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Theoretical model for moisture adsorption on ionic liquids: A modified Brunauer–Emmet–Teller isotherm approach
Fluid Phase Equilibria 301 (2011) 118–122
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Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Theoretical model for moisture adsorption on ionic liquids: A modified Brunauer–Emmet–Teller isotherm approach J. Carrete a , M. García a , J.R. Rodríguez a , O. Cabeza b , L.M. Varela a,∗ a
Grupo de Nanomateriales y Materia Blanda, Departamento de Física de la Materia Condensada, Universidad de Santiago de Compostela, E-15782, Santiago de Compostela, Spain Facultad de Ciencias, Universidad de A Coru˜ na, Campus A Zapateira s/n, E-15008, A Coru˜ na, Spain
b
a r t i c l e
i n f o
Article history: Received 9 August 2010 Received in revised form 8 November 2010 Accepted 15 November 2010 Available online 1 December 2010 Keywords: Ionic liquids Pseudolattice theory Adsorption isotherm Brunauer–Emmet–Teller Lateral interactions Hygroscopic Hydrophobic adsorption Hydrophilic adsorption
a b s t r a c t The adsorption of atmospheric water on the air–liquid interface of ionic liquids is analyzed by means of a modified version of the Brunauer–Emmet–Teller (BET) multilayer adsorption isotherm including lateral interactions between adsorbed molecules, treated in a mean-field fashion. Recently reported experimental results of water adsorption on hydrophobic ionic liquids of the 1-alkyl-3-methyl imidazolium tetrafluoroborate (Cn MIM-BF4 ) family are analyzed in the present theoretical framework. The calculated values of the lateral interaction are seen to be compatible with the Keesom dipole–dipole interaction in water, confirming the validity of the multilayer assumption hypothesis. A somehow surprisingly ordered hydrophilic-like adsorption of atmospheric water is suggested to take place in the free surface of hydrophobic ILs of the imidazolium family. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The highly hygroscopic nature of most ionic liquids (ILs) is well-known, and it is conventionally reported in almost all papers concerning these systems. The extreme affinity of these compounds for water provokes that almost every paper in the field considers the amount of water its systems contain. Water is known to be present even in apparently hydrophobic ILs, saturating with the significant amount of about 1.4% mass of water, and water uptake from the atmosphere can be much greater for hydrophilic ILs [1]. Particularly, ILs of the imidazolium family are known to be extremely hygroscopic. This means that all commercial and synthesized ILs may contain some variable amount of atmospheric water unless they are stored in an inert atmosphere. This fact causes problems in some applications of ILs because even small amounts of water induce large changes in the physicochemical properties of the pure IL. However, despite the deep impact of the presence of water in the properties of these systems, very limited work has been reported in quantitative evaluations of the amount of water present in ILs. The first reported study seems to be that of Tran et al. [2], in
∗ Corresponding author. E-mail address: [email protected] (L.M. Varela). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.11.011
which near-infrared (NIR) spectrometry was used for the determination of concentrations and structure of water absorbed by room-temperature ILs. The authors were able to prove that ILs based on 1-butyl-3-methylimidazolium, are hygroscopic and can quickly absorb water when they are exposed to air. This absorbed water interacts with the anions of the ILs, and these interactions lead to changes in the structure of the absorbed water. More recently, another quantitative evaluation of the amount of water present in ILs has been reported [3]. In this contribution, measurements of the variation of mass suffered by eight different ionic liquids (ILs) of the 1-alkyl-3-methyl imidazolium tetrafluoroborate (CnMIM-BF4) and 1-ethyl-3-methyl imidazolium alkyl sulfate (EMIM-Cn S) families, according to the atmosphere humidity grade were done. The authors prove that water is reversibly adsorbed in the free surface of the sample, and the amount of water adsorbed in the surface is quantified in terms of the level of humidity of the atmosphere. Up to our knowledge, these studies continue to be the only results of this kind reported in literature. Particularly, no theoretical investigation of the essentials of the sorption of water in ILs has been reported as far as we know. Despite the fact that that water is both absorbed in the bulk and adsorbed at the free surface of the ILs, the results in Ref. [3] indicate that the water incorporation is essentially reversible, so the latter mechanism seems to be dominant. Moreover, as we can see in Fig. 1, the water adsorbed from the atmosphere in ILs clearly
J. Carrete et al. / Fluid Phase Equilibria 301 (2011) 118–122
Nomenclature A C6 k kB m M ¯ N qi r T ˇ 0
total free surface of the ionic liquid Keesom coefficient ratio of the partition functions of adsorbed particles in successive layers Boltzmann’s constant total mass of the system number of adsorption sites in the ionic liquid surface mean number of adsorbed particles partition function of a molecule adsorbed in the i-th adsorption layer relative humidity degree absolute temperature inverse thermal energy lateral interaction energy between water molecules adsorbed in adjacent stacks surface coverage chemical potential of the adsorbate reference chemical potential of the adsorbate surface density of adsorption nodes
forms a macroscopic layer over the free surface of the system. However, it is by no means clear nowadays in which cases this adsorption process is associated to a hydrophobic-like mechanism leading to the formation of nonordered adsorption layers, or to a hydrophilic-like mechanism where the adsorbate molecules form hydrogen bonds with the substrate giving rise to ordered layers. If hydrogen-bonding with the substrate is the dominant mechanism, localized adsorption will be the expected. Contrarily, for hydrophobic adsorption nonordered layers are expected, and so the adsorbate is expected to adopt a microscopic structure similar to the corresponding bulk phase. The substrates analyzed in these papers are normally considered to be hydrophobic in nature [4], and they are known to form micelles and other self-association nanoaggregates in aqueous solutions, so the adsorption of water in these systems is expected to be of hydrophobic nature. This conclusion must be tested with a theoretical analysis of the experimental adsorption data, and this is one of the objectives of this paper. The formation of ordered structures of particles adsorbed on surfaces and phase separation into regions of different coverage due to lateral interactions have been observed in a number of multilayer experimental situations [5,6]. However, very limited work has been done to theoretically describe multilayer cooperative binding of particles onto solids, liquid interfaces, colloid particles or macromolecules, particularly when compared with the efforts tributed to monolayer cooperative adsorption. Up to our knowledge, the first cooperative multilayer adsorption framework based on the Brunauer–Emmet–Teller (BET) adsorption isotherm was reported
