MPa. Study of the specific solvation interactions of CO2 + IL.

RK

[43], [45], [52]

[45,46], [48], [52]

[44,45], [50–52], [54]

[45], [47], [49], [52,53]

Solubility of CO2 , SO2 , H2 , ammonia, trifluoromethane in selected ionic liquids at various thermodynamics conditions. Binary and ternary phase diagrams with C6 H6 and C6 F6 . Global phase behavior of CF3 H. VLE for the ternary systems with SO2 /CO2 and CO2 /H2 .

RST

[55–57], [59,60]

[60–62]

[55–62]

[59–62]

Gas solubilities of CO2 and hydrocarbons in imidazolium ionic liquids at low pressures.

[71–74], [76], [81], [87], [96,97], [103–105], [107]

[74–76] [81], [86], [100], [103–105] [107]

[71,72], [74], [77–79], [87–89], [104–107]

[68], [73,74], [80], [82–85], [87], [90–95], [97–99], [101], [103–105], [107]

SLE and LLE of alcohols, water, ketones, thiophene and hydrocarbons from 275 K till the boiling point of the solvent. Ternary mixtures of aromatics, alkanes, alkenes, alcohols and ketones at temperatures in the range 293–323 K. Activity coefficients of aliphatic and aromatic esters and benzylamine. Phase stability analysis in a binary mixture with water.

[108–113] [117]

[108–113] [114], [117]

[108–113]

[108–113] [115–117]

Density, surface tension, viscosity, speed of sound, liquid heat capacity and transport properties of a wide variety of ionic liquids. Solubility of hydroflorocarbons. Solubility of [C(12)mim][Cl] in numerous alcohols. Structural and interactions parameters for dialkylimidazolium-based ILs.

[125–126]

[121], [125,126], [129]

[125,126], [129]

[118], [120–130]

VLE and LLE of alcohols, hydrocarbons, ethers, ketones and water.

[133,134], [137], [140,141]

[133,134], [137], [140,141]

[134,135], [137], [140]

[134,135], [138]

Solubility of CO2 and CF3 H at high pressure up to 100 MPa. VLE and LLE equilibrium diagrams at temperatures from 313 to 343 K. Solubility of C7 hydrocarbons with ethylsulfate anions.

[144], [147]

[144]

[148,149]

[148,149]

[148,149]

Hetero-SAFT

[150]

[150]

[150]

PHST

[154]

[154]

[154]

Henry’s constants for 20 different solutes (hydrocarbons and gases).

tPC-PSAFT

[155–158]

[155–157]

[155–157]

Molar volumes, solubility of CO2 , O2 , CF3 H, benzene, benzaldehyde. VLE for ternary mixtures of water–CO2 –[NO3 ] and 1-hexane-trans-3-hexene-[PF6 ]

Soft-SAFT

[160]

[160]

NRTL E-NRTL NRTL1 NRTL-SAC UNIQUAC

GC + correlations UNIFAC

COSMO-RS

Lattice models

GC-EoS

SWCFs Homonuclear [148] and heteronuclear [149]

Solubility of CO2 . Phase diagrams and phase transitions of ternary mixtures of an organic compound, CO2 and bmim[BF4 ]

[161]

[148]

Molar volumes; solubility of CO2 , propane, propylene, butane, benzene, cyclohexane, propanol, 2-propanol, acetone and water in a range of temperatures from 298 in a range of temperatures from 298 till 368 K and pressures up to 15 MPa. Densities of pure ionic liquids from 293 to 415 K and up to 650 bar.

Solubility of CO2 , H2 and Xe in the range 293–473 K and pressures up to 100 MPa.

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

19

Table 1(Continued). EoS

[BF4 ]

[PF6 ]

[Tf2 N]

Other anions

Properties discussed

[68], [87], [105,106]

[68,69], [98]

SLE, LLE and VLE of binary mixtures between several pyridiniums with alcohols and thiophene. Activity coefficients of quaternary ammonium salts in water. Phase stability analysis in a binary mixture with water.

[110]

[110], [112,113]

[110], [113], [117]

Viscosity, liquid heat capacity and transport properties of a wide variety of pyridinium ionic liquids.

[129]

[129]

[129]

Mutual solubilities of water and hydrophobic ionic liquids.

+

Cation: [Py ] Pyridinium [87], [105] NRTL E-NRTL UNIQUAC

[110], [113], [117]

GC + correlations UNIFAC COSMO-RS

Cation: [H4 S+ ] Sulfonium NRTL

[68]

SLE and LLE of the mixture [EtS3 ][Tf2 N] with thiophene.

NRTL electrolytes

Cation: [H4 P+ ] Phosphonium RST NRTL NRTL1 NRTL2 UNIQUAC

[105]

[98]

Activity coefficients of quaternary ammonium salts in water.

[62]

[62]

CO2 gas solubility in room-temperature ionic liquids.

[105]

[102], [105]

SLE and LLE with alcohols, benzene and alkylbenzenes.

[131]

Solubility of water in tetradecyltrihexylphosphoniumbased ionic liquids.

COSMO-RS

Cation: [H4 N+ ] Ammonium RST NRTL

[62]

CO2 gas solubility in room-temperature ionic liquids.

