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Pseudolattice Theory of the Surface Tension of Ionic Liquid-Water Mixtures L. M. Varela,*,† J. Carrete,† M. Turmine,‡ E. Rilo,§ and O. Cabeza§ Grupo de Nanomateriales y Materia Blanda, Departamento de Fı´sica de la Materia Condensada, UniVersidad de Santiago de Compostela, E-15782, Santiago de Compostela, Spain, UniVersite´ Pierre et Marie Curie-PARIS6, Laboratoire Interfaces et Syste`mes Electrochimiques, CNRS, UPR15-LISE, Paris, F-75005 France, and Facultad de Ciencias, UniVersidad de A Corun˜a, Campus A Zapateira s/n, E-15072, A Corun˜a, Spain ReceiVed: June 17, 2009; ReVised Manuscript ReceiVed: July 20, 2009
A theoretical model for ionic liquids (ILs) based on a pseudoreticular structural model for the bulk mixture is reported. The original Bahe-Varela pseudolattice theory of concentrated ionic solutions is modified and the short-range interactions modeled by a Lennard-Jones potential. In this framework, the surface tension of the pure IL is calculated and the correct dependence of this magnitude on the density of the liquid, as provided by the parachor, is recovered. The anions in the mixture are assumed to form a continuum structureless neutralizing background, and that the organic cations and water molecules are placed in the nodes of the pseudolattice. The surface pressure of IL-water mixtures is calculated using a localized model for the adsorption of particles in the surface of the mixture and a mean-field Bragg-Williams approximation for the chemical potential of the adsorbed particles in the pseudolattice. The theoretical predictions are tested with experimental data of several ionic liquid aqueous mixtures. Introduction Ionic liquids (ILs) or room temperature molten salts are optimal candidates for replacing conventional organic solvents and providing a new category of “green solvents” due to their unique characteristics: negligible vapor pressure, nonflammability, ability to dissolve an enormous range of inorganic, organic, and polymeric materials at very high concentrations, noncorrosiveness, low viscosity, etc. The contemporary age of IL science can be said to start with the seminal contribution of Wilkes et al.1 synthesizing the air- and water-stable ionic liquid (IL) C2MIMBF4 (1-ethyl-3-methylimidazolium tetrafluoroborate) which allowed a much wider range of applications than its homologous systems reported since the final decades of the XIXth century (ethylammonium nitrate, alkylimidazolium and alkylpyridinium mixtures of ILs, etc.). Since then, the physicochemical characterization of ILs is among the most active research areas from both the industrial and academic perspectives. Many industrial and academic applications involving ILs take place in mixtures, so recently, the interest of many researchers has been focused on properties such as liquid-liquid and vapor-liquid equilibria of mixtures containing ionic liquids. The reported amount of data concerning bulk mixture properties such as densities, activity coefficients, viscosities, and conductivities has increased continuously. However, surface properties of IL mixtures have received much more limited attention2-11 despite their importance in catalysis processes between IL and overlying organic phases.11,12 Particularly, to our knowledge, no satisfactory molecular theory of surface adsorption in pure IL or IL mixtures has been reported to date. Surface properties are determined by the structure of the system and the interactions between the adsorbate and the adsorbent species. To date, the structural and transport properties * Corresponding author. E-mail:
[email protected]. † Universidad de Santiago de Compostela. ‡ Universite´ Pierre et Marie Curie-PARIS6. § Universidad de A Corun˜a.