119
by Dreyfus et al. [7]. In 2002, another theoretical approach to the problem of pattern formation on a multilayer of adsorbates with lateral interactions was reported by Casal et al. [8]. Dreyfus et al. allowed for the existence of constant interactions between particles adsorbed in a layer with their nearest neighbors in a molecular-field approximation, so they successfully explained the cooperative multilayer adsorption of nitrogen on non-porous samples of silica and butane on an Iceland Spaar substrate. In the first half of this decade, a theoretical treatment of the multilayer localized adsorption with lateral interactions has been reported by Varela et al. [9,10]. In these papers a cooperative multilayer adsorption model is introduced, combining the BET adsorption isotherm with surface interactions among adsorbed molecules treated in a mean-field fashion. For this purpose, a term proportional to the surface coverage in the chemical potential of the adsorbate with a constant of proportionality equal to the lateral interaction energy between bound molecules is introduced. Using the modified BET adsorption isotherm, the authors of Ref. [10] were able to analyze the adsorption of penicillin molecules on the surface of human serum albumin and to prove the role of van der Waals interactions in this process, since the interaction energies obtained from the empirical binding isotherms were of the order of tenths of the thermal energy. In this contribution, we provide a theoretical explanation of the adsorption isotherms of water on ILs on the hypothesis that reversible multilayer adsorption is the dominant mechanism of water incorporation in the substrate. Particularly, we assume that the adsorbed water molecules form multilayers in localized adsorption nodes in the free surface of the IL, where they suffer lateral interactions with other adsorbed molecules. As a consequence of the fitting procedure, we are able to prove that this interaction energy is compatible with the interaction between permanent dipoles in bulk water. 2. Theoretical section The existence of a “loose” lattice structure (hereinafter referred to as a pseudolattice) in electrolyte solutions for concentrations beyond 0.01 M is a well-known fact [11–18]. This structural scheme is gradually reinforced up to the highest possible concentrations of the ionic solutions or, in the case of mixtures of ILs, up to the limit of pure IL. From the experimental point of view, Katayanagi et al. [11] have recently verified the existence of long-range correlations in liquid 1-n-butyl-3-methylimidazolium iodide ([bmim]I) and that the structure of cations and anions are similar to those in the crystal by means of wide angle X-ray scattering and Raman spectroscopy. Moreover, the assumption of this structural model—the basic hypothesis of the so called Bahe–Varela formalism [12–17] allows the interpretation of both equilibrium [18] and transport properties [19] of ILs. According to this structural scheme, we can assume that the free surface of the IL is formed by 2D-cells which provide potential wells that act like adsorption nodes. In these
Fig. 1. Two views of a macroscopic film of adsorbed water on EMIM-BF4 . The IL was put in contact with an atmosphere of 100% humidity degree during 4 days.
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nodes, adsorbate molecules are either anchored (localized adsorption) or wander around. Consider the free surface of an IL in contact with the atmosphere containing a certain concentration of water molecules (humidity grade). Given the formation of a macroscopic layer in the analyzed systems, we shall model the free surface of the IL as made up of M equivalent, distinguishable and independent sites, on each one of which a macroscopically large number of water molecules can be adsorbed forming a multilayer. Moreover, we consider that the adsorption of any other type of molecules can be completely neglected. Let us assume that humid air can be considered to be an ideal gas under the experimental conditions. This is reasonable at ambient temperatures and for the typical water contents of atmospheric air in the laboratory, so the chemical potential of the adsorbate molecules is = 0 + kB T ln r, with kB being Boltzmann’s constant, T the absolute temperature and r the relative humidity degree, equivalent to the concentration of the adsorbate molecules in air. Additionally, we shall assume that the canonical partition function of a water molecule adsorbed in the first layer (next to the surface) is q1 and q2 the partition function in successive layers. Under these circumstances, the surface coverage or mean number ¯ adsorbed per binding site of the free surface of the of molecules N IL is given by the BET adsorption isotherm [20]: =
¯ N kx = , M (1 − x + kx)(1 − x)
(1)
where x = q2 exp(ˇ), k = q1 /q2 and ˇ being the inverse of the thermal energy as usual. As it is well-known, this equation is capable of describing a condensation of the adsorbate on the adsorption surface, since for x → 1, the surface coverage diverges [20]. Contrarily to what happens in the gas phase in contact with the surface, water molecules are expected to interact strongly when adsorbed at the surface of the liquid, so a cooperative adsorption model must be considered, where lateral interactions between neighbor stacks exist. However, an exact treatment of cooperative adsorption is a formidable task because of the extreme operational complexity of the polynomial expressions derived from the grand canonical partition function of the adsorption sites [10]. Fortunately, a mean-field approximation is usually enough to capture the essentials of the adsorption processes, while leading to considerable simplification. In this formalism, the adsorbed particles are assumed to fluctuate as free particles in the effective (or molecular) field created by the rest of the medium, analogously to the Weiss field in the theory of spin lattices [21]. In this fashion, the coverage is an extensive magnitude (the homologue of the magnetization) coupled to an ‘external’ field that drives the binding process. The equivalent role to that of the magnetic field in the magnetization problem is played here by the chemical potential of the adsorbate, which in equilibrium is equal to the chemical potential of the adsorbed particles. In a cooperative adsorption process, this chemical potential is made up of two contributions: (i) the ligand chemical potential, , and (ii) an excess (so-called pseudochemical [21]) potential associated to the interactions with the rest of the molecules in the substrate surface, . In a mean-field formalism, this latter contribution is proportional to the coverage itself (equivalently to the concentration of water molecules in the adsorbate), so the contribution of the interparticle interactions to the chemical potential can be written in the mean-field framework as =
(2)
where represents the interaction energy between adsorbed molecules in neighboring stacks. The case ␥ > 0 (␥ < 0) corresponds to attractive (repulsive) interactions and it induces positive (negative) cooperativity. The introduction in the formalism of a mean-field interaction like that in Eq. (2) is equivalent to a punctual
transformation of the concentration of the form x → x = xeˇ
(3)