[70], [89]

SLE of alkanes and aromatics. Activity coefficients at infinite dilution of hydrocarbons, alcohols, esters, and aldehydes.

a range of temperatures from 309 to 367.5 K and pressures up to 50 MPa [38]. The equation reads: P=

RT

v−b

−

a

(1)

v(v + b) + b(v − b)

P being the pressure, v the volume, T the temperature, and a and b the parameters obtained with quadratic mixing rules as modified by Adachi and Sugie [39]: a= b=





xi xj xi xj

 

ai aj [1 − kij − ij (xi − xj )]

bi + bj 2

(2)



(1 − lij )

(3)

x being the mole fraction, with kij , ij and lij as three binary temperature-dependent parameters. ai and bi were obtained by using the critical temperature, pressure and acentric factor of the pure compounds. No information about the way to obtain the critical parameters for [C2 mim][PF6 ] was given in the paper. In general, very good agreement in the whole range of temperatures and pressure was achieved, although the use of three binary parameters, temperature dependent, limits the predictive power of the procedure. Very recently, Carvalho et al. [40,41] also used the PR equation with the Wong–Sandler mixing rules, using the UNIQUAC model to

calculate the activity coefficients to describe Henry’s constants of CO2 with [C2 mim][Tf2 N] and with [C5 mim][Tf2 N], achieving good agreement with the measured data. In this case, the critical parameters were taken from the work of Valderrama and Robles [42], who proposed a group contribution method to estimate the critical properties of 50 different ionic liquids. Shifflet and Yokozeki have published a considerable amount of experimental contributions where they measured the solubilities of different compounds in a variety of ionic liquids and, later, they used the simple Redlich–Kwong (RK) equation of state to model them [43–54]: P= a=

b=

RT

v−b

−

 

a(T )

(4)

v(v + b) xi xj

ai aj

 xi xj





bi + bj 2

1 + ij T





lij lji (xi + xj )

1−

lji xi + lij xj

 (5)

 (1 − kij )(1 − mij )

(6)

In the above model, there are a maximum of four binary interaction (temperature-independent) parameters: lij , lji , mij and  ij for each binary pair. However, in most of the works, only two or three parameters were necessary. Pressure–temperature–composition

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L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

Table 2 Sketch of the different models, for a selected ionic liquid, following different theoretical approaches. Image of the molecule and COSMO-RS model taken from reference [186]. Sketch of GC-EoS taken from reference [144].

experimental data was used to obtain their values. The values of ai and bi were estimated, as in the previous case, by using the critical parameters of the pure compounds. In these works, critical information about ionic liquids was obtained through group contribution methods. The group of Noble and co-workers have published a very complete series of papers where they applied the Regular Solution Theory (RST) to estimate gas solubilities of CO2 and hydrocarbons in several imidazolium, ammonium and phosphonium ionic liq-

uids [55–62]. RST results from the premise that non-ideality in liquid solutions is due to differences in short-range attractive forces between the molecules present. Hence, the authors assumed that short-range forces dominate in ionic liquids because of their weak lattice energies, resulting from low Coulombic forces that are significantly delocalized, with a low degree of nucleophylicity [63]. The reader is referred to the work of Scovazzo et al. [55] and references therein for more details. The set of publications by Noble and co-authors differ from each other on the way used to estimate

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

the solubility parameter, directly related to the cohesive energy density ıS [55–61]: ıS =

 (H v − RT )z 1/2

(7)

VS

with z being the compressibility factor and Vs the molar volume of the solvent. For ionic liquids, as there are no measurable vapor pressures, the heat of vaporization is taken as the enthalpy of vaporization at the salt’s melting point: v = CT (Tmelt ) Hmelt

(8)

where Tmelt is the salt’s melting point temperature and CT is a salt category specific constant. A new parameter KT = (CT − R)z is assumed to be constant for each ionic liquid, so the cohesive energy density of the ionic liquids becomes: ıIL =

 K T 1/2 T melt

(9)

V IL

More recently, Scovazzo and co-workers [62] have estimated ıIL in ionic liquids from surface tension measurements. In summary, one of the main advantages of using these equations is that they are straightforward to use and they are present in any process simulator. However, several parameters, temperature and composition dependent are needed, in most of the cases, to make them readily accessible for ionic liquids’ calculations. This is due to the fact that in general classical equations are missing an important part of the physical nature of ionic liquids. Even if the anion and cation are considered as a neutral pair, they exhibit polarity and hydrogen bonding ability, two facts not taken into account in an explicit manner in cubic EoSs. In addition, a major drawback in the use of cubic EoSs is the fact they require critical parameters of the ionic liquids, which can only be obtained indirectly and with large uncertainties. This fact limits the predictive ability of these equations, and they are used for correlation purposes. 2.2. Activity coefficient models and group contribution methods As a consequence of the limitations already highlighted for cubic EoS, several authors have decided to model ionic liquids using several excess Gibbs energy models, such as the Wilson’s equation [64], the nonrandom two-liquid model (NRTL) [65] and UNIQUAC [66], or group contribution methods as UNIFAC [67]. NRTL and UNIQUAC are well known activity coefficient models, very useful for liquid–liquid equilibria calculations. The binary activity coefficients of the NRTL model are given by



ln 1 = x22 21

 ln 2 = x12 12

 

G21 x1 + x2 G21 G12 x2 + x1 G12

2 2

+

+

12 G12 (x2 + x1 G12 )2 21 G21 (x1 + x2 G21 )2



(10)

(11)

where G12 = exp (−˛12 )

and G21 = exp (−˛21 )

(12)

 12 ,  21 and ˛ are three parameters adjusted to experimental solubility data of the ionic liquids. In general,  12 and  21 are temperature dependent, while ˛ is usually set to a constant unique value. Concerning UNIQUAC, parameters ri and qi, related to the number of segments and external contacts of the molecule of type i, respectively, are usually related to the molar volumes of the ionic liquids following relatively simple expressions. Among the classical excess Gibbs models, Domanska and coworkers have published several works, where they have used NRTL to model the solid–liquid and liquid–liquid equilibria of several

21

ionic liquids and different solvents, like alkanes, alkanols, tiophene or water [68–71]. For solid–liquid equilibria (SLE), they used a simplified general thermodynamic equation relating temperature, TSLE , and the mole fraction of the ionic liquid, x1 , in the respective solvent: ln xi =

