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Interfacial Properties of Water/CO2: A Comprehensive Description through a Gradient Theory-SAFT-VR Mie Approach Thomas Lafitte,† Bruno Mendiboure,† Manuel M. Pin˜eiro,‡ David Bessie`res,† and Christelle Miqueu*,† UMR 5150 - Laboratoire des Fluides Complexes, UniVersite´ de Pau et des Pays de L’Adour, B.P. 1155, Pau, Cedex 64013, France, and Departamento de Fı´sica Aplicada, Facultade de Ciencias, UniVersidade de Vigo, 36310 Vigo, Spain ReceiVed: April 13, 2010; ReVised Manuscript ReceiVed: July 12, 2010
The Gradient Theory of fluid interfaces is for the first time combined with the SAFT-VR Mie EOS to model the interfacial properties of the water/CO2 mixture. As a preliminary test of the performance of the coupling between both theories, liquid-vapor interfacial properties of pure water have been determined. The complex temperature dependence of the surface tension of water can be accurately reproduced, and the interfacial thickness is in good agreement with experimental data and simulation results. The water/CO2 mixture presents several types of interfaces as the liquid water may be in contact with gaseous, liquid, or supercritical CO2. Here, the interfacial tension of the water/CO2 mixture is modeled accurately by the gradient theory with a unique value of the crossed influence parameter over a broad range of thermodynamic conditions. The interfacial density profiles show a systematic adsorption of CO2 in the interface. Moreover, when approaching the saturation pressure of CO2, a prewetting transition is highlighted. The adsorption isotherm of CO2 is computed as well in the case of a gas/liquid interface and compared with experimental data. The good agreement obtained is an indirect proof of the consistency of interfacial density profiles computed with the gradient theory for this mixture and confirms that the gradient theory is suitable and reliable to describe the microstructure of complex fluid interfaces. Introduction The study of phase equilibria and interfacial tension of water/ CO2 mixture has received much attention due to its relevant industrial and environmental applications. The most popular application is the crucial problem of CO2 storage in aquifers and depleted hydrocarbon reservoirs, which is believed to be one of the most promising alternatives for reducing CO2 emissions, and is the focus of a remarkable research effort. Its practical implementation in a real reservoir requires a previous assessment of safety assurance. One possible cause of leakage is the capillary failure of the caprock. These leaks are to a large extent controlled by the CO2/water interfacial tension (IFT). There are also many other areas of industrial relevance involving processes where CO2 is used as a solvent, including for instance separation in food industry, coatings, polymer production, and dry-cleaning.1 In most of these processes, where carbon dioxide plays a key role, water is also involved to a certain extent. The water + carbon dioxide binary mixture exhibits type III phase behavior according to the Scott and van Konynenburg classification,2 with a characteristic large region of liquid-liquid separation. This involves a rich interfacial behavior where, for temperatures below that of the Upper Critical End Point (UCEP), the isotherms of interfacial tensions are divided in two branches corresponding to a region of vapor-liquid equilibrium (VLE) at low pressure and a region of liquid-liquid equilibrium (LLE) at high pressure. Among the numerous studies devoted to the phase behavior of the water/CO2 binary mixture, the reader is * Corresponding author. Tel.: +33 5 59 57 44 15. Fax: +33 5 59 57 44 09. E-mail:
[email protected]. † Universite´ de Pau et des Pays de L’Adour. ‡ Universidade de Vigo.