The combination of the above chemical potential with the Langmuir adsorption isotherm for a monolayer adsorption of an ideal adsorbate leads to the so called Frumkin adsorption isotherm [22]. The cooperative BET adsorption isotherm, (x ) is obtained straightforwardly transforming Eq. (1) according to Eq. (3). Upon adsorption of water from the humid atmosphere, the total mass of the system suffers a change m → m = m + m. If one takes into account the mass of an adsorbate particle, ma , the total free surface area A and the surface density of adsorption nodes, , and Eq. (1), we can write the total mass of the adsorbed particles as m =
k2 xeˇ (1 − xeˇ
+ kxeˇ )(1 − xeˇ )
(4)
where k2 = ma Ak. This equation allows the analysis of the experimental data of Ref. [3] in terms of the relative humidity grade of the atmosphere, and the results of this analysis are the object of the next section. 3. Results and discussion Fig. 2 represents the total amount of water adsorbed after about 100 h at 20 ◦ C in the free surface of four ILs of the 1-alkyl3-methylimidazolium tetrafluoroborate for different humidity degrees, together with the numerical predictions of both the noncooperative and the cooperative BET adsorption isotherms. As one can see in Fig. 2, the amount of adsorbed molecules progressively decreases as the chain length of the cation of the IL increases. This clearly reflects the hydrophobic nature of the cation. Moreover, saturation in the amount of adsorbed water molecules takes place for humidity degrees close to 100% in the atmosphere in all studied cases. The parameters in the expressions for the isotherms were fitted to the experimental data using a modified Levenberg–Marquardt algorithm, using a broad range of initial values in order to try to avoid local minima. Multiplying Eq. (1) by ma A it was recast in the same form as Eq. (4) so that both can be written as m =
k2 y , (1 − y + ky)(1 − y)
(5)
where y = k1 r and y = k1 rek3 m in the non-cooperative and cooperative cases, respectively, k1 = q2 eˇ0 and k3 = (ˇ/ma A). In both models, k1 is theoretically independent of the ionic liquid under study (depending only on the properties of water) and was thus not considered an ordinary fitting parameter, but fixed for all liquids based on a previous fit to the set of all experimental data. However, the k1 used is different in the non-cooperative and cooperative cases, since in the regular BET isotherm it has to include part of the lateral interactions which are not explicitly considered. Some numerical difficulties can arise from the fact that the increment of mass upon adsorption, m, appears on both sides of Eq. (4) in the cooperative case. The naive approach of first fitting k and k2 to the experimental data and then solving that implicit equation numerically for each individual value of m results in a strongly discontinuous isotherm. This clearly unphysical result was avoided by the more time-consuming procedure of iteratively fitting k and k2 to the experimental data and then the whole vector of m values to k and k2 , until convergence was achieved. This method afforded the smooth theoretical isotherms which can be observed in Fig. 2. The non-cooperative BET adsorption isotherm is a good qualitative prediction of the average behavior of the experimental adsorption data. This confirms the localized multilayer hypothesis. However, the non-cooperative BET isotherm is unable to fit these data in the saturation region, predicting instead a divergence of
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Fig. 2. Increment of the mass of the IL upon adsorption as a function of the humidity degree of the atmosphere for four different ILs of the imidazolium family at T = 293.15 K: (a) EMIM-BF4 , (b) BMIM-BF4 , (c) HMIM-BF4 , and (d) MOIM-BF4 . The solid dots correspond to experimental data in Ref. [3], the dotted line to the predictions of a non-cooperative BET adsorption isotherm, and the solid line to the cooperative BET calculations for a lateral interaction parameter ␥ = 2.08 × 10−20 J.
the number of adsorbed molecules per node. However, considerably better fits of the experimental data are obtained if the lateral interactions among adsorbed molecules are taken into account. Particularly, the results shown in Fig. 2 are obtained if the cooperative version of the BET adsorption isotherm in Eq. (4) with a value of these lateral interactions fixed a priori to the well-known value of the Keesom interaction between permanent water dipoles for bulk water at T = 293 K, = 2.08 × 10−20 J„„, is used. This value of the interaction parameter is obtained if one uses the Keesom coefficient for water dipole-dipole interaction, C6 = 1.2 × 10−4 eV nm−6 [23]. In this sense, it is noteworthy to be mentioned that we have verified that when is left as a fitting parameter, we obtained the values 1.44 × 10−20 J, 1.63 × 10−20 J, 1.26 × 10−20 J, and 1.26 × 10−20 J for 1-ethyl-3-methylimidazolium, 1-butyl-3-methylimidazolium, 1-hexyl-3-methylimidazolium and 1-octyl-3-methylimidazolium tetrafluoroborates, respectively. So, the statistical average of the obtained values of the interaction parameters is ␥ = 1.44 × 10−20 J, a value slightly lower than that of bulk water probably due to the presence of the substrate and the consequent modification of the intermolecular potential of the adsorbed phase. It is noteworthy that this interaction parameter is expected to be temperature dependent. Moreover, one should notice that no additional fitting parameter is included with respect to the BET non-cooperative adsorption scheme because k3 is completely determined by the values of and k2 . Thus, both schemes imply bidimensional fitting procedures with two fitting parameters (k and k2 ). This confirms both the multilayer nature of the adsorption scheme and also the role of the lateral interactions in this surface process. Besides, given the value of the lateral interactions in the bulk, we must conclude that the water adsorbed at the interface of the IL forms a macroscopic film in which the properties of the adsorbed molecules are very similar to those in bulk pure water. Given the previously mentioned fitting parameters, we can obtain the values of the surface densities of adsorption nodes in each analyzed case. Taking into account that the experimental measurements reported in Ref. [3] correspond to Petri dishes of area A = 0.0035 m2 and the molecular masses of the different compounds, the obtained fitting parameters lead to surface densities of adsorption nodes of 3.0 × 1028 , 1.6 × 1028 and 7.3 × 1028 nodes m−2 respectively for 1-ethyl-3methylimidazolium, 1-butyl-3-methylimidazolium and 1-hexyl-3-