−Hmelt R

 1

TSLE

−

1 Tmelt



+ ln 

(13)

where Hmelt is the enthalpy of melting of the pure ionic liquid and  is the activity coefficient of the ionic liquid in the solution, calculated by means of the excess Gibbs free energy (GE) by using the Gibbs–Duhem equation. Shiflett and Yokozeki, already mentioned for their use of the Redlich–Kwong equation, have also used NRTL in some of their contributions, in order to correlate their measured data and to model the LLE of refrigerants and fluorinated compounds in imidazolium ionic liquids [72–79]. Several other groups [80–97] have also employed the NRTL equation to study the vapor–liquid and liquid–liquid equilibria of hydrocarbons and water in ILs, being able to correlate the data with a good degree of accuracy. Furthermore, some modified versions from the original NRTL model have also been employed. Belzève et al. [98] have used the electrolyte-NRTL version for modeling imidazolium and pyridinium salts. Domanska et al. also used Wilson, UNIQUAC, NRTL, NRTL1 and NRTL2 equations with parameters derived from the solid–liquid equilibrium to describe the solubility of alcohols, aromatics, alkanes and cyclohydrocarbons in ionic liquids [99–102]. Banerjee et al. [103] have used the polarizable continuum model (PCM) to estimate the parameters for the UNIQUAC model. This approach was used to calculate the structural parameters for 25 dialkylimidazolium-based ionic liquids and the interaction parameters for seven ionic liquid-based ternary systems. Simoni et al. [104,105] have studied the liquid–liquid equilibrium of ionic liquid systems using NRTL, electrolyte-NRTL, NRTL-SAC and UNIQUAC methods. The same authors have later refined their approach considering an asymmetric framework in which different phases have different degrees of electrolyte dissociation, and are thus represented by different Gibbs free energy models [106,107]. Concerning the use of group contribution methods, also known as group additivity relationships, they are useful for correlating a material property with the chemical composition and state of matter of a substance and are becoming more popular for modeling ionic liquids. The basic assumption made is that the physical property of a material is a sum of contributions from each of the material’s component parts. Then, the properties of known materials are correlated with their chemical structure, in order to identify the basic groups and their additive molar quantities (AMQs), while the properties of unknown materials are estimated through direct addition of AMQs from constituent chemical groups, or through the use of additive quantities to estimate parameters in more accurate correlations. This was the idea followed by Gardas and Coutinho, who published a series or articles [108–113], where they developed a second-order group additivity method, whose parameters were applied to different correlation equations for the estimation of thermodynamic and transport properties of ionic liquids. In particular, predictive methods for density [108], surface tension [109], viscosity [110], speed of sound [111], liquid heat capacity [112] and transport properties [113] were applied to imidazolium-, pyridinium-, and pyrrolidinium-based ionic liquids (ILs) containing [PF6 ], [BF4 ], [Tf2 N], [Br], [EtSO4 ], or [CF3 SO3 ] as anions. The UNIFAC model has also been used by other authors [114–118] to model mixtures of ionic liquids with alkanes, cycloalkanes, alcohols, water and other solvents. One of the most valuable features of all these methods is their applicability to multicomponent systems under the assumption

22

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

that local compositions can be described in this case by a relationship similar to that obtained for binary systems. However, one of the main disadvantages of these methods is that they depend on an extremely large amount of experimental data. Furthermore, the absence of the volume and surface parameters poses a hindrance in the calculation of the binary interaction parameters for UNIQUAC and UNIFAC models.

for the molecule describing the physics of the system is related to a major predictive ability, hence enhancing the possibility of extending the range of application of the equation. In the next subsections, we outline the latest and more refined efforts towards the modeling of ionic liquids and their solubilities. We have selected some physical approaches that have been recently used for this purpose due to their potential for predicting other properties.

2.3. Quantum chemistry calculations Unimolecular quantum mechanical calculations, by means of the conductor-like screening model for real solvents COSMORS [119], have also been used by several authors to predict vapor–liquid and liquid–liquid equilibria for systems involving ionic liquids and a wide variety of solvents [118,120–131]. In contrast to the group contribution methods, COSMO-RS calculates the thermodynamic data from molecular surface polarity distributions (Table 2c), resulting from quantum chemical calculations of the individual compounds in the mixture, without the need of experimental data. Although time-consuming, one advantage of this procedure is that the quantum chemical calculations have to be performed just once for each molecule or ion of interest. Most of these works were done by the use of the COSMOtherm program [132]. This program describes all the interactions between molecules as contact interactions of the molecular surfaces, and relates them to the screening charge densities,  and ’, the most significant descriptors of the interacting surface pieces. The COSMO output provides the total energy of a molecule in its conductor environment and the 3D polarization density distribution on the surface of each molecule. This information acts as an input for the statistical thermodynamic calculations therein after and it is independent of the solvent dielectric constant and temperature. The COSMO-RS method depends only on a small number of general or, at most, element-specific adjustable parameters (predetermined from known properties of a small set of molecules) that are not specific for functional groups or type of molecules. For ionic liquids, the cation and the anion are treated as separated compounds with the same mole fraction during all the procedure. In the work of Banerjee et al. [127,128], a sequential geometry optimization approach was used, and the charge densities were constructed starting with the imidazolium ring and adding successive alkyl groups until the desired cation was built. Then, the anion was added to the cation via a dummy atom reoptimized to obtain the minimum energy configuration. Concerning the application of this methodology, several authors have focused in the description of LLE systems involving ionic liquids and alcohols, hydrocarbons, ethers or ketones [120–126], while other researchers studied the VLE of ionic liquids with the same solvents and water [125–131]. In most of the cases, this approach provided qualitative predictions of the thermodynamic properties of these systems, although some model limitations for LLE predictions were found, especially for the anions influence [126]. As outlined, a main advantage of this method is that no experimental data is needed, being their main limitation the extensive computational time and also that in some cases the comparison with experimental data is only qualitative. In spite of this, having a truly predictive tool is a great step towards the characterization of un-synthesized ionic liquids. 2.4. Statistical mechanics-based approaches The capabilities of lattice models, chain fluid theories and other molecular-based approaches are attracting more and more scientists and engineers until the point that they are becoming standard tools for engineering purposes. The advantage of building a model