referred to the work of Dos Ramos et al.3 where the global phase behavior is described using the Statistical Associating Fluid Theory for chain molecules with attractive potential of Variable Range equation of state (SAFT-VR EOS).4,5 While the CO2-H2O interfacial tension experiments are well documented in the literature,6-11 the modeling of IFT has received much less attention. It is worthwhile to mention the recent work of Li et al.12 who computed the interfacial tension of the mixture through an approach based on the gradient theory. However, their study focused solely on the interfacial tension and suffered some limitations due to the choice of the equation of state used to evaluate the Helmholtz free energy of the homogeneous fluid. The gradient theory of fluid interfaces plays a central role in the interfacial tension modeling of the fluid mixture. The foundations of this theory were established in the seminal work of van der Waals,13 and the theory was reformulated later in 1958 by Cahn and Hilliard.14 The underlying basis of the theory is the conversion of the statistical mechanics of inhomogeneous fluid into a nonlinear boundary condition problem that, once solved, provides access to density and stress distributions through the fluid interface. This approach has already been applied with remarkable success to a wide variety of fluids: hydrocarbons and their mixtures,15-31 polar compounds and their mixtures,22,32-35 polymer and polymer melts,36-39 near critical interfaces,40-43 and other liquid-liquid interfaces.44,45 If this theory is combined with a thermodynamic model giving a reliable description of both phase equilibria and densities in the bulk phase, the gradient theory becomes a very efficient approach46 to better understand interfacial behavior. Additionally, it must be emphasized that the gradient theory is appropriate not only for modeling macroscopic properties, such as the interfacial tension, but also for providing an insight into the
10.1021/jp103292e 2010 American Chemical Society Published on Web 08/10/2010
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microscopic phenomena occurring in the interface. As an example, density profiles or/and surface segregation can be estimated to visualize the local composition fluctuations at the interface of multicomponent mixtures. In this work, we propose a comprehensive description of interfacial properties of water/CO2 through a microscopic model that combines the gradient theory with the SAFT-VR Mie equation of state. The water/CO2 mixture is studied within temperature and pressure conditions of CO2 geological storage. Three isotherms were investigated at 287, 298, and 318 K for pressures ranging from 0.1 to 20 MPa. Those thermodynamic conditions involve three different types of interface, namely, vapor CO2/water, liquid CO2/water, and supercritical CO2/water. Special attention is devoted to the estimation of the IFT step due to the crossing of the three-phase line (close to the pure CO2 liquid-vapor transition). The wetting of CO2 on water is analyzed from the CO2 density profile evolution and surface segregation estimated at the interface. Finally, the relative adsorption of CO2 is computed via two thermodynamic routes, through either the estimated IFT slope or the integration of the surface segregation. This allows discussing the consistency of the global modeling. The work is organized as follows. In the Theoretical Basis section, a brief recall of the gradient theory is given, together with the most relevant features of the SAFT-VR Mie equation of state. Results and Discussions are presented in the following section, and finally Conclusions are summarized. Theoretical Basis The gradient theory has been described extensively by several authors. Therefore, the discussion on this theory will be limited to its most significant features. For further details, the reader is referred for instance to Bongiorno et al.,47,48 Davis et al.,18,49 Carey’s thesis,16 or the Davis monography.50 The gradient theory is based on the assumption that the molecular gradients in the interface are small compared with the reciprocal of the intermolecular distance. This hypothesis allows the density n and its derivatives to be handled as independent variables. Using this assumption, the Helmholtz free energy can be expanded in a Taylor series around the homogeneous state and truncated after the second-order term to give
F)
∫V
[
f0(n) +
]
∑ ∑ 21 cij∇ni∇nj dV i
j
(1)
∇ni represents the local density gradient of component i. Thus, in the absence of an external potential, the Helmholtz free energy density of a heterogeneous fluid can be expressed as the sum of two contributions: the Helmholtz free energy f0(n) of a homogeneous fluid at local composition, n, and a corrective term which is a function of the local density gradients. The coefficients cij denote the so-called influence parameters, whose physical interpretation is related to the molecular structure of the interface, and essentially determine the density gradient response to the local deviations of the chemical potentials from their bulk value.48 According to the minimum free energy criterion applied in eq 1, the equilibrium densities must satisfy the following Euler-Lagrange equations