methylimidazolium tetrafluoroborates, respectively. However, the corresponding value for 1-octyl-3-methylimidazolium tetrafluoroborate is extremely sensible to fitting details, so we do not consider it significant and it is not reported. As expected, the surface density of adsorption nodes decreases as the chain length of IL’s cation increases. Interestingly enough, the measurements of the adsorbed amount of water on 1-octyl-3-methylimidazolium tetrafluoroborate reported in Ref. [3] coincide with the data of water solubility on this compound reported in Fig. 3 of Anthony et al. [24]. This is somehow surprising, since on a first approximation one would expect that both phenomena were different because the first one (adsorption) concerns the free surface of the IL and the other one (absorption) is a bulk phenomenon. The well-known low miscibility of water with this compound (0.065 mol kg−1 at room temperature [24]) is a possible cause of this behavior. If the alkyl side chain of the cation becomes sufficiently long, the IL–water system can present phase split, as is the case of this compound [25]. Under our experimental circumstances (T = 293.15 K, P = 1 atm.), a macroscopically observable layer of water is formed for this compound. However, it is possible that an undetermined amount of the liquid is absorbed in the bulk IL, but clearly the mass determination method is not the most appropriate method for elucidating this question, and we need to perform more experimental work in this point. This latter fact may give us some insight in the structure of water at IL interfaces. Apart from some simulations of Lynden-Bell et al. [26], very limited results concerning the peculiar structure of the adsorbed water on the free interface of ILs seem to have been reported in literature. However, a detailed image of the structure and dynamics of water at interfaces is fundamental to the understanding of several phenomena, such as molecular recognition, macromolecular folding or wetting. As it is well-known, the crucial factor for the structure at interfaces is the balance of hydrogen-bonding among water molecules and the comparable interactions with the substrate, defining the extreme hydrophobic and hydrophilic behavior. Long-range ordered layers are usually associated to hydrophilic subtrates while nonordered (long-range) behavior is associated to hydrophobic substrates (see for example Ref. [27] for water ice on a hydrophilic surface like silicon, chlorine terminated). In this latter case randomly oriented crys-
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tallites are expected to form with no interfacial long-range order because stronger water–water interactions are registered when compared with those of water–substrate. However, one must recall that anomalously ordered adsorption of water in vertically stacked bilayers has been also registered on some hydrophobic substrates like highly oriented pyrolytic graphite [28]. In our case, the experimental data and the theoretical analysis suggest that a partially disordered hydrophilic-like adsorption takes place for all the analyzed ILs, no matter the degree of hydrophobicity of the cation (it increases with the chain length of this ion; on the other hand, BF4 is normally considered to be an ion of intermediate hydrophobicity among the most common anions [29]). The validity of a BET localized adsorption scheme suggests a hydrophilic-like adsorption in which the interactions with the substrates are dominant, while the lateral interactions associated to the random order of a disordered phase indicate the absence of long-range structure in the adsorbed water. Thus, the adsorption of water in ILs of the imidazolium tetrafluoroborate family shows a short-range order typical of localized adsorption on a hydrophilic substrate, and a gradual transition to the long-range disorder typical of the microscopic structure of bulk water. This conclusion is surprising since, as we mentioned previously, the substrate is conventionally considered to be hydrophobic in nature. 4. Conclusions By means of a numerical analysis in the framework of a BET adsorption scheme, we have proved that the adsorption of water in the free surface of ILs can be described as a cooperative multilayer adsorption process with lateral interactions between the adsorbed molecules. These interactions between adsorbed molecules can be treated in a mean-field formalism, assuming that the interactions are equivalent to an effective chemical potential. Finally, the energy of these lateral interactions has been shown to be that of the Keesom interactions between permanent dipoles in water at the same temperature. From all these facts, a somehow surprisingly ordered hydrophilic-like adsorption of atmospheric water is suggested to take place in the free surface of hydrophobic ILs of the imidazolium family. The analysis of the influence of both the cation and anion on the nature of the adsorption of water in ILs is now in progress. Acknowledgements This work was supported by the Dirección Xeral de I+D+i de la Xunta de Galicia and the European Regional Development Fund
(Grant no. INCITE07PXI206076ES), and by the Spanish Ministry of Education and Science and the European Regional Development Fund (Grants no. FIS2007-66823-C02-01 and no. FIS2007-66823C02-02). J. Carrete wishes to thank the Spanish Ministry of Education for a FPU grant. References [1] P. Wasserscheid, T. Welton (Eds.), Ionic Liquids in Synthesis, Wiley–Verlag, Weinheim, 2003. [2] Chieu D. Tran, S.H. de Paoli Lacerda, D. Oliveira, Appl. Spectrosc. 57 (2003) 152–157. [3] S. Cuadrado-Prado, M. Domínguez-Pérez, E. Rilo, S. García-Garabal, L. Segade, C. Franjo, O. Cabeza, Fluid Phase Equilibr. 278 (2009) 36–40. [4] C. Jungnickel, J. Łuczak, J. Ranke, J.F. Fernández, A. Müller, J. Thöming, Colloids Surf. A 316 (2008) 278–284. [5] A. Mikhailov, G. Ertl, Chem. Phys. Lett. 238 (1995) 104–109. [6] D. Batogkh, M. Hildebrandt, F. Krischer, A. Mikhailov, Phys. Rep. 288 (1997) 435–456. [7] B. Dreyfus, D. Deloche, M. Laloë, Phys. Rev. B 9 (1974) 1268–1276. [8] S.B. Casal, H.S. Wio, S. Mangioni, Physica A 311 (2002) 443–457. [9] L.M. Varela, M. García, M. Pérez-Rodríguez, P. Taboada, J.M. Ruso, V. Mosquera, J. Chem. Phys. 114 (2001) 7682–7687. [10] L.M. Varela, M. Pérez-Rodríguez, M. García, Mol. Phys. 102 (2004) 79– 84. [11] H. Katayanagi, S. Hayashi, K. Hiro-o Hamaguchi, Nishikawa, Chem. Phys. Lett. 392 (2004) 460–464. [12] L.W. Bahe, J. Phys. Chem. 76 (1972) 1062–1071. [13] L.W. Bahe, J. Phys. Chem. 76 (1972) 1608–1611. [14] L.W. Bahe, D.J. Parker, J. Am. Chem. Soc. 97 (1975) 5664–5670. [15] L.M. Varela, M. García, F. Sarmiento, D. Attwood, V. Mosquera, J. Chem. Phys. 107 (1997) 6415–6419. [16] I. Bou Malham, P. Letellier, A. Mayaffre, M. Turmine, J. Chem. Thermodyn. 39 (2007) 1132–1143. [17] S. Bouguerra, I. Bou Malham, P. Letellier, A. Mayaffre, M. Turmine, J. Chem. Thermodyn. 40 (2008) 146–154. [18] L.M. Varela, J. Carrete, M. Turmine, E. Rilo, O. Cabeza, J. Phys. Chem. B 113 (2009) 12500–12505. [19] L.M. Varela, J. Carrete, M. García, L.J. Gallego, M. Turmine, E. Rilo, O. Cabeza, Fluid Phase Equilibr. 298 (2010) 280–286. [20] T.L. Hill, An Introduction to Statistical Thermodynamics, Dover, New York, 1986. [21] E. di Cera, J. Chem. Phys. 96 (1992) 6515–6522. [22] A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th ed., Wiley, New York, 1997, p. 392. [23] V. Magnasco, M. Battezzati, A. Rapallo, C. Costa, Chem. Phys. Lett. 428 (2006) 231–235. [24] J.L. Anthony, E.J. Maginn, J.F. Brennecke, J. Phys. Chem. B 105 (2001) 10942–10949. [25] M.G. Freire, L.M.N.B.F. Santos, A.M. Fernandes, J.A.P. Coutinho, I.M. Marrucho, Fluid Phase Equilibr. 261 (2007) 449–454. [26] R.M. Lynden-Bell, J. Kohanoff, M.G. del Popolo, Faraday Discuss. 129 (2005) 57–67. [27] C.Y. Ruan, V.A. Lobastov, F. Vigliotti, S. Chen, A.H. Zewail, Science 304 (2004) 80–84. [28] Yang Ding-Shyue, A.H. Zewail, Proc. Natl. Acad. Sci. 106 (2009) 4122–4126. [29] J.G. Huddleston, A.E. Visser, W.M. Reichert, H.D. Willauer, G.A. Broker, R.D. Rogers, Green Chem. 3 (2001) 156–164.