2.4.1. Lattice models As ionic liquids are usually composed of cations and anions with long alkyl chains or a long chainlike structure and, considering their negligible vapor pressures, they seem to share features of the polymer systems. In that sense, several authors have tried to model ionic liquid by the use of lattice models, assuming that the fluid structure can be approximated by a solid-like structure (see Table 2d). Several scientific groups have followed this approach: Ally et al. [133] used an irregular ionic lattice model, consisting of a two parameter model in which CO2 is considered the “solute” and the ionic liquids is the “solvent”, to predict carbon dioxide vapor pressure and solubility in several imidazoliums. Yang et al. [134] have successfully correlated infinite diluted activity coefficients, vapor pressures of solvents and liquid–liquid equilibria of ionic liquids solutions with a modified lattice model, which takes into account the hydrogen bonding through an exchange energy function. Tome et al. [135] have modeled high pressure experimental densities of imidazolium ionic liquids by means of the SanchezLacombe equation of state [136]. Kim et al. [137] have used a group contribution form of a nonrandom lattice-fluid model to predict solubilities of carbon dioxide in ionic liquids. Bendova and Wagner [138] have described the liquid–liquid equilibria in systems 1ethyl-3-methylimidazolium ethylsulfate + C7 hydrocarbons using a molecular-thermodynamic lattice model proposed by Qin and Prausnitz [139]. In a recent work, Hu and co-authors [140] have extended and applied a recently developed lattice-fluid equation that included holes into the close-packed lattice model [141], to the description of thermodynamic properties of the [Cn mim][BF4 ] and [PF6 ] families and the solubility of CO2 in them. Based on the chemical association theory, the resulting lattice fluid equation was expressed as p˜ = T˜



−ln(1 − ) ˜ +

 1

z 2 ln 2 z

r





˜ +1 −1 

−

z  ˜ 2

−

z 1 z 1 (3 ˜ 4 −4 ˜3 +  ˜ 2 )− (10 ˜ 6 −24 ˜ 5 +21 ˜ 4 − 8 ˜3 +  ˜ 2) 4 T˜ 12 T˜ 2

+

r−1+ 2 T˜  ˜ r

where D = exp

1 T

−1



[1 + D(1 − )] ˜ 2−1 [1 + D(1 − )][1 ˜ + D(1 ˜ − )] ˜



(14)

(15)

z being the coordination number, T˜ the reduced temperature, p˜ the reduced pressure and  ˜ the reduced density. With this approach, each pure fluid is characterized by four molecular parameters: the coordination number (z), the segmental volume (v∗ ), the segment number (r) and the temperaturedependent interaction energy (ε). For vapor–liquid binary mixture calculations, mixing rules are applied and a binary adjustable parameter kij appear in the calculations. Xu et al. applied this model to the calculation of 48 different pure ionic liquids and 40 binary mixtures [140]. z and v∗ have a fixed and constant value for all fluids, while the others were adjusted to experimental data. For liquid–liquid calculations, a parameter Cr describing the effect of the mixture composition on the chain length parameter r was further used, and a satisfactory correlation was obtained.

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

2.4.2. Group contribution EoS The group contribution equation of state approach (GC-EoS) was developed in the 80s by Skjold-Jorgensen [142]. This method differs from previous GC methods (UNIFAC) from the way the GC concept is applied. In Section 2.2, an activity coefficient model with a builtin group contribution method was used for calculation of mixture parameters of an EoS through appropriate mixing rules. Here, EoS parameters are evaluated using a built-in group contribution rule. In this case, the EoS is based in the generalized van der Waals function combined with the local composition principle. The residual Helmholtz energy of the system is evaluated through the sum of a free-volume term and an attractive term: ares (T, ) af v (T, ) aattr (T, ) = + RT RT RT

23

system was calculated as ahscf (T, ) aattr (T, ) aassoc (T, ) ares (T, ) = + + RT RT RT RT

(17)

where the superscripts hscf, attr and assoc refer to hard-sphere chain fluid, attractive interaction and association interaction, respectively. The hscf contribution is expressed by the effective two-particle cavity correlation function: ar (a = 0)  ahscf 2e xi rjk(i) ln yjk = r˜ K − RT RT i ri i=1 j=1 k=j K

(16) −

M M M K   

M

M

2e xi rjkl(i) ln yjkl

(18)

The free-volume term is described by the Mansoori and Leland expression for hard spheres and includes the hard-sphere temperature dependent diameter parameter [143]. The attractive part is a group contribution version of a density dependent NRTL type expression. In 2007, Breure et al. [144] studied the solubilities of the homologous families [Cn mim][PF6 ] and [BF4 ] with CO2 . Ionic liquids were modeled as a sum of different functional groups, containing one CH3 group, nCH2 groups (depending on the length of the alkyl chain) and a new functional group consisting of the methylimidazolium cation plus the anion (see Table 2e). The GC-EoS contains an important amount of pure group and group-group interaction parameters that need to be parameterized. The free-volume term includes the critical hard-sphere diameter (dc ). Breure et al. have followed the work of Espinosa et al. [145] and used a correlation using information of the van der Waals molecular volume (r). In the free-volume term, the volume Rk and surface Qk parameters of [Cn mim][PF6 ] and [Cn mim][BF4 ] were calculated by dividing the anion in constituent units whose information could be found in the literature [146]. Concerning the attractive term, pure group and binary interaction parameters for many gases and solvent groups were already reported in the literature. The attractive parameter (gii ) for the anions and the binary interaction parameters kij and ˛ij between the anion and the paraffin groups were calculated by fitting infinite dilution coefficients of alkanes in the ionic liquids. The binary parameters between the anions and CO2 were fitted to bubble point data of CO2 in the ionic liquid. Very recently, Kuhne et al. [147] have used the same approach and extended it to the calculation of ternary mixtures of [C4 mim][BF4 ] + organic solute + CO2 , being able to qualitatively predict liquid–vapor and liquid–liquid phase transitions. This methodology presents similar advantages and disadvantages than the other group contribution methods. On one hand, once the functional groups have been established, it provides a straightforward way to calculate multicomponent mixtures in a quite accurate way. On the other hand, a significant number of parameters must be fitted and, as a consequence, a large set of experimental data is required.