of pure ILs have been analyzed using mainly two theoretical formalisms: (i) the hole theory, originally formulated by Fu¨rth in the 1940s13 and later developed by Bockris,14,15 and (ii) the interstice model due to Yang and co-workers.16 In the first model, the low-density regions in the bulk IL are assumed to form an ideal gas of relatively small “holes”, and it provides a highly accurate model for the understanding of both structural and transport properties of ILs (see ref 17 and references therein). The main purpose of hole theory is the description of the spontaneous density fluctuations occurring in bulk liquids during the thermally driven motion of their constituents. On the other hand, the core assumption of the so-called interstice model is the existence of inherent cavities in the bulk fluid associated with the large size and great asymmetry of the ions that remain as distinguishable stable entities during thermal motion.16 Despite their undoubted virtues, none of the theories is well adapted for the study of IL mixtures in which either holes or cavities are occupied by molecules of the other species. In particular, the transition of the pure IL regime to IL mixtures with the limit in an electrolyte solution is not adequately described by the previous existing formalisms. Moreover, in both formalisms, the experimental fact of the existence of a pseudoreticular structure in aqueous solutions of these systems from very low concentrations, well-known for many decades,18-23 is ignored. Solutions of strong electrolytes are known to exhibit statistical order even at low to moderate concentrations. The model of an ionic solution in which each ion moves freely within one cell of a statistical lattice, hereafter referred to as a pseudolattice, is supported, up to concentrations of about 4 M, by X-ray measurements and by the dependence of activity coefficients on both c and c1/3. Besides, Katayanagi et al.24 have recently verified the existence of long-range correlations in liquid 1-n-butyl-3-methylimidazolium iodide ([bmim]I) and that the structures of cations and anions are similar to those in the crystal by means of wide-angle X-ray scattering and Raman spectros-
10.1021/jp9057065 CCC: $40.75 2009 American Chemical Society Published on Web 08/20/2009
Surface Tension of Ionic Liquid-Water Mixtures copy. Thus, it becomes necessary to incorporate these results into the current structural formalism of ILs. The pseudolattice structural model of concentrated ionic solutions due to Bahe25-27 and later extended by Varela and co-workers28 will be used for this purpose, modified with respect to its original formulation to include soft-core short-range repulsions, supposed to be predominant in the pure IL regime. The thermodynamic properties of 1:1 and 1:2 concentrated electrolyte solutions were successfully explained in this theoretical framework (see ref 28 and references therein). Moreover, this structural model has been recently employed to account for the thermodynamic properties of IL mixtures up to pure IL,29,30 showing excellent agreement with the experimental evidence. However, it has not been applied to the prediction of the surface tension of IL mixtures yet. In the present paper, we formulate a theoretical model of localized adsorption for the surface tension of IL mixtures, combining a pseudolattice structural model for the ions in the bulk with a mean-field Bragg-Williams-like treatment of their interactions, and we apply the results to analyze the experimental data for different IL-water mixtures. Experimental Section Products. To test the theory, in the present paper, we employ previously reported data of 1-ethyl-3-methyl imidazolium tetrafluoroborate (EMIM-BF4) and 1-butyl-3-methyl imidazolium tetrafluoroborate (BMIM-BF4),31 as well as of ethylammonium nitrate (EAN).32 The surface tensions of these pure ILs are 53.04, 45.33, and 49.15 mN m-1, respectively. The fourth studied substance, ethanolammonium formate (EOAF), was prepared by reacting equimolar amounts of ethanolamine (cooled aqueous concentrated solution) and pure formic acid to produce an aqueous solution as described elsewhere.33 Water was then removed by rotary evaporation at 35 °C, and the last vestiges of water were removed by lyophilization. The measured surface tension of this synthesized substance was 64.3 mN m-1. Techniques. Surface tensions were measured using a Kru¨ss tensiometer (K11 model) by the Wilhelmy plate method. The solutions were maintained at a constant