Contents lists available at ScienceDirect
Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Theoretical model for moisture adsorption on ionic liquids: A modified Brunauer–Emmet–Teller isotherm approach J. Carrete a , M. García a , J.R. Rodríguez a , O. Cabeza b , L.M. Varela a,∗ a
Grupo de Nanomateriales y Materia Blanda, Departamento de Física de la Materia Condensada, Universidad de Santiago de Compostela, E-15782, Santiago de Compostela, Spain Facultad de Ciencias, Universidad de A Coru˜ na, Campus A Zapateira s/n, E-15008, A Coru˜ na, Spain
b
a r t i c l e
i n f o
Article history: Received 9 August 2010 Received in revised form 8 November 2010 Accepted 15 November 2010 Available online 1 December 2010 Keywords: Ionic liquids Pseudolattice theory Adsorption isotherm Brunauer–Emmet–Teller Lateral interactions Hygroscopic Hydrophobic adsorption Hydrophilic adsorption
a b s t r a c t The adsorption of atmospheric water on the air–liquid interface of ionic liquids is analyzed by means of a modified version of the Brunauer–Emmet–Teller (BET) multilayer adsorption isotherm including lateral interactions between adsorbed molecules, treated in a mean-field fashion. Recently reported experimental results of water adsorption on hydrophobic ionic liquids of the 1-alkyl-3-methyl imidazolium tetrafluoroborate (Cn MIM-BF4 ) family are analyzed in the present theoretical framework. The calculated values of the lateral interaction are seen to be compatible with the Keesom dipole–dipole interaction in water, confirming the validity of the multilayer assumption hypothesis. A somehow surprisingly ordered hydrophilic-like adsorption of atmospheric water is suggested to take place in the free surface of hydrophobic ILs of the imidazolium family. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The highly hygroscopic nature of most ionic liquids (ILs) is well-known, and it is conventionally reported in almost all papers concerning these systems. The extreme affinity of these compounds for water provokes that almost every paper in the field considers the amount of water its systems contain. Water is known to be present even in apparently hydrophobic ILs, saturating with the significant amount of about 1.4% mass of water, and water uptake from the atmosphere can be much greater for hydrophilic ILs [1]. Particularly, ILs of the imidazolium family are known to be extremely hygroscopic. This means that all commercial and synthesized ILs may contain some variable amount of atmospheric water unless they are stored in an inert atmosphere. This fact causes problems in some applications of ILs because even small amounts of water induce large changes in the physicochemical properties of the pure IL. However, despite the deep impact of the presence of water in the properties of these systems, very limited work has been reported in quantitative evaluations of the amount of water present in ILs. The first reported study seems to be that of Tran et al. [2], in
∗ Corresponding author. E-mail address: [email protected] (L.M. Varela). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.11.011
which near-infrared (NIR) spectrometry was used for the determination of concentrations and structure of water absorbed by room-temperature ILs. The authors were able to prove that ILs based on 1-butyl-3-methylimidazolium, are hygroscopic and can quickly absorb water when they are exposed to air. This absorbed water interacts with the anions of the ILs, and these interactions lead to changes in the structure of the absorbed water. More recently, another quantitative evaluation of the amount of water present in ILs has been reported [3]. In this contribution, measurements of the variation of mass suffered by eight different ionic liquids (ILs) of the 1-alkyl-3-methyl imidazolium tetrafluoroborate (CnMIM-BF4) and 1-ethyl-3-methyl imidazolium alkyl sulfate (EMIM-Cn S) families, according to the atmosphere humidity grade were done. The authors prove that water is reversibly adsorbed in the free surface of the sample, and the amount of water adsorbed in the surface is quantified in terms of the level of humidity of the atmosphere. Up to our knowledge, these studies continue to be the only results of this kind reported in literature. Particularly, no theoretical investigation of the essentials of the sorption of water in ILs has been reported as far as we know. Despite the fact that that water is both absorbed in the bulk and adsorbed at the free surface of the ILs, the results in Ref. [3] indicate that the water incorporation is essentially reversible, so the latter mechanism seems to be dominant. Moreover, as we can see in Fig. 1, the water adsorbed from the atmosphere in ILs clearly
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Nomenclature A C6 k kB m M ¯ N qi r T ˇ 0
total free surface of the ionic liquid Keesom coefficient ratio of the partition functions of adsorbed particles in successive layers Boltzmann’s constant total mass of the system number of adsorption sites in the ionic liquid surface mean number of adsorbed particles partition function of a molecule adsorbed in the i-th adsorption layer relative humidity degree absolute temperature inverse thermal energy lateral interaction energy between water molecules adsorbed in adjacent stacks surface coverage chemical potential of the adsorbate reference chemical potential of the adsorbate surface density of adsorption nodes
forms a macroscopic layer over the free surface of the system. However, it is by no means clear nowadays in which cases this adsorption process is associated to a hydrophobic-like mechanism leading to the formation of nonordered adsorption layers, or to a hydrophilic-like mechanism where the adsorbate molecules form hydrogen bonds with the substrate giving rise to ordered layers. If hydrogen-bonding with the substrate is the dominant mechanism, localized adsorption will be the expected. Contrarily, for hydrophobic adsorption nonordered layers are expected, and so the adsorbate is expected to adopt a microscopic structure similar to the corresponding bulk phase. The substrates analyzed in these papers are normally considered to be hydrophobic in nature [4], and they are known to form micelles and other self-association nanoaggregates in aqueous solutions, so the adsorption of water in these systems is expected to be of hydrophobic nature. This conclusion must be tested with a theoretical analysis of the experimental adsorption data, and this is one of the objectives of this paper. The formation of ordered structures of particles adsorbed on surfaces and phase separation into regions of different coverage due to lateral interactions have been observed in a number of multilayer experimental situations [5,6]. However, very limited work has been done to theoretically describe multilayer cooperative binding of particles onto solids, liquid interfaces, colloid particles or macromolecules, particularly when compared with the efforts tributed to monolayer cooperative adsorption. Up to our knowledge, the first cooperative multilayer adsorption framework based on the Brunauer–Emmet–Teller (BET) adsorption isotherm was reported