∂c
∑ ∇(cij∇nj) - 21 ∑ ∑ ∂nkji ∇nk∇nj ) ∂Ω ∂ni j
k
j
for i,j,k ) 1...N (2) with the grand thermodynamic potential Ω ≡ f0(n) - ∑iniµi. In the case of a planar interface, McCoy and Davis,51 Carey et al.,17 and Miqueu et al.26 showed that the density dependence of the influence parameters can be neglected, so that the positiondependent densities ni(z) obey the following equilibrium conditions
∑
cij
j
d2nj dz2
) µi0(n1, ..., nN) - µi ≡ ∆µi(n1, ..., nN) for i,j ) 1...N (3)
where µ0i ≡ (∂f0/∂ni)T,V,nj and µi stands for the chemical potential of component i in the coexisting bulk phases. This set of equations that allow us to compute the density profile of each component across the interface is solved by a Galerkin method.50 Multiplying eq 3 by dni/dz, summing over i, and integrating the result yields
dn dn
∑ ∑ 21 cij dzi dzj ) ∆Ω(n) ) Ω(n) - ΩB i
(4)
j
where ΩB ) -P, P being the equilibrium pressure. The boundary conditions for a planar interface are for instance ni(z f -∞) ) nVi and ni(z f +∞) ) nLi where nVi and nLi are the equilibrium densities of component i in the vapor and liquid bulk phases, respectively. The interfacial tension γ is thus given by
γ)
dn dn
∫-∞+∞ ∑ ∑ cij dzi dzj dz ) ∫-∞+∞ 2[f0(n) - ∑ niµi + i
j
i
P]dz )
∫-∞ 2[Ω(n) - ΩB]dz +∞
(5)
Thus, once the phase equilibrium is obtained, the only inputs of the gradient theory are the free energy density of the homogeneous fluid and the influence parameters of the inhomogeneous fluid. A single equation of state is used to model both the equilibrium properties and the free energy density in the interface. Hence, given an EOS and a set of influence parameters, the gradient theory provides a unified theory for predicting both the fluid phase behavior and interfacial properties (surface tension, density profiles, ...). In this work, the Helmholtz free energy of the water-CO2 binary mixture is calculated with the SAFT-VR Mie EOS.52 This equation of state is a modified version of the original SAFT-VR equation4 in which an n-6 variable Mie potential is used as a reference intermolecular potential to describe chain and associating molecules. The concept of using generalized Lennard-Jones potentials for the elementary building block of homonuclear chains of spherical segments has proven52 to be very fruitful to model simultaneously the fluid phase behavior and second-derivative properties of real fluids. This equation can be written in terms of the residual Helmholtz free energy as follows
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FRES FMONO FCHAIN FASSOC ) + + NkT NkT NkT NkT
Lafitte et al.
(6)
TABLE 1: SAFT-VR Mie Molecular Parameters for CO2 and Water and Influence Parameters Used in the Gradient Theory σ
where N is the number of molecules; T is the temperature; and k is the Boltzmann constant. In the case of mixtures containing both chain and associating molecules, the free energy can be expressed as a sum of three microscopic contributions: a monomer term FMONO, which takes into account the attractive and repulsive forces between the segments that form the molecules, a chain contribution FCHAIN, which accounts for the connectivity of the molecules, and an association term FASSOC, which takes into account the contribution to the free energy due to hydrogen-bonding interactions. For a detailed description of these contributions, the reader is referred to earlier publications.52,53 As in the case of all SAFT EOSs, the SAFT-VR Mie approach requires the determination of the intermolecular potential parameters which are typically obtained by fitting to experimental macroscopic data. The procedure used for the optimization of these parameters is described in the next section. As proposed originally by Carey,16 the crossed influence parameters cij are related to the geometric mean of the pure component influence parameters ci and cj by
cij ) βij√cicj
(7)
where βij stands for adjustable binary interaction parameters. The stability of the interface requires that the values of βij are comprised between 0 and 1.16 It is important to point out that the influence parameter of the pure fluid can be obtained through the use of a rigorous theoretical expression41,48 which requires the knowledge of the direct correlation function of the homogeneous fluid. However, this quantity is not readily available for most of the systems of practical interest so that the influence parameters are generally estimated from measurable or computable quantities such as the surface tension. In this work (see Results and Discussion), they are derived from the experimental surface tension, as previously described in ref 26.