temperature ((0.1 °C) in a circulating water bath. Theoretical Section Let us consider an IL-solvent mixture. In the limit of low IL concentration, the mixture can be regarded as a strong z+:zelectrolyte solution. As previously mentioned, a pseudolattice exists in the bulk solution with lattice parameter R (cell size), and whose structure depends on the lattice type of the salt (fcc for a 1:1 electrolyte, bcc for a 1:2 electrolyte). In what follows, we will only consider 1:1 electrolytes for simplicity purposes. Moreover, as the concentration increases, this short-range order is reinforced, so the pseudolattice structure is expected to extend over the whole concentration range up to the pure IL regime. The successful interpretation of thermodynamic properties of IL-water mixtures previously cited29,30 reinforces this picture. Thus, the pseudolattice model is expected to provide the basic structural model for IL mixtures almost over the whole concentration range (excluding the limit of infinitely dilute solutions where the conventional mean-field Debye-Hu¨ckel formalism is recovered). The thermodynamics of these ionic systems is determined by both electrostatic and short-range dispersive ion-ion interactions, and also by ion-solvent interactions.25-28 Despite the fact that in the original version of the theory a hard-core potential was used to model the short-range interionic repulsions,28 the importance of these interactions in the pure IL regime and in
J. Phys. Chem. B, Vol. 113, No. 37, 2009 12501 extremely concentrated solutions leads us to use now a more refined version of the repulsive part of the potential. Thus, we assume that the short-range non-Coulomb interactions are well described by a Lennard-Jones (LJ) potential with an r-12 repulsion term
uijLJ(rij) )
A Λ - 6 12 rij rij
(1)
where the constant Λ comprises the effect of van der Waals forces.28 Following the same line of reasoning employed in this latter reference, it can be straightforwardly proved that the free energy of the electrolyte solution in the pseudolattice theory of solutions is given by
F ) A''
| |
q+q- 1 q+2 dε 1 Λ A Vsea 3 - D'' 6 + E'' 12 + B'' 4πε R (4πε0)2 dR R R R (2)
where R is the size of the cells of the pseudolattice (rij ) aijR, with aij being the lattice parameters), ε the dielectric constant of the bulk solvent, ε0 the dielectric constant of the medium at the ion surface, Vsea the volume of the region around each ion in which the dielectric constant varies from ε0 to ε, and q+ and q- are the cation and anion charges, respectively. On the other hand, A′′, B′′, D′′, and E′′ are Madelung-like constants whose values depend on the particular structure of the ionic solution. The first term on the right-hand side of eq 2 is associated with the Coulomb interactions in the bulk solution, the second one to the interaction of ions with the dielectric gradient around them, the third one to the attractive part of the LJ potential of dispersive origin, and the fourth one the contribution of the short-range repulsive part of the LJ potential associated with the repulsion of the electron clouds of the ions due to the Pauli exclusion principle. Once the free energy is known, the thermodynamics of the systems can be obtained by the conventional procedures. Due to their structural characteristics, the surface tension of the pseudolattice ionic systems can be calculated as that of solids. According to Shuttleworth,34 the surface tension of a crystal face is related to the surface free energy by the relation γ ) F + S(dF/dS), where S represents here the area of the surface. In our case, the surface area can be related to the pseudolattice cell size, since it scales as S ∼ R2, so
γ ) A''
| |
q+q- 1 q+2 dε 1 Λ 1 Vsea 3 + 2D'' 6 6 - B'' 8πε R 2(4πε0)2 dR R aA R 1 5E'' 12 (3) R
Obviously, although all of the interactions contributing to the free energy in eq 2 are permanently switched on throughout the whole range of concentration, in each regime, one of them is dominant over the others. In the pure IL limit, ion-solvent interactions are not present and the short-range repulsive term is expected to be overwhelmingly dominant due to the abundance of ionic contacts. Thus, the main contribution to the surface tension of pure ILs is expected to scale like R-12 ∼ F4, where F is the IL density. This is in agreement with the predictions of McLeod35 and Sugden36 for the parachor of liquids
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FWγ1/4 FWγ1/4 P ) CFW ) = F - Fv F
Varela et al.