M being the number of segment types and ar (a = 0) the contribution from the monomers calculated from the Mansoori–Carnahan–Starling–Leland equation [143]. As each heteronuclear chain consists of two types of segments, there are three different nearest-neighbor pairs and six different nextnearest-neighbor pairs, that are used to calculate the effective 2e and y2e . More details can be found cavity correlation function yjk jkl in Ref. [149]. Each segment is modeled using three parameters: the chain length r, the attractive energy ε/k and the segment diameter . The model parameters for the alkyl chain were taken from those for hydrocarbons, available in literature, while the imidazolium ring-anion block as well as the binary interaction parameters AB between segments, were fitted to experimental volumes of ionic liquids. The work includes several mixtures of compounds with ionic liquids (alkanes, alcohols, aromatics, water, etc.). In this case, three adjustable binary parameters are needed ( AB, AS, BS ), corresponding to the interaction between segments AB and between each segment with the solvent S. The parameter AB was refitted and included with the rest using the fugacity coefficients of the solvent in the liquid and the vapor phases. This approach is physically well sounded and provides very good agreement with experimental data when tested. However, it requires at least three adjustable parameters for binary mixtures that do not follow any trend, losing predictive power when trying to use it at other conditions or for other similar systems. Lately, Ji and Adidharma [150] have used a similar approach to estimate the density of several [BF4 ], [PF6 ] and [TfN2 ] imidazolium ionic liquids. In particular, they have employed a heterosegmented SAFT, referred as SAFT1 [151]. They have divided the ionic liquid in four different parts, considering the anion, the cation head, the methyl group and the alkyl chain as independent compounds with their molecular parameters. The spherical segments representing the cation head and the anion have one association site, which can only cross-associate. This complex model provides excellent agreement with experimental data, although a considerable amount of adjustable parameters are required even for pure compounds.

2.4.3. The square-well chain fluids (SWCF) equation of state The SWCF EoS has also been used for the modeling and study of ionic liquids. Wang et al. [148] correlated the pVT behavior of several ionic liquids and some mixtures with gases with only one temperature-independent adjustable parameter, using the homonuclear version of this equation. In a more recent contribution [149], the hetero-SWCF EoS was presented, and the molecule was divided in two parts: the alkyl group composed of A segments and the ring anion that included B segments (see Table 2f). The residual Helmholtz energy of the

2.4.4. The truncated Perturbed Chain Polar Statistical Associating Fluid Theory (tPC-PSAFT) tPC-PSAFT [35] is a refined version of the original PC-SAFT equation [152], developed by the group of Economou and co-workers to deal with systems as complex as ionic liquids. In the tPCPSAFT equation, dipolar interactions of the ionic liquid, quadrupolar interactions for CO2 , and the Lewis acid–base type of association between the anion of the ILs and the CO2 molecule are explicitly considered. As in any SAFT approach the total Helmholtz energy of the system is evaluated as a sum of different contributions:

i=1 j=1 k=1 l=j

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L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

ares (T, ) ahs (T, ) achain (T, ) adisp (T, ) aassoc (T, ) = + + + RT RT RT RT RT +

apolar (T, ) RT

(19)

where ahs accounts for the hard-sphere contribution, achain for the chain formation, adisp for the dispersion interaction, aassoc for the association effects and apolar for the dipole–dipole, quadrupole–quadrupole and cross dipole–quadrupole interactions. The hard-sphere, chain, association and dispersion terms are identical to those proposed for the original PC-SAFT [152], while the polar term corresponds to a Padé approximant proposed by Stell and co-workers [153], also used in other versions of SAFT: polar

a2 apolar (T, ) =m polar polar RT 1 − (a /a ) 3

(20)

2

The higher order terms of the polynomials are truncated; as a consequence, a new effective polar interaction diameter is used to take the spatial extent of the polar interactions into account. As we can see, polar contributions were explicitly considered in this approach, in order to capture the polarity effects in the ionic liquid. In a similar manner, Qin and Prausnitz [154] also reflected contributions from induction, dipole–charge and quadrupole–charge interactions using the Perturbed–Hard–Sphere Theory to correlate the behavior of volatile nonelectrolytes, obtaining the Henry constant for 20 different solutes. No association was considered in their case. Economou and co-authors first modeled the ionic liquid as a combination of two different species (anion and cation) (see Table 2g). The size and energy of the cations could be obtained using the PC-SAFT molecular parameters obtained from alkylbenzene family (those corresponding to the same alkyl chain length in the molecules than for the imidazolium ring). For the size and shape of the anions available literature parameters were used, while the dispersive energy was estimated with the Mavroyannis and Stephen equation. Finally, they used the standard combining rules in order to calculate the volume and the energy parameters of the molecule, and the segment number was fitted to ionic liquid density experimental data. For the dipole moment of ionic liquid they used the value from methanol. Additionally, they also modeled the Lewis acid–base association between the CO2 molecules and the anion of the IL as a specific cross-association interaction. These cross-association parameters (energy and volume of association) between the ionic liquid and the CO2 were estimated through the experimental enthalpy and entropy of dissolution of CO2 in ionic liquids [155]. When studying the solubility in water, crossassociation was also calculated from the experimentally measured hydrogen bonding, while the association volume was assumed to be the same as the volume between water molecules. Economou and co-workers have published several works [155–158] where they applied this methodology to the calculation of thermodynamic properties and solubilities of several ionic liquids with different solutes, obtaining very good agreement when comparing to experimental data. In the most recent contributions [157,158], they calculated ternary mixtures of water–CO2 and two different ionic liquids ([bmim][NO3 ] and [HOPmim][NO3 ]). The different binary kij interaction parameters were fitted to ternary data, making them temperature dependent. This is clearly a sound and robust method to describe the behavior of ionic liquids and the mixtures of several compounds with them, with great potential for predictions. The main limitation is the complexity of the model itself, which requires a high number of different (physical, though) parameters, and the fact that an explicit cross-association between the CO2 molecule and the ionic liquids is required.