m
(Å)
ε/k (K)
AB
λrep
κ
7.0579 CO2 2.738 2.5387 100.15 water 1 3.0856 177.6851 6.1232 0.046477
ΨΑΒ/κ
c
(Κ)
(J · m · mol-2)
1406
2.54 × 10-20 1.36 × 10-20
5
resulting molecular parameters of carbon dioxide and water are summarized in Table 1. Once the VLE is computed, we can determine the influence parameters suitable for fitting the surface tension data by rewriting eq 5 for a pure component. We obtain the following expression
c)
1 2
[∫
γexp nL
nV
√∆Ωdn
]
2
(8)
The classical trend, previously observed for n-alkanes,26 i.e., a slight increase with temperature, was obtained for both components. Considering this behavior, the influence parameters will be approximated in this work by a constant taken at the lowest temperature value (see Table 1). The surface tension was then estimated with this constant influence parameter. The results for water are plotted in Figure 1 together with the experimental data. This figure shows that
Results and Discussion Pure Components. In this work, the hydrogen bonding interactions between water molecules are mediated by introducing four off-center short-range square-well association sites (model initially proposed by Nezbeda et al.54). This simple approach has been successfully used in many studies3 and has been shown to be very efficient for the accurate description of the global phase behavior of aqueous mixtures. The originality of the work relies on the modeling of the isotropic repulsive and dispersion attractive interactions which are incorporated using a variable Mie potential. Following the framework of Galindo and Blas,55 we represent carbon dioxide as a nonassociating molecule. It is relevant to mention here that the polar and quadrupolar moments of water and carbon dioxide are not included explicitly but are treated in an effective way through the variable range of the Mie potential. The molecular parameters of carbon dioxide and water are adjusted to best fit the experimental data of vapor pressures, saturated liquid densities, and speed of sound in the condensed liquid phase. Note that in the case of the water molecule isobaric heat capacity data are also used in the fitting to obtain a realistic balance between dispersion and association energies.53 More information on the fitting procedure can be found in ref 52. The
Figure 1. Surface tension vs temperature for water. (9) Experimental data.63 (---) Gradient theory of fluid interfaces with the SAFT-VR Mie EOS and constant influence parameter (given in Table 1).
Figure 2. Water density profiles across the interface for temperatures ranging from 350 to 500 K: (s) computed with eq 4, (---) MC simulations.57
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Figure 3. Interfacial thickness of water versus temperature. (9) Experimental data.56 (b),58 (2)59 Molecular dynamics. ([) MC simulations.57 (s) Gradient theory.
the complex temperature dependence of the surface tension is perfectly described, demonstrating the consistency of the model. Once the surface tension is adequately represented, the gradient theory may be further applied to determine other interfacial properties such as density distributions and interfacial thickness. Figure 2 depicts the predicted density profiles of pure water at different temperatures. They are computed by numerical integration of eq 4. As expected, the density profiles display the traditional tanh shape. Figure 2 shows also the good agreement between these profiles and the ones obtained from MC simulations (TIP4P/2005 model for water)57 especially at high temperature. Figure 3 represents the interfacial thickness computed with the model (10-90 criterion) together with experimental data56 and values obtained by MC57 and MD58,59 simulations. Even for these microscopic scale properties, the results are in good agreement with experiments or simulations. Mixture. In Figure 4(a) and (b), we present the P-x isotherms of the water + CO2 system at T ) 298.15 K and T ) 323.15 K. The solid curves represent the SAFT-VR Mie calculations, using the kij value adjusted to best fit the calculations to the experimental data, represented by the symbols. It should be emphasized that Haslam et al.60 demonstrated that one cannot generally describe both the water-rich phase and the gas-rich phase of a water + gas mixture using the same binary interaction parameter. However, with the use of the SAFT-VR Mie EOS, and for the temperature range considered in this work, a unique value of kij ) -0.1 is seen to provide a satisfactory agreement for both coexisting phases in a wide range of pressure conditions. The experimental CO2-H2O interfacial tensions are well documented in the literature.6-10,61 The available data cover a temperature range of 277-344 K at pressures ranging from atmospheric to 28 MPa. The isotherms below that of the UCEP (approximately, at temperature below the critical temperature of CO2, T ) 304 K) are separated in two parts corresponding to a region of vapor-liquid equilibrium (VLE) at low pressure and a region of liquid-liquid equilibrium (LLE) at high pressure. These two regions are separated by the three-phase line pressure which is almost indistinguishable from the vapor pressure of CO2. Isotherms measured above that of the UCEP continuously decrease with increasing pressure. Three isotherms (287, 298, and 313 K) are studied to span over all types of interfaces and are plotted in Figure 5 together with the computed interfacial tensions. The optimal value β12 ) 0.915 for the interaction coefficient of the influence parameter allows us to satisfactorily reproduce the interfacial tensions for all types of interfaces (supercritical CO2/liquid water, vapor CO2/liquid
Figure 4. VLE of CO2/H2O at (a) 298.15 K and (b) 323.15 K. (s) SAFT-VR Mie EoS. (O) Experimental data.70
Figure 5. Surface tension versus pressure at several temperatures for the CO2/water mixture. ([) Experimental data60 at 287 K. (b) Experimental data60 at 298 K. (×) Experimental data60 at 318 K. (s) Gradient theory with β12 ) 0.915.
water, and liquid CO2/liquid water) and over a wide range of pressure conditions. This result highlights the consistency of the gradient theory when it is combined with an equation of state based on a rigorous statistical mechanics foundation and a realistic intermolecular potential model for each substance.