(4)
where FW stands for the formula weight, C is a constant characteristic of the liquid, and Fv is the density of the air. This result establishes the proportionality γ ∝ F4 between the surface tension and the density of the liquid, a prediction which has been recently verified for ILs by Deetlefs et al.37 It is to be noted that this prediction stands exclusively on the basis of a pseudolattice model for the bulk liquid, and that it is in marked contrast with the hole and interstice models where surface tension is treated as a purely experimental input necessary for the determination of the hole or interstice average radius, respectively. To our knowledge, this is the first theoretical evidence of the existence of a pseudolattice structure in a pure IL. Now that we have established the validity of the pseudolattice theory for the description of the surface properties of ILs, we turn to the topic of surface adsorption in IL mixtures. For these systems, we will analyze the pseudolattice model capability of describing the surface adsorption of one of the species in the air-liquid interface and we will verify the predictions with direct experimental data for IL-water mixtures in which one of the species in the mixture exhibits a surfactant-like behavior characterized by a rapid formation of the adsorption monolayer at low concentrations and an approximately constant concentration in the bulk at moderate and high concentrations. For this purpose, we must take into account that in IL mixtures (particularly in IL-water mixtures) we have three species (cations, anions, and water neutral molecules) and that two of them are adsorbed at the surface. Due to its hydrophobic nature, the positively adsorbed species in the water mixtures is normally the organic cationic species. The main effect of the anions on the surface tension is the formation of an electric double layer in the neighborhood of the interface, and their binding is usually described by the Stern isotherm.38 Despite the fact that, for a detailed description of the surface charge density, anions must be considered a second surface active species, in the present paper, we will assume as a simplifying hypothesis that they can be averaged over the whole volume of the mixture in the same fashion as in the description of the wellknown one-component plasma. Thus, their exclusive role is the preservation of both global and local electroneutrality. This is equivalent to consider the structure of the bulk mixture as a pseudolattice whose nodes are occupied by amphiphilic cations and water molecules immersed in a neutralizing background. The adsorption of cationic molecules (species A) at the surface of an IL mixture is determined by the migration of the adsorbate molecules through the bulk medium and for the peculiar features of the monolayer itself. In what the first influence is concerned, it is determined by the interactions of the amphiphilic cations with the water molecules (species B). In the following, we will treat them in the Bragg-Williams approximation, as a first approximation. On the other hand, studies of the surface tension of aqueous surfactant solutions are usually based either on localized adsorption models employing the Langmuir isotherm itself or some kind of Langmuirlike isotherm that include interactions between adsorbed particles (Frumkin isotherm, etc.), or in nonlocalized adsorption of the adsorbate leading to the van der Waals adsorption model, termed also the Hill-de Boer model (see ref 38 and references therein). Since both schemes are known to provide similar results in what adsorption of surfactants is concerned,38 for simplicity purposes, we will employ a localized adsorption scheme for IL mixtures.