2.4.5. The soft-SAFT equation of state The soft-SAFT equation of state [34] is another successful version of the original SAFT [159] that uses a Lennard–Jones intermolecular potential as the reference fluid. As other SAFT-type equations, it is written in terms of the total Helmholtz energy of the system. When applied to ionic liquids, the residual Helmholtz energy is written as aLJ (T, ) achain (T, ) aassoc (T, ) apolar (T, ) ares (T, ) = + + + RT RT RT RT RT (21) where the superscript LJ accounts for attraction and repulsion forces between monomers, following a Lennard-Jones intermolecular potential, and chain, assoc and polar have the same meaning that the one explained in the previous subsection. Using the soft-SAFT EoS, Andreu and Vega were able to accurately describe the solubility of carbon dioxide, hydrogen, carbon monoxide and xenon in the alkylimidazolium family [Cn mim]+ with n = 2, 4, 6 and 8 with different fluorinated anions such [BF4 ]− , [PF6 ]− , and [Tf2 N]− , achieving quantitative agreement with experimental data [160,161]. The objective of these works was to recover the original spirit of the SAFT approach and keep the model as simple as possible while retaining the main physics of the problem. In this sense, the model used in soft-SAFT for ionic liquids is simpler than the one used in the other versions of SAFT mentioned here (SWCF and tPC-PSAFT). Based on the concept of ion pairing of these systems, as observed in molecular dynamics [162–164], ionic liquids were modeled as LJ chains with one (for the [BF4 ] and [PF6 ] anions) and three (for the [Tf2 N] anion) associating sites in each molecule. This model mimics the neutral pairs (anion plus cation) as a single chain molecule with an only association site describing the specific interactions because of the charges and the asymmetry (see Table 2h). In this sense, only associating forces between ionic liquids pairs, as well as van der Waals forces, were taken into account for the ionic liquids, while the quadrupolar term was included for molecules showing quadrupolar interactions. In particular, no specific association between the CO2 molecule and the ionic liquids was needed to quantitatively reproduce the solubility of this compound in these ionic liquids, as in the work of Qin and Prausnitz [154] and contrary to the assumption of the tPC-PSAFT EoS. Given the excellent results obtained with soft-SAFT with no CO2 –ionic liquid association, it can be inferred that the specific interaction between the anion and CO2 is already taken into account within the specificity of the model. However, this is not the case of the solubility of BF3 in ionic liquids, where a clear BF3 –ionic liquid 1:1 weak interaction is observed, taken into account into the model with a cross-association interaction (shown later in the present work). The chain length, size and energy parameters of the ionic liquids were obtained by fitting to available density–temperature data, obtaining a correlation with the molecular weight of the compounds. The association parameters were transferred from those of the alkanols [165], thus avoiding further fitting. The correlation of the three first parameters and the fixed value of the association parameters allow the calculation of ionic liquids of the family not included in the fitting procedure or for which experimental data is not available. When calculating the solubility with carbon dioxide, quadrupolar interactions were taken into account through the molecular parameters Q and xp , defined as the fraction of segments in the chain that contain the quadrupole. Q was fitted but similar values were obtained from those of literature [166,167] and xp , it was fixed to 1/3 for carbon dioxide, thus mimicking the molecule as three segments with a quadrupole in one of them. Fig. 3a shows the temperature–density of [Cn mim]+ with n = 2, 4, 6 and 8 with [BF4 ]− as obtained with soft-SAFT and the model just

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

25

Table 4 Optimized influence parameters for the compounds studied in this work. See text for details.

Fig. 3. Thermodynamic characterization for some [BF4 ] imidazoliums. (a) Temperature-density diagram for [C2 mim][BF4 ] (circles), [C4 mim][BF4 ] (squares), [C6 mim][BF4 ] (diamonds) and [C8 mim][BF4 ] (triangles). Experimental data from [168]. (b) Surface tension as a function of temperature for the same ionic liquids (excepting [C2 mim][BF4 ]). Symbols represent experimental data [175,176] while the lines are soft-SAFT with density gradient theory calculations.

described. The selected molecular parameters for each ionic liquid are taken from Ref. [160] and reported here in Table 3 for completeness. As it can be observed, excellent agreement is achieved between the theory (lines) and the experimental data (symbols) [168]. We present here new results completing the thermodynamic characterization of these compounds by adding the calculation of their surface tension. In this case, the soft-SAFT Eos is coupled to density gradient theory [169,170], following the work by different authors [165,171–174], in order to obtain interfacial properties. The required molecular parameters are those obtained from the density–temperature data, used in a transferable manner, with the additional influence parameter c (see Table 4). Results are presented in Fig. 3b, where solid lines correspond to the calculations and symbols represent experimental data [175,176]. In agreement with the experimental data, a decreasing interfacial tension value with alkyl chain length has been obtained. This behavior is very

Ionic liquid

1019 c (J m5 mol−2 )