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by Dreyfus et al. [7]. In 2002, another theoretical approach to the problem of pattern formation on a multilayer of adsorbates with lateral interactions was reported by Casal et al. [8]. Dreyfus et al. allowed for the existence of constant interactions between particles adsorbed in a layer with their nearest neighbors in a molecular-field approximation, so they successfully explained the cooperative multilayer adsorption of nitrogen on non-porous samples of silica and butane on an Iceland Spaar substrate. In the first half of this decade, a theoretical treatment of the multilayer localized adsorption with lateral interactions has been reported by Varela et al. [9,10]. In these papers a cooperative multilayer adsorption model is introduced, combining the BET adsorption isotherm with surface interactions among adsorbed molecules treated in a mean-field fashion. For this purpose, a term proportional to the surface coverage in the chemical potential of the adsorbate with a constant of proportionality equal to the lateral interaction energy between bound molecules is introduced. Using the modified BET adsorption isotherm, the authors of Ref. [10] were able to analyze the adsorption of penicillin molecules on the surface of human serum albumin and to prove the role of van der Waals interactions in this process, since the interaction energies obtained from the empirical binding isotherms were of the order of tenths of the thermal energy. In this contribution, we provide a theoretical explanation of the adsorption isotherms of water on ILs on the hypothesis that reversible multilayer adsorption is the dominant mechanism of water incorporation in the substrate. Particularly, we assume that the adsorbed water molecules form multilayers in localized adsorption nodes in the free surface of the IL, where they suffer lateral interactions with other adsorbed molecules. As a consequence of the fitting procedure, we are able to prove that this interaction energy is compatible with the interaction between permanent dipoles in bulk water. 2. Theoretical section The existence of a “loose” lattice structure (hereinafter referred to as a pseudolattice) in electrolyte solutions for concentrations beyond 0.01 M is a well-known fact [11–18]. This structural scheme is gradually reinforced up to the highest possible concentrations of the ionic solutions or, in the case of mixtures of ILs, up to the limit of pure IL. From the experimental point of view, Katayanagi et al. [11] have recently verified the existence of long-range correlations in liquid 1-n-butyl-3-methylimidazolium iodide ([bmim]I) and that the structure of cations and anions are similar to those in the crystal by means of wide angle X-ray scattering and Raman spectroscopy. Moreover, the assumption of this structural model—the basic hypothesis of the so called Bahe–Varela formalism [12–17] allows the interpretation of both equilibrium [18] and transport properties [19] of ILs. According to this structural scheme, we can assume that the free surface of the IL is formed by 2D-cells which provide potential wells that act like adsorption nodes. In these
Fig. 1. Two views of a macroscopic film of adsorbed water on EMIM-BF4 . The IL was put in contact with an atmosphere of 100% humidity degree during 4 days.
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nodes, adsorbate molecules are either anchored (localized adsorption) or wander around. Consider the free surface of an IL in contact with the atmosphere containing a certain concentration of water molecules (humidity grade). Given the formation of a macroscopic layer in the analyzed systems, we shall model the free surface of the IL as made up of M equivalent, distinguishable and independent sites, on each one of which a macroscopically large number of water molecules can be adsorbed forming a multilayer. Moreover, we consider that the adsorption of any other type of molecules can be completely neglected. Let us assume that humid air can be considered to be an ideal gas under the experimental conditions. This is reasonable at ambient temperatures and for the typical water contents of atmospheric air in the laboratory, so the chemical potential of the adsorbate molecules is = 0 + kB T ln r, with kB being Boltzmann’s constant, T the absolute temperature and r the relative humidity degree, equivalent to the concentration of the adsorbate molecules in air. Additionally, we shall assume that the canonical partition function of a water molecule adsorbed in the first layer (next to the surface) is q1 and q2 the partition function in successive layers. Under these circumstances, the surface coverage or mean number ¯ adsorbed per binding site of the free surface of the of molecules N IL is given by the BET adsorption isotherm [20]: =
¯ N kx = , M (1 − x + kx)(1 − x)
(1)
where x = q2 exp(ˇ), k = q1 /q2 and ˇ being the inverse of the thermal energy as usual. As it is well-known, this equation is capable of describing a condensation of the adsorbate on the adsorption surface, since for x → 1, the surface coverage diverges [20]. Contrarily to what happens in the gas phase in contact with the surface, water molecules are expected to interact strongly when adsorbed at the surface of the liquid, so a cooperative adsorption model must be considered, where lateral interactions between neighbor stacks exist. However, an exact treatment of cooperative adsorption is a formidable task because of the extreme operational complexity of the polynomial expressions derived from the grand canonical partition function of the adsorption sites [10]. Fortunately, a mean-field approximation is usually enough to capture the essentials of the adsorption processes, while leading to considerable simplification. In this formalism, the adsorbed particles are assumed to fluctuate as free particles in the effective (or molecular) field created by the rest of the medium, analogously to the Weiss field in the theory of spin lattices [21]. In this fashion, the coverage is an extensive magnitude (the homologue of the magnetization) coupled to an ‘external’ field that drives the binding process. The equivalent role to that of the magnetic field in the magnetization problem is played here by the chemical potential of the adsorbate, which in equilibrium is equal to the chemical potential of the adsorbed particles. In a cooperative adsorption process, this chemical potential is made up of two contributions: (i) the ligand chemical potential, , and (ii) an excess (so-called pseudochemical [21]) potential associated to the interactions with the rest of the molecules in the substrate surface, . In a mean-field formalism, this latter contribution is proportional to the coverage itself (equivalently to the concentration of water molecules in the adsorbate), so the contribution of the interparticle interactions to the chemical potential can be written in the mean-field framework as =
(2)
where represents the interaction energy between adsorbed molecules in neighboring stacks. The case ␥ > 0 (␥ < 0) corresponds to attractive (repulsive) interactions and it induces positive (negative) cooperativity. The introduction in the formalism of a mean-field interaction like that in Eq. (2) is equivalent to a punctual
transformation of the concentration of the form x → x = xeˇ
(3)