Consequently, let us consider that the free liquid-air interface of the IL-water mixture is composed of M localized independent adsorption sites formed by the negatively adsorbed species, in which either zero or one molecule of the positively adsorbed compound can be bound. Let µA be the chemical potential of an adsorbate molecule. The grand-canonical partition function of the adsorption surface is therefore
ln Ξ ) M ln(1 + ξA)
(5)
where ξA ) qAλA. qA represents the canonical partition function of an adsorbed particle and λA ) exp(βµA), with β ) 1/kBT being the inverse of the thermal energy, is the fugacity of the adsorbate. Using well-known thermodynamic relations, one can obtain the average number of molecules per adsorption site recovering the conventional Langmuir adsorption isotherm:
( )
jsA ) kBT
∂ln Ξ ∂µA
) T
ξA 1 + ξA
(6)
At this point, one must provide an explicit form for the chemical potential of the adsorbate. In our particular case, given the pseudolattice structure of the adsorbate, a lattice statistics problem must be solved in order to get an expression for the chemical potential of the adsorbate. Considering that in concentrated ionic liquids the Coulomb interaction is highly screened and that above the Kirkwood crossover a transition from monotonic charge to damped oscillatory charge dominated decay takes place,39 the main thermodynamic properties of these systems are presumably describable in terms of conventional lattice statistics terms. The fundamental statistical analysis of ionic systems in a lattice formalism is due to Kobelev et al.,40 who reported a theoretical analysis of Coulomb systems on lattices in general dimensions. Moreover, they developed the thermodynamics using Debye-Hu¨ckel theory with ion-pairing and dipole-ion solvation. Nevertheless, for the purposes of a first understanding of surface adsorption in ILs, we will employ simplified lattice models such as the quasichemical or the Bragg-Williams approximations. In fact, the quasi-lattice, quasi-chemical theory of preferential solvation of ions in mixed solvents has been used to explain the properties of the electrolyte solutions in solvent mixtures.41,42 Moreover, quasichemical-like models have been used to analyze all of the condensed phases of salt mixtures including their liquid solutions.43 As a first approximation to the surface adsorption problem, in the present paper, we will adopt a Bragg-Williams meanfield-like form for the chemical potential of the adsorbate (either IL in water or a molecular species in IL), so from the lattice theory of solutions,44 we get
βµA ) ln xA - ln qAe-βcωAA/2 -
βcω(1 - xA)2 2
(7)
where c is the number of neighbors in the pseudolattice, ωAA (ωBB) represents the interaction between molecules of the A (B) species in the pseudolattice, ω ) ωAA + ωBB - 2ωAB, and xA is the molar fraction of the cation species. These interaction parameters comprise the contributions of several interactions (electrostatic, dispersive van der Waals interactions, hydrophobic interactions, ...). This chemical potential results from the
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assumption of a random distribution of particles in the pseudolattice, and it is recovered from the quasichemical approximation in the limit βω f 0. Combining eqs 5 and 6, one gets
ξA xA exp(e-βcωAA/2)e-βcω(1 - xA) /2 jsA ) ) 2 1 + ξA 1 + xA exp(e-βcωAA/2)e-βcω(1 - xA) /2 2
(8) Obviously, for ω f 0, the Langmuir isotherm is recovered. On the other hand, one can refine the model introducing higher order terms in the expansion of the quasichemical expression for the chemical potential of the adsorbate for higher interaction strengths.44 Once we know the fraction of molecules in the surface, we can use the conventional relation for the surface tension of nonideal mixtures, γ ) xAsγA + xBsγB, to get a reduced surface pressure:45,46
π* )
γ - γΒ ) xAS γΑ - γΒ
(9)
and considering that the mole fraction of A in the surface can be related to the fraction of surface occupied by the adsorbate as xAs ) jsA,47 we finally get π* ) jsA. In Figure 1, we show the recently reported values of the reduced surface pressure of aqueous mixtures of EMIM-BF4 (a) and BMIM-BF4 (b) as a function of the IL molar fraction at 298.15 K.31 As we can see in that figure, both compounds exhibit a similar amphiphilic behavior: a very fast formation of a monolayer in the free interface of the system in the low concentration region, followed by an almost constant bulk concentration beyond the critical micelle concentration limit. As it is well-known, the surface pressure of an ionic solution is lowered (equivalently, the surface tension is increased) by the addition of ions. Conventional ions are usually negatively adsorbed in water, and they tend to go to the bulk. However, in our case, the positively adsorbed species in both the EMIM and BMIM cases is the cation (EMIM+ or BMIM+, respectively) due to the hydrophobic character of both the hydrocarbon chain and the imidazolium group. They simply act as cationic surfactants in water in the low concentration region. Moreover, as we can see in the representation, the longer the chain of the hydrophobic moiety, the higher the rate of increase of the surface pressure, which indicates the leading role of hydrophobic interactions in the surface adsorption of these compounds in water mixtures. This effect can be naturally accounted for by the Bragg-Williams approximation with ω < 0 giving rise to a phase separation. The experimental data have been fitted by means of eq 9 combined with eq 8, and the best fitting parameters where, respectively, cωAA ) -1.29kBT and cω ) -7.24kBT for BMIM-BF4 and cωAA ) -1.41kBT and cω ) -0.42kBT for EMIM-BF4. In both cases, ω < 0, so the species in the pseudolattice tend to separate, mimicking the formation of the surface monolayer. On the other hand, the pair interaction energies between the hydrophobic chains, ωAA, are very similar for both compounds. In Figure 2, we show the experimental values of the surface pressure of EAN-water mixtures throughout the whole concentration range.32 Once again, the Langmuir isotherm combined with the chemical potential of a pseudolattice mixture is in good agreement with the experimental results. In contrast to the previous substances, the EAN exhibits a slow, progressive
Figure 1. Reduced surface pressure of (a) EMIM-water and (b) BMIM-water mixtures.31 The solid lines correspond to the predictions of eq 9 with the parameters mentioned in the text.