[C4 mim][BF4 ] [C6 mim][BF4 ] [C8 mim][BF4 ]

13.468 15.034 14.067

surprising as it is contrary to the other chemical families where the interfacial tension increases with the chain length. Deetlefs et al. [177] suggested that the increase of the cation size and the increasing diffuse nature of the anion negative charge results in a more delocalized charge leading to a decreasing ability to form hydrogen bonds. Besides, these tendencies can also be affected by entropic effects leading to distortions on the expected results [176]. Another example of the capabilities of a simple, physically sound and reliable approach is presented next. Although a great effort has been devoted to develop ionic liquids for capturing CO2 , most of the published work on ionic liquids for this purpose show higher solubilities than other gases, but still very low solubilities compared to the industrial amines process used today to capture CO2 . Through the different experimental, and also simulation and modeling works, it has been shown that the solubility of CO2 in these known ionic liquids is mainly due to van der Waals forces, and no weak chemical interactions have been observed so far. However, the weak complexation of ionic liquids with some compounds has been observed experimentally. Tempel et al. [178] demonstrated that a reversible complexation of Lewis basic gases like BF3 with ionic liquids is possible, permitting the use of ionic liquids as a transport media that allows the reversible absorption and desorption of this dangerous compound. They investigated the absorption of PH3 and BF3 in two different ionic liquids, the 1-butyl-3-methylimidazolium trichlorodicuprate [C4 mim][Cu2 Cl3 ] and the 1-butyl-3-methylimidazolium tetrafluoroborate [C4 mim][BF4 ], respectively. The experimental results showed the possibility to store and transport large amounts of gases in small volumes at low pressures. We have applied the same methodology described previously (soft-SAFT with a simple model) to see if the general model was able to capture this behavior as well. For this purpose, a BF3 model was built based on liquid density and vapor pressure experimental data taken from the DIPPR® database [179]. In this case, the BF3 was modeled as a non-associating compound with four molecular parameters: m, the length of the chain, , the Lennard–Jones diameter of each spherical segment forming the molecule, ε, the LJ energy for each segment, and Q, the quadrupole. The quadrupole–quadrupole interactions between BF3 molecules were taken into account by fixing the quadrupole term according to the experimental value reported elsewhere [180]. Following previous approaches for molecules with non-zero quadrupole, like CO2 and N2 , the fraction of the molecule (xp ) associated to the quadrupole was set to 0.25. Molecular parameters are summarized in Table 3 and the phase diagram of this molecule is presented in Fig. 4a and b. The molecular parameters and L correspond

Table 3 Molecular parameters for the [Cn mim][BF4 ] family and the BF3 model used in the soft-SAFT EoS. The fourth and fifth columns for the BF3 case correspond to the values adjusted to the associating site for cross-association, also added to the modeled ionic liquid.

[C2 mim][BF4 ] [C4 mim][BF4 ] [C6 mim][BF4 ] [C8 mim][BF4 ] BF3 a a

m

 [Å]

ε/kB [K]

kHB [Å3 ]

εHB /kB [K]

Q [C m2 ]

L/



3.980 4.495 5.005 5.570 2.321

3.970 4.029 4.110 4.170 2.889

415.0 420.0 423.0 426.0 128.7

3450 3450 3450 3450 8000

2250 2250 2250 2250 4730

– – – – 3.15 × 10−40

– – – – 1.29

– – – – 6.90

Parameters L/ and are crossover parameters, only used when the crossover term is included in the soft-SAFT equation (dashed line in Fig. 4).

26

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

Fig. 5. VLE diagram for the system BF3 + n-butane at 195.49 K obtained from the soft-SAFT EoS. The dotted line corresponds to the prediction without fitting, the dashed line to the description with a binary energy parameter and the full line with two binary parameters. The symbols correspond to experimental data taken from reference [179]. See text for details.

several other mixtures, this is due to the asymmetry in size (the  parameter) and van der Waals attractive energy (the parameter) between the two different compounds integrating the mixture. In fact, similar results were obtained for the same system with a similar EoS [184], although in the soft-SAFT case the binary parameters are slightly closer to unit. The good agreement obtained for the BF3 + n-butane mixture with experimental data validates the molecular parameters of BF3 for other binary systems. Note also that soft-SAFT was able to capture the behavior of this mixture without assuming any specific interaction between BF3 and n-alkane. Unfortunately, the data for BF3 mixtures is very scarce in the literature and no further comparFig. 4. Phase diagram for the BF3 compound. (a) Vapor–liquid equilibria diagram. (b) Pressure–temperature diagram. Symbols are experimental data taken from [179] while the solid line corresponds to the soft-SAFT model.

to the inclusion of a specific renormalization-group treatment in order to consider the inherent density fluctuations in the critical region into soft-SAFT, in a version known as crossover soft-SAFT [181,182]. The dashed line of the figure corresponds to the calculation using the crossover soft-SAFT equation of state, while the solid line corresponds to the original soft-SAFT equation. Quantitative agreement is found for the whole range of liquid temperature and pressure when the crossover term is used, while the original equation describes well both densities, except in the near critical region, as expected. In order to test the BF3 model and parameters we checked its performance in mixtures with other well known compounds, nbutane in this case. As at these conditions the mixture is far away from its critical point, the soft-SAFT version used is the one without crossover, making the calculations faster. The parameters for pure n-butane were taken from reference [183]. Results for the mixture BF3 + n-butane at 195.49 K are depicted in Fig. 5. The dotted lines correspond to predictions from pure component parameters, and, as shown in the figure, the equation is unable to capture the behavior of the mixture unless two binary parameters, close to unity, are taken into account ( = 0.925 and = 0.95). As already observed with

Fig. 6. Absorption isotherms for the system [C4 mim][BF4 ] + BF3 at three different temperatures. Symbols represent the experimental data from Ref. [178]. Comparison between original calculations with no association (dashed lines) and association (solid lines). See text for details.