The combination of the above chemical potential with the Langmuir adsorption isotherm for a monolayer adsorption of an ideal adsorbate leads to the so called Frumkin adsorption isotherm [22]. The cooperative BET adsorption isotherm, (x ) is obtained straightforwardly transforming Eq. (1) according to Eq. (3). Upon adsorption of water from the humid atmosphere, the total mass of the system suffers a change m → m = m + m. If one takes into account the mass of an adsorbate particle, ma , the total free surface area A and the surface density of adsorption nodes, , and Eq. (1), we can write the total mass of the adsorbed particles as m =
k2 xeˇ (1 − xeˇ
+ kxeˇ )(1 − xeˇ )
(4)
where k2 = ma Ak. This equation allows the analysis of the experimental data of Ref. [3] in terms of the relative humidity grade of the atmosphere, and the results of this analysis are the object of the next section. 3. Results and discussion Fig. 2 represents the total amount of water adsorbed after about 100 h at 20 ◦ C in the free surface of four ILs of the 1-alkyl3-methylimidazolium tetrafluoroborate for different humidity degrees, together with the numerical predictions of both the noncooperative and the cooperative BET adsorption isotherms. As one can see in Fig. 2, the amount of adsorbed molecules progressively decreases as the chain length of the cation of the IL increases. This clearly reflects the hydrophobic nature of the cation. Moreover, saturation in the amount of adsorbed water molecules takes place for humidity degrees close to 100% in the atmosphere in all studied cases. The parameters in the expressions for the isotherms were fitted to the experimental data using a modified Levenberg–Marquardt algorithm, using a broad range of initial values in order to try to avoid local minima. Multiplying Eq. (1) by ma A it was recast in the same form as Eq. (4) so that both can be written as m =
k2 y , (1 − y + ky)(1 − y)
(5)
where y = k1 r and y = k1 rek3 m in the non-cooperative and cooperative cases, respectively, k1 = q2 eˇ0 and k3 = (ˇ/ma A). In both models, k1 is theoretically independent of the ionic liquid under study (depending only on the properties of water) and was thus not considered an ordinary fitting parameter, but fixed for all liquids based on a previous fit to the set of all experimental data. However, the k1 used is different in the non-cooperative and cooperative cases, since in the regular BET isotherm it has to include part of the lateral interactions which are not explicitly considered. Some numerical difficulties can arise from the fact that the increment of mass upon adsorption, m, appears on both sides of Eq. (4) in the cooperative case. The naive approach of first fitting k and k2 to the experimental data and then solving that implicit equation numerically for each individual value of m results in a strongly discontinuous isotherm. This clearly unphysical result was avoided by the more time-consuming procedure of iteratively fitting k and k2 to the experimental data and then the whole vector of m values to k and k2 , until convergence was achieved. This method afforded the smooth theoretical isotherms which can be observed in Fig. 2. The non-cooperative BET adsorption isotherm is a good qualitative prediction of the average behavior of the experimental adsorption data. This confirms the localized multilayer hypothesis. However, the non-cooperative BET isotherm is unable to fit these data in the saturation region, predicting instead a divergence of
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Fig. 2. Increment of the mass of the IL upon adsorption as a function of the humidity degree of the atmosphere for four different ILs of the imidazolium family at T = 293.15 K: (a) EMIM-BF4 , (b) BMIM-BF4 , (c) HMIM-BF4 , and (d) MOIM-BF4 . The solid dots correspond to experimental data in Ref. [3], the dotted line to the predictions of a non-cooperative BET adsorption isotherm, and the solid line to the cooperative BET calculations for a lateral interaction parameter ␥ = 2.08 × 10−20 J.
the number of adsorbed molecules per node. However, considerably better fits of the experimental data are obtained if the lateral interactions among adsorbed molecules are taken into account. Particularly, the results shown in Fig. 2 are obtained if the cooperative version of the BET adsorption isotherm in Eq. (4) with a value of these lateral interactions fixed a priori to the well-known value of the Keesom interaction between permanent water dipoles for bulk water at T = 293 K, = 2.08 × 10−20 J„„, is used. This value of the interaction parameter is obtained if one uses the Keesom coefficient for water dipole-dipole interaction, C6 = 1.2 × 10−4 eV nm−6 [23]. In this sense, it is noteworthy to be mentioned that we have verified that when is left as a fitting parameter, we obtained the values 1.44 × 10−20 J, 1.63 × 10−20 J, 1.26 × 10−20 J, and 1.26 × 10−20 J for 1-ethyl-3-methylimidazolium, 1-butyl-3-methylimidazolium, 1-hexyl-3-methylimidazolium and 1-octyl-3-methylimidazolium tetrafluoroborates, respectively. So, the statistical average of the obtained values of the interaction parameters is ␥ = 1.44 × 10−20 J, a value slightly lower than that of bulk water probably due to the presence of the substrate and the consequent modification of the intermolecular potential of the adsorbed phase. It is noteworthy that this interaction parameter is expected to be temperature dependent. Moreover, one should notice that no additional fitting parameter is included with respect to the BET non-cooperative adsorption scheme because k3 is completely determined by the values of and k2 . Thus, both schemes imply bidimensional fitting procedures with two fitting parameters (k and k2 ). This confirms both the multilayer nature of the adsorption scheme and also the role of the lateral interactions in this surface process. Besides, given the value of the lateral interactions in the bulk, we must conclude that the water adsorbed at the interface of the IL forms a macroscopic film in which the properties of the adsorbed molecules are very similar to those in bulk pure water. Given the previously mentioned fitting parameters, we can obtain the values of the surface densities of adsorption nodes in each analyzed case. Taking into account that the experimental measurements reported in Ref. [3] correspond to Petri dishes of area A = 0.0035 m2 and the molecular masses of the different compounds, the obtained fitting parameters lead to surface densities of adsorption nodes of 3.0 × 1028 , 1.6 × 1028 and 7.3 × 1028 nodes m−2 respectively for 1-ethyl-3methylimidazolium, 1-butyl-3-methylimidazolium and 1-hexyl-3-