Figure 2. Reduced surface pressure of EAN-water mixtures. The solid line corresponds to the predictions of eq 9 with the parameters mentioned in the text. Data taken from ref 32.
increment of the surface pressure up to the pure IL, and no indication of such a marked surfactant-like self-assembly is apparent in the experimental results. In this case, the best fitting parameters for the interaction energies were cωAA ) -1.12kBT and cω ) 1.76kBT. Such a low but positive value of ω reflects
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Figure 3. Reduced surface pressure of EOAF-water mixtures. The solid line corresponds to the predictions of eq 9 with the parameters mentioned in the text, and the dotted one indicates the ionic behavior of EOAF in the preaggregated region.
that EAN is not acting in the mixture as a strongly amphiphilic system. Finally, in Figure 3, the experimental measurements of the reduced surface pressure of EOAF are shown, and the interaction energies obtained by the fitting procedure were cωAA ) -0.88kBT and cω ) 6.14kBT. One notable difference between the behavior of EOAF-water mixtures is that we can observe a low-concentration region (up to xIL ) 0.15) where the EOAF exhibits a clearly ionic behavior, dominant over the hydrophobic one. In fact, a decrease of the surface pressure is observed in that region, showing that EOAF molecules go into the bulk solution at low concentrations instead of going to the free air-liquid interface. The relatively high value of ω in this case reflects this tendency to mix up to moderate concentrations of the EOAF. As we can observe in the figures presented above, the presently reported pseudolattice formalism in the Bragg-Williams approximation in eq 9 accurately fits the reduced surface pressure data in the low to moderate concentration regimes. Besides, it represents a surprisingly good average prediction of the behavior of this magnitude at high IL concentrations. The deviations of theoretical predictions from the experimental values are presumably associated with the theoretical assumptions of one cation adsorbed per adsorption site, the averaging out of the anions, and the approximation of the surface concentration of adsorbate by its bulk value, assumptions which are expected to progressively break down as the concentration of IL increases. All of these effects must be taken into account for an accurate prediction of the observed surface pressure in the IL intermediate and high concentration regimes. Conclusions We have introduced a theoretical model for IL mixtures based on the Bahe-Varela pseudolattice theory of concentrated ionic solutions. Assuming a pseudolattice structure for the bulk mixture throughout the whole range of concentrations from the limit of infinitely dilute solutions (linearized Debye-Hu¨ckel regime) up to the pure IL, the correct density dependence of the surface tension of the pure IL liquid was predicted. Thus, a γ ∝ F4 dependence was recovered, associated with the contribution of the short-ranged part of the Lennard-Jones potential, providing promising evidence of the existence of a pseudolattice short-range order in pure ILs.