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Figure 6. Surface tension versus pressure at 287 K for the CO2/water mixture. ([) Gradient theory of fluid interfaces with β12 ) 0.915 for pressures just below and above the saturation pressure of CO2.
Indeed, in a previous work62 we had coupled the GT with the PR-SW EOS63 to perform the same modeling. This EOS is a modified version of the original PR EOS dedicated to describe the VLE of CO2, H2S, or CH4 with water. Hence, this EOS is very efficient to provide an accurate description of the equilibrium densities for the above-mentioned systems. However, when combined with the GT, the approach showed two shortcomings: first a single value for β12 could not be used for all types of interfaces, and its value was found to be very low, underlining the limitation of the EoS. Below the critical temperature of CO2, the interfacial tension isotherms present a jump close to the saturation pressure of CO2 (cf. Figure 6 for T ) 287 K and T ) 298 K). The isotherm reflects the interfacial tension of water with either gaseous or liquid CO2. For both isotherms, the difference between the computed liquid/vapor and liquid/liquid interfacial tensions at the three-phase point is nearly equal to the experimental CO2 surface tension. At 287 K, this difference is 3.1 mN/m, and the experimental CO2 surface tension is 2.1 mN/m.64 The slight difference between these values can be partially explained by the mutual solubilities effect. Finally, for temperatures above that of the UCEP, the discontinuity in the interfacial tension isotherms disappears. One of the main advantages of the gradient theory is that it can also be used for computing other interfacial properties such as density profiles, which are hardly accessible to experimental observation. Here, the gradient theory + SAFT-VR Mie approach yields a satisfactory prediction of the surface tension of the CO2/H2O mixture, so it is justified to consider that the density profiles obtained for the energy minimization are correct. Figure 7 depicts, as an example, the structure in the CO2/H2O interface at 298 K for two pressures: 2 MPa, which corresponds to a gas/liquid interface, and 10 MPa, a liquid/liquid interface. The following typical behavior, appearing at these conditions, has actually been obtained for each thermodynamic condition considered in this work: (1) The density of water increases monotically from the CO2rich phase to the water-rich phase. The density profile of water has the traditional tanh shape. No change in the shape of the water profile is evidenced from 0.1 to 25 MPa. (2) The free energy of the system is minimized when a substantial amount of CO2 is adsorbed at the interface. Indeed, Figure 7 shows an enhancement of CO2 density highlighting an adsorption of CO2 molecules on the water surface. The adsorption peak is located on the CO2-rich side of the interface. The same behavior has been obtained by Biscay et al.65 for the
Lafitte et al.
Figure 7. Density profiles across the interface computed with eq 4 for the CO2/water mixture at 298 K. (---) 2 MPa (gas/liquid interface). (s) 10 MPa (liquid/liquid interface). Gray profiles: CO2. Black profiles: water.
Figure 8. Density profiles of CO2 across the interface computed with eq 4 at 287 K. From down to up: 4 MPa, 4.5 MPa, 5 MPa, 5.5 MPa.
CH4/water mixture with MC simulations. The gradient theory is in this case able to capture this behavior. Figure 8 depicts the density profile of CO2 in the interface just below and after the saturation pressure of CO2 at 287 K. For the gas/liquid interface, CO2 accumulates and forms a thin liquid film that perfectly wets the interface between the gaseous CO2 and the liquid water. As the pressure increases, this adsorbed layer grows thicker continuously just below the saturation pressure of CO2. This change in the surface coverage from a thin to a thick liquid film represents a prewetting transition,66 which means that Tcw < T < Tc, Tcw being the critical wetting temperature. The equilibrium partial density profiles contain the information about the enrichment of one component in the interface. However it is important to note that, as the pressure changes, not only the height of the adsorption peak is modified but also the densities in the bulk phases. To better visualize the local enrichment, Telo da Gama and Evans67 as well as Wadewitz and Winkelmann68 suggested the definition of a local relative concentration of the two species called symmetric interface segregation which corresponds to an extension of Gibbs enrichment.