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

27

3. Conclusions and future directions

Fig. 7. Absorption isotherms for the system [C4 mim][BF4 ] + BF3 at three different temperatures. The solid lines are the prediction from soft-SAFT when the crossassociation model is used. Same data and symbols as in Fig. 6, but with a different scale.

isons could be made. Hence, the next step was to test the validity of the model for the mixture with ionic liquids. When applying the model used to describe the [Cn mim][PF6 ] and [BF4 ] families with CO2 and other gases, and also the BF3 + n-butane mixture we found that the model is not able to reproduce the behavior of the [bmim][BF4 ] + BF3 mixture (dashed lines in Fig. 6). Note that a logarithm scale has been used in the plot to highlight that the model provides several orders of magnitude deviations with respect to experimental data (symbols). It is thought that the higher absorption of BF3 in [C4 mim][BF4 ] is actually due to a weak 1:1 interaction [178]. Within the soft-SAFT model this type of interaction (stronger than van der Waals forces, but weaker than chemical bonds, and with specific directions) is modeled as a cross-association interaction between the two components. In this case two additional parameters are required, those concerning the cross-association. In order to check if the cross-association model was able to capture the behavior of this mixture, we have transferred the volume cross-association parameters from our previous work on perfluoroalkanes + O2 mixtures [185], where a cross-association of this kind was found. The energy cross-association parameter was used as a fitting parameter to the BF3 + ionic liquid mixture. In the absence of experimental data, these association parameters could be obtained with the help of ab initio calculations. Results with these parameters are shown in Fig. 6 (solid lines) and in Fig. 7. An excellent agreement between experimental data and results obtained from soft-SAFT with crossassociation is observed at all temperatures, with the same accuracy than other mixtures with CO2 . These results reinforce the fact that there is association in the mixture, and that the model is able to capture it in a simple manner. Although the results obtained from soft-SAFT are also very encouraging, given the relative simplicity of the model, its predictive power and the agreement obtained with experimental data, the main limitation of the approach is that it considers the anion and cation as a unique molecule. Considering the cation and anion as two independent species will empower the equation with more predictive power, especially for conditions at which the ionic liquid can dissociate, such as aqueous mixtures.

A summary of recently published modeling works in ionic liquids has been presented here, emphasizing some of the main advantages and limitations of each of them for predictive purposes. Particular attention has been paid to the most refined approaches involving a physical description of the compound. A detailed list of the works published, with the systems investigated has been provided in a tabular form, providing a general overview of the available techniques and systems investigated up to now. As inferred from the amount of work published up to now and revised in this work, several steps have been taken towards developing tools for advancing in the fundamental understanding of ionic liquids, relating their structure to their properties. This is particularly true in the case of molecular-based methods, as their physical background allows for extrapolations and predictions at other conditions and for similar systems. In particular, we have shown that a SAFT-based equation provides not only solubility data, but also interfacial properties. In addition, it has also been shown that soft-SAFT, with a simple model, is able to describe the weak chemical complexation found in BF3 with ionic liquids, with a simple cross-association model, while no specific association between CO2 and ionic liquids is needed to quantitatively describe the solubility of CO2 in them. Further developments are still needed in order to extend the predictive power of the model, focusing on the parameters transferability and the ability of the equations to explore all regions of the phase diagram, as well as other properties. This should include an independent model for the anion and the cation in order to make the model more transferable, in a group contribution manner. In addition, the models should be simple enough for engineering calculations. List of symbols a Helmholtz free energy density CN carbon number Cr binary parameter to measure the effective chain length for LLE (lattice model) CT salt category specific constant c influence parameter critical hard-sphere diameter dc attractive parameter gii kHB volume of association kB Boltzmann constant kij binary interaction adjustable parameter Gibbs energy Gij H enthalpy KT parameter (in the RST model) L cutoff length (crossover parameter) m number of LJ segments, chain length Mw molecular weight Avogadro’s number NA P pressure p˜ reduced pressure Q quadrupole Qk group volume parameter (GC method) R ideal gas constant Rk group surface parameter (GC method) r segment number (lattice model) r vdW molecular volume (GC method) r chain length (SWCF) T temperature T˜ reduced temperature Z compressibility factor z coordination number (lattice model)

28

x y

L.F. Vega et al. / Fluid Phase Equilibria 294 (2010) 15–30

mole fraction cavity correlation function

Greek letters ˛ij binary interaction parameters ı cohesive energy density ε interaction energy εHB association energy  size parameter of the generalized Lorentz–Berthelot combining rules

average gradient of the wavelet function (crossover parameter) compressibility

chemical potential energy parameter of the generalized Lorentz–Berthelot combining rules  density  ˜ reduced density i activity coefficient  surface tension segmental volume (lattice model) v∗  segment diameter  ij binary interaction parameter (NRTL model) Superscripts assoc association term attr attraction term chain chain term cross crossover disp dispersion e effective fv free-volume term hs hard-sphere term hscf hard-sphere chain fluid term id ideal term LJ Lennard–Jones polar polar term ref reference term res residual term Subscripts melt melting i,j the specific compound i or j in a mixture Acknowledgements Discussions with J.A.P. Coutinho, C. Rey-Castro, B. Peterson, D. Tempel and C.J. Peters are gratefully acknowledged. F. Llovell acknowledges a JAE-Doctor fellowship from the Spanish Government. This work has been partially financed by the Spanish government, Ministerio de Ciencia e Innovación, under projects CTQ2008-05370/PPQ, NANOSELECT and CENIT SOST-CO2 CEN2008-01027 (a Consolider project and a CENIT project, respectively, both belonging to the Ingenio 2010 program). Additional support from the Catalan government, under projects SGR200500288 and 2009SGR-666, is also acknowledged.

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