methylimidazolium tetrafluoroborates, respectively. However, the corresponding value for 1-octyl-3-methylimidazolium tetrafluoroborate is extremely sensible to fitting details, so we do not consider it significant and it is not reported. As expected, the surface density of adsorption nodes decreases as the chain length of IL’s cation increases. Interestingly enough, the measurements of the adsorbed amount of water on 1-octyl-3-methylimidazolium tetrafluoroborate reported in Ref. [3] coincide with the data of water solubility on this compound reported in Fig. 3 of Anthony et al. [24]. This is somehow surprising, since on a first approximation one would expect that both phenomena were different because the first one (adsorption) concerns the free surface of the IL and the other one (absorption) is a bulk phenomenon. The well-known low miscibility of water with this compound (0.065 mol kg−1 at room temperature [24]) is a possible cause of this behavior. If the alkyl side chain of the cation becomes sufficiently long, the IL–water system can present phase split, as is the case of this compound [25]. Under our experimental circumstances (T = 293.15 K, P = 1 atm.), a macroscopically observable layer of water is formed for this compound. However, it is possible that an undetermined amount of the liquid is absorbed in the bulk IL, but clearly the mass determination method is not the most appropriate method for elucidating this question, and we need to perform more experimental work in this point. This latter fact may give us some insight in the structure of water at IL interfaces. Apart from some simulations of Lynden-Bell et al. [26], very limited results concerning the peculiar structure of the adsorbed water on the free interface of ILs seem to have been reported in literature. However, a detailed image of the structure and dynamics of water at interfaces is fundamental to the understanding of several phenomena, such as molecular recognition, macromolecular folding or wetting. As it is well-known, the crucial factor for the structure at interfaces is the balance of hydrogen-bonding among water molecules and the comparable interactions with the substrate, defining the extreme hydrophobic and hydrophilic behavior. Long-range ordered layers are usually associated to hydrophilic subtrates while nonordered (long-range) behavior is associated to hydrophobic substrates (see for example Ref. [27] for water ice on a hydrophilic surface like silicon, chlorine terminated). In this latter case randomly oriented crys-
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tallites are expected to form with no interfacial long-range order because stronger water–water interactions are registered when compared with those of water–substrate. However, one must recall that anomalously ordered adsorption of water in vertically stacked bilayers has been also registered on some hydrophobic substrates like highly oriented pyrolytic graphite [28]. In our case, the experimental data and the theoretical analysis suggest that a partially disordered hydrophilic-like adsorption takes place for all the analyzed ILs, no matter the degree of hydrophobicity of the cation (it increases with the chain length of this ion; on the other hand, BF4 is normally considered to be an ion of intermediate hydrophobicity among the most common anions [29]). The validity of a BET localized adsorption scheme suggests a hydrophilic-like adsorption in which the interactions with the substrates are dominant, while the lateral interactions associated to the random order of a disordered phase indicate the absence of long-range structure in the adsorbed water. Thus, the adsorption of water in ILs of the imidazolium tetrafluoroborate family shows a short-range order typical of localized adsorption on a hydrophilic substrate, and a gradual transition to the long-range disorder typical of the microscopic structure of bulk water. This conclusion is surprising since, as we mentioned previously, the substrate is conventionally considered to be hydrophobic in nature. 4. Conclusions By means of a numerical analysis in the framework of a BET adsorption scheme, we have proved that the adsorption of water in the free surface of ILs can be described as a cooperative multilayer adsorption process with lateral interactions between the adsorbed molecules. These interactions between adsorbed molecules can be treated in a mean-field formalism, assuming that the interactions are equivalent to an effective chemical potential. Finally, the energy of these lateral interactions has been shown to be that of the Keesom interactions between permanent dipoles in water at the same temperature. From all these facts, a somehow surprisingly ordered hydrophilic-like adsorption of atmospheric water is suggested to take place in the free surface of hydrophobic ILs of the imidazolium family. The analysis of the influence of both the cation and anion on the nature of the adsorption of water in ILs is now in progress. Acknowledgements This work was supported by the Dirección Xeral de I+D+i de la Xunta de Galicia and the European Regional Development Fund
(Grant no. INCITE07PXI206076ES), and by the Spanish Ministry of Education and Science and the European Regional Development Fund (Grants no. FIS2007-66823-C02-01 and no. FIS2007-66823C02-02). J. Carrete wishes to thank the Spanish Ministry of Education for a FPU grant. References [1] P. Wasserscheid, T. Welton (Eds.), Ionic Liquids in Synthesis, Wiley–Verlag, Weinheim, 2003. [2] Chieu D. Tran, S.H. de Paoli Lacerda, D. Oliveira, Appl. Spectrosc. 57 (2003) 152–157. [3] S. Cuadrado-Prado, M. Domínguez-Pérez, E. Rilo, S. García-Garabal, L. Segade, C. Franjo, O. Cabeza, Fluid Phase Equilibr. 278 (2009) 36–40. [4] C. Jungnickel, J. Łuczak, J. Ranke, J.F. Fernández, A. Müller, J. Thöming, Colloids Surf. A 316 (2008) 278–284. [5] A. Mikhailov, G. Ertl, Chem. Phys. Lett. 238 (1995) 104–109. [6] D. Batogkh, M. Hildebrandt, F. Krischer, A. Mikhailov, Phys. Rep. 288 (1997) 435–456. [7] B. Dreyfus, D. Deloche, M. Laloë, Phys. Rev. B 9 (1974) 1268–1276. [8] S.B. Casal, H.S. Wio, S. Mangioni, Physica A 311 (2002) 443–457. [9] L.M. Varela, M. García, M. Pérez-Rodríguez, P. Taboada, J.M. Ruso, V. Mosquera, J. Chem. Phys. 114 (2001) 7682–7687. [10] L.M. Varela, M. Pérez-Rodríguez, M. García, Mol. Phys. 102 (2004) 79– 84. [11] H. Katayanagi, S. Hayashi, K. Hiro-o Hamaguchi, Nishikawa, Chem. Phys. Lett. 392 (2004) 460–464. [12] L.W. Bahe, J. Phys. Chem. 76 (1972) 1062–1071. [13] L.W. Bahe, J. Phys. Chem. 76 (1972) 1608–1611. [14] L.W. Bahe, D.J. Parker, J. Am. Chem. Soc. 97 (1975) 5664–5670. [15] L.M. Varela, M. García, F. Sarmiento, D. Attwood, V. Mosquera, J. Chem. Phys. 107 (1997) 6415–6419. [16] I. Bou Malham, P. Letellier, A. Mayaffre, M. Turmine, J. Chem. Thermodyn. 39 (2007) 1132–1143. [17] S. Bouguerra, I. Bou Malham, P. Letellier, A. Mayaffre, M. Turmine, J. Chem. Thermodyn. 40 (2008) 146–154. [18] L.M. Varela, J. Carrete, M. Turmine, E. Rilo, O. Cabeza, J. Phys. Chem. B 113 (2009) 12500–12505. [19] L.M. Varela, J. Carrete, M. García, L.J. Gallego, M. Turmine, E. Rilo, O. Cabeza, Fluid Phase Equilibr. 298 (2010) 280–286. [20] T.L. Hill, An Introduction to Statistical Thermodynamics, Dover, New York, 1986. [21] E. di Cera, J. Chem. Phys. 96 (1992) 6515–6522. [22] A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th ed., Wiley, New York, 1997, p. 392. [23] V. Magnasco, M. Battezzati, A. Rapallo, C. Costa, Chem. Phys. Lett. 428 (2006) 231–235. [24] J.L. Anthony, E.J. Maginn, J.F. Brennecke, J. Phys. Chem. B 105 (2001) 10942–10949. [25] M.G. Freire, L.M.N.B.F. Santos, A.M. Fernandes, J.A.P. Coutinho, I.M. Marrucho, Fluid Phase Equilibr. 261 (2007) 449–454. [26] R.M. Lynden-Bell, J. Kohanoff, M.G. del Popolo, Faraday Discuss. 129 (2005) 57–67. [27] C.Y. Ruan, V.A. Lobastov, F. Vigliotti, S. Chen, A.H. Zewail, Science 304 (2004) 80–84. [28] Yang Ding-Shyue, A.H. Zewail, Proc. Natl. Acad. Sci. 106 (2009) 4122–4126. [29] J.G. Huddleston, A.E. Visser, W.M. Reichert, H.D. Willauer, G.A. Broker, R.D. Rogers, Green Chem. 3 (2001) 156–164.