∆C(z) )
n2(z) - nII2 n1(z) - nII1 R2 R1
with symmetric concentrations given by
(9)
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Ri )
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niII - niI (nII1 + nII2 ) - (nI1 + nI2)
(10)
where I denotes the CO2-rich phase and II the water-rich phase. As an example, Figure 9 shows the pressure dependence of the enrichment of CO2 calculated using eq 9 at 298 K for pressures below the saturation pressure of CO2 (i.e., in the case of the interface between a gaseous CO2-rich phase and a liquid water-rich phase). The local enrichment of CO2 is clearly visible. When the pressure increases, the adsorption of CO2 decreases, the interfacial thickness increases, and the enrichment of CO2 moves to the “water side” of the interface. In Figure 10, ∆C(z) is presented at the same temperature but for higher pressures, i.e., for liquid/liquid interfaces. The behavior is quite different as the interface is separated in two areas: one enriched with CO2 followed by an area impoverished in CO2. As pressure increases, the interfacial thickness remains nearly constant, around 20 Å. The density profiles in the interface and hence the surface segregation are microscopic local properties that are not accessible to experiment. Nevertheless, one can obtain the relative adsorption of CO2 (as defined by Gibbs) from the integration of ∆C(z)69
Γ12 ) -R1
∫-∞+∞ ∆C(z)dz
(11)
This property can also be computed from bulk densities and experimental surface tension68 with the following expression
Γ12 ) -
nI1nII2 - nII1 nI2 ∂γ nII - nI ∂P 2
2
Figure 9. Surface segregation computed with eq 9 at 298 K for pressures corresponding to the gas/liquid interface. From left to right: 1, 2, 4, 5, 6 MPa.
( )
T
Figure 10. Surface segregation computed with eq 9 at 298 K for pressures corresponding to the liquid/liquid interface. From left to right: 12, 4, 10, and 8 MPa.
(12)
The latter route is of interest since the adsorption isotherms computed from the surface segregation with eq 11 can then be compared with experimental data. The slope of interfacial tension versus pressure in use in eq 12 can have a large uncertainty especially in the case of liquid/ liquid interfaces. Hence, in this work it has been computed only in the case of the vapor/liquid interface. At 298 K, for pressures ranging from 0.1 to 6 MPa, the slope computed from experimental data is nearly constant and equal to -6.014 nm. The relative adsorption of CO2 computed with eq 11 is compared with the one obtained from experimental data using eq 12. The good agreement between computed and “experimental” relative adsorption of CO2 observed in Figure 11 is an indirect proof of the consistency of interfacial density profiles computed with the gradient theory for the CO2/H2O mixture. Conclusions The gradient theory of fluid interfaces has been combined with the SAFT-VR Mie EOS to model the interfacial properties of the water/CO2 mixture. The model was first applied to pure water. With a constant influence parameter, the complex temperature dependence of the surface tension of water could be perfectly reproduced. The interfacial thickness computed from the density profiles is in good agreement with experimental data and simulation results. The interfacial tension of the water/CO2 mixture could be modeled with a unique value of the crossed influence parameter
Figure 11. Adsorption isotherm at 298 K for pressures corresponding to gas/liquid interface. (9) Computed from experimental data via eq 12. (s) Computed from density profiles obtained with gradient theory via eq 11.