Varela et al. The surface pressure of IL-water mixtures was also a matter of study in the present paper. This magnitude was calculated in the pseudolattice structural framework assuming that the organic cations are the only species adsorbed in the surface of the aqueous mixture, and that the anions are averaged out to form a neutralizing background in the spirit of the one-component plasma model of ionic solutions. The calculations were made combining a localized model for the adsorption of cations in the air-liquid interface with a Bragg-Williams approximation for the thermodynamic properties of the cation-water composite pseudolattice. The theoretical predictions allowed highly accurate fittings of experimental results of several IL mixtures for ILs of different hydrophobic strengths in the low to intermediate concentration range, and a remarkably good average prediction of the behavior of this magnitude at high IL concentrations. The inclusion in the pseudolattice theoretical framework of the possibility of adsorption of more than one particle per site, the detailed role of the anion, or an accurate description of the surface concentration of ions are expected to improve the theoretical predictions for highly concentrated IL-water mixtures and are now under consideration, together with the application of the theory to mixtures of ILs for which no significant solvophobic effect is present. Acknowledgment. This work was supported by the Direccio´n 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 (Grant Nos. FIS2007-66823-C02-01 and FIS2007-66823-C02-02). J. Carrete wishes to thank the Spanish Ministry of Education for a FPU grant. References and Notes (1) Wilkes, J. S.; Zaworotko, J. J. Chem. Soc., Chem. Commun. 1992, 965. (2) Heintz, A. J. Chem. Thermodyn. 2005, 37, 525–535. (3) Freire, M. G.; Carvalho, P. J.; Fernandes, A. M.; Marrucho, I. M.; Queimada, A. J.; Coutinho, J. A. P. J. Colloid Interface Sci. 2007, 314, 621–630. (4) Wertz, Ch.; Tschersich, A.; Lehmann, J. K.; Heintz, J. Mol. Liq. 2007, 131-132, 2–6. (5) Halka, V.; Tsekov, R.; Freyland, W. J. Phys.: Condens. Matter 2005, 17, 3325–3331. (6) Wandschneider, A.; Lehmann, J. K.; Heintz, A. J. Chem. Eng. Data 2008, 53, 596–599. (7) Martino, W.; Ferna´ndez de la Mora, J.; Yoshida, Y.; Saito, G.; Wilkes, J. Green Chem. 2006, 8, 390–397. (8) Fitchett, B. D.; Rollings, J. B.; Conboy, J. C. Langmuir 2005, 21, 12179–12186. (9) Yang, Jia-Zhen; Tong, J.; Li, Jing-Bin; Li, Ji-Guan; Tong, Jian J. Colloid Interface Sci. 2007, 313, 374–377. (10) Domanska, U.; Pobudkowska, A.; Rogalski, M. J. Colloid Interface Sci. 2008, 322, 342–350. (11) Dong, Bin; Li, Na; Zheng, Liqiang; Yu, Li; Inoue, T. Langmuir 2007, 23, 4178–4182. (12) Law, G.; Watson, P. R. Langmuir 2001, 17, 6138–6141. (13) Fu¨rth, R. Proc. Cambridge Philos. Soc. 1941, 37, 252–275; ibid. 1941, 37, 276-280. (14) Bockris, J.O’M.; Hooper, G. W. Discuss. Faraday Soc. 1961, 32, 218–236. (15) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry. Vol. 1 Ionics, 2nd ed.; Plenum Press: New York, 2000; Chapter 5. (16) Yang, J.-Z.; Lu, X.-M.; Gui, J.-S.; Xua, W.-G. Green Chem. 2004, 6, 541–543. (17) Abbott, A. P.; Harris, R. C.; Ryder, K. S. J. Phys. Chem. B 2007, 111, 4910–4913. (18) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolyte Solutions, 3rd ed.; Reinhold: New York, 1958. (19) Robinson, R. A., Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959. (20) Kirkwood, J. G. Chem. ReV. 1936, 19, 275–307. (21) Desnoyers, J. E.; Conway, B. E. J. Phys. Chem. 1964, 68, 2305– 2311.
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