whatever the thermodynamic conditions, i.e., for various types (gas/liquid, liquid/liquid, or supercritical/liquid) of interfaces. One of the attractive features of the gradient theory is that it also provides the interfacial density profiles of each component in the mixture. A systematic adsorption of CO2 on the interface was observed. Moreover, when approaching the saturation pressure of CO2, a prewetting transition was detected as the thin-liquid CO2 film appearing at lower pressures grows thicker in the immediacy of the saturation pressure of CO2. The surface segregation was also computed to better understand the adsorption behavior of CO2 as the pressure increases. This property was used to compute the adsorption isotherm of
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this mixture at 298 K. This latter was compared with the one deduced from the slope of the experimental surface tension data. The good agreement between the predicted and “experimental” adsorption isotherms is an indirect proof of the consistency of interfacial density profiles computed with the gradient theory for the CO2/water mixture and thus confirms that this theoretical approach is efficient to describe the microstructure of complex fluid interfaces. Acknowledgment. The authors acknowledge financial support from project ANR 06 JCJC-0070-01186000 of the French National Agnecy (ANR) and Consellerı´a de Educacio´n e Ordenacio´n Universitaria (Xunta de Galicia, Spain) and Ministerio de Ciencia e Innovacio´n (Proj. ref. FIS2009-07923, Spain). References and Notes (1) DeSimone, J. M. Science 2002, 297 (5582), 799–803. (2) Scott, R. L.; Van Konynenburg, P. H. Discuss. Faraday Soc. 1970, 49, 87–97. (3) dos Ramos, M. C.; Blas, F. J.; Galindo, A. Fluid Phase Equilibria, Properties and Phase Equilibria for Product and Process Design. 11th International Conference on Properties and Phase Equilibria for Product and Process Design, 2007; Vol. 261, 1-2, pp 359-365. (4) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. J. Chem. Phys. 1996, 106 (10), 4168–4186. (5) Galindo, A.; Davies, L. A.; Gil-Villegas, A.; Jackson, G. Mol. Phys. 1998, 93 (2), 241–252. (6) Jho, C.; Nealon, D.; Shogbola, S.; King, A. D. J. J. Colloid Interface Sci. 1978, 65 (1), 141–154. (7) Kvamme, B.; Kuznetsova, T.; Hebach, A.; Oberhof, A.; Lunde, E. Computational Materials Science, Selected papers from the International Conference on Computational Methods in Sciences and Engineering 2004 - ICCMSE-2004, International Conference on Computational Methods in Sciences and Engineering 2004; 2007; Vol. 38, 3, 506-513. (8) Chun, B.-S.; Wilkinson, G. T. Ind. Eng. Chem. Res. 1995, 34 (12), 4371–4377. (9) Massoudi, R.; King, A. D. J. Phys. Chem. 1974, 78 (22), 2262– 2266. (10) da Rocha, S. R. P.; Harrison, K. L.; Johnston, K. P. Langmuir 1998, 15 (2), 419–428. (11) Chiquet, P.; Daridon, J.-L.; Broseta, D.; Thibeau, S. Energy ConVers. Manage. 2007, 48 (3), 736–744. (12) Li, X.-S.; Liu, J.-M.; Fu, D. Ind. Eng. Chem. Res. 2008, 47 (22), 8911–8917. (13) Rowlinson, J. S. J. Stat. Phys. 1979, 20 (2), 197–244. (14) Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1958, 28 (2), 258–267. (15) Carey, B. S.; scriven, L. E.; Davis, H. T. AIChE J. 1978, 24 (6), 1076–1080. (16) Carey, B. S. The gradient theory of fluid interfaces; University of Minnesota: Mineapolis, MN, 1979. (17) Carey, B. S.; Scriven, L. E.; Davis, H. T. AIChE J. 1980, 26 (5), 705–711. (18) Davis, H. T.; Scriven, L. E. AdV. Chem. Phys. 1982, 49, 357–454. (19) Falls, A. H.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1983, 78 (12), 7300–7317. (20) Perez-Lopez, J. H.; Gonzalez-Ortiz, L. J.; Leiva, M. A.; Puig, J. E. AIChE J. 1992, 38 (5), 753–760. (21) Cornelisse, P. M. W.; Peters, C. J.; De Swaan Arons, J. Fluid Phase Equilib. 1993, 82, 119–129. (22) Cornelisse, P. M. W.; Peters, C. J.; De Swaan Arons, J. Mol. Phys. 1993, 80 (4), 941–955. (23) Cornelisse, P. M. W. The squared gradient theory applied. Simultaneous modelling of interfacial tension and phase behaViour; TU Delft: Delft, 1997. (24) Sahimi, M.; Davis, H. T.; Scriven, L. E. Soc. Pet. Eng. J. 1985, 235–254. (25) Zuo, Y.-X.; Stenby, E. H. Fluid Phase Equilib. 1997, 132, 139– 158. (26) Miqueu, C.; Mendiboure, B.; Graciaa, A.; Lachaise, J. Fluid Phase Equilib. 2003, 207, 225–246. (27) Miqueu, C.; Mendiboure, B.; Graciaa, A.; Lachaise, J. Fluid Phase Equilib. 2004, 218, 189–203. (28) Miqueu, C.; Mendiboure, B.; Graciaa, A.; Lachaise, J. Ind. Eng. Chem. Res. 2005, 44 (9), 3321–3329. (29) Miqueu, C.; Mendiboure, B.; Graciaa, A.; Lachaise, J. Fuel 2008, 87 (6), 612–621.
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